Thesis

Wesleyan University Physics Department

The Effects of Non-Universal Large Scales onConditional Statistics in Turbulence

by

Daniel Brian Blum

A dissertation submitted to the

faculty of Wesleyan University

in partial fulfillment of the requirements for the

Degree of Doctor of Philosophy

Middletown, Connecticut April, 2011

Dedication

For Karen Blum, who packed me lunch every day of school for the first 12 years, and called

every week for the remaining 10.

Acknowledgements

As I near the end of my time at Wesleyan I feel many things, but two emotions that stand our

are relief, and more strongly, thankful. This has been a great undertaking, (great, both in terms

of magnitude, and in positive impact) and it absolutely would not have been possible without

many people. First and foremost I would like to thank my advisor, Professor Greg Voth. Having

a knowledgeable advisor is common, but having such a patient, helpful and enthusiastic advisor

seems to be particular to Greg, and for this I am very grateful. I would also like to express my

gratitude to the other members of my thesis committee, Professors Fred Ellis and Tsampikos

Kottos for their support over the years. The Wesleyan physics department as a whole has been

very supportive, and I owe them particular gratitude for helping to fund my class at Yale. For

their collaboration, which makes up chapter 5, I would like to thank Eberhard Bodenschatz,

Mathieu Gibert, Armann Gylfason, Laurent Mydlarski, Haitao Xu, and P.K. Yeung. In addition,

I’d like to express my gratitude to Zellman Warhaft, Mark Nelkin, and Nick Ouellette who have

contributed their expertise to this work. My fellow Voth lab researchers have helped a great

deal over the years, Jim Johnson helped construct the tank, Dominic Stitch, Dennis Chan, and

Susantha Wijesinghe created the image compression circuits, Shima Parsa Moghaddam, Tom

Glomann, Emmalee Reigler, Rachel Brown, Nick Rotile, Reuben Son, and Surendra Kunwar

have all helped both with technical projects and camaraderie. This research was financially

supported by NSF Grant No. DMR-0547712 and the Alfred P. Sloan Foundation. It is common

to thank ‘the machine shop guys’ for their work on the apparatus, but I can truly say that

Dave Boule, and Bruce and Dave Strickland are the finest machine shop guys I have worked

with. I can also safely say that I would not have passed the qualifying exams without studying

ii

intensely with Joshua Bodyfelt. I also appreciate the camaraderie of my classmates over the

years, particularly Luis Fernando Vargas. I would also like to thank those at the beginning

of my physics career, Professors Sean Washburn, Lloyd Carrol, and Daniel Lathrop, as well as

Josh LaRocque, Adam Brooks, Rachel Rosen, Dan Zimmerman, Greg Bewley, and Santiago

Traiana.

Graduate school can be difficult, but it would have been impossible for me without the support

of my friends and family. For his years of dinner ‘collaborations,’ rides to New York, and dear

friendship I am very grateful to Joe Fera. There have been times when my best buddy, Joanna

Tice has carried me on her broad swimmer’s shoulders, and brought great joy, companionship and

strength into my life, and for this I thank her sincerely. I have shared in a condensed American

dream with the Bravos, Daniel, Felipe, Valentina, Laura, and Claudia Antonio, starting with

little and getting so much. It has also been a lot of fun, and I’d like to thank everyone for being

a part of it, Anna Haensch, Nathan Boon, Annie Rorem, Hiram Navarrete, Eric and Shannon

Paul, Charlie Mcintosh, James Ricci, Eliz Cox, Nicole Bobitski Olcese, Eran and Marmo Bugge,

and Evan Jones. As well as the MVC and the eternal email chain, Mike Shea, Mike Martin, Dave

Kanner, Seamus Scott, Beth Weil, Justin Hillman, Kevin Leahy, and Danny Shekhtman.

And of course my family, the bedrock of my support, Shirlie, Theodore, Karen, Michael, Andrew,

Natalie, and David Blum, Samuel and Estelle Ginsberg, and Diane, Eddie, Jordan and Rebecca

Steinberg.

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Contents

1 Introduction 1

1.1 Basic Turbulence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Apparatus and Experiment 10

2.1 The turbulence tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 3D Particle Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Particle Tracking and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Effects of Non-Universal Large Scales on Conditional Structure Functions 18

3.1 Characterizing the Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Structure Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Energy Dissipation Rate Measurement . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Phase Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Dependence on Large Scale Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.1 Eulerian structure functions conditioned on the large scale velocity: Cen-

ter region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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CONTENTS CONTENTS

3.5.2 Lagrangian Structure Functions Conditioned on the Large Scale Velocity:

Center Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5.3 Eulerian Structure Functions Conditioned on the Large Scale Velocity:

Near Grid Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5.4 Third order Eulerian Structure Functions Conditioned on the Large Scale

Velocity: Center Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.6 A Powerful Method for Plotting Conditional Structure Functions . . . . . . . . . 31


Center Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


Higher Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


Center Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


Near Grid Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


Near Grid Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.6.6 Third Order Eulerian Structure Functions Conditioned on the Large Scale

Velocity: Center Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.6.7 Second Order Eulerian Structure Functions Conditioned on the Velocity

Magnitude: Center Region . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 Properties of the Large Scales and Their Effects . . . . . . . . . . . . . . . . . . . 39

3.7.1 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.7.2 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7.3 Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7.4 Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7.5 Large Scale Intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.8 Kinematic Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Effects of Large Scale Intermittency on Conditional Structure Functions 47

4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1.1 Driving Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . . . 51

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CONTENTS CONTENTS

4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Signatures of Non-Universal Large Scales in Conditional Structure Functions

Compared in Various Turbulent Flows 73

5.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Conclusions 87

A Measurement Error Considerations 90

B Tank Preparation 93

C Calibration 96

C.1 Acquiring Calibration Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

C.2 Running the Calibration Programs . . . . . . . . . . . . . . . . . . . . . . . . . . 97

C.3 Dynamic Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

D Data Acquisition 101

D.1 Filling the tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

D.2 Using the laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

D.3 Using the image compression circuits . . . . . . . . . . . . . . . . . . . . . . . . . 103

D.4 Using external trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

D.5 Running the motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

D.6 Water cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

D.7 Adding particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

D.8 Recording the grid position (optional) . . . . . . . . . . . . . . . . . . . . . . . . 108

D.9 Recording data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

E Data Processing 110

E.1 Stereomatching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

E.2 After Stereomatching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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List of Figures

2.1 Experimental apparatus diagram. Two oscillating grids were held 56.2 cm apart

in an 1,100 l octagonal prism Plexiglas tank. Four high speed cameras were used

to stereoscopically image an illuminated volume in order to record 3D particle

positions. Illumination was provided by a Nd:YAG laser with 50 W average power. 11

2.2 Diagram of angle discrimination definitions. The red + is the particle seen on

camera 1. The red o can be placed anywhere along the ray, it does not effect the

angle θ2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Mean and variance of the vertical velocity along the central vertical axis of the

tank. Grid frequency is 3 Hz and grid separation distance is 56.2 cm. The dot-

dash line represents the grid height at maximum amplitude. We will focus on

measurements in the two regions designated by the vertical dashed lines: one at

the center of the tank and one near the grid. . . . . . . . . . . . . . . . . . . . . 19

3.2 Scale diagram of 56 cm × 100 cm area between grids showing the mean circula-

tion torii which are nearly rotationally symmetric about the central vertical axis.

Center (C) and near grid (NG) overreaction volumes are drawn in dashed lines,

which shows the relative size and position of the observation volume in Fig. 3.1.

Horizontal dot-dashed lines represent the range of motion of the top and bottom

grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

vii

LIST OF FIGURES LIST OF FIGURES

3.3 Eulerian second order longitudinal velocity structure function shown as a function

of pair separation r normalized by the Kolmogorov length η. The inset shows this

data compensated by Eq.3.1 for p=2. . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Eulerian third order longitudinal velocity structure function. The inset shows

this data compensated by Eq.3.1 for p=3. . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Second order compensated velocity structure functions conditioned on grid phase.

The collapse shows the very weak phase dependence: (a) center of the tank, (b)

near the grid. Zero and 2π phase represents grid at lowest possible amplitude. φ:

+ = 0 - 2π/5, ∗ = 2π/5 - 4π/5, ⋄ = 4π/5 - 6π/5, △ = 6π/5 - 8π/5 , � = 8π/5 -

2π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Second order velocity structure function conditioned on particle pair velocity

(vertical component) in the center of the tank. (a) Uncompensated structure

function. (b) Individually compensated by the energy dissipation rate for each

conditional data set. Symbols represent the following dimensionless vertical ve-

locities, Σuz/√

〈u2z〉: + = 4.2 to 2.5, ∗ = 2.5 to 0.84, ⋄ = 0.84 to -0.84, △ = -0.84

to -2.5, � = -2.5 to -4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.7 Second order Lagrangian velocity structure function conditioned on instantaneous

velocity (vertical component) in the center of the tank. (a) Uncompensated struc-

ture function. (b) Individually compensated to have the peak values match. Sym-

bols represent the following dimensionless vertical velocities, Σuz/√

〈u2z〉: + =

3.1 to 1.9, ∗ = 1.9 to 0.62, ⋄ = 0.62 to -0.62, △ = -0.62 to -1.9 , � = -1.9 to -3.1. 28

3.8 Second order velocity structure function conditioned on particle pair vertical ve-

locity (z direction) in the region near the bottom grid. The condition with

the largest downward velocity has been eliminated due to lack of statistical

convergence. Symbols represent the following particle pair vertical velocities

Σuz/√

〈u2z〉: + = 3.8 to 2.3, ∗ = 2.3 to 0.75, ⋄ = 0.75 to -0.75, △ = -0.75

to -2.3, a) Uncompensated structure functions b) Individually compensated by

the energy dissipation rate for each conditional data set. . . . . . . . . . . . . . . 31

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3.9 Third order velocity structure function plots conditioned on particle pair vertical

velocity and individually compensated for each conditional data set. Data are

taken in the center region of the tank, and the extreme vertical velocity plots

have been eliminated due to lack of statistical convergence. Symbols represent

the following vertical velocities Σuz/√

〈u2z〉: ∗ = 2.5 to 0.84, ⋄ = 0.84 to to -0.84,

△ = -0.84 to -2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.10 Eulerian second order conditional structure function versus large scale velocity.

Data taken in the center region. Each curve represent the following separation

distances r/η: + = 0 to 40, ∗ = 40 to 70, ⋄ = 70 to 110, △ = 110 to 140, � =

300 to 370, × = 370 to 440. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


Data taken in the center region at higher grid frequency, 5Hz, resulting in higher

Taylor Reynolds number 380. Symbols represent the following separation dis-

tances r/η: + = 0 to 50, ∗ = 50 to 100, ⋄ = 100 to 150, △ = 150 to 200, � =

310 to 420, × = 420 to 520. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


The thin plots are from atmospheric boundary layer data [1] r/η: ∗ ∼ 100, △∼ 400, � ∼ 1000, × ∼ 1250. The thick line is from fig. 3.10, which has been

overlaid for comparison, r/η: ⋄ = 70 to 110. . . . . . . . . . . . . . . . . . . . . . 35

3.13 Lagrangian second order conditional structure function versus large scale vertical

velocity. Data taken in the center region. Symbols represent the following τ/τη:

+ = 0.42 , ∗ = 1.3, ⋄ = 3.5, △ = 10. . . . . . . . . . . . . . . . . . . . . . . . . . 35


Data taken in the near grid region of the tank. The structure function is heavily

influenced by the bottom grid which has skewed the symmetry of the plot min-

ima in the negative direction. Symbols represent the following non-dimensional

separation distances r/η: + = 0 to 50, ∗ = 50 to 110, ⋄ = 110 to 160, △ = 270

to 320, � = 330 to 450, × = 450 to 560. . . . . . . . . . . . . . . . . . . . . . . . 37

3.15 Lagrangian second order conditional structure function versus large scale vertical

velocity. Data taken in the near grid region of the tank. Symbols represent the

following τ/τη: + = 0.94, ∗ = 2.8, ⋄ = 8.0. . . . . . . . . . . . . . . . . . . . . . 38

ix


3.16 Eulerian third order conditional structure function versus large scale vertical ve-

locity in the center region. Symbols represent the following non-dimensional sep-

aration distances r/η: + = 0 to 40, ∗ = 40 to 70, ⋄ = 70 to 110, △ = 110 to 140,

� = 220 to 300, × = 300 to 370. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.17 Eulerian second order conditional structure function versus magnitude of the ve-

locity pair in the center region. Symbols represent the following non-dimensional

separation distances r/η: + = 0 to 40, ∗ = 40 to 70, ⋄ = 70 to 110, △ = 110 to

140, � = 300 to 370, × = 370 to 440. . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Eulerian second order conditional structure function versus large scale velocity,

plotted in a similar manner to figure 3.10. The grid oscillation frequency was

modulated such that the frequency would switch from high to low at 15 second

intervals. The frequencies modulated were: a) 3 Hz continuous, b) 3-2 Hz, c) 3-1

Hz, d) 3-0 Hz. Each curve represents the following separation distances r/η: +

= 2.67 to 5.33, ◦ = 5.33 to 10.67, ∗ = 10.67 to 21.33, × = 21.33 to 42.67, � =

42.67 to 85.33, ⋄ = 85.33 to 170.67, △ = 170.67 to 341.33, ▽ = 341.33 to 682.67. 53

4.2 Second order conditional structure functions vs. large scale velocity (the same

data as figure 4.1). All curves represent one separation distance, r/η = 10.67 to

21.33 measured from the different driving frequency modulations. Each curve is

measured from the following driving frequency modulations: + = 3 Hz continu-

ous, ◦ = 3-2 Hz, ∗ = 3-1 Hz, × = 3-0 Hz . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Curves in figure 4.1 were fit to au4 + bu2 + c. The coefficient b is a measure

of the dependence of the conditional structure function on the large scales. The

coefficient b is shown here versus the separation distance r/η. The drive frequency

period was 30 seconds (15 seconds with the motor on, and 15 seconds with the

motor off). Each line represents the following driving frequency modulations:

+ = 3 Hz continuous, ◦ = 3-2 Hz, ∗ = 3-1 Hz, × = 3-0 Hz . . . . . . . . . . . . . 55

4.4 Conditional structure function vs. large scale velocity showing one separation

distance, r/η = 10.67 to 21.33 (similar to figure 4.2). The grid oscillation fre-

quency was kept constant. Each curve represents the following grid oscillation

frequencies: + = 1 Hz, ◦ = 2 Hz, ∗ = 3 Hz, × = 4, Hz � = 5 Hz . . . . . . . . . 57

x


4.5 The dependence of the conditional structure function on the large scales is quan-

tified. The coefficient b (where the polynomial au4 + bu2 + c was fit to the

conditional structure function) is shown here as a function of r/η. This is plotted

in the same way as figure 4.3. Each curve represents the fittings for the following

grid oscillation frequencies: + = 1 Hz, ◦ = 2 Hz, ∗ = 3 Hz, × = 4 Hz, � = 5 Hz. 58

