Citation
Drag Forces on Pile Groups

Material Information

Title:
Drag Forces on Pile Groups
Creator:
Crowley, Raphael
Place of Publication:
[Gainesville, Fla.]
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (143 p.)

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Coastal and Oceanographic Engineering
Civil and Coastal Engineering
Committee Chair:
Sheppard, Donald M.
Committee Members:
Thieke, Robert J.
Sheremet, Alexandru
Graduation Date:
5/1/2008

Subjects

Subjects / Keywords:
Average velocity ( jstor )
Cylinders ( jstor )
Drag coefficient ( jstor )
Flumes ( jstor )
Pile groups ( jstor )
Pressure distribution ( jstor )
Reynolds number ( jstor )
Velocity ( jstor )
Velocity distribution ( jstor )
Vorticity ( jstor )
Civil and Coastal Engineering -- Dissertations, Academic -- UF
cylinder, drag, force, group, pile, piv, velocity
Escambia Bay ( local )
Genre:
Electronic Thesis or Dissertation
bibliography ( marcgt )
theses ( marcgt )
Coastal and Oceanographic Engineering thesis, M.S.

Notes

Abstract:
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. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (M.S.)--University of Florida, 2008.
Local:
Adviser: Sheppard, Donald M.
Statement of Responsibility:
by Raphael Crowley

Record Information

Source Institution:
UFRGP
Rights Management:
Copyright by Raphael Crowley. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
7/11/2008
Classification:
LD1780 2008 ( lcc )

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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.









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).













AL










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
































X (mm)


Figure 4-12. Re


2.57x103, 9 Piles Average Velocity


-5 0 5 10 15 20 25 30 35
V (cm/sec)

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


I .


I*










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).






U Approximate
................... ..... Extent of
Wake




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).


=18/C -d 1- ,] (3-1)
U 18/7 CDd b

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










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-


x 10-4


0 0.5 1 1.5 2 2.5 3
Frequency (Hz)


3.5 4 4.5 5


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









Where r 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:


r = pl2 (2-23)


Since the y-velocity component gradient is small, after substituting into the governing

equations, the following is obtained:


U = 212 1 2 (2-24)
ax y Oy2

Schlichting assumes that the mixing length is constant over the wake half-width and

proportional to it so that /= pb(x). The ratio r =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:

b =B(C x)2
1 (2-25)
u1 = Um fC(x


Inserting into the governing equation, a differential equation is obtained forf(r).


S(f + f') = 2 ff" (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 7 = 1,f f' = 0. Integrating twice and applying these boundary conditions yields


f =22 1- 72 (2-27)
922p2









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.









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









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









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

I S I S I




x







Figure 2-4. Olsson Definition Sketch

where s is the spacing between rows, U. is the velocity if there were no bars, and u; is the

reduced velocity, Olsson assumes that in a fully developed flow, the velocity distribution is

expected to be a periodic function iny:
-1
u, = UOA x cos 2rY- (2-31)


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

Schlichting cites an extension of Prandtl's mixing length theory


2 u 2u .82Uf .
r, = p12 2 (2-32)


and states that it seems reasonable to assume that 1I = s/2m. Taking the derivative and

dividing both sides by p gives


IO 2 u- A2 cos 23 r (2-33)
p sy s ) \ s









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 SD =

3.0 & 5.0. Figure 2-9 shows his results. St is the Strouhal Number which is defined as St =

fD/U, where, is the vortex shedding frequency, D is the pile's diameter, and Uis the free-

stream velocity. His results are presented in Figure 2-9.



*7 i 0
*5 3 S/

) 06 St 1 3 4



FIig 9 C ci I


[ .
gz--




*r ... .. i -
_0|74 6 B 2 6 81 ic

Re

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



































2 4 6 8 10 12 14 16
V (cm/sec)


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


-2 0 2 4 6 8 10 12 14
V (cm/sec)

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


-- _cyl
-3_cyl
--9_cyl


2.57x103


2.57x103









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:



P OA "Point Al"



"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











0.08


0.06


0.04


0.02





o -0.02
U-

-0.04


-0.06


-0.08


-0.1
0 10 20 30 40
Time (sec)

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


x 10-4
2.4

2.2

2


0.4 i I
0 0.5 1 1.5 2 2.5 3 3.5
Frequency (Hz)

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


50 60


3.83x103






























4 4.5 5
-









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

B = l0/ (2-28)

With Schlichting assuming that


SI12 2d=9 (2-29)
-1 10

The final solution to flow past a single pile then becomes


b = lO/(xCd)2

u xC 2 2 (2-30)
U1 18 2CDd) b


A plot of results from Equation 2-30 are shown below. This plot shows u /Uinf



U0.7
u1lUlnf
30 0.8


20
0.6
10
a, 10
0.5

I 0 0.4

i -0.3
-10

_5 0.2
-20
0.1


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

































-5 0 5 10 15 20 25
V (cm/sec)

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


3.85x103


10 15 20 25
V (cm/sec)


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


3.85x103








Parameters for computation of this velocity field are as follows: d = 1.905cm, CD= 1.0,

and b = 30cm. Schlichting determined the constant from measured values. According to

measurements by Schlichting and H. Reichardt (Schlichting 1979),/ = 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.


F


low Direction


Column 1

--------------------- 1-----------------


Column 2




Column 3


___ --- -----J___ --___----- ----- _1------------- -
Row 1 Row 2 Row 3


Figure 2-3. Column and Row Definition Sketch


0

1-------------------1____


0


0


0









(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.





















X (mm)


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-












St fd (4-1)
Uo


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


U (cm/sec)
30 00
28 00
26 00
24 00
22 00
20 00
18 00
16 00
14 00
S~ 1200
10 00
8 00
6 00
4 00
E 2 00
E I0 o















50 10 10 260
X (mm)









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. D rag C efficient for 1 P ile ........................................................................ .................. 100

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

4-59. R results for Tw o 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 Arrangem ents ................................................ ............... 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).................13

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............. .................19









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/dvalue, 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.










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


around tandem cylinders. His results are non-dimensionalized such thatCp = P where po
S 0.5pV2


is the pressure in the free stream, p is the new pressure, p is the density of water, and Vis the free

stream velocity.

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

LIi




S/D
S 1.03
v 1.18


v ji 0
S.09 I









30 60 90 120 150 ~0 0 30 60 90 120 150 180
0 6
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









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, IN, 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 IN 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



















0.8

C0.

0.4


-=- /th

--- 2rd

2nd



S 20 3-0 D4O 5.0
S/D


-0.4




-0.8


SID


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


rrrt.~



~~l ~ ~ ~ aL -$lk~B;-~*P;r~$i


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



























Figure 4-2. Re


3.85x103, 1 Pile Average Velocity


X(mm)


2.57x103, 1 Pile Average Velocity


I


X (mm)


j I.-i li c,: I


Figure 4-3. Re









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?










10

8

6

4

2

0

-2

-4

-6

-8

-10-
0


5 10 15
Time (sec)


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

25



20 -


*f 15

703
0
5 10
cn


5


0 1 2 3 4 5 6
Frequency (Hz)

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


7 8


k









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).




















Vorticity Magnitude
360
340
320
300
280
260
240
220
200
1 80
1 60
140
1 20
1 00
080
S060
040
020


X (mm)


Figure 4-46. Re


5.13x103 Average Vorticity for Two Piles


Vorticity Magnitude
S280
2 70








70
260
250
240
230
220
210
200
1 90
1 80
1 70
1 60
1 50
140
1 30
1 20
1 10
1 00
0 90
080
070
060
0504

030
0 20


X (mm)


3.85x103 Average Vorticity for Two Piles


Figure 4-47. Re









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 (T op V iew ) ..........................................................................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 arrangem ents used in this study........................................ ............... ............... 50

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

3-6. Typical PIV im age pair................................. .. .. ........ .. ............53

3-7. TFHRC PIV Flum e in M cLean, VA ............................................. ............................ 54

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

3 -9 S o loP IV laser.......................................................................... 5 5

3-10. M egaPlus Camera used in the Experiments ............................................. ............... 55

3-11. P hoto of cam era-m irror setup ................................................................... .. ...................56

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

3-13. P IV w ith 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. AD V Probe M easurem ents .................................................................... ............... 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. M asked P IV Im age (9 P iles)........................................................................ ..................62

3-20. Exam ple of a Correlation Im age (no piles) .............................................. ............... 63

3-21. A average V elocity Im age Exam ple............................................................ .....................64










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

By

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.



























0------------------- ----------------9---------------L----------------------------------




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










3 M A TER IA L S A N D M E TH O D S...................................................................... ...................47

Schlichting' s Far W ake Theory ............. ................................................ .............. ..... .... 47
N ear W ake R region ............ .......................................................... .............. .......... ......49
Pile Configurations ....................... .................................... ........50
Particle Im age V elocim etry .................. ... ............................ ........ .............. ............... 51
Problems with Traditional Measuring Techniques........................................................51
Particle Im age V elocim etry .................................................................. ............... 52
Turner Fairbanks Highway Research Center PIV Flume..............................................54
S e tu p ................................................................................................................... 5 4
PIV m easurem ent ................ ................. ...... .......... ............. .............. .. 57
Potential errors in PIV measurements ....................................... ...............59
V verification w ith A D V Probe........................... .................. .................. ............... 60
PIV D ata A analysis ......... ......................... ........ .............. .... 62
Force M easurem ents .......... ....... ...... ........ .......... ........... ............. 64
TFH R C Force B balance Setup ............................................... ............................... 64
M measurements in the Force Balance Flume................................... ....... ............... 68
M methods Summary ......... .. .. .......................................... ................. 69

4 EX PER IM EN TA L R E SU L TS.......................................................................... ...................70

P IV D ata ...........................................................................7 0
Velocity Fields .............................................. 70
V velocity Profiles from PIV data .................. ......... ........................ ............... 70
Vorticity Data ......... ....................... .................................. 70
Strouhal N um ber Com parison........................................ ..... ................. ............... 70
M measured Force D ata ......... ....... ....................................99
O n e P ile ..................................................................................................................... 9 9
A lig n ed P iles ...............................................................10 0
Side-by -Side P iles ...............................................................10 1
Pile Group Configurations............................................................ ...... .. 02
R e su lts S u m m ary ....................................................................................................10 4

5 DISCUSSION ........... ......................... ........ .. ............ ............. 105

P IV D ata A n aly sis ....................................................................................10 5
Average Velocity Field M easurements ........................................ .. ........ 106
O ne pile arrange ent .............. ....................................................106
Three pile arrange ent......... ...... .................................... .... ........ ............. 107
N ine pile arrange ent .................................... .... ...... .. .. ............. 108
Tw o Pile A rrangem ent ................... ............... ............................................ 109
Demorphing PIV Data .............................................................................109
V velocity Profile M easurem ents ........................................................ .............. 112
V orticity M easurem ents ................................................................ ... .....................112
F force B balance D ata A naly sis............................................................................... ...... .. 113
O ne Pile A rrangem ent ...................................................... .... .. ........ .. 113



6












Point A
Point B
Point C
Point D


1 2 3 4
Frequency (Hz)


5 6 7 8


Figure 5-27.


U-Velocity Spectrum for 3 Piles, Re


25




20




15

(.0
0

I 10
U)


5




0
0 1 2


3 4
Frequency (Hz)


5 6 7 8


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


3.85x103


Point A
Point B
Point C
Point D









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









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 1-10 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,

































-2 0 2 4 6 8 10 12 14 16 18
V (cm/sec)


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


2.57x103


2 4 6 8 10 12 14 16


V (cm/sec)

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


2.57x103









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











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.







Z, -------------------------------------------------------------------------








First Velocity
Second Velocity
-Third Velocity





0 2 4 6 8 10 12 14 16 18 20
Time (sec)


Figure 3-15. ADV Probe Measurements
















Data
-y=x
S- Linear Data Fit


-M -----------+--------------------- ---------- ^-------- --------- ----------

Sy = 0.9783x
R2 = 0.9352







I
o------------ ----------------------------- ---------- --------------------




0 0.5 1 1.5 2 2.5 3 3.5

0.5pdV2


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.












Point A
Point B
Point C
Point D


20



15
03
0a

10


5-



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


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

35
Point A
Point B
30 -
30- Point C
Point D


20


5 15
Q.