4.6 The number of samples shown as a function of large scale velocity for frequency

modulation 3-0 Hz, duty cycle 50%, and period 384 seconds. The curves represent

the following separation distances r/η: + = 2.67 to 5.33, ◦ = 5.33 to 10.67, ∗ =

10.67 to 21.33, × = 21.33 to 42.67, � = 42.67 to 85.33, ⋄ = 85.33 to 170.67, △ =

170.67 to 341.33, ▽ = 341.33 to 682.67. . . . . . . . . . . . . . . . . . . . . . . . 59

4.7 Second order conditional structure functions vs. large scale velocity. All curves

represent one separation distance, r/η = 10.67 to 21.33 measured from the dif-

ferent driving frequency modulations. Each curve is measured from the following

driving frequency modulations: + = 4 Hz continuous, ◦ = 4-2.67 Hz, ∗ = 4-1.33

Hz, × = 4-0 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.8 Curves in figure 4.1 were fit to au4 + bu2 + c. The coefficient b is a measure

of the dependence of the conditional structure function on the large scales. The

coefficient b is shown here versus the separation distance r/η. The drive frequency

period was 30 seconds (15 seconds with the motor on, and 15 seconds with the

motor off). Each line represents the following driving frequency modulations:

+ = 4 Hz continuous, ◦ = 4-2.67 Hz, ∗ = 4-1.33 Hz, × = 4-0 Hz . . . . . . . . . 62

4.9 Conditional structure function vs. large scale velocity showing one separation dis-

tance, r/η = 10.67 to 21.33 (similar to figure 4.2). The grid oscillation frequency

was modulated between 3 and 0 Hz, where the motor was in the high state for

different percentages of the period, also called duty cycles. Each curve represents

the following duty cycles: + = 100%, ◦ = 75% Hz, ∗ = 50%, × = 25% . . . . . . 64

4.10 The dependence of the conditional structure function on the large scales is quanti-

fied. The coefficient b (where the polynomial au4+bu2+c was fit to the conditional

structure function) is shown here as a function of r/η. This is plotted in the same

way as figure 4.3 Each curve represents the fittings for the following grid duty

cycles: + = 100%, ◦ = 75% Hz, ∗ = 50%, × = 25% . . . . . . . . . . . . . . . . . 65

xi


4.11 The phase average energy shown as a function of cycle phase. The motor was

halted at 2pi and turned on at pi. Each curve represents the following cycle

periods r/η + = 1.5/3”, ◦ = 3/6”, ∗ = 6/12”, × = 12/24” � = 24/48” ⋄ =

192/384” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.12 Conditional structure function vs. large scale velocity showing one separation dis-

tance, r/η = 10.67 to 21.33 (similar to figure 4.2). The grid oscillation frequency

was modulated between 3 and 0 Hz and the period was varied. Each curve rep-

resents the following time the motor was on/total period (seconds): + = 1.5/3”,

◦ = 3/6”, ∗ = 6/12”, × = 12/24”, � = 24/48”, ⋄ = 192/384” Note all have a

duty cycle of 50%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.13 The dependence of the conditional structure function on the large scales is quanti-

fied. The coefficient b (where the polynomial au4+bu2+c was fit to the conditional

structure function) is shown here as a function of r/η. This is plotted in the same

way as figure 4.3 Each curve represents the fittings for the following cycle periods:

+ = 1.5/3”, ◦ = 3/6”, ∗ = 6/12”, × = 12/24” � = 24/48” ⋄ = 192/384” Note

all have a duty cycle of 50% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1 Eulerian second order longitudinal velocity structure functions compensated by

(εr)2/3. a) Direct Numerical Simulation b)Passive Grid Wind Tunnel c) Active

Grid (Synchronous Driving) d) Active Grid (Random Driving) e) Counter Rotat-

ing Disks f) Oscillating Grids g) Lagrangian Explorer Module (Constant Driving)

h) Lagrangian Exploration Module (Random Driving). . . . . . . . . . . . . . . . 77

5.2 (Color Online) The Eulerian second order longitudinal structure functions are

conditioned on the transverse velocity sum (Σu⊥), and plotted versus Σu⊥. Sym-

bols represents the following separation distances: r/η:+ = 4, ◦ = 8, ∗ = 16,

× = 32, � = 64, ♦ = 128, △ = 256, ▽ = 512 ⊲ = 1024. a) Direct Numerical

Simulation b)Passive Grid Wind Tunnel c) Active Grid (Synchronous Driving)

d) Active Grid (Random Driving) e) Counter Rotating Disks f) Oscillating Grids

g) Lagrangian Explorer Module (Constant Driving) h) Lagrangian Exploration

Module (Random Driving). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

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5.3 (Color Online) Structure functions conditioned on the vertical component of the

velocity sum, Σuz. Symbols represents the following separation distances r/η:+

= 4, ◦ = 8, ∗ = 16, × = 32, � = 64, ♦ = 128, △ = 256, ▽ = 512 ⊲ = 1024. a)

Counter Rotating Disks b) Oscillating Grids . . . . . . . . . . . . . . . . . . . . . 79

5.4 (Color Online) The Eulerian second order longitudinal structure functions are

conditioned on the longitudinal velocity sum (Σu‖), and plotted versus Σu‖.

Symbols represents the following separation distances r/η:+ = 4, ◦ = 8, ∗ =

16, × = 32, � = 64, ♦ = 128, △ = 256, ▽ = 512 ⊲ = 1024. a) Direct Numerical

Simulation b)Passive Grid Wind Tunnel c) Active Grid (Synchronous Driving)

d) Active Grid (Random Driving) e) Counter Rotating Disks f) Oscillating Grids

g) Lagrangian Explorer Module (Constant Driving) h) Lagrangian Exploration

Module (Random Driving). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.5 Conditional structure functions for Gaussian random fields. The Eulerian second

order longitudinal structure function is conditioned on the longitudinal velocity

sum. Symbols represents the following separation distances r/η:+ = 4, ◦ = 8, ∗= 16, × = 32, � = 64, ♦ = 128, △ = 256, ▽ = 512 ⊲ = 1024. . . . . . . . . . . 81

A.1 Eulerian second order conditional structure function versus large scale velocity,

plotted in a similar manner to figure to 4.1a, with error bars representing the

statistical error. The following separation distances are shown in separate plots

for clarity: r/η: ◦ = 5.33 to 10.67, × = 21.33 to 42.67, ⋄ = 85.33 to 170.67, ▽ =

341.33 to 682.67. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

C.1 An example calibration image that is an average of 10 calibration images taken

from camera 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

C.2 An example of a calibration image that has been edited so all that remains is the

dots which are viewed by each camera . . . . . . . . . . . . . . . . . . . . . . . . 99

D.1 A flowchart for the path of the camera trigger. . . . . . . . . . . . . . . . . . . . 105

E.1 A data processing flowchart, rounded edges represent programs, rectangles rep-

resent data files, and shaded areas occur on the computer cluster . . . . . . . . . 111

E.2 A flowchart for the functions used in stereomatching. . . . . . . . . . . . . . . . . 115

E.3 The meaning of each column in the vel3d2d files. . . . . . . . . . . . . . . . . . 116

xiii

Chapter 1Introduction

In a reductionist sense most of fluid dynamics was solved with the Navier-Stokes equations in the

early 1800’s. The fundamental equations that govern fluid dynamics are essentially conservation

laws for a continuous fluid, deterministic equations that can be written down rather simply.

Once the equations are solved with the proper boundary conditions the entire velocity field of a

system can be found. However, once one tries to describe a real world flow, it quickly becomes

apparent that the Navier-Stokes equations are analytically solvable for only the most basic,

hypothetical systems. Real world flows generally have complex boundary conditions, and move

quickly enough to have the complex non-linear terms become important, and the velocity field

quickly becomes unpredictable, and turbulent.

In a way this is what makes this field exciting: the fundamental equations have been known

for centuries, but attempts to predict individual trajectories are almost by definition futile in

most cases. Instead, we must invent different approaches to even hope to shed some light on

to the field. It quickly became clear that since predicting individual trajectories can not be

done, a statistical description of turbulence is needed to progress forward. There has been some

wonderful successes in searching for a statistical description of turbulence. Certain statistical

quantities have been shown to be very robust, appearing in a wide range of flows, which have

very real engineering benefits. The field has grown very wide encompassing many perspectives

and affecting other branches of research and engineering for instance: turbulent mixing, insect

flight, polymer drag reduction, to name just a few. The list of fields where turbulence research

1

Chapter 1 - Introduction 2

can help guide progress will certainly grow. It has become a model problem for digging into an

unpredictable, highly nonlinear system with a wide range of length and time scales- it is the

model problem for problems that are difficult to model.

1.1 Basic Turbulence Theory

After we have accepted the limitations of attempting to directly solve the Navier-Stokes equa-

tions we can begin to explore other insights in to turbulence. I will now give a basic background

to the turbulence theory needed to understand this thesis. A student is hard pressed to find

a better text than Turbulent Flows by S.B. Pope [2], which is the major reference for this

section.

In 1922 L.F. Richardson introduced the idea of an energy cascade in turbulence. The idea

begins with kinetic energy entering a fluid at a large scale through some stirring mechanism.

This stirring energy will create an eddy at a large scale. An eddy can be loosely defined as a

turbulent motion localized within a certain region with a definite size, that is at least moderately

coherent within that region. As an example, the stirring mechanisms in our experiment are the

two oscillating metal grids. In the notion of the energy cascade they are injecting energy at a

length scale of approximately the mesh size of the grids (the spacing between the bars of the grids

which is 8 cm). This begins the energy cascade with a large scale eddy that is approximately

8 cm. Large scale eddies that are created by the stirring mechanism are unstable, and break

up. Energy is thus transferred from the large eddies to smaller eddies. The smaller eddies are

themselves unstable and chaotically break up in to even smaller eddies. This process continues

until, at the smallest length scales the energy is dissipated into heat by the viscous action of the

fluid. Richardson even wrote a little poem about it [3]:

Big whirls have little whirls;

That feed on their velocity;

And little whirls have lesser whirls,

And so on to viscosity

(in the molecular sense).

Hopefully, the energy cascade makes some intuitive sense. We can progress further by seeing

how such a notion can be quantitative and what predictions can be made.


The Reynolds number is a key number in almost all of fluid dynamics and represents the ratio of

the inertia to the viscous forces, Re = UL/ν where U is a characteristic velocity (usually the rms

velocity), L is a characteristic length scale (usually can be approximated by the energy input

length scale, in our case 8 cm), and ν is the kinematic viscosity of the fluid (the resistance of

the fluid to motion scaled by the fluid density), note that the Reynolds number is dimensionless.

The Reynolds number is important because it can immediately describe the qualitative motion

of a fluid. If the Reynolds number is very small the viscous forces dominate, the fluid velocity is

relatively low, and the flow is laminar. If the Reynolds number is very large the inertial forces are

dominant, the viscosity plays a relatively small role in the flow, and the flow is unpredictable

and turbulent. To give a feel for its importance, the Navier-Stokes equations have only one

free parameter, and that is the Reynolds number. We will not go into great depth about the

Reynolds number here, but if it is helpful it can be thought of as a ’turbulence intensity’. As an

example when the oscillating grids in our experiment are moving at 5 Hz the Reynolds number

is 9,400, when they are moving at 3 Hz the Reynolds number is 5,400, when they are moving at

1 Hz the Reynolds number is 1,300, and as they move slower the Reynolds number will continue

to decrease until the flow is completely laminar.

One can assign a Reynolds number to the eddies in a turbulent flow. We can write Re0 = u0l0/ν,

where u0 is the characteristic velocity of an eddy, l0 is the length scale of an eddy, and these can

be related in the simple definition τ0 ≡ l0/u0 where τ0 is the characteristic timescale associated

with the eddy. So at the largest eddies Re0 is approximately the same as Re. As we examine

smaller and smaller eddies l0 gets smaller as does u0 (more on why they both decrease later),

and ν grows in importance until it is completely effective at dissipating the kinetic energy.

This is important because one can hypothesize that at large enough Reynolds number the large

length scales would be unaffected by the viscosity, all of the energy of one large length scale

can be transferred directly into the somewhat smaller ones with negligible loss due to viscosity.

These eddies have energy of order u20 and a timescale τ0. So the rate of energy transfer is just

u20/τ0 = u3

0/l0. Since essentially no energy is lost in the energy cascade until l0 is small, once the

energy finally reaches the energy dissipation scales it has the same amount of energy as when

the energy started to be transferred. This means that at sufficient Reynolds number the energy

transfer rate is equal to the energy dissipation rate per unit mass, independent of ν, ε = U3/L

where ε is the energy dissipation rate. One can already see how knowing the rate at which

energy is dissipated in a flow by only knowing U and L could be very useful, especially from an


engineering perspective.

At this point the idea of the energy cascade may seem plausible, and possibly intriguing, but

still has many questions unanswered. We still need to answer why when l0 decreases u0 also

decreases, and try to better define ’small’ scales.

In 1941 Kolmogorov posited 3 hypotheses which further the energy cascade idea, and have been

very useful in turbulence theory. All assume a sufficiently large Reynolds number and are stated

here less formally than in Kolmogorov’s original work.

· Local Isotropy - Small scale turbulent motions are statistically isotropic.

Directional information of the large scales is lost as the energy is transferred down the cascade.

In fact, all information about the the large eddies, geometry, direction, boundary conditions,

etc., is lost as the eddies are chaotically broken apart into smaller and smaller scales. This is

a crucial point that this thesis hopes to address, to what extent is all information about the

large scales lost? If all large scale information is lost the small scale motions have some amount

of universality, they should be similar in every turbulent flow with sufficient Reynolds number.

This leads to the next hypothesis:

·The First Similarity Hypothesis - Small scale motions have a universal form that is uniquely

determined by ν and ε.

Since all large scale information is lost, all small scales -independent of flow and therefore

universally- depend on two parameters, ν and ε. One result of this hypothesis is that we can

now define unique time, length, and velocity scales, which are called Kolmogorov scales:

τη ≡ (ν/ε)1/2

η ≡ (ν3/ε)1/4

uη ≡ (νε)1/4

(1.1)

These are derived from the two parameters which characterize the smallest, dissipative eddies.

Derivation is a simple matter of dimensional analysis, arranging the two parameters so that

they have the proper units for time, length and velocity respectively. This creates a definition

for the smallest scales. A consistency check shows the Reynolds number for the Kolmogorov

scales shows ηuη/ν = 1, which agrees with the earlier notion that smaller eddies have smaller

Reynolds numbers.


To arrive at the next hypothesis, one can further suppose that given a sufficiently high Reynolds

number there is a special range of scales, which are called the inertial range, that are larger

than the dissipative scales (where ν and ε play a role). The inertial range includes length scales

that are larger than the dissipative range, and large enough so that ν will no longer play a

role, yet smaller than the largest length scales where the boundary conditions and other non-

universal attributes play a role. They exist (again given sufficiently high Reynolds number) in

a intermediate range. This leads to the third hypothesis:

·Kolmogorov’s Second Similarity Hypothesis - There exists a certain range of length scales such

that there is a universal form which is uniquely determined by only ε.