10


5


0
0 1 2


3 4
Frequency (Hz)


5 6 7 8


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


=5.13x103


















































2.57x103 Average Vorticity for Nine Piles


LA

s c
Li-


U1



0 3 C


REGION OF T..PJl.LIT ,f.TEI TRAIL
AND LAMINAR BOUNDARY LAYER ON THE
CYLINDER


UJz-



i -

<
a:
ec, o,
-1Z-a
g,
/,


REGION -
CYLINDER OSCILLATING AT / :rTE. SHE: =
ITS r;"TIJFAAL FREQUENCY (NPL) FPFijU':; -
i[LIliH AS =_ _
DOMINANT -:.--
IN A SPEC2 -


m1 102 101 101 10s li


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






98


*


Figure 4-54. Re


#












Point A
Point B
Point C
Point D


0.5-


0 1 2 3 4
Frequency (Hz)


Figure 5-30. U-Velocity Spectrum for 3 Piles, F


5 6 7 8



e = 2.57x103


/

VV


0 1 2
0 1 2


3 4 5
Frequency (Hz)


S Point A
Point B
Point C
Point D


















6 7 8


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


S6
o

5-



3-

2

1


2.57x103









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).



z 8 + 1-63 x 105
U x 1-14 x 105
S .4 o 065 x 10

U a

40

U

uj --'- /
_:a
I, --iW-N

t- D

2 3 4 5 6 T/D


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









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).









LIST OF REFERENCES


Aiba, S., H. Tsuchida, & T. Ota (1981). "Heat Transfer Around a Tube in a Bank." Bulletin of
Japan Society of Mechanical Engineers, 23, 311-19. Quoted in M.M. Zdravkovich, Flow
Around Circular Cylinders Volume 2: Applications, 1097.

Ball, D. J. & C. D. Hall (1980). "Drag of Yawed Pile Groups at Low Reynolds Numbers."
ASCE Journal of Waterways and Harbors Coastal Engineering Division, 106, 229-38.

Batham, J.P (1973). "Pressure Distributions on In-Line Tube Arrays in Cross Flow."
Proceedings British Nuclear Engineering Society Symposium Vibration Problems in
Industry, ed. Wakefield J. Keswick, 28. Quoted in M.M. Zdravkovich, Flow Around
Circular Cylinders Volume 2: Applications, 1137.

Bearman, P.W. & A. J. Wadcock (1973). "The Interaction Between a Pair of Circular Cylinders
Normal to a Stream." Journal ofFluidMechanics, 61, 499-511. Quoted in M.M.
Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1027.

Bierman, D. & W. H. Herrnstein (1934). "The Interference Between Struts in Various
Combinations." National Advisor Committee for Aeronautics TR 468. Quoted in M.M.
Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1024.

Chen, S. S. & J. A. Jendrzejczyk (1987). "Fluid excitation Forces Acting on a Square Tube
Array." Journal of Fluids Engineering, 109, 415-23. Quoted in M.M. Zdravkovich,
Flow Around Circular Cylinders Volume 2: Applications, 1138.

Crowe, C. T., D. F. Elger & J. A. Robinson (2001). Engineering Fluid Mechanics. John Wiley
& Sons, Inc., New York.

Dean, R. G. & R. A. Dalrymple (2002). Coastal Processes n/ ith Engineering Applications.
Cambridge University Press, New York.

Dean, Robert G. & R. A. Dalrymple (2000). Water Wave Mechanics for Engineers and
Scientists. World Scientific Publishing Company, London.

Hori, E (1959). "Average Flow Fields Around a Group of Circular Cylinders." Proc. 9th Japan
National Congress ofAppliedMechanics, Tokyo, Japan, 231-4. Quoted in M.M.
Zdravkovich, Flow Around Circular Cylinders, Volume 2 Applications, 1002.

Lam, K. & X. Fang (1995). "The Effect of Interference of Four Equal-Distanced Cylinders in
Cross Flow on Pressure and Force Coefficients." Journal of Fluids and Structures, 9,
195-214. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders Volume 2:
Applications, 1100.

Igarashi, T (1981). "Characteristics of a Flow Around Two Circular Cylinders in Tandem."
Bulletin ofJapan Society of Mechanical Engineers, 323-31. Quoted in M.M.
Zdravkovich, Flow Around Circular Cylinders, Volume 2 Applications, 1003.









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

































o
0 --------- ------------ ------ ---------------- -----------------^-----



0 5 10 15 20
V (cm/sec)

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


-2 0 2 4 6 8 10 12 14 16
V (cm/sec)

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


25


3.85x103









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.



































o
0 5 10 15 20 25 30
V (cm/sec)

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

0


0 5 10 15 20 25 30
V (cm/sec)

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


35


5.13x103


35


5.13x103



















Vorticity Magnitude
400
380
360

320
300
280
260
240
220
200
180
160


X (mm)


Figure 4-52. Re


5.13x103 Average Vorticity for Nine Piles


Vorticity Magnitude
300
280
260
240
220
200
1 80
160
1 40
1 20
1 00
080
060


X (mm)


3.85x103 Average Vorticity for Nine Piles


Figure 4-53. Re


































200E+03 400E+03 600E+03 800E+03 100E+04 120E+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
Plle
Figure 4-59. Result Piles Side-by-Side Pi
Lowest PIV Reading

S--Highest PIV Reading








-- --- -- --- --






20OE+03 4 OOE+03 6 OOE+03 8 OOE+03 1 OOE+04 1 20E+04
Re


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






























0 --- ______-_________-_ _-_-________
LO



o
0 5 10 15 20
V (cm/sec)

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


25


3.85x103


10 15 20 25
V (cm/sec)


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


3.85x103
















I:nllsecr
I '

I a~


.1.
I


50 100 150 200
x(mm)


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)









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 yl, the slower one from

(yi-1) preceding the faster one from (y1+1). 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 yl... This

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




I v'|= const I u' = cost I-d (2-12)
Sdy

Another way of writing this is that v' I du/dy. Therefore,

Db u
-I1 (2-13)
Dt 9y

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 ul/b, the following expression is also true:

Db 1
= const -u = const /3u (2-15)
Dt b

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









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).


(a) -,


s/d= 2.06


(c) ,-- -;r. (d)




s/d = 3.24


s/d= 3.53


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


13,000


C D D




Re : 2.2xiO4


S/D


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


1.2


0.8

U
0,.4





-0,4


-0.8


13,000









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.













LJ I-II S i: I


0 O

Op


22
20
18
16
14
12
10
8
6

I


X (mm)



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







o=-so ,,,12D,

8 00















X(mm)



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


76









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 sid 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.

















IJeSTriM rt LINDEM DONSTK~ r :.u -


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









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
































-5 0 5 10 15 20 25
V (cm/sec)

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


-5 0 5 10 15 20 25
V (cm/sec)

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


3.85x103


3.85x103










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:


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



6-


5

0)
C
4






2-
1 V\




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









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.


x 10-4






3-


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













0 0 0 00








0 0 0 00

(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.


2 00E+03


4 00E+03 6 00E+03 8 OE+03 1 00E+04


120E+04


Re

Figure 4-61. Results for Complex Pile Arrangements










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.


(a) Column
-21 + I


!L
(pg ^ 05
cIL A 7 -"
0.4 L 9



0.4+

(b) Column
12-
I o 5












Batham, like most authors of research involving multi-tube heat exchangers, is mostly
O.4I- +9 + ;I
0.4 +

I I I 1*1. 11* I*



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 Uis 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








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..




0 00



u 00 O 0




(1.) (2.)




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


















U (cm/sec)
1 15 00
14 00
13 00
12 00
11 00
10 00
9 00
8 00
7 00
6 Do
S00


I


1o 260 '


X (mm)


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






U (cm/sec)

26 00
24 00





20 00
8 (mm)
16 O0
14 00

8000
S-00


















5 1 1 0

X (mm)


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

































5 10 15 20 25 30 35
V (cm/sec)


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


5.13x103


-5 0 5 10 15 20 25
V (cm/sec)

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









5-9. 0-Pile Spectrum in M iddle of PIV W indow, 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 W indow, 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 Al, Re = 5.13x103 ............................124

5-15. Velocity Spectrum for 1 Pile at Point Al, 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 Al, 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 ............................. .... ....................... 12

5-27. U-Velocity Spectrum for 3 Piles, Re = 3.85x103 ....................... .................... 13

5-28. V-Velocity Spectrum for Pile Re =3.85x103.................................. ..... ............... 13
5-26. Force Spectru m for 3 Piles, R e = 3.85x103 .............. ...................................................... 131





5-2 Force Spectrum for 3 Piles, Re = .8x103 ..................................... ....................... 13
5-29. Force Spectrum for 3 Piles, R e = 2.86xl03 ....................................................................... 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









Igarashi, T (1986). "Characteristics of the Flow Around Four Circular Cylinders Arranged In-
Line." Bulletin ofJapan Society of Mechanical Engineers, 29, 751-7. Quoted in M.M.
Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1097.

Igarashi, T. & K. Suzuki (1984). "Characteristics of the Flow Around Three Circular
Cylinders." Bulletin ofJapan Society of Mechanical Engineering 27, 2397-404, quoted
in M. M. Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1080.

Ishigai, S., S. Nishikawa, K. Nishimura, & K. Cho (1972). "Experimental Study on Structure of
Gas Flow in Tube Banks With Tube Axes Normal to Flow." Bulletin ofJapan Society of
Mechanical Engineers 15, 949-56. Quoted in M.M. Zdravkovich Flow Around Circular
Cylinders Volume 2: Applications, 1027.

Oshkai, P. (2007). "Experimental System and Techniques." University of Victoria, Victoria,
BC, Canada, (June 28, 2007).

Pearcey, H. H., R. F. Cash, & I. J. Salter (1982). "Interference Effects on the Drag Loading for
Groups of Cylinders in Uni-Directional Flow." National Maritime Institute, UK Rep.
NMI R130, 22, 27. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders
Volume 2: Applications, 1099.

Raffel, M., C. Willert, & J. Kompenhans (1998). Particle Image Velocimetry, a Practical Guide.
Springer, New York.

Schlichting, H (1979). McGraw-Hill Series in Mechanical Engineering: Boundary Layer-
Theory. Frank J. Cerra, ed., McGraw Hill, Inc., New York.

Sarpkaya, T. & M. Isaacson. Mechanics of Wave Forces on Offshore Structures. Litton
Educational Publishing Inc., New York.

Spivack, H. M. (1946). "Vortex Frequency and Flow Pattern in the Wake of Two Parallel
Cylinders at Varied Spacing Normal to an Air Stream." Journal ofAero. Science, 13,
289-301. Quoted in M.M. Zdravkovich Flow Around Circular Cylinders Volume 2
Applications, 1018.

Sumer, M. B. & J. Fredsoe (1997). Hydrodynamics Around Cylindrical Structures. World
Scientific Publishing Company, London.

Sumer, M. B. & J. Fredsoe (2002). The Mechanics of Scour in the Marine Environment. World
Scientific Publishing Company, London.

Wardlaw, R. L., K. R. Cooper, R. G. Ko, & J. A. Watts (1974). "Wind Tunnel and Analytical
Investigations into the Aeroelastic Behavior of Bundled Conductors." Institute of
Electrical and Electronics Engineering Summer Meeting Energy Resources Conference,
Anaheim, CA, 368-77.









































0 1 2 3 4
Frequency (Hz)


5 6 7 8


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


U-
E
0

0

-1



-2



-3


Time (sec)



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


4.5

4

3.5

3
U)

2.5


Q 2
C-


1.5

1

0.5


2.57x103









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 sd 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

I tLCO W -,


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









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









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/dof 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









Werle, H (1972). "Flow Past Tube Banks." Revue Francais de Mecanique, 41, 28. Quoted in
M.M. Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1078.

Williamson, C. H. K. (1985). "Evolution of a Single Wake Behind a Pair of Bluff Bodies."
Journal ofFluid Mechanics 159, 1-18. Quoted in M.M. Zdravkovich, Flow Around
Circular Cylinders Volume 2 Applications, 1018.

Zdravkovich, M. M (1997). Flow Around Circular Cylinders Volume I, Fundamentals, Oxford
University Press, New York.

Zdravkovich, M. M (2003). Flow Around Circular Cylinders, Volume II, Applications. Oxford
University Press, New York.