This means that if a flow has sufficient Reynolds number some range of length scales will

depend only on ε, and will do so in a way that is universal, independent of the unique large

scale boundary conditions, and the viscosity which dominates the small scales.

An important note about language, it is commonplace to categorize all scales in a turbulent

system in to “large” and “small.” Of course there is energy in the turbulent flow at length

scales which range continuously from near the dimensions of the system to below the Kolmogorov

lengths. In this work, the large scales refer to any scale which is size L or larger. Conversely, the

small scales refer to any scale which is smaller than L, either in the inertial or dissipative ranges,

where the large scale information is predicted to have been lost in the energy cascade.

Now, using this Second Similarity Hypothesis and some more dimensional arguments we can find

the functional form in the inertial range that only depends on ε and is predicted to be universal.

Testing the universality of these predictions is the basic motivation for this thesis.

1.1.1 Scaling Laws

Let’s begin by considering the easily measurable quantity, the velocity difference, ∆ur = [u(x) - u(x+r)]L

which is the instantaneous velocity difference between two points where x is the position, r is

the separation distance, and the subscript L denotes the longitudinal component of the veloc-

ity difference vector (the component of the velocity difference vector projected on the the ray

connecting the two points). The measurement of ∆ur is useful because as a velocity difference

between two points it can easily describe the dependence on length scale, r, while not being as

susceptible to the effects of non-universal large scales that would be inherent in a single point


statistic. We will commonly raise the velocity difference to a power and take the ensemble

average, this is called the structure function Dp = 〈(∆ur)p〉.

We see that ∆ur depends on r, and if we assume homogeneity for a moment, the Second

Similarity Hypothesis implies that in the inertial subrange ∆ur = F (ε, r), the functional form

only depends on ε and r. Now some simple dimensional analysis, r has units of length, ε has

units of energy per unit mass per unit time, and ∆ur has units of velocity,

r → [m] ,

ε →[

m2

s3

]

,

〈(∆ur)p〉 →

[m

s

]p

.

We can use these to find the functional form of F (ε, r). We set the structure function propor-

tional to some combination of ε and r

〈(∆ur)p〉 ∝ εαrα

We now use dimensional analysis to determine this relation

[m

s

]p

∝[

m2

s3

]α

[m]α.

This directly leads to α = p/3. So now, by using dimensional analysis we have

〈(∆ur)p〉 = Cp(εr)

p/3 (1.2)

where Cp is a universal constant which is generally found experimentally. This is a testable

result, and has been the beginning for much of the experimental turbulence research since its

inception in 1941. Experiments have examined all aspects of the relation, the universality of

Cp, the robustness of the p/3 exponent, and as we will test the degree to which the relation

is universal. An additional very nice check is that the relation for p = 3 can be been derived

directly from the Navier-Stokes equations. This gives great confidence in the result, as well

as gives C3 = −4/5. Since this can be shown analytically, the p = 3 case is the standard for

evaluating an experiment.


1.1.2 Motivation

The theories of Kolmogorov I have just sketched have been very successful. Granted, there have

been significant modifications to some of the specific predications, but I classify the theory as a

success because of its utility as a framework from which progress has been made. Not only has

the inertial range been shown to exist in countless systems, there are statistics in the inertial

range which adhere very closely to Kolmogorov’s predictions. It has been carefully shown that

there are statistics that are nearly identical in different flows, the non-universal properties of the

large scales which are unique in each flow do not affect them [4–6]. This work is by no means an

attempt to dispute these findings. Instead we hope to shed some light on small scale statistics

which are affected by the non-universal properties of the large scales within a flow, and perhaps

what differentiates them from the universal small scale statistics. Other work has shown that

the coefficients of scaling laws [7, 8], or the scalar derivative skewness [9] are not universal. We

will be primarily focused on structure functions with p = 2 or 3, as defined equation 1.2.

A traditional approach to determine if a small scale statistic is universal is to first categorize

flows such as: free shear flows (jets, mixing layers, etc.), wall bounded shear flows (boundary

layers, channel flows, etc.), or isotropic turbulence (wind tunnel grid turbulence, or numerical

simulations in a box with periodic boundary conditions, etc.). Then a careful empirical compar-

ison of small scale statistics between different flows can show which statistics are independent

of the large scales.

However, we will not be comparing different flows to determine if a statistic is universal. Instead,

we will use a more direct method, the small scale measurements in a flow will be conditioned1

on a measurement of the state of the large scales. Any dependence of a small scale statistic

on the non-universal large scales within the flow will be obvious, and can even be quantified.

Three previous studies have done just that, and shown strong dependence of the small scales of

Eulerian structure functions on the instantaneous velocity in the flow, which is dominated by

the large scales.

Praskovsky et al. conducted experiments in two high Reynolds number wind tunnel flows, a

1Conditioned statistics will be used heavily throughout this work. The concept is simple, when a variable is

measured the condition of a different variable is also measured. The conditional statistic can then show how the

two measurements are related. For example, measure the distribution of marble sizes, and condition it on marble

color. You could see that the biggest marbles are red, for instance.


return channel and a mixing layer. The Eulerian structure functions were conditioned on the

large scale velocity and showed a strong correlation. All length scales were significantly affected,

including the small length scales in both flows. In their discussion Praskovsky et al. discounted

the possibility of having a Reynolds number that was too small, their apparatus achieved a

very large Reynolds number. They also discounted that Kolmogorov’s theory was incorrect.

Instead, they concluded their observations were consistent with Kolmogorov’s theory when it is

correctly applied to a flow with a fluctuating energy injection at large scales. Sreenivasan and

Dhruva [1] measured Eulerian velocity structure functions from atmospheric boundary layer

data for Reλ > 104, some of the largest Reynolds numbers ever measured. They find that the

structure functions conditioned on the large scale velocity show a strong dependence, and they

also show that direct numerical simulation in a periodic box (DNS) and passive grid turbulence

measurements show almost no dependence. They attribute the dependence in the atmospheric

boundary layer to large scale shear, which is not present in the DNS and passive grid turbulence.

They also show how to remove the large scale dependence through experimentally found fit

parameters in order to improve power law scaling. Kholmyansky and Tsinober [10] also measured

conditional structure functions in a high Reynolds number atmospheric boundary layer and

found a dependence that is weaker than Sreenivasan and Dhruva found. They attribute the

strong dependence they observe to a direct and unavoidable coupling between the large and

small scales in turbulence [11].

A natural initial objection when discussing how the small scales depend on the large scales is that

the Reynolds number is not sufficiently large enough. One recalls how important a sufficiently

high Reynolds number is to Kolmogorov’s hypotheses. However, the atmospheric boundary layer

data is measured at some of the highest Reynolds numbers ever recorded. We cannot discount

the possibility that any infinite Reynolds number system has small scale statistics that are always

independent from the large scales. However, this is an asymptotic limit, whose value is doubtful

in the real world. Our aim is to better understand turbulent flows as they actually are, and to

some extent move away from analyzing the idealized constituent parts. Upon accepting that

real world flows almost always have small scales which depend on the state of the large scales

our work tries to quantify what properties of the large scales are responsible for the observed

dependence.

One challenge in discussing interactions between large scales and small scales is the very non-


universal nature of the large scales. Each flow has a unique set of large scales, which may

depend on time, geometry, or driving parameters. So it has been difficult to isolate the aspects

of the large scale flow that are affecting the small scales. Anisotropy is the aspect that is best

understood. Extensive work has identified persistent anisotropy at small scales even at very

high Reynolds numbers [9, 12], and analysis using spherical tensor decomposition has placed

this problem on solid footing [13, 14]. However, this is not the only effect of the large scales.

Here we have found two additional aspects of the large scales that are particularly important.

Inhomogeneity is the spatial variation of statistics, and will be discussed in chapter 3. Large

scale intermittency is temporal fluctuations on time scales longer than the eddy turnover time,

L/U , and will be discussed in chapter 4. Both inhomogeneity and large scale intermittency

often occur together in real flows, but are distinct properties since flows can be conceived that

have each without the other. For example, a homogeneous turbulent flow in DNS can have large

scale intermittency by having the energy injection varied in time.

It is my hope that this work can be helpful to better understand the dependence of small scale

statistics in turbulence on non-universal large scales which will help in the identification of

universal statistics, and the comparison of different flows.

Chapter 2Apparatus and Experiment

This work1 is based on optically tracking passive tracer particles seeded in a turbulent flow. Es-

sentially, 3D positions are determined by the 4 cameras in a similar fashion to 2 eyes determining

the 3D position of an object in nature. Tracking each particle is accomplished by comparing

adjacent frames and identifying a particle by its possible trajectory as it moves through the

frames. Once a particle has been identified over several frames its velocity is then determined

by calculating the distance traveled over time. After we have acquired a significant amount of

velocity data we can begin our analysis which this thesis is based on. Our system has the distinct

advantage of real time image compression; with compression factors of 100-1000 data could be

acquired continuously and nearly endlessly. This allowed for greatly increasing the amount of

data that could easily be acquired using a relatively simple apparatus. In later chapters we will

see how the key contributions of conditional statistics require very large amounts of data.

This chapter will describe the experimental apparatus and the protocols used in the experiments.

I will describe the turbulence tank in section 2.1, the detection of the particles in section 2.2,

the calibration method in section 2.3, 3D particle finding in section 2.4, and the method for

tracking and velocity determination in section 2.5.

1Note this chapter is largely adapted from work that has been published as Blum et al. “Effects of nonuniversal

large scales on conditional structure functions in turbulence” Physics of Fluids 22, 015107 (2010)

10

Chapter 2 - Apparatus and Experiment 11

1 m

1.5 m

Figure 2.1: Experimental apparatus diagram. Two oscillating grids were held 56.2 cm apart in an 1,100

l octagonal prism Plexiglas tank. Four high speed cameras were used to stereoscopically image an illuminated

volume in order to record 3D particle positions. Illumination was provided by a Nd:YAG laser with 50 W average

power.

2.1 The turbulence tank

The turbulence tank is a 1 × 1 × 1.5 m3 Plexiglas prism and is filled with approximately 1,100

l (300 gallons) of filtered, degassed water. Two grids generate the turbulence, the grids have 8

cm mesh size, 36% solidity, and are evenly spaced from the top and bottom of the tank with

a 56.2 cm spacing between the grids, and a 1 cm gap between the grids and the tank walls as

shown in figure 2.1. The stroke was 12 cm peak to peak, powered by an 11 kW motor. A typical

grid frequency was 3 Hz, but could be raised up to 5 Hz safely. Water cooling maintains the

temperature at ± 0.1◦ C during each run. Grid position is determined by a simple photogate

placed underneath one of the rods such that the oscillation of the grids breaks the photogate’s

beam. The grids were at their bottom position when the photogate beam has been broken for

half of the total time it was broken. This position was assigned the phase angle 0=2π.

The water used for the experiment was degassed overnight and filtered through 2 filters, the first

had a 50µm pore size and the second 0.2 µm. Keeping the water and the tank free from debris

and air bubbles was a challenge. To address debris in the tank (often composed of biological


material) we found degassing did an adequate job inhibiting biological growth, replacing the need

for adding antibacterial chemicals. However, some debris and biological material still found its

way into the tank via the hoses running into the tank and times when a port is open in the tank.

Another possible source of debris was corrosion in the tank, although only anodized aluminum

and plastic come in contact with the water to try to minimize corrosion, it still persisted. One

possible reason for the corrosion was the a chemical reaction between the different aluminum

alloys used in the grids and the top and bottom surfaces of the tank. This could be minimized by

resurfacing or replacing the grids. To remove as much debris as possible the water was filtered

continuously while the motor occasionally stirred the water. This was moderately successful,

however this introduced air bubbles in to the tank which should be avoided so as not to be

confused with tracer particles when recording. An additional source of air bubbles in the tank

was air being drawn in to the water past the seals. To minimize this source the seal housing was

modified so that both sides of the seal have a reservoir of water. This effectively eliminated air

bubbles entering the water while the motor was running. Overall, degassing overnight had the

benefit of minimizing the effect of any air bubbles that were introduced in to the tank. Although

no completely satisfactory solution was found to eliminate debris and air bubbles in the tank,

these contaminants were estimated to represent less than 1% of the particles found, and usually

not tracked.

Neutrally buoyant 136 µm diameter polystyrene tracer particles were added to the flow. While

2 cameras were in place the seeding density could be up to 50 particles per frame without

significant stereomatching errors. After 4 cameras were in place the seeding density could be

raised to over 180 particles per frame without significant stereomatching errors. This has not

been limit tested due to the expense of particles.

One difficulty inherent in the oscillating grid experiment is the vibrations introduced from the

oscillatory driving. The benefits of imaging particles with high spatial resolution could be lost

if the imagers themselves are vibrating significantly. To minimize the source of vibrations the

flywheel was milled to precise tolerances, and lightweight Aluminum was used in the construction

of the grids. To minimize vibrations coupling to the cameras, a lightweight custom built camera

support was used to hold the cameras rigidly with respect to each other. In addition, the camera

support is mounted on an optical table that is not connected to the tank, and rests on rubber

vibration reducing pads.


It should be noted the Plexglas tank did rupture once, before I began working on the project.

The tank was filled with cold water, then the all-thread rods were tightened. The tank was then

drained and refilled with warmer water which caused the Plexiglas to thermally expand under

compression. The Plexiglas joints failed, spilling 300 gallons of water onto the floor. The tank

was repaired and the all thread rods were buffeted with springs, so the Plexiglas could thermally

expand more easily, which has proven successful to the present date.

2.2 Detection

These data were acquired using three dimensional particle tracking velocitmetry measurements.

The data in chapter 3 were acquired using 2 Bassler A504K video cameras capable of 1280 ×1024 pixel resolution at 480 frames per second (a data rate of approximately 625 Mbyte per

second per camera). Later, 2 newer cameras were added, model Mikrotron MC1362 which have

similar pixel resolution and data rates, but have the advantage of greater sensitivity. The noise

(frame to frame deviations from the mean) depend more on brightness level than camera. The

key difference in camera model is the sensitivity; given the same light on each pixel the Mikrotron

cameras can give a much higher pixel value, thus allowing them a greater dynamic range.

Recording such high data rates with 4 cameras is a significant technological hurdle. A well

equipped desktop system could store data in 4 Gbyte of video RAM, so that one run could last

just 7 s before waiting approximately 7 min for the data to download to the hard disk. We

have developed an image compression circuit to threshold images in real time so that only pixels

above a user defined brightness limit are regarded as particle data and retained, while the dark

background pixels are discarded [15]. This technique produces a dynamic data compression

factor of 100-1000, which enables continuous data collection and storage to hard disk.

Our first implementation of the image compression circuit faced two major challenges. First, the

simple thresholding compression reduces particle center accuracy. However, particle finding is

typically degraded by only 0.1 pixel, which is typically less than the uncertainty in the particle

finding from unthresholded images. To help this, plans are underway to implement a nearest

neighbor algorithm which will record pixels adjacent to the bright pixels. Second, because

frame number information was created and recorded separately on each computer, any operating

system delay can lead to frames lost and timing mismatch between the cameras. Some frame


number errors were corrected in postprocessing. Updated versions of the image compression

circuit solved this problem by including camera frame number in the data stream the computers

record.

Particles are illuminated using a 532 nm pulsed Nd:YAG laser with 50 W average power. The

beam was expanded to create an illumination volume approximately 7 × 4 × 5 cm3. The Condor

model laser manufactured by Quantronix corporation has had some operational issues that

required a return shipment to the manufacturer. The issues were: a faulty transient absorber on

the power control circuit, loose wires, and biological growth in the deionized cooling water line.

These issues were addressed, and the maintenance procedure now includes powering down the

laser from the circuit breaker, and running the laser for at least 1 hour each month to inhibit

biological growth.

2.3 Calibration

The goal of the calibration is to set up a real space coordinate system with a known origin and

camera positions. This will be used to find the 3D particle positions in real space. The calibration

procedure involved placing a calibration mask into the tank at the desired observation volume,

filling the tank with water, and recording calibration images of the well illuminated calibration

mask. The calibration mask (manufactured by Applied Image Inc.) is simply a 50 × 100 mm

sheet of glass with an array of 0.2 mm dots placed 2 ± 0.001 mm apart. The perspective each

camera has of the known calibration mask distorts the regular array of dots slightly, the columns

farthest away seem contracted. This distortion contains enough information for the cameras 3D

position to be calculated. A detailed description of the procedure and algorithm used for the

measurements in chapter C can be found in Ref. [16] and [17]. A similar procedure, although

different algorithm was used for the measurements in chapter 4, greater detail can be found in

refs. [18],

This traditional calibration method can obtain a particle finding accuracy of approximately 2 pix-

els, or 100 µm. Particle finding accuracy can be greatly increased by using known stereomatched

pairs from the cameras and running a nonlinear optimization to minimize the stereomatching

error and find optimal camera position parameters. This can increase particle finding accuracy

to less than half of a pixel.


2.4 3D Particle Finding

The procedure for finding particles in 3D is essentially creating a 3D ray which starts at the

known camera position and goes through the particle which that camera sees. With multi-

ple cameras all seeing the same particle the rays should intersect at one point in 3D space,

which is the position of the particle. This technique, referred to as stereomatching, has 3 chal-

lenges.

First, in real world experiments the rays never exactly intersect. There is always some mea-

surement error, both in the 2D images and in tracing the real space ray pointing to the particle

(having a straight ray is an approximation of how the light travels to the cameras). Fortu-

nately, these effects are small, and the stereomatching intersection error can be less than half a

pixel.

Second, there are pathologies which can yield false particles. One can imagine two rays that

intersect, but do not have a particle where they intersect. Instead, there are two particles each

out of the frame of the other camera, but they are positioned in such a way that the rays

intersect in the observation volume, and report a false particle. Fortunately this pathology is

rare, and becomes exceedingly rare when more than two cameras are used.

The third, and most challenging problem, is that of particle density. Imagine a particle seen

on one camera is a ray seen from another camera. If the particle density is low, and each

camera sees only one particle in its frame there is no ambiguity as to which which ray on one

camera is pointing to which particle seen on a different camera. If each camera sees a handful

of particles calculating the position of a particle would require taking one ray on one camera

and calculating which rays on the other cameras it comes closest to intersecting. The correct set

of rays will be much closer to intersecting than any other combination of ray sets. This works

fine for low particle densities. The problem occurs when an experiment calls for a high particle

density, the brute force method of trying every ray set combination quickly breaks down. For

example if there are 600 particles in a frame, the brute force method requires one ray from

one camera to be matched to every ray from each other camera. With 4 cameras this requires

6004 or 1.3×1011 combinations, even at 1 millisecond computation time per combination to

evaluate the 3D matching, (which is a generously fast estimate) each frame would take over 4

years of computation. This is for only 1 frame, our measurements require millions of frames.


Camera 1’s view

Camera 2

[x1,y

1] Camera coordinates of

particle seen in camera 1

[xc,y

c] Coordinates of camera 2

if it could be seen by camera 1

[x2,y

2]

+

θ1

θ2

θ1=tan-1( )

yc-y

1

xc-x

1

θ2=tan-1( )

yc-y

2

xc-x

2

Figure 2.2: Diagram of angle discrimination definitions. The red + is the particle seen on camera 1. The red

o can be placed anywhere along the ray, it does not effect the angle θ2.

We have developed a solution to this problem called angle discrimination which, although more

conceptually difficult, greatly reduces the computation time.

The key to solving this problem is to limit the number of ray set combinations to try. Consider

two cameras on a horizontal plane pointed at the same observation volume. If one camera sees

a particle at the very top of its frame we can safely choose only rays originating near the top of

the frame of the second camera to try for an intersection. Even if we eliminate half of the rays,

the number of combinations to try has greatly decreased. Taking this concept further, and using

some simple geometry, we can do better still. We can accurately predict which particles seen on

a 2d frame are viable candidates to be matched to particles seen on other cameras, using only

their position and the position of the cameras.

Figure 2.2 shows a diagram for a method of determining which rays are good candidates for

3D matching. Camera 1’s field of view is shown, it sees only one particle denoted by a red +.

Camera 2 cannot be seen by camera 1, but since we know it’s 3D coordinates from the calibration

we can project it on to camera 1 coordinate space. In other words, if camera 2 is a point particle

camera 1 sees it at [xc, yc]. In this diagram camera 2 is directly above camera 1. The ray which

travels through the red circle is the most important part of this diagram. Camera 2 can point


a 3D ray in real space to the particle it sees, here the particle it sees is drawn as a red circle.

The ray shown in the diagram is the 2D projection of that ray on to camera 1 coordinate space.

Note the red circle can be placed anywhere along the ray without affecting θ2. The angles θ1

and θ2 can now be compared. Only rays that point to the same particle will have similar angles,

typically within small fractions of a degree. Each frame needs to be sorted by angle only once,

and the angle can quickly be used to discriminate against unlikely rays. This comparison can

be calculated very quickly and can greatly reduce the number of combinations of rays needed

to find the ray set that intersects at a real particle position. The increase in speed depends on

the tolerances used for the angle discrimination and the particle density, but an average frame

can now be processed in under 1 second, far better than 4 years.

2.5 Particle Tracking and Velocity

After the 3D positions of the particles have been determined a particle tracking program (writ-

ten by John C. Crocker and Erik Weeks 2) is run which identifies particles through multiple

frames. This works by recording the particle positions on one frame, then considering all pos-

sible new positions on the next frame. It then calculates all possible identifications of the old

positions with the new positions and minimizes the total displacement. Parameters such as

maximum displacement between frames and gaps is a track are considered. This program has

proven adequate for the experiment described here. An improved version (which was started)

could include the probable particle trajectories and splice particle tracks together that became

separated by a particle absent over several frames.

After particles tracks were identified, particle velocities were then calculated. The straightfor-

ward derivative calculation ~v = ∆(~x)/∆(t) is too noisy in this experiment. Instead, a particle

trajectory was fit to a second order polynomial function. The derivative of this function was

then found, and the velocity was determined. This method reduces the noise in the velocity

measurement but can artificially smooth over velocity features if the fit is applied over too many

frames. Details of finding the best duration to fit over can be found in Voth et al. [19].

2http://www.physics.emory.edu/ weeks/idl/

Chapter 3Effects of Non-Universal Large Scales on

Conditional Structure Functions

This chapter1 reports measurements of conditional Eulerian and Lagrangian structure functions

in order to assess the effects of nonuniversal properties of the large scales. As stated in the

introduction, a foundational concept in the study of turbulence is that of the energy cascade.

A process where kinetic energy enters the system at a large scale, and cascades down to smaller

and smaller scales through the chaotic break up of eddies. It is supposed that in this chaotic

process large scale information is lost by the time the kinetic energy is at the small, universal

scales and should be independent of the large scales. The purpose of these experiments is to

explore this notion of the small scale independence from the large scales.

We begin in section 3.1 by characterizing the flow. Section 3.5 discusses conditioning structure

functions in order to assess large scale influence. In section 3.7 specific properties of the large

scales are considered.

1Note this chapter is largely adapted from work that has been published as Blum et al. “Effects of nonuniversal

large scales on conditional structure functions in turbulence” Physics of Fluids 22, 015107 (2010) [20]

18

Chapter 3 - Effects of Non-Universal Large Scales on Conditional Structure Functions 19

Mea

n V

ertic

al V

eloc

ity (

cm/s

) V

ertic

al V

eloc

ity V

aria

nce

(cm

2 /s2 )

-6

-4

-2

0

2

4

6

-20 -15 -10 -5 0

300

250

200

150

100

50

0

Distance From Center (cm)

a

b

Center

Region

Near

Grid

Region

Figure 3.1: Mean and variance of the vertical velocity along the central vertical axis of the tank. Grid

frequency is 3 Hz and grid separation distance is 56.2 cm. The dot-dash line represents the grid height at

maximum amplitude. We will focus on measurements in the two regions designated by the vertical dashed lines:

one at the center of the tank and one near the grid.

3.1 Characterizing the Flow

With the large scales being such an important part of this discussion it is important to adequately

characterize them in this particular flow. Through a good deal of effort a flow profile was created


-28.0

-16.8

-5.6

5.6

16.8

28.0D

ista

nce

From

Cen

ter

(cm

)

C

NG

Top Grid

Bottom Grid

. .

Figure 3.2: Scale diagram of 56 cm × 100 cm area between grids showing the mean circulation torii which

are nearly rotationally symmetric about the central vertical axis. Center (C) and near grid (NG) overreaction

volumes are drawn in dashed lines, which shows the relative size and position of the observation volume in Fig.

3.1. Horizontal dot-dashed lines represent the range of motion of the top and bottom grids.

that spans from the middle of the tank to bottom grid as shown in figure 3.1.

We define a characteristic velocity by U = (〈uiui〉/3)1/2 and a characteristic length scale by

L = U3/ε where ε is the energy dissipation rate per unit mass defined in section 3.3. For the

center region U = 6.0 cm/s, L = 9.0 cm, and for the near grid region U = 8.3 cm/s, L = 4.5 cm.

The Taylor Reynolds number, Reλ = (15UL/ν)1/2, (where ν is the kinematic viscosity) ranges

from 285 for 3 Hz grid frequency to 380 for 5 Hz grid frequency in the center. Near the grid

at 3Hz Reλ = 230. The Kolmogorov length and time scales are η = 140µm, τη = 20 ms in the

center region, and η = 94 µm, τη = 8.8 ms in the near grid region.

Figure 3.1a shows the mean vertical velocity as a function of the vertical position along the

central axis of the tank. The top and bottom grids are separated by 56.2 cm, approximately 7L.

In Fig. 3.1 the dot-dashed line indicates the maximum amplitude of the bottom grid, 22.1 cm


below the center of the tank. Data were collected at five separate heights in order to measure the

complete flow profile from the center to the bottom grid. Mapping the bottom half of the talk is

sufficient because the geometrical symmetry produces a mirror image above the midplane. The

two volumes which we will focus on are bounded by the dashed lines and will be referred to as

the center and near the grid observation volumes. At this grid separation distance the mean flow

traces four torii, two above and two belove the center plane of the tank, as shown in the sketch

in Fig. 3.2 (drawn to scale). In the large central region, the effect of the mean flow is to pump

highly energetic fluid from the region near the grid toward the center of the tank. In Fig. 3.1(a),

there are two points where the mean vertical velocity approaches zero: one near the center; the

other 18 cm from the center just below the near grid observation volume. The existence of this

second stagnation point and reverse circulation region depicted in Fig. 3.2 is a common feature

in mean flows generated by oscillations [21]. In all of the following measurements, the mean

velocity field has been subtracted so that we study the fluctuating velocity.

Figure 3.1(b) shows the vertical velocity variance along the central axis as a function of the

vertical position. The velocity variance is large near the grid and quickly falls off toward the

center where it is nearly homogeneous. The center and near grid observation volumes were chosen

to provide a contrast between the large homogeneous region in the center and the much more

inhomogeneous region near the grid. In the center, the variance of the velocity is homogeneous

for several L in either direction. The velocity variance ranges moderately in the near grid

observation volume and enormously with one L below this region. In Fig. 3.1 deviations from a

smooth curve are not due to statistical uncertainty, but are a result of patching five calibrated

regions together with the majority of error coming from measuring absolute position in the

tank.

It is interesting to note that we made measurements in a flow with smaller grid separation of

35 cm and found the Reynolds number in the center was lower. The characteristic velocity in

the center did increase due to the closer proximity of the grids, but L was reduced by a larger

amount resulting in approximately 8% decrease in Reλ. The reason for the unexpected decrease

in Reynolds number is a reversal of the mean velocity compared with larger grid separations.

For larger grid separation distances, energetic fluid from near the grids is carried to the center

by the mean flow. However, at 35 cm grid separation the mean velocity reverses which results

in a lower Reynolds number in the center.


10 1000.1

1.0

10.0

< ∆ur >2

(εr)2/3

r/η

r/η10 100

0.5

1.0

1.5

2.0<

∆u r

>2

(cm

/s)2

Figure 3.3: Eulerian second order longitudinal velocity structure function shown as a function of pair separation

r normalized by the Kolmogorov length η. The inset shows this data compensated by Eq.3.1 for p=2.

3.2 Structure Functions

We focus on the second order structure function because it is a easily measurable quantity that

has been studied thoroughly in the literature, is scale dependent, is clearly predicted in turbu-

lence theory, and requires an amount of statistics that is accessible to our measurements. To

measure the Eulerian structure functions we first find the instantaneous longitudinal velocity

difference between two particles a distance r apart ∆ur = [u(x)-u(x+r)]L, where the L sub-

script denotes the longitudinal component, found by projecting the 3D velocity difference vector

onto the vector connecting the two particles. The longitudinal structure functions are defined as

Dp = 〈(∆ur)p〉 where p represents the order of the structure function and the brackets represent

the ensemble average. In the inertial range, Kolmogorov (1941) gives

〈(∆ur)p〉 = C(E)

p (εr)p/3 (3.1)

where C(E)p are the Eulerian Kolmogorov constants and ε is the energy dissipation rate.

Figures 3.3 and 3.4 show measured second and third order longitudinal velocity structure func-

tions with the straight thin lines representing Kolmogorov’s prediction from Eq.(3.1). The insets

show the structure functions compensated by Eq.(3.1). At Reλ = 285, any scaling range is very


10 1000.01

0.10

1.00

10.00

r/η

< ∆ur >3

(εr)

r/η10 1000.0

0.2

0.4

0.6

0.8

1.0<

∆u r

>3(c

m/s

)3

Figure 3.4: Eulerian third order longitudinal velocity structure function. The inset shows this data compen-

sated by Eq.3.1 for p=3.

limited, but the plateaus can be used to estimate the inertial range.