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: o) = V x U, where Uis the velocity vector defined as U = ui + vj + wk and o) 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:



















Vorticity Magnitude
1 70
1 60
1 50
1 40
1 30
1 20
1 10
1 00
090
080
070
060
050
040
030
020
010


X (mm)


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


Vorticity Magnitude
S340
320
300
280
260
240
220
200
1 80
1 60
1 40
1 20
1 00



1


X (mm)


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










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










tUj




't LAMINAR BOUNDARY LA ON THE
z Co




in 0.3 > <11
0.54 W o. I








Figure 5-5. Strouhal Number vs. Reynolds Number from Sarpkaya and Issacsson (1981)
SW REGION OF TiJLOneET VORTEX TRAIL
z 5 M flD LAMINAR BOUNDARY LA"fEF ON THE
Sr r r
z 0.3 l l 9 3 l


















Drag forces on the two side by side pile arrangement seem to indicate that the PIV
^ g C(ilDlE OSCILLATING AT VORTEX ":7. |
0.1a ITis not 1PAL FPE(ilearr (NPL) lightly greater t
DOMINANT
IN A P,-' *

in 10 101 10

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











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, ul is small


compared to U., so the two dimensional boundary layer equation (equation 3-20) reduces to


-U., 1 (2-34)
d p dy


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

yields


A =(/)2 (2-35)
8z3

and the solution for velocity reduction becomes



U 3 cos2;r J (2-36)
8;-3 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, 1, was assumed to be constant

where = 0.4cm. In this plot, piles are located at (0,0) and (0,+/-6).

U1 IUlnf
1 10






I-
2
0



-10
2

2 3-4
-6
-8


1 2 3 4 5 6
Distance Downstream From Pile (cm)


Figure 2-5. Plot of Equation 2-36



































-5 0 5 10 15 20 25 30
V (cm/sec)

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

0


0 5 10 15 20 25 30
V (cm/sec)

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


35


5.13x103


35


5.13x103











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











TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ..............................................................................................................4

LIST OF FIGURES ................................. .. ..... ..... ................. .8

ABSTRAC T ............................... ..................... 14

1 INTRODUCTION ................................ .. ...... ...... .... .................. 15

M otiv action for R research .............................................................................. ..................... 15
The 2004 H hurricane Season.................................................. ............................... 15
The 2005 H hurricane Season.................................................. ............................... 16
S c o p e o f R e se a rch ............................................................................................................. 17
M e th o d o lo g y ..................................................................................................................... 1 8

2 L IT E R A TU R E R E V IE W .............................................................................. ......................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
E extension to M multiple P iles ..................................................................... ..................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
D rag on side-by-side cylinders........................................... .......................... 35
Origins of the bistable flow phenom enon ............................................................. 36
P ile G rou p s ......................................................... ...............................3 7
Shedding patterns behind three in-line cylinders ............................................. 37
Pressure fields behind three in-line cylinders .................................. ............... 38
F our in-line cylinders ..................... .. ............................ ..... ............... 40
Square cylinder clu sters ........................... ...................... ........ .. ...... ............42
Flow in heat exchangers ........................................................................ 43
Literature R review Sum m ary ............................................................................. 45

































0 --------- --------------------------- --- ------- -- ------ -
LO





0 5 10 15 20
V (cm/sec)

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


-5 0 5 10 15 20
V (cm/sec)

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


25


3.85x103


3.85x103










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.



--. .


.i ... "- .., -.



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.


2.06






II
-zJ \ 1 t 2 nd 3 rd _. ,






Ss[-

-li 90 90 0 0 0 90 180


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









db 1 (2-16)
U u-1 = u 1 (2-16)
dx b

Where / is a constant. Rewritten this gives

db (2-17)
dx U.

From the expression obtained from the momentum equation

u1 CDd (2-18)
U, 2b

and substituting, the following expression is obtained:

db
2b db CDd (2-19)
dx

or,


b (xCDd)2 (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.


l_ (~CDd) 2 (2-21)


In summary, b ~ x'12 and ul ~ x-12

Substitution into Boundary Layer Equations

In two-dimensional incompressible flow, the governing equations are

au au au 1 Or
--+U--+v -
at x 8y p ay
SP Y (2-22)
u 8v
-+-=0
OU'

































2008 Raphael Crowley









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
















U (cm/sec)
S22 00
20 00
18 00
16 00


S


X (mm)


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





U (cm/sec)
13 00
12 00
11 00
10 O0
9 00
8 O0
7 00
6 D0
5 00
4 00
300
2 00
0000
E I-1 00
EE


X(mm)


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


d ~









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










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.

(a) (b)
1.-ji r I i


0 0


0


0


a 90 0
et


90 0 9O
e e


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


1st 2rd 3rd 4 th



0-S =2.













1.32


1 1.91


0 90 0 90 0 90 0 90 18


0








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 dis 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).


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


A O

O U3d

0 .--O O'
B D


-o000 -oto
UO























































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


J (cm/sec)
24 60
24 40
24 20
*\24 00
23S
23 0
2340
23 20
23 00
22 SO
22 60
2240
22 20
2200
S21 80
21 60
21 40


X (mm)


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










110



















Vorticity Magnitude
260
240
220
200
1 80
1 60
1 40
1 20
1 00
080
060
6


X (mm)


Figure 4-50. Re


3.85x103 Average Vorticity for Three Piles


Vorticity Magnitude
S1 60
1 50
1 40
1 30
1 20
1 10
1 00
090
080
070
060
050


X (mm)


2.57x103 Average Vorticity for Three Piles


Figure 4-51. Re









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:

Fr = F, + FD

FD -CD pAU U
2
1 AU
F, = -CpA
2 at

Where CD is the drag coefficient, p the mass density of water, A the projected area, Uthe

fluid velocity just upstream of the structure, and CI the inertial coefficient. As stated above, this

work focuses on drag forces in steady flows.









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




















U (cm/sec)
24 60
24 40
24 20
24 00
23 80
23 60
23 40
23 20
23 00
22 80
22 60
2240
22 20
2200
21 80
21 60
21 40


X (mm)


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







U (cm/sec)
18 30
18 20
18 10
18 00
17 90
17 80
17 70
17 60
17 50
17 40
17 30
17 20
17 10
17 00
E 16 90
E le so
6 70

16 50
16 40
16 30
16 20
16 10
16 00
15 90
15 80


X(mm)


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










wI =' --j --.---


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.


















c,-


,-,









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

plot of Fxh vs. pdV2, will equal the drag coefficient. This plot is shown in Figure 4-57.




Full Text

PAGE 1

1 DRAG FORCES ON PILE GROUPS By RAPHAEL CROWLEY A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008

PAGE 2

2 2008 Raphael Crowley

PAGE 3

3 To my Dad.

PAGE 4

4 ACKNOWLEDGMENTS I would like to thank my committee (Dr. Al ex 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 Res earch Center (TFHRC) in McLean, Va. Thanks also go to the TFHRC staff, especially Matthia s Poehler, Thies Stange, Dan Brown, and Jesse Coleman. This thesis would not have been possibl e without the lab and its staff. Thanks also go to all the professors IÂ’ve had th rough the years--especially Richard Crago at Bucknell University for giving me my start in research.

PAGE 5

5 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..........................................................................................................1 9 SchlichtingÂ’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.................................................................................................................2 8 HoriÂ’s pressure fields around tandem cylinders.......................................................29 IgarashiÂ’s pressure fiel d around tandem cylinders...................................................30 Drag coefficients at lower Reynol ds Numbers for tandem cylinders......................31 Drag coefficients for tandem cyli nders at higher Reynolds Numbers.....................32 WilliamsonÂ’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

PAGE 6

6 3 MATERIALS AND METHODS...............................................................................................47 SchlichtingÂ’s Far Wake Theory..............................................................................................47 Near Wake Region............................................................................................................... ...49 Pile Configurations............................................................................................................ .....50 Particle Image Velocimetry....................................................................................................5 1 Problems with Traditional Measuring Techniques..........................................................51 Particle Image Velocimetry.............................................................................................52 Turner Fairbanks Highway Re search Center PIV Flume................................................54 Setup.........................................................................................................................5 4 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.................................................................................................................1 00 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

PAGE 7

7 Two Side by Side Pile Arrangement.............................................................................113 Three In-Line Pile Arrangement...................................................................................114 Force Decrease at High Reynolds Number s 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

PAGE 8

8 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 Ba sed 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 Distri bution Around Tandem Cylinders.......................................................29 2-7. Mean Pressure Di stribution for Re = 35,000.........................................................................30 2-8. Compilation of Drag Coe fficients 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 sc attered 108 Hz and 47 Hz fre quencies 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 Coeffici ent 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-Lin e Cylinders. AibaÂ’s data is on the left and IgarashiÂ’s data is on the right.............................................................................................41 2-19. Smoke Visualization past Four Cy linders for Four Different Values of s/d .......................41 2-20. Drag Coefficient for a 2x2 Matrix....................................................................................... 43

PAGE 9

9 2-21. Pressure Coefficient on a 5x9 Cylinder Ma trix (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 Coefficien ts for Circular Tubes in a 7x7 Matrix............................45 3-1. Definition sketch for Schlichting’s 1917 th eory for flow downstream from a circular pile (Top View)................................................................................................................ ..47 3-2. Definition sketch for scenar io 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 sche matic 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 la rge arrows represen t “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

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10 3-22. TFHRC Force Balance Setup..............................................................................................6 5 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 Ce nter of the First Pile for Re = 5.13x103...........................77 4-14. Velocity Profiles 20 mm from the Center of the First Piles for Re = 5.13x103...................78 4-15. Velocity Profiles 40 mm from the Center of the First Pile for Re = 5.13x103.....................78 4-16. Velocity Profiles 60 mm from the Center of the First Pile for Re = 5.13x103.....................79 4-17. Velocity Profiles 80 mm 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

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11 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 Ce nter of the First Pile for Re = 3.85x103...........................82 4-24. Velocity Profiles 20 mm from the Center of the First Pile for Re = 3.85x103.....................83 4-25. Velocity Profiles 40 mm from the Center of the First Pile for Re = 3.85x103.....................83 4-26. Velocity Profiles 60 mm from the Center of the First Pile for Re = 3.85x103.....................84 4-27. Velocity Profiles 80 mm 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 Ce nter of the First Pile for Re = 2.57x103...........................87 4-34. Velocity Profiles 20 mm from the Center of the First Pile for Re = 2.57x103.....................88 4-35. Velocity Profiles 40 mm from the Center of the First Pile for Re = 2.57x103.....................88 4-36. Velocity Profiles 60 mm from the Center of the First Pile for Re = 2.57x103.....................89 4-37. Velocity Profiles 80 mm 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

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12 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 Da ta 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 Configura tions 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 Sar pkaya and Issacsson (1981)...................113 5-6. Deduced Drag Forces Based on Measur ements on a Three Inlin e 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

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13 5-9. 0-Pile Spectrum in Mi ddle 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

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14 Abstract of Thesis Presen ted 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 By 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 s howed good agreement with existing data. With the force balance, drag coefficients for a single pile seemed to le vel 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 pileÂ’s wake. A significant zero velocity zone exists in the wake of the first pile inline, which encompasses the second pile. The appr oach velocity in front of the third pile in alignment is significantly larger than the velocity approaching the second p ile. In fact, PIV data shows that a negative drag coefficient should be expected for the second p ile 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 predicti ng the drag force on a group of piles. Instead, a more complex method is needed to accurately predict these forces.

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15 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 activit y 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 de stroyed during these stor ms. The goal of this research is to investigate a possible mode of bri dge failure so that bridges in the future can be built to withstand stronger-intensity storms. The 2004 Hurricane Season In 2004, there were fifteen trop ical storms, nine hurricanes, and six major hurricanes a hurricane category three or greate r – in the North Atlantic Ocean. Five hurricanes impacted the United States that year. Florid a received the worst of the 2004 season as four hurricanes made landfall on the peninsula – Charley, Frances, Iv an, and Jeanne. According to The National Oceanic and Atmospheric Admini stration (NOAA), the 2004 hurri cane season cost the United States an estimated forty-two bil lion dollars. At the time, this was the most expensive hurricane season on record. The second-most expensiv e season was in 1992 when Hurricane Andrew struck the Miami-Dade area, and cost taxpa yers an estimated $35 billion (NOAA 2007). Hurricane Ivan, which made landf all 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

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16 bridge girder elevation for most of the bridge spans. Waves, superimposed on the storm water level, battered the bridge deck. The surge a nd 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 cau sed by Katrina, the le sser-publicized storms also played their role in ravaging the Gulf Coast. In 2005, twenty-five named storms developed ov er 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. Th e previous record for number of hurricanes in a season, which was set in 1969, was twelve. Duri ng 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, Fl orida received more th an 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 US90 Biloxi-Ocean Springs Bridge in Biloxi,

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17 Mississippi, and The US-90 Bay St Louis Bridge in Bay St. Loui s, 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 worki ng on problems associated with bridge failure during hurricanes since the Escambia Bay Bri dge 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 br idge substructures th at are not understood. Many of the older bridge substructu res 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 vertic al 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 indu ced forces on these types of structures. On single, smaller structures, where diffraction forces are small, wave forces can be computed using the Morison Equation: t U A C F U AU C F F F FI I D D D I T 2 1 | | 2 1 Where CD is the drag coefficient, the mass density of water, A the projected area, U the fluid velocity just upstrea m of the structure, and CI the inertial coefficient. As stated above, this work focuses on drag forces in steady flows.