Lagrangian structure functions were measured from temporal velocity differences along a particle

trajectory. The velocity difference now becomes ∆uτ = u(t) − u(t + τ), where τ is the time

interval between measurements. We use the vertical velocity for Lagrangian velocity differences

throughout this thesis, although results for the other components are similar. For Lagrangian

structure functions Kolmogorov (1941) predicts

〈(∆uτ )p〉 = C(L)

p (ετ)p/2. (3.2)

3.3 Energy Dissipation Rate Measurement

The energy dissipation rate ε is an important value throughout this analysis, let us discuss

how it is determined. Limitations in particle density preclude direct measurement via the

definition

ε = 2ν〈sijsij〉


with

sij =1

2

(

∂ui

∂xj+

∂uj

∂xi

)

.

Instead we utilize Kolmogorov’s 4/5 law: Eq. 3.1 with p = 3 where the coefficient C(E)3 = −4/5

can be derived from the Navier-Stokes equations. We identify the inertial range with the plateau

in the compensated third order structure function (Fig. 3.4 inset). The inertial range is chosen

to be 25 to 91 r/η (.35 to 1.3 cm). If the same inertial range is used in the second order

structure function, the energy dissipation rate determined from it (using the empirical coefficient

C(E)2 = 2.0) [2] is within 3% of the value calculated from the third order.

3.4 Phase Dependence

A simple energy cascade has constant energy input at the largest length scales. An obvious

departure from constant energy input is the oscillating grid driving mechanism. The sinusoidal

motion of the grid directly corresponds to energy with periodic time dependence. It seems likely

that such a strongly periodic energy input would have a signature throughout the whole energy

cascade.

The method we employ throughout this work to detect signatures of the large scales is to

condition structure functions on some measurement of the state of the large scales. This utilizes

the structure functions described in 3.2 and determines how they depend on the large scales.

It is using this method we can comment on how the large scales effect all scales in a turbulent

flow, from near the characteristic length scale L, down to near the dissipation scales η. In this

case, we condition of the phase of the grid motion, φ. Conditioning instantaneous single particle

statistics such as the mean and variance of the velocity shows some sinusoidal dependence on

grid phase. For example, in the center the conditional variance, 〈(u − 〈u〉)2|φ〉, varies by 1%

over the cycle of the gird. In the near grid region, the conditional variance varies by 10%. The

mean vertical velocity in the center 〈u|φ〉, varies by 0.8 cm/s over a cycle of the grid which is

10% of the standard deviation. Near the grid, the conditional mean velocity varies by 2 cm/s

which is 20% of the standard deviation at that location.

Figure 3.5 shows the second order longitudinal structure functions conditioned on phase, 〈∆u2r|φ〉.

In the center of the flow (fig.3.5(a)) the structure functions have essentially no change with

phase. Near the grid (3.5(b)) there is a slight dependence on grid phase. To emphasize the


10 1001.0

1.2

1.4

1.6

1.8

2.0

< ∆ur | φ >2

(ε3r)2/3

r/η3

10 100r/η

3

1.0

1.2

1.4

1.6

1.8

2.0

< ∆ur | φ >2

(ε3r)2/3

a

b

Figure 3.5: Second order compensated velocity structure functions conditioned on grid phase. The collapse

shows the very weak phase dependence: (a) center of the tank, (b) near the grid. Zero and 2π phase represents

grid at lowest possible amplitude. φ: + = 0 - 2π/5, ∗ = 2π/5 - 4π/5, ⋄ = 4π/5 - 6π/5, △ = 6π/5 - 8π/5 , � =

8π/5 - 2π.

differences between structure functions at different phases, we compensated the structure func-

tions by a single energy dissipation rate in each figure, ε3 = 2.46 × 103 mm2/s3 in the center

and ε3 = 1.31 × 104 mm2/s3 near the grid. These values were determined when the grid is


in mid-amplitude (the third bin). The good collapse of the structure functions at all phases

across the entire range of r shows the minimal dependence of the small scales on the large scale

periodicity of the flow created by the oscillating grids. One possible source of dependence of

small statistics on the large scales has been shown to be minimal.

3.5 Dependence on Large Scale Velocity

3.5.1 Eulerian structure functions conditioned on the large scale ve-

locity: Center region

A more revealing dependence on the large scales of the flow is found by conditioning the velocity

structure functions directly on the large scale velocity. A convenient measurable quantity that

reflects the local instantaneous state of the large scales is the average velocity of the particle

pair used for the structure function, defined as Σu = (u(x)+u(x+ r))/2. This velocity sum can

also be decomposed into longitudinal (Σu‖) transverse Σu⊥) or directional Σui) components.

Alternatively, conditioning on the average velocity of many particles, not just one pair, was

studied and found to have similar results, but we choose to focus on Σu because it can be more

easily measured and does not depend on the observation volume and seeding density. Additional

conditioning quantities will be discussed in Sec.3.7.

Figure 3.6(a) shows the second order Eulerian velocity structure function conditioned on Σuz.

The smallest values of the structure function correspond to pair velocities near zero, represented

by ⋄, while large |Σuz| results in larger values of the structure function. For the bins we chose, the

structure function conditioned on large values of Σuz, is nearly twice the value when conditioned

on Σuz near zero.

Figure 3.6(b) shows the data in Fig. 3.6(a) compensated by Kolmogorov inertial range scaling.

The functional forms are quite similar, confirming the impression from Fig. 3.6(a) that all length

scales are affected similarly by the instantaneous state of the large scales. In Fig. 3.6(b) we

used a different energy dissipation rate εuzto compensate each of the five individual large scale

vertical velocity bins. This insures all conditions plateau at approximately the same value and

allows for direct comparison of the functional forms of the conditional structure functions.

The strong dependence of the conditional structure functions on the large scale velocity at all


10

100

10 1000.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

r/η

a

b

< ∆

u r |

Σu z

>2

(εΣ

u r)

2/3

z

< ∆

u r |

Σu z

>2

(cm

/s)2

Figure 3.6: Second order velocity structure function conditioned on particle pair velocity (vertical component)

in the center of the tank. (a) Uncompensated structure function. (b) Individually compensated by the energy

dissipation rate for each conditional data set. Symbols represent the following dimensionless vertical velocities,

Σuz/√

〈u2z〉: + = 4.2 to 2.5, ∗ = 2.5 to 0.84, ⋄ = 0.84 to -0.84, △ = -0.84 to -2.5, � = -2.5 to -4.2.

scales reveals that the small scales are not statistically independent of the large scales in this

flow. There is no detectable trend toward the smaller scales becoming less dependent on the

large scale velocity than somewhat larger scales.


10-1

100

101

102

1

2

3

4

5

< ∆

u τ | Σ

u z >2 (εΣ

u τ

)z

10-1 100 101τ/τ

η102

a

b

< ∆

u τ | Σ

u z >

2(c

m/s

)2

Figure 3.7: Second order Lagrangian velocity structure function conditioned on instantaneous velocity (vertical

component) in the center of the tank. (a) Uncompensated structure function. (b) Individually compensated to

have the peak values match. Symbols represent the following dimensionless vertical velocities, Σuz/√

〈u2z〉: + =

3.1 to 1.9, ∗ = 1.9 to 0.62, ⋄ = 0.62 to -0.62, △ = -0.62 to -1.9 , � = -1.9 to -3.1.

3.5.2 Lagrangian Structure Functions Conditioned on the Large Scale

Velocity: Center Region

In much the same way we can evaluate the conditional Lagrangian structure functions. Figure

3.7(a) shows the second order Lagrangian structure function conditioned on the vertical compo-


nent of the large scale velocity, Σuz. Here we find Σuz by averaging the velocity of the particle

at the two times used to determine ∆uτ . The conditional structure functions for different large

scale velocity are different by a factor of about 2.5 and they remain nearly parallel throughout

the entire time range.

Figure 3.7(b) shows the second order conditional Lagrangian structure function compensated by

Eq. 3.2, where ε is individually chosen so the maxima of all the conditioned structure functions

coincide. This aids comparison of the functional forms of the conditioned structure functions.

Again, the functional form is nearly identical for different large scale velocities, indicating that

the large scales affect all time scales in the same way. There may be a small trend toward larger

values of the compensated Lagrangian structure functions at small times when the magnitude

of the large scale velocity is large.

It should be noted that there is a bias present in Lagrangian measurements that is not present

in Eulerian measurements. A sample of measured Lagrangian trajectories is biased toward low

velocity particles since the high velocity particles are more likely to have left the measurement

volume. This bias becomes larger for larger τ . Berg et al. [22] studied this bias and find

that it can be quite large for typical experimental conditions. We quantified this bias in our

data by measuring the Lagrangian structure functions using trajectories that remained inside

artificially restricted measurement volumes. From a simple extrapolation of the dependence

on the size of the artificial detection volume, we estimate that our experimental Lagrangian

structure functions underestimate the true value by 17% for τ = 8τη. This is roughly consistent

with the size of the error we expect based on the critical time lag defined in Ref. [22]. Note that

we have not performed the compensation they recommend and we are roughly translating their

uncompensated results. Because of this bias, we will focus attention on τ < 10τη. As we will

discuss in Sec.3.6.3, the dependence of the conditioned Lagrangian structure functions on the

large scale velocity does not seem to be significantly influenced by this bias.

3.5.3 Eulerian Structure Functions Conditioned on the Large Scale

Velocity: Near Grid Region

By comparing separate regions of the tank we are able to explore the effects of inhomogeneity

on this conditional dependence. Figure 3.8(a) shows the Eulerian structure functions, similar


to Fig. 3.6(b), but with data collected near the inhomogeneous region near the bottom grid.

The separation between the Eulerian structure function conditions doubled to approximately a

factor of 4. Note the different ordering of the structure functions. The up-down symmetry is

now broken since these data were not taken in the center of the tank. Fluid traveling upwards (∗symbols) has a large structure function, while fluid traveling downward with the same magnitude

vertical velocity (△ symbols) has the lowest value of the structure function. We interpret this

as highly energetic fluid originating near the bottom grid and being turbulently advected into

the observation volume. Similarly, fluid carried down from the more quiescent region above the

detection volume has low energy and a smaller structure function.

Figure 3.8(b) shows the compensated Eulerian structure functions, similar to Fig. 3.6(b), but

reveals a novel insight. Stepping through the vertical velocity bins is equivalent to stepping

through the energy cascade. Fluid coming directly upward from the bottom grid (symbol +)

carries energy that was recently injected into the large scales. As a result, the compensated

structure function for upward moving fluid is biased toward the large scales. Fluid that has

downward vertical velocity (symbol △) comes from the center region far away from the grid. It

has had more time to mature, and in this process the energy is transported to smaller length

scales. Conditional structure functions appear to be effective tool to evaluate whether or not a

turbulent flow is fully developed and has established a stable cascade.

3.5.4 Third order Eulerian Structure Functions Conditioned on the

Large Scale Velocity: Center Region

Figure 3.9 shows the third order structure function individually compensated and conditioned

on Σuz in the center of the tank. Convergence of third order statistics was more difficult, so

elimination of the two extreme conditions was required. The third order structure function

proves to be similar to the second order in separation, symmetry and collapse to a single func-

tional form. The energy dissipation rates found for the three conditions are ε∗ = 25.2cm2/s3,

ε⋄ = 21.7cm2/s3,ε△ = 28.7cm2/s3.


3.6 A Powerful Method for Plotting Conditional Struc-

ture Functions



An alternative, and in many ways more powerful, method of visualizing the data is presented

in Fig. 3.10. Here we show the second order Eulerian structure function conditioned on the

1001.2

1.4

1.6

1.8

2.0

2.2

2.4

10

100

a

b

r/η

< ∆

u r |

Σu z

>2

(εΣ

u r)

2/3

z

< ∆

u r |

Σu z

>2

(cm

/s)2

Figure 3.8: Second order velocity structure function conditioned on particle pair vertical velocity (z direction)

in the region near the bottom grid. The condition with the largest downward velocity has been eliminated due

to lack of statistical convergence. Symbols represent the following particle pair vertical velocities Σuz/√

〈u2z〉:

+ = 3.8 to 2.3, ∗ = 2.3 to 0.75, ⋄ = 0.75 to -0.75, △ = -0.75 to -2.3, a) Uncompensated structure functions b)

Individually compensated by the energy dissipation rate for each conditional data set.


10 100

0.2

0.4

0.6

0.8

r/η

< ∆

u r |

Σu z

>3

(εΣ

ur)

z

Figure 3.9: Third order velocity structure function plots conditioned on particle pair vertical velocity and

individually compensated for each conditional data set. Data are taken in the center region of the tank, and the

extreme vertical velocity plots have been eliminated due to lack of statistical convergence. Symbols represent the

following vertical velocities Σuz/√

〈u2z〉: ∗ = 2.5 to 0.84, ⋄ = 0.84 to to -0.84, △ = -0.84 to -2.5.

vertical component of the large scale velocity (the same data as Fig. 3.6). However, the scaled

vertical pair velocity is plotted on the horizontal axis with the conditioned structure functions

on the vertical axis. When the structure functions are scaled by their value at Σuz = 0, we

find very good collapse of the data. The fact that these curves for different r/η collapse so well

is a striking demonstration that the large scales affect all length scales in the same way. The

fact that the conditional structure functions vary by a factor of 2.5 demonstrates the strong

dependence on the large scales. Note that for Gaussian random fields, Fig.5.5 is flat (this will

be discussed further in section 3.9, and again in 5.3). A nearly flat result is also observed in

DNS and grid turbulence [1] and reproduced in figures 5.2a and 5.2b. Note there is a significant

difference in the structure functions when the large scale velocity is larger than two standard

deviations away from the mean between these two simulations, this will be discussed in section

5.3. We will also discuss the measurement error in this type of plot in appendix A.

In Fig. 3.10 we see the dependence is a steep parabola. This plotting method makes clear the

extent to which all the smaller scales are affected by the large scale velocity; in fact, all length

scales collapse nearly perfectly onto one parabola. The large scale velocity affects all length


2

<Δur|Σu

z>

<Δur|Σu

z>

Σu = 0

2

Z

Σuz

√<uz>2

-4 -2 0 2 40.5

1.0

1.5

2.0

2.5

3.0

Figure 3.10: Eulerian second order conditional structure function versus large scale velocity. Data taken in

the center region. Each curve represent the following separation distances r/η: + = 0 to 40, ∗ = 40 to 70, ⋄ =

70 to 110, △ = 110 to 140, � = 300 to 370, × = 370 to 440.

scales in nearly the exact same way, all the way down to the dissipative range.


Velocity: Higher Reynolds Number

Figure 3.11 shows the effect of increasing Reynolds number. These data are at the center of the

tank with the grids oscillating at 5 Hz instead of 3 Hz which increases Reλ from 285 to 380. The

collapse of the structure function remains. The curvature in the this figure is not significantly

different from the lower Reynolds number data in Fig. 3.10 indicating that if there is a Reynolds

number dependence it is weak.