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18 Storm surge and wind induced currents can be ve ry large, thus producing large drag forces. A better understanding of these forces also pr ovides a basis for unde rstanding one of the components of wave induced forces. A first step in understanding and pred icting these forces is to measure the flow field in the vicinity of a p ile group and to measure the forces for a range of flow velocities. The work reported on in this th esis addresses these issu es for a limited range of conditions. Methodology This study approaches the problem from the st andpoint 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 be tter understand they flow hydrodynamics in the vicinity of pile groups. A force balance is used to measure drag fo rces 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.

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19 CHAPTER 2 LITERATURE REVIEW SchlichtingÂ’s Far Wake Boundary Layer Theory H. Schlichting (1979) provides a means for estima ting the time mean flow velocities in the far-field wake region for steady flow around a singl e circular pile. SchlichtingÂ’s theory is summarized below: Relationship for Wake Half-Width and Velocity Reduction Factor Figure 2-1. Definition Sketch for Flow around a Single Pile In Figure 2-1, U is the free stream velocity, d is the pileÂ’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 yy Ddy u U u h dy u U u h F1 1) ( (2-1) Because u1 is small, O(u1 2) ( terms on the order of u1 2) [i.e.] can be neglected, and the result is dy u U h Fy D 1 (2-2) The drag force is typically expressed as hdU C FD D2 1 (2-3) U x y b Approximate Extent of Wake u

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20 Equating the expressions in 2-2 and 2-3 yields dy u U h hdU C FD D 12 1 (2-4) Therefore dy u dU CD 1 22 1 (2-5) According to Schlichting, there is a dire ct relationship between change in linear momentum and the drag on a pile, dA u U u FD (2-6) He assumes that the control surface has been pl aced 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: 1 1 1u U u u U u U u (2-7) Substituting this into the momentum integral: dA u U FD1 (2-8) For a wake behind a pile, dA is simply hb where h is the pileÂ’s height and b is the halfwidth of the wake. This can be rewritten as follows hb u U FD 1~ (2-9) Using the expression for drag force, FD = 1/2 CD U 2hd, and equating according to the expression above, one obtains b d C U uD2 ~1 (2-10)

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21 Schlichting (1979) cites Pra ndtl’s work, and Prandlt said that “the following rule has withstood the test of time” ~ v Dt Db (2-11) Where v’ is the transverse velocity and D/Dt represents the total derivative. Schlichting (1979) says the following about the origins of the transverse ve locity 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 di rections with respec t to the layer at y1…This argument implies that th e transverse component v’ is of the same order of magnitude as u’ ….” dy u d l const u const v | | | | (2-12) Another way of writing this is that v’ ~ l du/dy Therefore, y u l Dt Db ~ (2-13) At the wake boundary, we have dx db U Dt Db (2-14) and if it is assumed that the mean value of du/dy taken over the half-wid th of the wake is proportional to u1/b the following expression is also true: 1 1* 1 u const u b const Dt Db (2-15) where, again, u1 = U – u. Equating these two expr essions yields

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22 1 11 ~ u u b dx db U (2-16) Where is a constant. Rewritten this gives U u dx db1~ (2-17) From the expression obtained from the momentum equation b d C U uD2 ~1 (2-18) and substituting, the followi ng expression is obtained: d C dx db bD~ 2 (2-19) or, 2 1~ d C x bD (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. 2 1 1~ x d C U uD (2-21) In summary, b ~ x1/2 and u1 ~ x-1/2 Substitution into Boundary Layer Equations In two-dimensional incompressible fl ow, the governing equations are 0 1 y v x u y y u v x u u t u (2-22)

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23 Where is the turbulent shear stress. All pre ssure terms have been dropped because we are assuming that pressure remains constant. Pra ndtlÂ’s mixing length theory can be used in the above equations: y u y u l 2 (2-23) Since the y-velocity component gradient is small, after substitu ting into the governing equations, the following is obtained: 2 1 2 1 2 12 y u y u l x u U (2-24) Schlichting assumes that the mixing length is constant over the wake half-width and proportional to it so that l = b(x) The ratio y/b is introduced as th e independent variable that represents the similarity of the velocity profiles. In agr eement with the proportions obtained from the momentum equations, the following are assumed to be true: f d C x U u dx C B bD D 2 1 1 2 1 (2-25) Inserting into the governing equation, a differential equation is obtained for f() 2 2 12f f B f f (2-26) At the free surface, the velocity reduction s hould equal zero and the y-gradient of the reduction factor should also equal zero. In ot her words, the boundary conditions are defined as at = 1, f = fÂ’ = 0. Integrating twice and applying these boundary conditions yields 2 2 3 21 2 9 1 B f (2-27)

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24 From the momentum integral the integration constant, B can be determined. 10 B (2-28) With Schlichting assuming that 1 1 2 2 310 9 1 d (2-29) The final solution to flow past a single pile then becomes 2 2 3 2 1 1 2 11 18 10 10 b y d C x U u d xC bD D (2-30) A plot of results from Equation 2-30 are shown below. This plot shows u1/UInf: Distance Downstream From Pile (cm)Lateral Distance From Pile (cm)u1/UInf 5 10 15 20 25 30 35 40 45 50 -30 -20 -10 0 10 20 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 2-2. Example of Reduced Velocity Past a Circular Pile Based on Equation 2-30

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25 Parameters for computation of this velo city field are as follows: d = 1.905cm, CD = 1.0, and b = 30cm. Schlichting determined the constant from measured values. According to measurements by Schlichting a nd H. Reichardt (Schlichting 1979) = 0.18. The pile’s locus in Figure 2-2 is at coordi nate point (0,0). Extension to Multiple Piles When looking at more complex pile configurat ions, 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 ydirection. Figure 2-3. Column a nd Row Definition Sketch Flow Direction Column 3 Column 2 Column 1 Row 1 Row 2 Row 3

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26 R Gran Olsson studied flow past an array of multiple in-line piles both experimentally and theoretically. Although OlssonÂ’s analysis is for an infinite row of bars, it will be used to represent a finite number of piles. Gi ven a row with the following configuration Figure 2-4. Olsson Definition Sketch where sis the spacing between rows, Uis the velocity if there were no bars, and u1 is the reduced velocity, Olsson assumes that in a full y developed flow, the ve locity distribution is expected to be a periodic function in y : s y s x A U u2 cos1 1 (2-31) with A being a free constant to be dete rmined from experimental data. Schlichting cites an ex tension of PrandtlÂ’s mixing length theory 2 2 2 2 1 2 y u l y u y u lt (2-32) and states that it seems reasonable to assume that l1 = s/2 Taking the derivative and dividing both sides by gives s y s A U s x l y 2 cos 2 13 2 2 2 2 (2-33) s s u1 U x y

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27 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 y x u U 11 (2-34) Substituting equation 3.31 into equation 3.32, and assuming the mixing length, l is constant yields 3 28 / l s A (2-35) and the solution for velocity reduction becomes s y x s l s U u 2 cos 82 3 1 (2-36) An example of a plot of Equation 2-36 is pr esented 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 ar e located at (0,0) and (0,+/-6). Distance Downstream From Pile (cm)Lateral Distance From Pile (cm)u1/UInf 1 2 3 4 5 6 -10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 Figure 2-5. Plot of Equation 2-36

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28 Olsson verified that this equati on (equation 2-36) is valid for xs > 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/sratio. M.M. Zdravkovich Work on flow beyond circular cylinders is ex tensive; over the year s, 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 Cyli nders: Volume I: Fundamentals and Flow Around Circular Cylinders Volume II: Applications. Each volume comprises over one thousand pages of material concerning several nuan ces regarding the flow around a ci rcular cylinder. Zdravkovich acknowledges that even these ex tensive 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 ci rcular 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 anal ysis of previous work concerni ng 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 bi plane), space engineering (twin boost er rockets), civil engineers (twin chimney stacks) electrical engineering (transmissi on lines), and even chemical engineering (pipe racks). Zdravkovich (2003) divi des previous work into thr ee classification – experiments

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29 performed on tandem cylinders (one cylinder behi nd the other), side-by-si de arrangements, and staggered arrangements (not covered in this thesis). HoriÂ’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 cylindersÂ’ diameters. Hi s results are presented in Figure 2-6. Figure 2-6. Mean Pressure Di stribution Around Tandem Cylinders HoriÂ’s figure is backwards from what one woul d normally think. On the first cylinder, the negative pressures are plotted on the axis behi nd the cylinder while the positive pressures are presented in front of and within the first cylinder. On the seco nd cylinder, the negative pressures are plotted in front of it and the positive pressures are plotted behind it. Interestingly, HoriÂ’s results show that th e pressure distributi on around each of the two cylinders is significantly different from one another. Most no tably, pressure around the first cylinder is positive in the stagnation region. On the downstream cylinder, the gap pressure is

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30 lower than the base pressure – thus inducing a ne gative drag coefficient on the second cylinder. Based on this pressure distribution, one should expect significantly differe nt forcing and drag coefficients on the first and sec ond 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) conducte d extensive pressure field measurements around tandem cylinders. His result s are non-dimensionalized such that 2 05 0 V p p Cp where p0 is the pressure in the free stream, p is the new pressure, is the density of water, and V is the free stream velocity. Igarashi’s results are pres ented in Figure 2-7: Figure 2-7. Mean Pressure Distribution for Re = 35,000 Of note in this study: the pr essure on the front cylinder in the two-cylinder tandem is different than it would have been with a singl e cylinder. In other words, the downstream Downstream Pile Upstream Pile

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31 cylinder induces significant changes on the fl ow patterns on the upstr eam cylinder – thus changing the pressure coeffici ent 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 decrease s, pressure coefficient on the second cylinder obeys a certain pattern.” Instead, the fluctuations of pressure co efficient behave uniquely for the different spacings studied in his experi ments 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 Coope r (1973), Taneda et al. (1973), and Cooper (1974). The single plot is shown in Figure 2-8. Cl osed symbols represent drag coefficients on the first cylinder, and open symb ols represent drag coe fficients on the second cylinder. Figure 2-8. Compilation of Drag Coefficients on Tandem Cylinders

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32 Zdravkovich says that from this plot, a few tr ends can be determined. First, there is negligible Reynolds Number effect on drag coef ficients 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 pileÂ’s diameter, and U is the freestream velocity. His results are presented in Figure 2-9. Figure 2-9. Drag Coefficien t and Strouhal Number Variati on at Higher Reynolds Numbers

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33 WilliamsonÂ’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 pres ented in Figure 2-10. As seen, two vortex streets transform into a large-sc ale 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

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34 Hot-wire tests of flow the field be hind two side-by-side cylinders In 1946, Spivack used hot-wires to explore the fl ow 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 frequenc y 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 Equati on was replaced with 2 d 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 ex isted 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 frequencie s were found outside th e wakes, but these frequencies occurred at differe nt positions in the field of fl ow. Sometimes, these different frequencies would be found simultaneously in the same place. An example is given in Figure 211, which shows unrelated vortex fr equencies of 47 Hz and 108 Hz s cattered in the flow field. Zdravkovich says that these two fr equencies 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 a lthough 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. Fi rst, 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 w ith narrow and wide wakes separated by the

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35 “biased gap flow.” Secondly, th e notion that only stable flow w ould 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). 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 re gion between two cylinders, different drag forces are observed for the side-by-side cylinder confi gurations. Biermann and Herrnste in (1934) measured the drag force on two side-by-side cylinders with a spaci ng of s/d ranging from 1.0 to 5.0. Reynolds Numbers ranged from 65,000 from 163,000. Bier mann and Herrnstein defined a new term called an interference drag coe fficient to account for this. The inte rference drag coefficient is defined Flow Direction

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36 as the drag coefficient expected for a single cy linder minus the observed drag coefficient when two side-by-side cylinders ar e 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 th ere was a bistable flow pattern involved. Origins of the bistable flow phenomenon Recall the bistable gap flow phenomenon wher e two strange paradoxes form. First, an entirely symmetrical oncoming flow into an entirely symmetrical configuration leads to asymmetric narrow and wide wake s behind to identical side-by-si de cylinders. Second, uniform and stable flow induces a non-uniform and ra ndom 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 defl ected when following the surface. Bearman and

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37 Wadcock (1973) performed a series of experiment s to verify the Coanda hypothesis for bistable flow past two side by side cylinde rs, 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 coul d 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 cy linders became “side-by-side”), in terms of the wakes, one cylinder remained “upstream.” In ot her words, one cylinder’s wake remained larger than the other. When the cylinders were comp letely side-by-side, the asymmetric nature of preserved, but became bistable because neithe r 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 ther efore impossible” (Zdravkovich 2003). Pile Groups Previous work on pile groups in -line with the fluid flow is more limited than work on twopile arrangements. However, there have been some studies completed. Shedding patterns behind three in-line cylinders For flow behind two in-lin e cylinders, there are tw o flow regimes. If s/d is less than a critical s/d value, the shedding behind the upstream cy linder 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 tur bulence (Zdravkovich 2003). A third cylinder placed in-line w ith 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 critic al s/d for the second in-line cylinder.

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38 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 s econd and third cylinder. Figure 2-13. Flow Past Three In-Line Cy linders Showing Shedding Behind the Second Cylinder, Re = 2000 Pressure fields behind three in-line cylinders In 1984, Igarashi and Suzuki st udied flow past thr ee in-line cylinders. They observed two types of average pressure coeffici ent 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 involve s the reattachment peak between sixty and eighty degrees. The variation of pressure coefficien t 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 Th ree Cylinders for Different Values of s/d

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39 Igarashi and Suzuki provided smoke visualiza tion to supplement their pressure distribution measurements (Figure 2-15). They also computed the drag coefficients on each of the three cylinders based on pressure di stribution (Figure 2-16). Figure 2-15. Smoke Visualizati on with Three In-Line Cylinders, Re = 13,000 Figure 2-16. Variation in Drag Coef ficient for Three In-Line Cylinders, Re =13,000 s/d = 1.91 s/d = 2.06 s/d = 3.24 s/d = 3.53

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40 Four in-line cylinders In the early to mid 1980Â’s, Aiba (1981) and Igarashi (1986) carried out experiments for four cylinders of equal diameter arranged inline 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 coeffi cients. Figure 2-17 shows each studyÂ’s pressure distributions, Figure 2-18 shows each studyÂ’s drag coefficients, a nd Figure 2-19 shows IgarashiÂ’s accompanying smoke visualization. Figure 2-17. Pressure Distribu tions Around Four In-Line Cylinders. (a) is mean and (b) is fluctuating

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41 Figure 2-18. Drag Coefficient Va riation for Four In-Line Cylinders AibaÂ’s data is on the left and IgarashiÂ’s data is on the right Figure 2-19. Smoke Visualization past Four Cylinders for Four Different Values of s/d

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42 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 coeffici ents 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 ha d 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, PearceyÂ’s Reynolds Number was fixed at 80,000 and his spacing ratio was 5.0. This study found that the maximum drag coeffici ent for flow without a skew angle occurred in the firs t row of cylinders, and the minimu m drag coefficient occurred in the third row. In 1995, Lam and Fang measured the pressure co efficient 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 co mplex cylinder arrays (Ball and Hall 1980, Wardlaw 1974), but their spacing ratios were much gr eater than the spacing ratios typically seen in pile groups. WardlawÂ’s spaci ng ratio was fixed at 10 and Ball and HallÂ’s ratio was fixed at 8. Most spacing ratios in pile groups for br idge foundations are between 3 and 5.

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43 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 multitube 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 fl ow around heat exchanger tubes and flow around bridge foundation pile groups. First, fl ow around heat exchanger tubes is confined between two walls while pile group flows have a free surface. Secondly, the spacing of multitube arrays is smaller than the spacing s een 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 word s, the longitudinal spacing ra tio 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 arra ys with symmetric spacing in the x and y directions.

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44 Only a few experiments with multi-tube arra ys could be found where the spacing was close to the spacing found in bridge foundation pile gro ups. In 1973, Batham measured the pressure field around a 5x9 matrix. His resu lts are presented in Figure 2-21. Figure 2-21. Pressure Coefficient on a 5x9 Cy linder 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 invol ving 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 fr ee-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-stre am turbulence is defined as the ratio between u’ and u times 100% In 1987, Chen and Jendrezejczyk measured the av erage drag forces on different rows of tubes in a 7x7 matrix. They took the entire co lumn 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 Column Column

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45 piles in the first row, “Tube 2” means that these are the averag e values for piles in the second column, etc. Their results are presented in Figure 2-22. Figure 2-22. Average Drag a nd Lift Coefficients for Circ ular Tubes in a 7x7 Matrix As evidenced by these plots from Chen and Je ndrzejczyk, the drag coefficients are very small at these Reynolds Numbers. For one cylinder, the drag coe fficient decreases dramatically when Reynolds Number is greater than 104. It therefore is not unreas onable to assume that the drag coefficient may decrease as well at highe r Reynolds Numbers when dealing with multi-tube arrays. Literature Review Summary First, work on complex pile arrays is seve rely 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 patte rns 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 wi thin a pile group, but no one has yet

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46 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 hi s anthology to formulate a method that will solve for drag forces on a pile group of arbitrary geometry.

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47 CHAPTER 3 MATERIALS AND METHODS SchlichtingÂ’s Far Wake Theory Analytic methods were first tried to predict th e flow field past a ci rcular pile. The hope was that experiments with pile groups would pr oduce 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 defini tion sketch in Fi gure 3-1). Figure 3-1. Definition sketch fo r SchlichtingÂ’s 1917 theory for fl ow 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 yposition normal to the flow, the original upstream velocity, U, the pileÂ’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). 2 2 / 3 2 / 11 18 10 b y d C x U uD, (3-1) 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 U x y b u Approximate Extent of Wake

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48 centerline of the pile, b is the wake half-width, d is the pile diameter, and 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 multip le piles downstream from one another. Suppose a line of piles is spaced along the x-axis in the a bove definition sketch. If the assumption is made that the downstream piles do not a ffect 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 ve locity 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 fluctuat e, one is not limited to aligning piles along the xaxis. In other words, imagine the following configuration: Figure 3-2. Definition sk etch for scenario of offset piles (Top View). One could say that pile 2 “experiences” a ve locity that is smalle r than the velocity “experienced” by pile 1 (pile 1 “experiences” a velocity of U. In fact, this reduced velocity should be given directly by Schlic hting’s equation. Likewise, one could also say that pile 3 “experiences” a velocity that is slightly smalle r than the velocity “expe rienced” by pile 2, and that this reduced velocity shoul d be given by Schlichting’s equati ons (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 enoug h so that pile 2 is still within pile 1’s wake zone. Therefore, U 1 2 3 4

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49 with a slight offset in the y-di rection, 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. Figure 3-3. Definition Sketch of Side-by-Side Piles (Top View). Schlichting cites R. Gran Olsson who proposed th e following equation for side-by-side piles: s y x s l s U u 2 cos 82 3 (3-2) 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/sequals 0.103 for s/d = 8 (Schlichting 1979). Near Wake Region These two formulae provide a good starting poin t for this study. The distance between most piles within a pile group on bridge piers is not large enough for the dow nstream pile to be in the “far-wake zone.” However, the question is how accurate are these equations in the nearwake region. If experimental studies can show that these equations are accurate to within say 1 2 A B U y x

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50 10% then they may be useful in estimating the flow field in the near-wake region. Perhaps they can be modified to improve their acc uracy in the near-wake region. Regardless of whether a combination of Sc hlichtingÂ’s Equation, OlssonÂ’s Equation, and MorisonÂ’s Equation can be used to estimate the drag forces on pile groups, a study of the hydrodynamics in the vicinity of pi le groups and the forces on th e pile groups from a coastal engineering perspective would be useful. A robust dataset of th e velocity at various Reynolds Numbers for various pile configur ations and a dataset of forces on various pile groups at various Reynolds Numbers would be useful for impr oving knowledge of flow patterns and forcing within a pile group. The goals of this thesis are to measure the velocity fi eld in the vicinity of various pile groups, measure the forces on variou s pile groups, and use this dataset to further understanding of fluid flow through a pile group. Pile Configurations Four different pile configurati ons were studied: a si ngle 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 3 d where s is the centerline distan ce 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 typi cal bridge piers ranges from 3 d to 5 d ). Figure 3-4. Pile arrangeme nts used in this study. A B C D U U U U 3 d 3 d 3 d 3 d

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51 All experiments in this study were conducted in a steady flow Measurements were made of the velocity field within each pile configuration Tota l 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 de veloped 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 waterÂ’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 ge neral 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 w ithin 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 cause d by the probesÂ’ 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 sit ting, and it would be di fficult to recreate the conditions within the test flume (water height, water velocity, te mperature within the lab, etc.) exactly for each series of measurements. It w ould be better if there was a method that would capture the entire flow field in one series of measurements.

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52 Particle Image Velocimetry Particle Image Velocimetry (PIV) allows a user to measure flow thr oughout a flow field at specified intervals in time. The technology for PIV has progresse d steadily over the past fifteen years with advances in optics, lasers, electr onics, video, and comput er processing power. According to Raffel (1998), PIVÂ’s ability to measur e an entire velocity fi eld is unique. Except for Doppler global Velocimetry (DGV), which is only appropriate for high-speed airflows, all other techniques for velocity measurement onl y allow measurement at a single point. PIV works by the user adding tiny reflective pa rticles to the fluid and using a high-speed digital camera to capture the fluid flow. These pa rticles must be illuminated in a plane at least twice within a short time interval. Each time light hits the fluid, and subs equently 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 pos ition 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 th e following (Figure 3-5): Figure 3-5. Typical PIV setup schematic drawing (Oshkai 2007).

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53 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 inte rrogation 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 betw een light bursts is known, the velo city 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

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54 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 Saft ronics 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 trum pet entrance section are installe d 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

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55 A high-powered New Wave Resear ch 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

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56 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 ta nk, 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 ta nk so that piles could be inserted into the flow field. The top-rig appara tus consists of two Plexiglas pl ates with holes drilled at the appropriate spacing. This plat e-on-plate design, combined w ith tight-fitting holes through the Plexiglas, minimizes the amount of wobble that th e piles will experience as the flow moves past them. Three-quarter inch Plexigla s piles are inserted into the rig in the desired configuration for each test (Figure 3-12). This pile configurati on was checked to ensure that maximum blockage

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57 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 ha d been installed, the test was initiated. All light in the lab was turned off to make sure that the partic les in the water were only reflecting the laser’s

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58 light. The camera’s time delay was adjusted so th at errors in image pairs would be minimized. This adjustment process was trial and error. Fo r example, a time delay of 10ms might be tried for a certain configuration. A test correlation would be run, and if it prod uced nonsensical results, the time delay was adjusted until the er rors were minimized. Nonsensical results mean that the velocity vectors produ ced by the test correlation di splayed no discernable pattern. Obviously velocity should look uniform upstream and downstream from the pile group. Within the cluster, velocity should sti ll 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.

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59 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 di vides the measurement area into boxes – each box measuring sixty-four pixels by sixty-four pixels, and the boxes overlap one another by seventyfive 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 fl ash 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 tim e 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 me thod. Each pixel on the screen represents a certain number of millimeters. A typical rati o between pixels and millimeters would be 247 pixels = 960mm. As far as the computer “knows,” fo r 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 brig htest pixel from the first image – therefore, no displacement is tracked, which al so is obviously an error, refe rred 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 mi nimized. Presently, there is no quantitative minimization techni que used to determine when the number of small and large errors is at a minimum. Instead, visual insp ection based on experience is used. Generally, an average pixel displacement of 4.00 – 6.00 pixels per image pair is a “good” time delay.