Figure 3.12 shows a comparison of our data with data taken in the atmospheric boundary

layer [1] with Reλ > 104. Atmospheric boundary layer turbulence shows a similar collapse

of conditional structure functions at all length scales. The curvature is also similar in both

data sets, indicating that the dependence on the large scales is similar even at these very large

Reynolds numbers.


2

<Δur|Σu

z>

<Δur|Σu

z>

Σu = 0

2

Z

Σuz

√<uz>2

-4 -2 0 2 40.5

1.0

1.5

2.0

2.5

3.0

Figure 3.11: Eulerian second order conditional structure function versus large scale velocity. Data taken in the

center region at higher grid frequency, 5Hz, resulting in higher Taylor Reynolds number 380. Symbols represent

the following separation distances r/η: + = 0 to 50, ∗ = 50 to 100, ⋄ = 100 to 150, △ = 150 to 200, � = 310 to

420, × = 420 to 520.



Figure 3.13 shows the Lagrangian structure functions plotted versus the large scale velocity,

comparable to the Eulerian data shown in Fig. 3.10. The parabolic shape remains, but the

curvature is greater for all Lagrangian time scales that it is in the Eulerian data. All time scales

are affected by the large scale velocity. To determine the effect of measurement volume bias, we

have done this analysis for artificially restricted measurement volumes. By decreasing the volume

by a factor of 2, we observe the large τ curves shift by approximately the deviations between

the curves. We conclude that the bias does not have a significant effect on the conditional

dependence shown in Fig.3.13 for the time differences presented (τ ≤ 10τη).


2

<Δur|Σu

z>

<Δur|Σu

z>

Σu = 0

2

Z

Σuz

√<uz>2

-4 -2 0 2 40.5

1.0

1.5

2.0

2.5

3.0

Figure 3.12: Eulerian second order conditional structure function versus large scale velocity. The thin plots

are from atmospheric boundary layer data [1] r/η: ∗ ∼ 100, △ ∼ 400, � ∼ 1000, × ∼ 1250. The thick line is

from fig. 3.10, which has been overlaid for comparison, r/η: ⋄ = 70 to 110.

-4 -3 -2 -1 0 1 2 3 4

1

2

3

4

5

Σuz

√<uz>2

2

<Δuτ|Σu

z>

<Δuτ|Σu

z>

Σu = 0

2

Z

Figure 3.13: Lagrangian second order conditional structure function versus large scale vertical velocity. Data

taken in the center region. Symbols represent the following τ/τη : + = 0.42 , ∗ = 1.3, ⋄ = 3.5, △ = 10.




Figure 3.14 shows a conditional Eulerian structure function similar to Fig. 3.10, but measured

in the inhomogeneous region near the grid. The structure function here is strikingly different

than in the center. The minimum is shifted by more than one standard deviation to the left.

The inhomogeneity breaks the up-down symmetry so that fluid coming directly up from the

bottom grid is markedly different than fluid coming down from the more quiescent region above

(analogous to the ∗ and △ separation in Fig.3.8). It follows that fluid with an upward velocity

has higher energy than fluid with the same velocity magnitude in the downward direction.

The atmospheric boundary layer data in Fig.3.12 also show this effect with a minimum at

Σuz/√

〈u2z〉 = −0.5, presumably as a result of weaker inhomogeneity. Also notable is that the

collapse of plots for various r values is not as complete as in the central region. This is consistent

with Fig. 3.8(b) which shows that the conditional structure functions have somewhat different

r dependence.



Figure 3.15 shows a Lagrangian structure function measured in the near grid region, similar to

the Eulerian data in Fig. 3.14. The minimum is shifted to the left here also as a result of the

inhomogeneity in this region of the flow. The conditional dependence on the large scale velocity

is again somewhat larger than in the Eulerian case, and the collapse at different time scales is

not as complete.

3.6.6 Third Order Eulerian Structure Functions Conditioned on the

Large Scale Velocity: Center Region

The third order Eulerian velocity structure function plotted versus the large scale vertical veloc-

ity is shown in Fig.3.16 using data from the center of the tank. Statistical convergence is weaker

than the second order which limits the large scale velocity range available for analysis. The

collapse seems similar to the second order case shown in Fig.3.10, although the measurement


-4 -2 0 2 40.5

1.0

1.5

2.0

2.5

3.0

2

<Δur|Σu

z>

<Δur|Σu

z>

Σu = 0

2

Z

Σuz

√<uz>2

Figure 3.14: Eulerian second order conditional structure function versus large scale velocity. Data taken in the

near grid region of the tank. The structure function is heavily influenced by the bottom grid which has skewed

the symmetry of the plot minima in the negative direction. Symbols represent the following non-dimensional

separation distances r/η: + = 0 to 50, ∗ = 50 to 110, ⋄ = 110 to 160, △ = 270 to 320, � = 330 to 450, × = 450

to 560.

uncertainties are larger here. The curvature seems to be slightly larger for the third order than

for the second order case.

3.6.7 Second Order Eulerian Structure Functions Conditioned on the

Velocity Magnitude: Center Region

The second order Eulerian structure function plotted versus the magnitude of the pair velocity

is shown in Fig.3.17 using data from the center observation volume. The magnitude of the pair

velocity is also a useful indicator of large scale activity. It has no preferred direction and it is

a more direct indicator of the instantaneous local energy. A similar dependence remains as in

Fig.3.10, the collapse seems similar, and the curvature is significantly larger.


-4 -3 -2 -1 0 1 2 3 4

2

3

4

5

6

Σuz

√<uz>2

2

<Δuτ|Σu

z>

<Δuτ|Σu

z>

Σu = 0

2

Z

Figure 3.15: Lagrangian second order conditional structure function versus large scale vertical velocity. Data

taken in the near grid region of the tank. Symbols represent the following τ/τη : + = 0.94, ∗ = 2.8, ⋄ = 8.0.

-2 -1 0 1 20.5

1.0

1.5

2.0

2.5

3.0

Σuz

√<uz>2

3

<Δur|Σu

z>

<Δur|Σu

z>

Σu = 0

3

Z

Figure 3.16: Eulerian third order conditional structure function versus large scale vertical velocity in the center

region. Symbols represent the following non-dimensional separation distances r/η: + = 0 to 40, ∗ = 40 to 70, ⋄

= 70 to 110, △ = 110 to 140, � = 220 to 300, × = 300 to 370.


0.0 1.0 2.0 3.00.5

1.0

1.5

2.0

2.5

3.0

2

<Δur|Σ|u|>

<Δur|Σ|u|>

Σ|u| = 0

2

Σ|u|2

√<|u| >

Figure 3.17: Eulerian second order conditional structure function versus magnitude of the velocity pair in the

center region. Symbols represent the following non-dimensional separation distances r/η: + = 0 to 40, ∗ = 40

to 70, ⋄ = 70 to 110, △ = 110 to 140, � = 300 to 370, × = 370 to 440.

3.7 Properties of the Large Scales and Their Effects

The measurements provided in this chapter show the dependence of all scales including the

inertial range and small scales on the current state of the large scales. In this section the

properties of the large scales that might be responsible for this dependence on the large scales

will be discussed. It is important to note that this list is not exhaustive, and involves some level

of speculation. However, once this dependence has been shown and quantified the next step of

identifying possible contributors to the large scale dependence is valuable. We begin to address

the effects of the properties of the large scales here, and continue by comparing different flows

with unique large scales in chapter5.

3.7.1 Reynolds Number

An immediate concern when discussing large scale effects is if the oscillating grid flow has

a Reynolds number insufficient for adequate scale separation, and it is this which leads to

contamination of the small scale statistics by the large scales. Evidence points to large scale


dependence not being caused by limited Reynolds number. The comparison in Fig. 3.12 shows

that atmospheric boundary layer data [1] with very large Reynolds number (Reλ > 104) have

nearly the same dependence on the large scales as our flow. Increasing the Reynolds number

in our flow makes very little difference. Additionally, all length scales collapse to nearly the

same functional form indicating that limed separation of scales is not the primary factor. Taken

together, these lead us to the conclusion that merely high Reynolds number alone is not enough

to create small scales that are statistically independent of the large scales.

3.7.2 Anisotropy

The flow between oscillating grids is somewhat anisotropic. The ratio of vertical to horizontal

velocity standard deviations is 1.5:1 in the center. The effects of large scale anisotropy on the

small scales have been studied extensively, [13] but we suspect it is not a major factor in the

conditional dependence studied here. We analyzed the data by averaging over particle pairs

with all orientations, so when the structure functions are conditioned on a quantity with no pre-

ferred direction like the velocity magnitude (Fig.3.17) there should be very little contribution

from anisotropy. In fact, we find that the conditional dependence on velocity magnitude is even

stronger than the dependence on the vertical velocity component. We also observe the condi-

tional dependence remains when conditioned on other quantities without preferred directions

like Σu‖ and Σu⊥. We believe anisotropy is not a significant cause of the conditional dependence

we observe, but anisotropic effects is an intriguing topic currently being investigated by another

graduate student in the lab.

3.7.3 Shear

Sreenivasan and Dhruva [1] attribute the strong conditional dependence of the Eulerian structure

functions on the large scale velocity to shear in the atmospheric boundary layer. In comparing

conditional structure functions from both the atmospheric boundary layer and the flow between

oscillating grids a similar dependence is found despite significantly different shear. The mean

velocity gradient normalized with the eddy turnover time is 1.2 in the center of the oscillating

grid flow, and is estimated to be in the range of 5 or greater for their atmospheric boundary layer

data. There must be some other properties that exist in the shear flow, but are also important


in our flow with small shear.

3.7.4 Inhomogeneity

In figure 3.1(b) the homogeneity in the center and the near grid observation volumes is similar,

the velocity r.m.s. varies only slightly within the defined regions. What varies greatly for the

two regions is the inhomogeneity in velocity r.m.s. directly adjacent to the observation areas.

The center observation volume is surrounded by homogeneous turbulence for several L on along

the central axis. Whereas the velocity r.m.s. varies greatly within one L. We observe that

the Eulerian and Lagrangian data near the grid in Figs. 3.14and 3.15 show that the structure

functions depend greatly on the origin of the fluid being swept into the observation volume.

Fluid coming from energetic regions of the tank has larger structure functions than fluid coming

from more quiescent regions. Inhomogeneity is directly responsible for the shift of the minima

in Figs. 3.14and 3.15 away from zero vertical velocity. In the center of the tank (Fig. 3.10), the

inhomogeneity is much smaller, but it could be responsible for some part of the curvature since

both fluid coming downward and fluid coming upward would be coming from more energetic

regions symmetrically.

3.7.5 Large Scale Intermittency

Inhomogeneity alone does not account for all of the large scale dependence observed. There is

also a significant contribution from large scale intermittency. Fernando and De Silva [23] show

large scale intermittency can exist in an oscillating grid flow depending on boundary conditions.

Although we use their recommended boundary conditions we observe clear signatures of large

scale intermittency. The velocity distribution in the center of the flow is bimodal, indicative of

switching between two flow states. This effect is more prominent in preliminary data we took

for grid spacings of 66 and 100 cm that it is in the present data taken at 56.2 cm.

Our measurements show a dependence of the conditional structure functions on the large scale

velocity that cannot be fully attributed to inhomogeneity, and large scale intermittency is the

most likely cause. Clear evidence for this comes from conditioning the structure functions

on the horizontal components of the large scale velocity, Σux and Σuy instead of the vertical

component, Σuz. The horizontal midplane (x and y directions) is much more homogeneous


than the vertical axis (Z direction). Yet, the structure functions conditioned on Σux or Σuy

show a large scale dependence that is only moderately smaller than the Σuz condition (72% and

85% of the dependence seen in Σuz). If the inhomogeneous direction shows similar conditional

dependence on the large scales as the homogeneous directions show, then it seems that a large

part of the conditional dependence must come from fluctuations of the large scales and not

directly from inhomogeneity. Praskovsky et al. [24] attribute large scale intermittency as a

crucial component of the large scale dependence they observe. To investigate this effect further

we devised an experiment described in chapter 4.

3.8 Kinematic Correlations

A reasonable suspicion might be that the observed dependence is a kinematic correlation, mean-

ing that particle pairs with large velocity may also have a large velocity difference simply because

the same measurements are used in both cases and they are intertwined in a mathematical sense.

Hosokawa [25] identified that Kolmogorov’s 4/5ths law requires that velocity sums and differ-

ences be correlated so that

〈u2+∆u−〉 =

ǫr

30(3.3)

where u− is half the longitudinal velocity difference and u+ is half the sum. (For comparison, we

have used ∆ur = 2u− and Σu‖ = u+.) Kholmyansky and Tsinober [10] provide an experimental

confirmation of this and in a more recent paper [26] present a list of kinematic relations. In

addition, several lines of evidence indicate that kinematic correlation does not account for the

majority of the dependence we observed.

First, note that two independent random samples with identical Gaussian distributions have a

difference that is uncorrelated with the sum, so that the conditional dependence seen in Fig.

3.10 would be flat. This remains true for velocity differences and sums from Gaussian random

fields. Both of these results can be obtained by considering the joint pdf of the two samples and

the the rotating 45◦ to the coordinate system of sums and differences. Because the samples are

interchangeable, the sum and difference axes have to be principle axes of the joint Gaussian pdf,

and the conditional variance of the difference is independent of the sum.

Of course, turbulent velocities are not joint Gaussian. However, from kinematic relations in the

literature we have not been able to derive predictions for the conditional structure functions


that we consider or for the correlation 〈(Σu)2(∆u)2〉 that would capture the main conditional

dependence we see.

An analysis of conditioning in Gaussian random fields shows that conditioning on the velocity

at either point alone produces a kinematic correlation that is avoided by our conditioning on

the velocity sum.

To make an experimental estimate of the effect of kinematic correlations, we conditioned the

velocity differences on several different quantities. For each particle pair, we calculated the longi-

tudinal and transverse components of the average velocity of the particles. We then conditioned

the longitudinal structure functions on the longitudinal and transverse pair velocities. The idea

here is that while conditioning on the longitudinal component (Σu‖) could have a kinematic cor-

relation, conditioning on the transverse component (Σu⊥) should have no kinematic correlation.

We found that conditioning on (Σu‖) had a roughly 30% larger effect on the structure functions

that conditioning on (Σu⊥). Conditioning on Σuz should have less kinematic correlation than

conditioning on Σu‖. So more than 70% of the effect remains unexplained by kinematic corre-

lation. We conclude that while kinematic correlation may possibly make a contribution to the

conditional dependence, the majority of the effect comes from the large scales.