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60 Verification with ADV Probe To ensure that the PIV was recording th e correct velocities, a Son-Tek Micro ADV velocity probe was inserted into the water to de termine the velocity in the free stream. Results from the ADV readings were compared with a PI V reading with no piles. Both the PIV and the ADV read at the same frequency – 15Hz. The A DV and the PIV are relatively close, with the error from the PIV compared to the ADV betwee n 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 wate r had reached the piles. 05101520253002468101214161820Time (sec)Velocity (cm/s) First Velocity Second Velocity Third Velocity Figure 3-15. ADV Probe Measurements

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61 Figure 3-16. Average Velocity Image in PIV at First Velocity Figure 3-17. Average Velocity Image in PIV at Second Velocity

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62 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 ar e blacked out, or “masked,” to prevent the crosscorrelation algorithm from calculating any velo city in these regions (Figure 3-19). Figure 3-19. Masked PIV Image (9 Piles)

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63 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 check ed for errors. First, the “large errors” are eliminated by checking each image for the case wh ere one velocity vector is significantly larger than the surrounding velocity vectors. If this happens, the large ve ctor 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 wr ong 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

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64 (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 consiste d of a metal carriage that moves freely laterally in the x-direction and vertically in the z-dire ction. 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 betw een 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 configuratio ns. The force balance is connected to an ELEKTAT Deck Force Analyzer DF2D am plifier and the amplifier is conn ected to a computer so that the computer can signal when the force bala nce should begin its measurement. The force-

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65 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 balanceÂ’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 we re fixed in the x-direct ion (the direction along the flume). The Plexiglas Force Balance Flume is fourt een inches wide by twenty inches deep by twenty-seven feet long. A com puter-controlled flapgate is instal led 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 ba lance experiments was varied to ensure that the

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66 amount of force on each pile c onfiguration 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).

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67 Figure 3-25. SonTek MICRO-ADV Robot This robot, like the ADV probe in the PIV flume, can be lowere d 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 Ve nturi 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 with in the flume to ensure that depth remains constant during the test. A top-rig mechanism, whic h is similar to the rig used in the PIV Flume, is attached to the force balance, and the same p iles used in the PIV experiments were used for the force measurements.

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68 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 confi guration, 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 m easurement was taken. Each experiment was repeated three times to ensure repeatability. The force balance only measures the force on th e pile group. Attempts were made to use the force balance to isolate the forces on each pi le by adding piles one-by-one. For example, to determine the force on the third pile in the thr ee-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 pileÂ’s indivi dual 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. U pon analyzing the results from e xperiments in the Force Balance and PIV flumes, and reviewing more literature, it became clear that this assumption was invalid.

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69 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 th e same pile if another pile is placed in the first pile’s wake. As a result, the Force Balan ce 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 Numb ers. Capturing data in the PIV is time consuming – each dataset take s about 40 hours to get from the “capture” point to the final product because of post-processing tim e. Measurements in the force balance flume are much easier – one simply has to adjust th e 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 ve locity 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 m easure the forces on the piles

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70 CHAPTER 4 EXPERIMENTAL RESULTS PIV Data Velocity Fields PIV average velocity color contour plots are sh own 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 ve locity profiles normal to the approach flow were extracted and presented in Figure 4-13 th rough 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: U where U is the velocity vector defined as wk vj ui U and is the vorticity. The velocity data was used to compute the vorticity for each image frame, and contour plots for vorticity are plotted fo r each frame. These frames are compiled into a series of animations that show how the vortices move down stream 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 enti re 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:

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71 U fd St (4-1) 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 Re ynolds number plot should look like that shown in Figure 4-43. Given these valu es 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

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72 Figure 4-2. Re = 3.85x103, 1 Pile Average Velocity Figure 4-3. Re = 2.57x103, 1 Pile Average Velocity

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73 Figure 4-4. Re = 5.13x103, 2 Piles Average Velocity Figure 4-5. Re = 3.85x103, 2 Piles Average Velocity

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74 Figure 4-6. Re = 2.57x103, 2 Piles Average Velocity Figure 4-7. Re = 5.13x103, 3 Piles Average Velocity

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75 Figure 4-8. Re = 3.85x103, 3 Piles Average Velocity Figure 4-9. Re = 2.57x103, 3 Piles Average Velocity

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76 Figure 4-10. Re = 5.13x103, 9 Piles Average Velocity Figure 4-11. Re = 3.85x103, 9 Piles Average Velocity

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77 Figure 4-12. Re = 2.57x103, 9 Piles Average Velocity 050100150200250-505101520253035V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-13. Velocity Profile s through the Center of th e First Pile for Re = 5.13x103

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78 050100150200250-50510152025303540V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-14. Velocity Profiles 20mm from the Center of the First Piles for Re = 5.13x103 050100150200250-50510152025303540V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-15. Velocity Profiles 40mm from th e Center of the First Pile for Re = 5.13x103

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79 050100150200250-505101520253035V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-16. Velocity Profiles 60mm from th e Center of the First Pile for Re = 5.13x103 05010015020025005101520253035V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-17. Velocity Profiles 80mm from th e Center of the First Pile for Re = 5.13x103

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80 05010015020025005101520253035V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-18. Velocity Profiles 100mm from th e Center of the First Pile for Re = 5.13x103 050100150200250-505101520253035V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-19. Velocity Profiles 120mm from th e Center of the First Pile for Re = 5.13x103

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81 05010015020025005101520253035V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-20. Velocity Profiles 140mm from th e Center of the First Pile for Re = 5.13x103 05010015020025005101520253035V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-21. Velocity Profiles 160mm from th e Center of the First Pile for Re = 5.13x103

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82 05010015020025005101520253035V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-22. Velocity Profiles 180mm from th e Center of the First Pile for Re = 5.13x103 050100150200250-50510152025V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-23. Velocity Profile s through the Center of th e First Pile for Re = 3.85x103

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83 050100150200250-5051015202530V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-24. Velocity Profiles 20mm from th e Center of the First Pile for Re = 3.85x103 050100150200250-5051015202530V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-25. Velocity Profiles 40mm from th e Center of the First Pile for Re = 3.85x103

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84 050100150200250-5051015202530V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-26. Velocity Profiles 60mm from th e Center of the First Pile for Re = 3.85x103 0501001502002500510152025V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-27. Velocity Profiles 80mm from th e Center of the First Pile for Re = 3.85x103

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85 0501001502002500510152025V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-28. Velocity Profiles 100mm from th e Center of the First Pile for Re = 3.85x103 050100150200250-50510152025V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-29. Velocity Profiles 120mm from th e Center of the First Pile for Re = 3.85x103

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86 0501001502002500510152025V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-30. Velocity Profiles 140mm from th e Center of the First Pile for Re = 3.85x103 0501001502002500510152025V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-31. Velocity Profiles 160mm from th e Center of the First Pile for Re = 3.85x103

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87 0501001502002500510152025V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-32. Velocity Profiles 180mm from th e Center of the First Pile for Re = 3.85x103 050100150200250-20246810121416V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-33. Velocity Profile s through the Center of th e First Pile for Re = 2.57x103

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88 050100150200250-4-2024681012141618V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-34. Velocity Profiles 20mm from th e Center of the First Pile for Re = 2.57x103 050100150200250-2024681012141618V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-35. Velocity Profiles 40mm from th e Center of the First Pile for Re = 2.57x103

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89 050100150200250-2024681012141618V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-36. Velocity Profiles 60mm from th e Center of the First Pile for Re = 2.57x103 0501001502002500246810121416V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-37. Velocity Profiles 80mm from th e Center of the First Pile for Re = 2.57x103

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90 0501001502002500246810121416V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-38. Velocity Profiles 100mm from th e Center of the First Pile for Re = 2.57x103 050100150200250-20246810121416V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-39. Velocity Profiles 120mm from th e Center of the First Pile for Re = 2.57x103

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91 0501001502002500246810121416V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-40. Velocity Profiles 140mm from th e Center of the First Pile for Re = 2.57x103 0501001502002500246810121416V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-41. Velocity Profiles 160mm from th e Center of the First Pile for Re = 2.57x103

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92 0501001502002500246810121416V (cm/sec)Y (mm) 1_cyl 3_cyl 9_cyl Figure 4-42. Velocity Profiles 180mm from th e Center of the First Pile for Re = 2.57x103 Figure 4-43. Re = 5.13x103 Average Vorticity for One Pile

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93 Figure 4-44. Re = 3.85x103 Average Vorticity for One Pile Figure 4-45. Re = 2.57x103 Average Vorticity for One Pile

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94 Figure 4-46. Re = 5.13x103 Average Vorticity for Two Piles Figure 4-47. Re = 3.85x103 Average Vorticity for Two Piles

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95 Figure 4-48. Re = 2.57x103 Average Vorticity for Two Piles Figure 4-49. Re = 5.13x103 Average Vorticity for Three Piles

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96 Figure 4-50. Re = 3.85x103 Average Vorticity for Three Piles Figure 4-51. Re = 2.57x103 Average Vorticity for Three Piles

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97 Figure 4-52. Re = 5.13x103 Average Vorticity for Nine Piles Figure 4-53. Re = 3.85x103 Average Vorticity for Nine Piles

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98 Figure 4-54. Re = 2.57x103 Average Vorticity for Nine Piles Figure 4-55. Published Strouhal Number Data (Sarpkaya 1981)

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99 01234500.511.522.533.544.55Theoretical Frequency (1/s)Measured 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 Re ynolds 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 plot of Fx/h vs. dV2, will equal the drag coefficient. This plot is shown in Figure 4-57.

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100 y = 0.9783x R2 = 0.935200.511.522.533.500.511.522.533.5 dV2Fx/h Data y = x Linear Data Fit Figure 4-57. Drag Coefficient for 1 Pile As can be seen in Figure 4-57, the slope of th e best-fit line is approxima tely equal to 1.0. This indicates that the force balance is accurate ly measuring forces in this range of Reynolds Numbers. Aligned Piles After the one-pile experiment was run, piles we re added behind the first pile to see what their effect would be on the total force on the pi le group. First, a second pile was added behind the first pile, and then a third. Forces on th e pile groups are presente d in Figure 4-58. The yellow line represents the force on one pile, the re d 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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101 0.001.002.003.004.005.002.00E+034.00E+036.00E+038.00E+031.00E+041.20E+04ReFx/h (System) 1 Pile 2 Aligned Piles 3 Aligned Piles Lowest PIV Reading Highest PIV Reading Figure 4-58. Results for One Row of Piles Side-by-Side Piles An experiment was run on a tw o side-by-side pile arrangeme nt to determine how flow between the two piles affected fo rcing on the pile group. The re sults are presented in Figure 459: 0.001.002.003.004.005.006.007.002.00E+034.00E+036.00E+038.00E+031.00E+041.20E+04ReFx/h (System) 1 Pile 2 Piles Side-by-Side Lowest PIV Reading Highest PIV Reading Figure 4-59. Results for Two Side-by-Side Piles.

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102 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 co mplex pile arrangements were studied with the hope that subtracting the result s of one experiment from anot her 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 downstr eam piles changes the pressure distribution and flow around the upstream piles. Unfortunately th en, it is not possible to determine the force on individual piles within th e group from the data obtained in this study. The total force on the pile group configurations investigated are, however, valid. The confi gurations tested are shown in Figure 4-60a and 4-60b.. Figure 4-60a. First Two Complex Conf igurations Run in the Force Balance U U (1.) (2.)

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103 Figure 4-60b. Second Two Complex Configur ations Run in the Force Balance Flume The total force on these pile group arrangeme nts 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 ni ne-pile arrangement. 0.002.004.006.008.0010.0012.0014.0016.002.00E+034.00E+036.00E+038.00E+031.00E+041.20E+04ReFx/h (System) 4 piles 6 piles 7 piles 9 piles Lowest PIV Reading Highest PIV Reading Figure 4-61. Results for Complex Pile Arrangements U U (3.) (4.)

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104 Results Summary PIV data was used to create the following: One-minute average velocity fields One minute average velocity profile s normal to the approach flow One minute average vorticity magnitude plots Animations of fifteen s econd averages of vorticity A comparison between observed vortex shed ding 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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105 CHAPTER 5 DISCUSSION PIV Data Analysis Recall that the original motiva tion for this research was to find a way to use an expression similar to the expressions given by Schlichting a nd Olsson to find a way to reduce velocity in the near-wake region. The hope was that because dr ag force on a pile is given by the following equation, | | 2 1U AU C FD D (5-1) accurate velocity measurements would yield a be tter way the approach velocities at downstream piles. The presumption was that if piles ar e 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 th e 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 pers ist along the en tire line of piles. This method assumes that the pressure fiel d around the first pile in the alignment and subsequent piles in the alignment are similar. Th erefore, 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 experiment s, and looking at exis ting data, it is clear that this entire line of thi nking does not work. Using a “reduc ed velocity” method for predicting

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106 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 expe cted. The presence of the pile induced a wake downstream from the pile. As the wake-re gion 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-wak e of the one-pile arrangement, velocity would eventually retu rn 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 th e highest Reynolds Number, the wake extended about 40mm (about 2 diameters) dow nstream from the pile, and at the lowest region, the extent of the wake was almost 60mm (about 3 di ameters) downstream from the pile. Recall that pile spacing was fixed at 3d where d is the diameter. Al so recall that threequarter 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 Numb ers) the front face of th e pile still penetrates the first pile’s wake. In the worst case (small Re ynolds 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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107 velocity. A pile in a negative velocity field likely has a much differe nt pressure gradient surrounding it than a pile in a positive velocity field. Velocities within the dead region are sim ilar 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 measurem ents. 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 wa ke 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 no t seem to be a function of upstream velocity because the extent of the wake behind the sec ond pile is similar for each Reynolds Number. Velocities leading into the th ird pile are similarly reduced at each Reynolds Number. Recall that what was expected was that th e 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 reduc tion 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 lo wer, 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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108 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 for ce on the second pile. Additionally, the wake behind the third pile is almost non-existe nt at all Reynolds Numbers. It is almost as if the wake beyond th e 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 pi le extends itself to beyond the third pile because of the presence of the two extra piles. Nine pile arrangement The nine pile arrangement reinforces th e observations made with the three pile arrangement. In the nine pile configuration, if one only looked at th e wake regions, it would appear that the three rows of p iles act almost independent of each other. The wake from the top three piles in the PIV image seem to act indepe ndently from those in the second and bottom rows of the PIV image. The wake behind the firs t pile in each row enco mpasses 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 –i ndicating a positive force on the third pile. A closer look at the nine pile arrangement doe s, however, reveal that there is some flow interaction between the pile rows This is due to the flow bei ng 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 arrangeme nt exhibit almost the same patterns as the wake regions in the three in-line pile arrang ement in that it almost looks like the wake from the front pile is just extending itself further downs tream because of the presence of the other two

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109 piles. The nine-pile configuration seems to conf irm 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 configurati on act independently from one anot her. The piles’ wakes in the two-pile arrangement appear to interfere fr om the front face of th e 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 wake s 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 came ra 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 loca tion 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 thes e experiments. The evidence for this comes from the velocity fields obtained without any p iles (Figures 5-2 – 5-4).

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110 Figure 5-1. Schematic Drawi ng of Light Bouncing Off Mirror Figure 5-2. Average Velocity Im age in PIV at First Velocity Light Sheet Camera Mirror

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111 Figure 5-3. Average Velocity Im age in PIV at Second Velocity Figure 5-4. Average Velocity Im age in PIV at Third Velocity

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112 Because the tank in which these experiments we re run had relatively parallel side walls, the velocity field across the horizon tal cross section of the tank s hould be relatively uniform. If morphing was an issue, the velocities at the t op 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 th e 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 woul d describe the average velocities within the wake regions. This would not, however, solve the prob lem 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 fiel d data and the results were used to validate observed Strouhal Numbers against published vort ex shedding frequency results. Observed shedding frequencies in these tests were consis tently higher than published data, but only slightly. At higher Reynolds Numbers, the obs erved 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 fo r observed vs. published frequencies assumed a Strouhal Number of 0.2, but using a value slight ly different than 0.2 would allow the observed shedding frequencie s to fall directly on the y=x line.

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113 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 indicat es that the force balance is measuring the correct forces because the drag coefficient meas ured with the force balance is approximately equal to published values. Previ ous studies have show n 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 p iles spaced at 3 diameters apart behave independently from one another is not 100% correct. The force on the tw o-pile arrangement is slightly greater than

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114 double that of the force on a single pile. If one assumes that th e force is evenly distributed between the two piles, one would b ack-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 cau sed 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 diffe rent 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 p ile induces changes in the wake regions of the first pile. Attempts were made to subtract a sing le 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 pile s, 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 subtra ction 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 da ta, which indicates that the force on the second pile is almost zero.

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115 0123452.00E+034.00E+036.00E+038.00E+031.00E+041.20E+04 ReFx/h System 1st Pile 2nd Pile 3rd Pile System Avg 1st Pile Avg 2nd Pile Avg 3rd Pile Avg Figure 5-6. Deduced Drag Forces Based on Me asurements on a Three Inline Pile Arrangement The problem is that much of the PIV data w ould also indicate that the force on the second pile should be negative, and computations us ing 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 incr eased by the presence of the second pile. Force Decrease at High Reynolds Numbers in the Three-Pile Configuration Also of note with the three-pi le 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 time s. This is also very counterintuitive because one would think that increasing the velocity shou ld increase the total fo rce 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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116 possible that the flow separates at a certain angle when there ar e three piles in line with one another. As the velocity increases, the separa tion point may change to the point where it results in a higher pressure in the wake region a nd thus a reduced pres sure drag component. The tail-off of forces under the three pile conf iguration may also be due to experimental error. The force balance has four different rang es 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 Reyn olds Number, the force might be out of range – 1.5N for example. When the e xperiments 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-pr ocessing, 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 arrangem ents exhibit a higher total force when more piles are added to the array. The original concep t for this test sequence was to obtain the forces on the individual piles in the group with as few test s 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 ca se. While not provi ding sufficient data to determine the forces on the individual piles with in 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-l ine arrangement. The consistency of the data between the data sets w ould seem to rule out experimental error because

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117 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 sepa ration point on the lead ing piles impacting the pressure distribution on these piles. A study th at includes the measurem ent 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 s econd pile in the three in-line pile arrangement must be negative, the hope wa s 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 vorte x shedding, and not the steady-state velocity in the flow domain, then the dominant velo city fluctuation frequency should approximately match the dominant force fluctuat ion frequency, at a gi ven Reynolds Number. Additionally, the force and velocity time series were analyzed to determine if values used throughout this paper (such as averages) are stat istically meaningful. All force and velocity measurements were “de-meaned.” In other word s, the average force a nd 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 meaningf ul statistic, then the de-meaned velocity or force signal should fluctuate around zero, and the sp ectrum should show equal energi es 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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118 2.) Discuss other possible methods for anal yzing the flow around the piles. Because the PIV measures velocity everywhere within the flow field, it would be possible to extract the flow spectra ever ywhere throughout the field. W ithin the flow window, a few key points that would isolate the vor tex effects were thor oughly 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: Figure 5-7. Labeling Scheme for PIV Spectral Analysis. Only the five flow conditions that allowe d 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 th e idea to perform a spectral analysis came “Point A1” “Point A” “Point B” “Point C” “Point D”

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119 long after experiments for this thesis were completed. However, the Reynolds Numbers are close enough to allow for an accurate co mparison between the measurements. No-Pile PIV experiments Before any pile configurations were analyzed an analysis was conduc ted 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 58 through 5-13 are timeseries 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. Ad ditionally, 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. 0 5 10 15 -4 -3 -2 -1 0 1 2 3 Time (sec)Velocity (cm/sec) Figure 5-8. 0-Pile Time-Series in Middle of PIV Window, Re = 5.13x103

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120 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 Frequency (Hz)Spectral Density Spectrum for 0 Piles Velocity in Middle of PIV Window Re = 5 13e3 Figure 5-9. 0-Pile Spectrum in Middle of PIV Window, Re = 5.13x103 0 5 10 15 -4 -3 -2 -1 0 1 2 3 Time (sec)Velocity (cm/sec) y Figure 5-10. 0-Pile Time Series in Middle of PIV Window, Re = 3.85x103

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121 0 1 2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (Hz)Spectral Density Spectr u m for 0 Piles Velocit y in Middle of PIV Windo w Re = 3 85e3 Figure 5-11. 0-Pile Spectrum in Middle of PIV Window, Re = 3.85x103 0 5 10 15 -3 -2 -1 0 1 2 3 Time (sec)Velocity (cm/sec) D eM eane d V e l oc it y vs. Ti me f or 0 Pil es a t i n Middl e o f PIV Wi n d ow, R e = 2 57 e 3 Figure 5-12. 0-Pile Time Series in Middle of PIV Window, Re = 2.57x103

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122 0 1 2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (Hz)Spectral Density Spectrum for 0 Piles Velocity in Middle of PIV Window Re = 2 57e3 Figure 5-13. 0-Pile Time Series in Middle of PIV Window, Re = 2.57x103 At first, it is unclear wh at causes this ~4.2Hz spike at all three Reynolds Numbers. However, if the tank dimensions are analyzed, th e 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 fl ow 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 dist urbance between 1.0Hz and 1.5Hz – depends on Reynolds Number. Th is disturbance does not match any of the fundamental frequencies for this flume, so res onating flume conditions cannot be its cause. The

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123 only other thing that could cause disturbances in the flow-field during this experiment is the pump that is pumping water th rough 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 onl y running at 25% capacity, it will have another frequency. Because this flow fluctuation depe nds 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 5.5H z disturbance, occurs only at the lowest Reynolds Number. It is unknown what causes th is disturbance as it does not match any fundamental frequencies that result s from the geometry of the flum e, nor does it appear at any of the other Reynolds Numbers. Further investigati on would be useful to determine the causes of this flow fluctuation. Single-Pile Experiments Re ~ 5x103 PIV and force balance time-series and spectral re sults 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 statistica lly 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 th e force balance’s measurement range (10Hz). Zooming into a five-second window of the time series (F igure 5-14), clearly exposes the saw-toothed nature of the signal.

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124 0 5 10 15 -15 -10 -5 0 5 10 15 Time (sec)Velocity (cm/sec) D eM eane d V e l oc it y vs. Ti me f or 1 Pil e a t P o i n t A1 R e = 5 13 e 3 Figure 5-14. De-Meaned Velocity vs. Ti me for 1 Pile at Point A1, Re = 5.13x103 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 14 Frequency (Hz)Spectral Density Spectrum for 1 Pile Velocity at Point A1 Re = 5 13e3 Figure 5-15. Velocity Spectrum for 1 Pile at Point A1, Re = 5.13x103

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125 0 10 20 30 40 50 60 -0.15 -0.1 -0.05 0 0.05 0.1 Time (sec)Force (N) Figure 5-16. De-Meaned Force vs. Time for 1 Pile, Re = 4.72x103 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 2 3 x 10-4 Frequency (Hz)Spectral DensitySpectrum for 1 Pile, Re = 5.13e3 Figure 5-17. Force Spectrum for 1 Pile, Re = 4.72x103

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126 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Time (sec)Force (N) D eM eane d F orce vs. Ti me f or 1 Pil e, R e = 5 13 e 3 Figure 5-18. Zoomed Force Time Series, Re = 5.13x103 Re ~ 4x103 At the middle Reynolds Number, the ~4.5Hz sp ike due to the resona nce of the flume is still present in the signal. However, there does not appear to be any ot her significant spikes in the velocity signal. The wave-like nature of th e force signal is also mi nimal as evidenced by the small magnitudes of the spectral density f unction, 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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127 0 5 10 15 -10 -8 -6 -4 -2 0 2 4 6 8 10 Time (sec)Velocity (cm/sec) De Meaned Velocity vs Time for 1 Pile at Point A1 Re = 3 85e3 Figure 5-19. De-Meaned Velocity vs. Ti me for 1 Pile at Point A1, Re = 3.85x103 0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 Frequency (Hz)Spectral Density py, Figure 5-20. Velocity Spectrum for 1 Pile, Re = 3.85x103

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128 0 10 20 30 40 50 60 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Time (sec)Force (N) Figure 5-21. De-Meaned Force vs. Time for 1 Pile, Re = 3.83x103 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 x 10-4 Frequency (Hz)Spectral DensitySpectrum for 1 Pile, Re = 3.85e3 Figure 5-22. Force Spectrum for 1 Pile, Re = 3.83x103