3.9 Discussion

An immediate concern when discussing large scale effects is if the oscillating grid flow has

a Reynolds number insufficient for adequate scale separation, and it is this which leads to

contamination of the small scale statics by the large scales. Evidence points to large scale

dependence not being caused by limited Reynolds number. The comparison in Fig. 3.12 shows

that atmospheric boundary layer data [1] with very large Reynolds number (Reλ > 104) has

nearly the same dependence on the large scales as our flow. Increasing the Reynolds number in

our flow makes very little difference. Additionally, all length scales collapse to nearly the same

functional form indicating that limited separation of scales is not the primary factor. Taken

together, these lead us to the conclusion that merely high Reynolds number alone is not enough

to create small scales that are statistically independent of the large scales.

Our flow is somewhat anisotropic. The ratio of vertical to horizontal velocity standard deviations

is 1.5:1 in the center. The effects of large scale anisotropy on the small scales has been studied


extensively [14], but it is not a major factor in conditional dependence studied here. We have

analyzed our data by averaging over particle pairs with all orientations, so when the structure

functions are conditioned on a quantity with no preferred direction like the velocity magnitude

(Figure 3.17) there should be very little contribution from anisotropy. In fact, we find that

the conditional dependence on velocity magnitude is even stronger than the dependence on

the vertical velocity component. We also observe the conditional dependence remains when

conditioned on other quantities without preferred directions like Σu‖, and Σu⊥. We conclude

that anisotropy of the large scales is not a significant cause of the conditional dependence we

observe.

Sreenivasan and Dhruva [1] attribute the strong conditional dependence of the Eulerian structure

functions on the large scale velocity to shear in the atmospheric boundary layer. In making

this argument, they show an important piece of information in their figure 6 which shows

conditioned structure functions in homogeneous turbulence from both DNS and wind tunnel

grid turbulence. The conditional statistics in these homogeneous and isotropic flows show no

apparent dependence on the large scale velocity (we will investigate these systems independently

in chapter 5). However, we conclude that shear is not the fundamental property responsible in

our flow since the oscillating grid flow has a much lower shear but produces much the same

dependence on the large scale velocity. The mean velocity gradient normalized with the eddy

turnover time is 1.2 in the center of our flow and we estimate it is in the range of 5 or greater

for their atmospheric boundary layer data. There must be some other properties that exist in

the shear flow, but also are important in our flow with small shear.

Our data clearly show the role that inhomogeneity plays in the observed large scale depen-

dence. Our Eulerian and Lagrangian data near the grid in Figs. 3.10 and 3.15 show that the

structure functions depend greatly on the origin of the fluid being swept into the observation

volume. Fluid coming from energetic regions of the tank have larger structure functions than

fluid coming from more quiescent regions. Inhomogeneity is directly responsible for the shift of

the minimum in Fig. 3.14 away from zero vertical velocity. In the center of the tank (Fig. 3.10),

the inhomogeneity is much smaller, but it could be responsible for part of the curvature since

both fluid coming downward and fluid coming upward would be coming from more energetic

regions symmetrically.

However, inhomogeneity alone does not account for all of the large scale dependence observed.


There is also a significant contribution from large scale intermittency, and it is possible that

this is the dominant contribution in the center of the tank. Large scale intermittency has been

difficult to quantify. It can be defined as any temporal fluctuations in the large scales that occur

on timescales longer than the eddy turnover time, L/U . We will discuss it briefly here, and it is

the focus for the next chapter.

Fernando and DeSilva [23] show large scale intermittency can exist in an oscillating grid flow de-

pending on boundary conditions. We have observed clear signatures of large scale intermittency

in our flow. Although we use their recommended boundary conditions, the velocity distribution

in the center of the flow is slightly bimodal indicative of switching between two flow states. This

effect is more prominent in preliminary data we took for grid spacings of 66 cm and 100 cm

than it is in the data for 56.2cm presented in this thesis.

Our measurements show a dependence of the conditional structure functions on the large scale

velocity that can not be fully attributed to inhomogeneity, and large scale intermittency appears

to be the most likely cause. The clearest evidence for this comes from conditioning the structure

functions on the horizontal components of the large scale velocity, Σux and Σuy instead of

on the vertical component, Σuz. The horizontal midplane (x and y directions) is much more

homogeneous than the vertical axis (z direction). Yet, the structure functions conditioned on

Σuy or Σux show a large scale dependence that is only moderately smaller than for the Σuz

condition (85% and 72% of the dependence seen in Σuz). If the inhomogeneous direction shows

similar conditional dependence on the large scales as the homogeneous directions show, then

it seems that a large part of the conditional dependence must come from fluctuations in the

large scales, and not directly from inhomogeneity. Praskovsky et al. [24] attribute large scale

intermittency as a crucial component of the large scale dependence they observe. This is not

enough to draw any conclusions about the role intermittency plays, so the next chapter is

dedicated to further investigation.

We have largely ignored considerations of power law scaling which has been a focus of much

of the previous work on this subject. Because of the relatively low Reynolds number of our

experiment, we can not make sensitive tests of scaling. However, our data provides a plausible

picture about how the large scales should affect power law scaling. If the data in Fig. 3.10

collapses to a single curve, then the dependence of the conditional structure functions on r and

uz are separable and the large scale dependence will have no effect on the scaling exponents of


unconditional structure functions. When this type of plot does not collapse as in Figs. 3.14 and

3.15, then the power law scaling will be affected by the large scales.

Chapter 4Effects of Large Scale Intermittency on

Conditional Structure Functions

The previous chapter gave experimental evidence that the large scale velocity affects all scales

in a turbulent flow in our oscillating grid experiment. This is an intriguing finding because it

contradicts a strict interpretation of Kolmogorov, that the small scales are completely indepen-

dent of the large scales. This result appears robust, as we will discuss in the next chapter, the

dependence is not an exception, but actually typical for all but the most controlled flows. Once

we have accepted the dependence exists a natural next step is to gain some understanding as

to what role various properties of the large scales play in this dependence. In section 3.7 we

discussed several properties of the large scales and what affects they may have on the observed

dependence. Two particular properties of the large scales that are likely candidates for influ-

encing the dependence small scales have on the large scales are inhomogeneity and large scale

intermittency. Whereas the experiments in the previous chapter focused on inhomogeneity by

measuring separate regions of the tank, this chapter will focus on experiments done to quantify

the effects of large scale intermittency.

As defined before, large scale intermittency refers to temporal fluctuations on time scales longer

than the eddy turnover time, τL = L/U , where U is the characteristic velocity U = (〈uiui〉/3)1/2

and L is the characteristic length scale L = U3/ε where ε is the energy dissipation rate per unit

47

Chapter 4 - Effects of Large Scale Intermittency on Conditional Structure Functions 48

mass. We can easily estimate a time scale associated with large scale intermittency in our flow.

For continuous driving at 3Hz the characteristic length scale L is 7.9 cm, and the characteristic

velocity U is 5.7 cm/s, making the eddy turnover time τL = 1.4 seconds. For an upper estimate

of the time scale of large scale intermittency we can approximate the length scale to be half the

width of the tank. This is an appropriate approximation because large scale intermittency can

include any fluctuations on the mean flow. As we saw in diagram 3.2 the mean flow consist of

large torii, where the size of one torus is approximately half the width of the tank; this is the

essentially the largest coherent structure allowed by the system dimensions. To calculate the

velocity scale we use (〈〈ui〉〈ui〉〉/3)1/2 where 〈ui〉 is the mean velocity which we had previously

subtracted off the velocity measurements to attain the fluctuating velocity. We arrive at a time

scale of roughly 49 seconds. In other words, we can estimate large scale intermittency to be

time scales from τL to 35τL.

Directly measuring length scales that are half the size of the tank is restricted by the maximum

size attainable of the observation volume. However, our apparatus has the advantage of being

able to measure very long durations, and can easily measure time scales over the duration of

interest. In fact, inspection of a velocity vs. time scatter plot for data taken in the center of

the tank reveals some long duration coherent structures. Due to the intermittent nature of this

phenomenon these structures are not immediately apparent, and elude simple quantification by

Fourier transform for example. However, close inspection reveals the velocity does show some

switching behavior, transitioning, for example from, predominately upward to predominately

downward velocity on the order of 30 seconds, which is in the range of interest. This can

also be seen in probability distribution function of velocity (particularly the vertical velocity

component), where a clearly bimodal distribution is not uncommon for data sets of several

hundred τL in duration.

Quantification of large scale intermittency is a challenge. The time scales may be long with

a broad range (as our example just showed), and there is no obvious way to control it. That

is to say, there is no obvious parameter than can be adjusted that can dial up or down the

large scale intermittency, and there is no accepted way to measure it. De Silva and Fernando

suggested the perimeter of the oscillating grid plays a role in the large scale intermittency in the

flow [23]. We have followed their suggestion to minimize large scale intermittency, yet as we will

see it remains to some extent. There does not seem to be a straightforward way to lessen the


large scale intermittency further. Faced with this, we instead measured the effect of large scale

intermittency by attempting to control the large scale energy injection. In essence, large scale

intermittency varies the amount of energy in the observation volume on time scales longer than

the eddy turnover time. Since we could not lessen the large scale intermittency, our strategy to

measure its effects was to augment the large scale intermittency in a typical continuous driving

case and measure the results. To accomplish this we varied the energy input into the system over

a range of time scales. For example, the energy can be injected into the system by running the

grids at a certain frequency, then periodically halting the grids suddenly, thus briefly stopping

any energy injection, and lowering the energy measured in the observation volume. This can

effectively augment the large scale intermittency which occurs with constant driving. There are

3 parameters that we varied in order to control the addition of large scale intermittency: the

amplitude of the energy injection, the duty cycle over which we varied the energy injection, and

the duration of the energy injection.

There has been some previous work on attributing the dependence of the small scales on specific

properties of the large scales. The seminal work of Praskovsky et al. was done in two high

Reynolds number wind tunnel flows, a return channel and a mixing layer. They observed

strong correlations between the large scales and the velocity structure functions. All length

scales were largely affected in the return channel; at two standard deviations greater than

the mean large scale velocity the conditioned structure function was 50% greater than that of

the unconditioned structure function. The mixing layer was only slightly less affected by the

conditioning on the large scales, and had a large difference in the spanwise and streamwise flow.

Praskovsky et al. concluded Kolmogorov theory was not contradicted, the small scales could

still be independent of the large scales. The observed correlation between the large scales and

the velocity structure functions ”... reflects the spatial and temporal variability of the energy

flux through a cascade” [24]. Although the terminology was different this equates to attributing

the observed dependence of the small scales on the large scales on the fluctuations of the energy

into or out of the large scales. These fluctuations occur on time scales longer than the eddy

turnover time, a characteristic of large scale intermittency.

In a slightly different manner turbulent shear flows also contain large scale intermittency. S. B.

Pope shows through an analysis of direct numerical simulations that a large amount of quiescent

ambient fluid is entrained in the turbulent fluid produced by round jets [2]. He gives the example


of a turbulent round jet, where at x/d = 100, where x is the position downstream, and d is the

jet diameter, 80% of the fluid in turbulent motion has been entrained between x/d = 20 and

x/d = 100. In such a manner the fluid within the jet simultaneously contains fluid elements

with very different origins, and different turbulent properties. In addition to fluctuations of the

energy flux which Praskovsky observes, entrainment of fluid with different turbulent properties

could possibly lead to fluctuations of energy longer than the eddy turnover time, a characteristic

of the large scale intermittency.

What we wish to accomplish in this chapter is to begin to quantify the effects the large scale

intermittency. Specifically, how it effects the dependence of the small scales on the large scales

which we have previously observed. We hypothesize the large scale intermittency can affect the

measured small scales in a mixture of the two ways exemplified by Praskovsky et al. and Pope:

fluctuations in the energy flux into and out of the largest scales, and the entrainment of fluid

with different turbulent properties present in the observation volume. Future studies may be

able to measure the relative weight of these to mechanisms of large scale intermittency, but in

this chapter, regardless of it’s cause, we will see large scale intermittency can have a significant

effect on small scale statistics.

4.1 Data

In this section we quantify experimentally how conditional structure functions depend on large

scale intermittency. Throughout this chapter we will use continuous driving of the oscillating

grids at 3 Hz as a base case. As we showed in figure 3.10 the structure function at all scales

(including the small scales) is affected by the large scale velocity. We attributed this dependence

in part to both inhomogeneity and large scale intermittency. To isolate the role that large scale

intermittency plays we attempt to augment it by varying the time dependence, and amplitude

of the energy injection.

In order to add large scale intermittency in a controlled way the motor driving frequency was

modulated, which created two distinct energy inputs. In an ideal experiment the motor could be

changed instantaneously from one motor frequency to another. However, in the real world ex-

periment the motor was attached to the grids and rods in the apparatus which have considerable

rotational inertia. This was most pronounced when halting the motor completely going from 3


to 0 Hz. Efforts were made to minimize this effect by replacing the flywheel with a much smaller

one, with less inertia, and studying how best to ramp down the motor current. Ultimately,

the time it took to fully start or stop the motor was less than the time of one revolution and

accounted for a small fraction of the data. For example, the single revolution before and after

the grid was halted amounted to 2% of the total data that could have possibly been affected

for the common case of alternating between 3 and 0 Hz at 30 second periods. A greater concern

in the analysis of motor phase dependence was the instrumentation used to record the motor

frequency. As described in section 2.1 a photogate was used to determine the frequency of the

motor. This added complexity to data acquisition; the motor would sometimes stop when the

beam was broken, sometimes stop when the motor was not broken, and sometimes stop when

the beam was just on the edge of being broken. In addition, the duration of a period could not

be precisely controlled, it may vary by a few tenths of a second. This caused difficulty in when

that variation in period meant the period would end with the beam broken, or not broken. Two

methods were employed to solve this problem. First, a modeling program was implemented to

determine the cycle phase at any time based on the recorded beam breaks. This worked well

for runs when the motor was not fully stopped such as frequency modulations between 3 - 2

Hz. For runs when the motor was fully stopped the non-consistent periods made this method

difficult. As a result a second method was implemented where no modeling took place, only

straightforward rules assigning long and short periods between beam breaks to different cycle

phases. Some accuracy was lost in this method, but the effect was very modest, and it primarily

affected runs involving very short periods, and only data near the motor on/off transition.

4.1.1 Driving Frequency Modulation

Varying the Energy Input

We modulated the frequency of the driving in order to artificially increase the large scale in-

termittency, so that in addition to any effects that large scale intermittent energy fluctuations

play on the continuous 3 Hz case there would be the additional effects of the modulated driving

frequency. For example, the driving frequency was modulated from 3 Hz to 0 Hz (the grids were

halted) at 15 second intervals. The grids would run for 15 seconds then be halted for 15 seconds,

and would repeat in this fashion for over an hour, until enough data was acquired. One could


rightly point out that this energy fluctuation is not intermittent, as it would be in the base case

and most experimental flows. We briefly considered randomly halting the grids, but concluded

it would make phase averaging difficult, and periodic halting is preferable because it allows us

to better control the energy input at no real loss of generality.