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129 Three-Pile Experiments As indicated in Figure 5-7, the three-pile a rrangements were sampled at four different points. Because vortex movies were made of the flow around a ll pile configurations, and the goal was to isolate the vortex effects on the pile groups, the po ints that were chosen were purposely chosen where vortex effe cts would be seen. The goal was to isolate the fundamental vortex frequencies and to see if the vortex fre quencies 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 pres ented from Figure 5-23 through Figure 5-25: 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 2 3 4 5 6 7 x 10-4 Frequency (Hz)Spectral DensitySpectrum for 3 Piles, Re = 4.76e3 Figure 5-23. Force Spectrum for 3 Piles, Re = 4.76x103

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130 0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 30 Frequency (Hz)Spectral Density Point A Point B Point C Point D Figure 5-24. U-Velocity Spectrum for 3 Piles, Re = 5.13x103 0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 30 35 Frequency (Hz)Spectral Density yp Point A Point B Point C Point D Figure 5-25. V-Velocity Spectrum for 3 Piles, Re =5.13x103

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131 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 e xperiment. Because the velocity spectra do not match the forcing spectra, it is reasonable to conc lude 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 Fi gure 5-26 through Figure 528: 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 2 3 x 10-4 Frequency (Hz)Spectral DensitySpectrum for 3 Piles, Re = 3.85e3 Figure 5-26. Force Spectrum for 3 Piles, Re = 3.85x103

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132 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 14 Frequency (Hz)Spectral Density Point A Point B Point C Point D Figure 5-27. U-Velocity Spectrum for 3 Piles, Re = 3.85x103 0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 Frequency (Hz)Spectral Density VVelocity Spectrum for Re 3.85e3 Point A Point B Point C Point D Figure 5-28. V-Velocity Spectrum for Re = 3.85x103

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133 Results at this Reynolds Numb er are very similar to re sults 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 forc ing spectrum, but it does correspond to a similar spike in the u-velocity spectra. The second spike in the v-veloci ty spectra does match a spike in the forcing spectra, but again, from the zero-pile anal ysis, it is likely that a portion of this spike is due to experimental resonance. Because the largest spike in ve locity does not correspond to any spike in forcing (in fact, it coin cides with a minimum at this Re ynolds Number!), this seems to indicate that once again, the vortices have a small effect on th e overall forcing on the pile group. Re ~ 3x103 Finally, spectral results at the lowest Reynolds Number ar e presented from Figure 5-29 through Figure 5-31. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 2 3 x 10-4 Frequency (Hz)Spectral DensitySpectrum for 3 Piles, Re = 2.86e3 Figure 5-29. Force Spectrum for 3 Piles, Re = 2.86x103

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134 0 1 2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 Frequency (Hz)Spectral Density U V e l oc it y S pec t rum f or R e = 2 57 e 3 Point A Point B Point C Point D Figure 5-30. U-Velocity Spectrum for 3 Piles, Re = 2.57x103 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 10 Frequency (Hz)Spectral Density VVelocity Spectrum for Re 2.57e3 Point A Point B Point C Point D Figure 5-31. V-Velocity Spectrum for 3 Piles, Re = 2.57x103

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135 Results at this Reynolds Nu mber 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 anothe r 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 proba bly be explained by the same resonant condition present in all the experiments, this signal is uniqu e in that the spectral density of this signal is higher than all the other experiments. Results at this Reynolds Number indicate that at this velo city, vortex effects have the greatest net-effect on forcing relati ve 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 forcin g 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 for ce 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-eff ects of vortex forcing on piles at even lower Reynolds Numbers would be valuable. Would th e magnitude of the forcing on a downstream pile, which is on the order of 0.02N, remain the sa me 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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136 CHAPTER 6 FUTURE WORK Additional Pressure Field Measurements It would be beneficial to c onduct drag force tests that incl ude 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 pile s, 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 di rection. 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 th e locations within the group. Other Drag Force Measurements It would be beneficial to c onduct experiments on other configur ations of pile groups. The most complex group studied in this thesis was a three-by-three ma trix 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 al ong the same line? Would the pattern of a large zero velocity zone behind the odd-numbered piles continue down the en tire line? Would velocity consistently be larger on the even-numbered piles in the line? From the experiments in this thesis, it seem s as though this pattern would persist. The wake behind the first pile would engulf the sec ond pile, the wake behi nd the third pile would

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137 engulf the fourth pile, and so-on. Measurements to back up this hypothesis would be useful so that a more comprehensive method for determini ng 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 in teraction when the flow is sk ewed 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 measuremen ts would also greatly enhance this study. Although the present TFHRC for ce balance setup does not allow for transverse force measurements, it would be possible to design a ne w 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 vorti ces 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 fo rces 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 experi ment would help to validate that the drag forcing data that was obtained in this thesis under stea dy flow conditions can be utilized to predict the drag component of the Morison Equation under unsteady flow conditions. Secondly, this

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138 experiment would give a value for the inertial for ce on the piles with in 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 great est wave force. Sarpkaya and Issacsson (1981) say that this is not necessari ly 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 construc tive interference of waves. As waves propagate past the first couple of columns of piles in a group, they are scatte red, 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 inertia l 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 sm all-scale laboratory testing of pile group configurations. After a reliable method for dete rmining 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 high est Reynolds Number possible in the TFHRC flume is on the order of 104. Under field conditions, pile s will be subject to average Reynolds Numbers two orders of magnitude high er. It has already been shown that drag coefficient varies considerab ly at higher Reynolds Number s. 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 si milar things happen during wave action – i.e., as

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139 Reynolds Number approaches the Reynolds Number that will be expected in the field, the behavior of the drag and inertial forcing com ponents of wave forcing will also change. Larger-scale tests are then need ed to verify the lab results. Ideally, new bridge piers that are built could include an instrumentation packag e 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 pr esent 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 a rrangement, pile group orientation to the flow, etc. that need to be inve stigated in the laboratory. Th e addition of waves increases the complexity of the problem by adding additional wa ter and wave parameters (i.e. the addition of water depth, and wave hei ght, period and direction).

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140 LIST OF REFERENCES Aiba, S., H. Tsuchida, & T. Ota (1981). “Heat Transfer Around a Tube in a Bank.” Bulletin of Japan Society of Mechanical Engineers, 23, 311-19. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1097. Ball, D. J. & C. D. Hall (1980). “Drag of Yawed Pile Groups at Lo w Reynolds Numbers.” ASCE Journal of Waterways and Har bors Coastal Engineering Division, 106, 229-38. Batham, J.P (1973). “Pressure Distributions on In-Line Tube Arrays in Cross Flow.” Proceedings British Nuclear Engineering Soc iety Symposium Vibration Problems in Industry, ed. Wakefield J. Keswick, 28. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1137. Bearman, P.W. & A. J. Wadcock (1973). “The In teraction Between a Pair of Circular Cylinders Normal to a Stream.” Journal of Fluid Mechanics, 61, 499-511. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinde rs Volume 2: Applications, 1027. Bierman, D. & W. H. Herrnstein (1934). “T he Interference Between Struts in Various Combinations.” National Advisor Committee for Aeronautics TR 468. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1024. Chen, S. S. & J. A. Jendrzejczyk (1987). “Flu id excitation Forces Acting on a Square Tube Array.” Journal of Fluids Engineering, 109, 415-23. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1138. Crowe, C. T., D. F. Elger & J. A. Robinson (2001). Engineering Fluid Mechanics. John Wiley & Sons, Inc., New York. Dean, R. G. & R. A. Dalrymple (2002). Coastal Processes with E ngineering Applications. Cambridge University Press, New York. Dean, Robert G. & R. A. Dalrymple (2000). Water Wave Mechanics for Engineers and Scientists. World Scientific Publis hing Company, London. Hori, E (1959). “Average Flow Fields Around a Group of Circular Cylinders.” Proc. 9th Japan National Congress of Applied Mechanics, Tokyo, Japan, 231-4. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders, Volume 2 Applications, 1002. Lam, K. & X. Fang (1995). “The Effect of Inte rference of Four Equal-Distanced Cylinders in Cross Flow on Pressure and Force Coefficients.” Journal of Fluids and Structures, 9, 195-214. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1100. Igarashi, T (1981). “Characteris tics of a Flow Around Two Circ ular Cylinders in Tandem.” Bulletin of Japan Society of Mechanical Engineers, 323-31. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders, Volume 2 Applications, 1003.

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141 Igarashi, T (1986). “Characteristics of the Fl ow Around Four Circular Cylinders Arranged InLine.” Bulletin of Japan Society of Mechanical Engineers, 29, 751-7. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinde rs Volume 2: Applications, 1097. Igarashi, T. & K. Suzuki (1984). “Charact eristics of the Flow Around Three Circular Cylinders.” Bulletin of Japan Society of Mechanical Engineering 27, 2397-404, quoted in M. M. Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1080. Ishigai, S., S. Nishikawa, K. Nishimura, & K. Cho (1972). ”Experiment al Study on Structure of Gas Flow in Tube Banks With Tube Axes Normal to Flow.” Bulletin of Japan Society of Mechanical Engineers 15, 949-56. Quoted in M.M. Zdravkovich Flow Around Circular Cylinders Volume 2: Applications, 1027. Oshkai, P. (2007). “Experiment al System and Techniques.” University of Victoria, Victoria, BC, Canada, (June 28, 2007). Pearcey, H. H., R. F. Cash, & I. J. Salter (1982). “Interference Effects on the Drag Loading for Groups of Cylinders in Uni-Directional Flow.” National Maritime Institute, UK Rep. NMI R130, 22, 27. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders Volume 2: Applications, 1099. Raffel, M., C. Willert, & J. Kompenhans (1998). Particle Image Velocimetry, a Practical Guide. Springer, New York. Schlichting, H (1979). McGraw-Hill Series in Mechanica l Engineering: Boundary LayerTheory. Frank J. Cerra, ed., McGraw Hill, Inc., New York. Sarpkaya, T. & M. Isaacson. Mechanics of Wave Forces on Offshore Structures. Litton Educational Publishing Inc., New York. Spivack, H. M. (1946). “Vortex Frequency and Flow Pattern in the Wake of Two Parallel Cylinders at Varied Spacing Normal to an Air Stream.” Journal of Aero. Science, 13, 289-301. Quoted in M.M. Zdravkovich Flow Around Circular Cylinders Volume 2 Applications, 1018. Sumer, M. B. & J. Fredsoe (1997). Hydrodynamics Around Cylindrical Structures. World Scientific Publishing Company, London. Sumer, M. B. & J. Fredsoe (2002). The Mechanics of Scour in the Marine Environment. World Scientific Publishing Company, London. Wardlaw, R. L., K. R. Cooper, R. G. Ko, & J. A. Watts (1974). “Wind Tunnel and Analytical Investigations into th e Aeroelastic Behavior of Bundled Conductors.” Institute of Electrical and Electronics Engineering Su mmer Meeting Energy Resources Conference, Anaheim, CA, 368-77.

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142 Werle, H (1972). “Flo w Past Tube Banks.” Revue Francais de Mecanique, 41, 28. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinde rs Volume 2: Applications, 1078. Williamson, C. H. K. (1985). “Evolution of a Single Wake Behind a Pair of Bluff Bodies.” Journal of Fluid Mechanics 159, 1-18. Quoted in M.M. Zdravkovich, Flow Around Circular Cylinders Volume 2 Applications, 1018. Zdravkovich, M. M (1997). Flow Around Circular Cylinders Volume I, Fundamentals, Oxford University Press, New York. Zdravkovich, M. M (2003). Flow Around Circular Cylinders, Volume II, Applications. Oxford University Press, New York.

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143 BIOGRAPHICAL SKETCH Raphael Crowley has always had a dual-interest in both fluids and structures. In 2004, Crowley graduated from Bucknell Un iversity 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 assist ed Crago with a project comparing the NRCS Curve Number method for computing runoff w ith the TOPMODEL method. Results were presented at Bucknell UniversityÂ’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 ye ar with McLaren, Crowle y began graduate school at the University of Florida to pursue his Ph.D. in coastal engineering. Under the tutelage of D. Max Sheppard, CrowleyÂ’s research focuses on coastalstructural interactions. Particular interest is paid to research surrounding the 2004-2005 hurrican e-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 st eady 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 engine ering in May 2008 and finish his Ph.D. by December 2010.