Of course modulating the driving frequency between 3 and 0 Hz is just one data point in

parameter space, and an extreme one at that. In addition to modulating between 3-0 Hz, we

also modulated between intermediate frequencies, namely 3-1 Hz and 3-2 Hz all with the same

cycle duration and duty cycles. These additional data points allow us to see a trend in the large

scale dependence as a function of driving modulation frequency.

Figure 4.1 shows the Eulerian second order conditional structure functions versus large scale

velocity. Note Fig. 4.1 is very similar to Fig. 3.10 where the component of the large scale

velocity and the normalization has changed. Here we condition the structure function on the

velocity component that is transverse to r denoted Σu⊥. We normalize the vertical axis by

the unconditioned structure function, and the horizontal axis by the characteristic velocity

(〈u2〉)1/2 = ((u2x + u2

y + u2z)/3)

1/2. We have made these changes so the conditional structure

functions can be more easily compared, this should not have any major affect on the appearance

or interpretation of the plots.

As before, we note the curvature of the conditional structure function and its collapse on to one

functional form. In Fig. 4.1(a) the curvature is significant, but relatively small; the structure

function does depend somewhat on the state of the large scales. The collapse of the structure

function at all length scales on to a single curve is not absolute, there is some r/η dependence,

but it is slight. Figure 4.1(b) shows the dependence of the conditional structure function in the

presence of grid frequency modulation. We see alternating the grid frequency between 3 and 2 Hz

every 15 seconds has a large effect on the curvature. The curvature has increased significantly,

and the collapse remains approximately the same. Comparing these two plots alone is enough

to derive the conclusions of this chapter. By artificially increasing the large scale intermittency

we see a striking increase in the dependence of the structure function at all length scales on

the instantaneous state of the large scales. We can continue increasing the level of large scale

intermittency by increasing the disparity in the two grid frequencies. Figure 4.1(c) shows the

conditioned structure function when the grid oscillation frequency modulates between 3 and

1 Hz every 15 seconds. We see the curvature increases further from 4.1(b), and the collapse


-2 -1 0 1 2 3

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

-3

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Σu┴

√<u2>

<(Δur)2|Σu

┴

<(Δur)2>

>

<(Δur)2|Σu

┴

<(Δur)2>

>

<(Δur)2|Σu

┴

<(Δur)2>

>

<(Δur)2|Σu

┴

<(Δur)2>

>

a

c

b

d

Figure 4.1: Eulerian second order conditional structure function versus large scale velocity, plotted in a similar

manner to figure 3.10. The grid oscillation frequency was modulated such that the frequency would switch from

high to low at 15 second intervals. The frequencies modulated were: a) 3 Hz continuous, b) 3-2 Hz, c) 3-1 Hz, d)

3-0 Hz. Each curve represents the following separation distances r/η: + = 2.67 to 5.33, ◦ = 5.33 to 10.67, ∗ =

10.67 to 21.33, × = 21.33 to 42.67, � = 42.67 to 85.33, ⋄ = 85.33 to 170.67, △ = 170.67 to 341.33, ▽ = 341.33

to 682.67.


-3 -2 -1 0 1 2 3

0.8

1

1.2

1.4

1.6

1.8

2

2.2

<(Δur)2|Σu

┴

<(Δur)2>

>

Σu┴

√<u2>

Figure 4.2: Second order conditional structure functions vs. large scale velocity (the same data as figure 4.1).

All curves represent one separation distance, r/η = 10.67 to 21.33 measured from the different driving frequency

modulations. Each curve is measured from the following driving frequency modulations: + = 3 Hz continuous,

◦ = 3-2 Hz, ∗ = 3-1 Hz, × = 3-0 Hz

on to one functional form remains approximately the same. In figure 4.1(d) we modulate the

grid between oscillating at 3 Hz and being completely stopped at 0 Hz, again alternating every

15 seconds. The curvature increases still further, but the jump in curvature is no longer as

dramatic as comparing (a) and (b), or (b) and (c). Again, the collapse on to one functional

form remains.

Since all length scales in figure 4.1(a, b, c, and d) essentially collapse onto one functional

form the 4 plots can easily be condensed into one. Figure 4.2 extracts only the 3rd r/η bin

where r/η = 10.67 to 21.33 from figure 4.1(a, b, c, and d), and plots them on one graph for

better comparison. The minima all fall on the mean large scale velocity, but at different values

vertically. The symmetry around the mean large scale velocity is due to the inherent isotropy

in the measurement of the transverse component of the large scale velocity. Since the particles


101

1020

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

r/η

b

Figure 4.3: Curves in figure 4.1 were fit to au4 + bu2 + c. The coefficient b is a measure of the dependence

of the conditional structure function on the large scales. The coefficient b is shown here versus the separation

distance r/η. The drive frequency period was 30 seconds (15 seconds with the motor on, and 15 seconds with

the motor off). Each line represents the following driving frequency modulations: + = 3 Hz continuous, ◦ = 3-2

Hz, ∗ = 3-1 Hz, × = 3-0 Hz

are randomly distributed r is randomly oriented, and therefore the component of Σu that is

transverse to r is isotropic. The result is Σu⊥ is isotropic, and therefore all dependence on it

must be symmetric. As discussed previously the curvature in +, ◦, ∗, and × is dictated by the

amount of large scale intermittency. The relation of curvature and the value at the minima

is due to the normalization. The low curvature lines are necessarily closer to unity. This is

because the less dependent the structure function is on the large scales the closer it is to the

unconditioned structure function, thus producing a ratio closer to unity.

Another convenient way to represent the data is shown in figure 4.3. The curves in figure 4.1

were all fit to the functional form au4 + bu2 + c, and the coefficient b is shown here versus

the separation distance r/η. The coefficient b gives the second derivative at the origin, a useful


measure of the dependence of the conditional structure function on the large scales. For example

when b is zero, there is no dependence on the large scales and the curve is flat, and when b is large

there is a steep parabola and a large dependence on the large scales. The coefficient a was not

used because it is dominated by data at the extreme large scale velocities, in addition measuring

the coefficient of the second order term is in keeping with previous studies [1]. As figure 4.3

shows there is a major increase in the dependence of the conditional structure function on the

large scales as the large scale intermittency increases. The dependence increases by a factor of

more than 5 when the intermittency is increased from 3 Hz continuously driven to alternating

between 3 and 0 Hz. Even at the most modest driving frequency modulation between 3 and 2

Hz the dependence increased by more than a factor of 2 over the 3 Hz continuously driven case.

The collapse of all length scales on to a single functional form can also be evaluated by figure 4.3.

The curvature is plotted as a function of separation distance r/η, so a flat line (no separation

distance dependence) would mean all of the length scales are affected by the large scales in the

same way, and they have all collapsed on to the same functional form. Figure 4.3 shows for the

low intermittency case of 3 Hz continuously driven there is only a modest dependence on the

separation distance. This dependence increases with increasing intermittency, where the most

intermittent case of 3-0 Hz has a curvature that ranges from about .35 to .45.

Constant Energy Input

It is worthwhile to compare the driving frequency modulation (figures 4.2-4.3) to the case where

the driving is not modulated. Figure 4.4 shows conditional structure functions versus large scale

velocity for constant grid frequencies from 1 to 5 Hz. In these cases no additional intermittency

has been added by modulating the driving frequency. It is important to note how the curvatures

in the various constant driving cases are all somewhat similar, and relatively low. We can draw

from this that the large scale effects, including large scale intermittency, are all somewhat similar

regardless of driving frequency and therefore Reynolds number. We can see clearly in figure 4.5

how low the curvature is compared to figure 4.3, all of the constant driving cases have curvature

between .05 and .14, which is lower than any of the cases in which the driving frequency was

modulated and the large scale intermittency was increased. This helps to illustrate the extent

to which large scale intermittency can affect the dependence on the large scales. Changing the

Reynolds number by a factor of approximately 4 results in a smaller difference in the dependence


-3 -2 -1 0 1 2 30.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Σu⊥√

〈u2〉

〈(∆ur)2|Σu⊥〉

〈(∆ur)2〉

Figure 4.4: Conditional structure function vs. large scale velocity showing one separation distance, r/η =

10.67 to 21.33 (similar to figure 4.2). The grid oscillation frequency was kept constant. Each curve represents

the following grid oscillation frequencies: + = 1 Hz, ◦ = 2 Hz, ∗ = 3 Hz, × = 4, Hz � = 5 Hz


101

102

0

0.05

0.1

0.15

r/η

b

Figure 4.5: The dependence of the conditional structure function on the large scales is quantified. The

coefficient b (where the polynomial au4 + bu2 + c was fit to the conditional structure function) is shown here as a

function of r/η. This is plotted in the same way as figure 4.3. Each curve represents the fittings for the following

grid oscillation frequencies: + = 1 Hz, ◦ = 2 Hz, ∗ = 3 Hz, × = 4 Hz, � = 5 Hz.


-3 -2 -1 0 1 2 310

2

103

104

105

106

107

108

109

Num

ber

of S

ampl

es

Σu┴

√<u2>

Figure 4.6: The number of samples shown as a function of large scale velocity for frequency modulation 3-0

Hz, duty cycle 50%, and period 384 seconds. The curves represent the following separation distances r/η: + =

2.67 to 5.33, ◦ = 5.33 to 10.67, ∗ = 10.67 to 21.33, × = 21.33 to 42.67, � = 42.67 to 85.33, ⋄ = 85.33 to 170.67,

△ = 170.67 to 341.33, ▽ = 341.33 to 682.67.

than from increasing the large scale intermittency.

Although the curvatures in Fig. 4.3 are relatively similar, there is a trend for higher constant

grid frequencies to have higher curvatures. It is very interesting to note that this is contrary to a

simple attribution of the observed dependence on insufficient Reynolds number. If it were simply

insufficient Reynolds number responsible for the dependence one would expect the dependence

to decrease with increasing Reynolds number. In addition, the dependence on the length scale

for the constant grid frequencies is fairly small with one notable exception. At 2 Hz constant

grid frequency (symbol ◦) the curvature does not follow the trend of the other constant grid

frequencies, and more prominently, has a much larger dependence on length scale. Mechanisms

for this anomalous behavior are unknown at this time.


An Illustrative Example

In order to interpret the results shown in figure 4.1, and throughout this chapter it is fruitful

to think of an illustrative example. Imagine the two modes of the motor rigorously correspond

to two modes in the fluid. When the motor is the high state the fluid is very energetic with a

high mean energy, and therefore large standard deviation of velocity. In the other mode, the

motor is in the low state and the fluid has a low mean energy, and therefore a small standard

deviation of velocity. After many periods it would become apparent the low energy fluid would

dominate the statistics near zero large scale velocity and would have a low structure function.

Quite simply, the low energy fluid would not have very many particle pairs that make it out

to 3 standard deviations large scale velocity. Instead, the structure function at the extrema of

large scale velocity would be dominated by the high energy fluid, and would have a much higher

value for the structure function.

In fact, we took experimental data that very clearly shows this illustrative example. Figure

4.6 shows the number of samples as a function of large scale velocity for different separation

distances. The frequency modulation was 3-0 Hz, the duty cycle 50%, and the period was a

very long 384 seconds. This long period allowed the weight of the transitions to be very small

so the motor on time was dominated by very energetic fluid with a large standard deviation

of velocity, and the motor off time was dominated by low energy fluid with small standard

deviation of velocity. The peculiar bumps in the center of each curve are essentially all the

additional samples of the low energy fluid to the high energy fluid.

Simply connecting the conditional structure functions of the high energy fluid at the extrema to

the low energy fluid near the center in a smooth way will create the curvature seen in figure 4.1

(and later we will see it even more clearly in figure 4.12 symbol ⋄). The greater the disparity

in energy between the fluids the greater the disparity between the structure function value of

the lower point in the center and the higher points at the extrema, and thus the higher the

curvature.

Varying the Energy Input (Large Amplitude)

Figures 4.7-4.8 are similar to figures 4.2 and 4.3 but with different grid modulation frequencies,

namely 4Hz continuous, 4-2.66 Hz, 4-1.33 Hz, and 4-0 Hz. These frequencies were chosen to


-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3

Σu⊥√

〈u2〉

〈(∆ur)2|Σu⊥〉

〈(∆ur)2〉

Figure 4.7: Second order conditional structure functions vs. large scale velocity. All curves represent one

separation distance, r/η = 10.67 to 21.33 measured from the different driving frequency modulations. Each

curve is measured from the following driving frequency modulations: + = 4 Hz continuous, ◦ = 4-2.67 Hz, ∗ =

4-1.33 Hz, × = 4-0 Hz


101

1020

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

r/η

b

Figure 4.8: Curves in figure 4.1 were fit to au4 + bu2 + c. The coefficient b is a measure of the dependence

of the conditional structure function on the large scales. The coefficient b is shown here versus the separation

distance r/η. The drive frequency period was 30 seconds (15 seconds with the motor on, and 15 seconds with the

motor off). Each line represents the following driving frequency modulations: + = 4 Hz continuous, ◦ = 4-2.67

Hz, ∗ = 4-1.33 Hz, × = 4-0 Hz


have the same ratios of high to low as those in figures 4.1-4.3, the duty cycle and period were

also chosen to be the same. Note the primary result is reinforced, increasing the large scale

intermittency increases the dependence all scales have on the large scales. Comparing the cases

3 and 3-1 Hz to 4 and 4-1.33 Hz shows the ratios of the curvature are not the same, as would

be expected, but have a moderate discrepancy in the ratio of curvature from 4.3 to 6.4. The

differences between Figs. 4.7 - 4.8 and Figs. 4.2 - 4.3 highlight the non-linear nature of this

discussion. Increasing the magnitude of the frequencies, even while maintaining the ratio of the

frequencies and all other parameters, we suspect, can have secondary effects such as increasing

the mean flow, anisotropy, or other large scale properties that can not easily be predicted, but

have a measurable affect on the observed large scale dependence. Unfortunately this must remain

as speculation at this point, we have made no direct measurements to confirm this suspicion

and can draw no hard conclusions.

Varying the Duty Cycle

Let us now examine the effects of the duty cycle parameter on the observed dependence. Here

the duty cycle is defined as the fraction of the period where the motor is in its high state.

This measurement sheds some light on the sensitivity of the dependence to fluctuations in the

energy input that may occur only seldomly. It is common in actively modulated turbulence to

have the driving mechanism change states some fraction of the time. In addition, in steadily

driven turbulence fluid that had recently been in a relatively quiescent region may sweep by the

observation volume occasionally. Our goal here is to measure the impact of the relative duration

of these fluctuations on the dependence of the small scales on the large scales.

Figures 4.9 and 4.10 are plotted similarly to figures 4.2 and 4.3 with the duty cycle varied. The

grid frequency was modulated from 3 to 0 Hz with a period of 30 seconds, but the duration

the motor was high was varied. The duration high was varied from 30, 22.5, 15, to 7.5 seconds

which corresponds to duty cycles of 100%, 75%, 50%, and 25%. These results are similar to

those shown in figures 4.2 and 4.3; an increase in large scale intermittency results in an increase

in the dependence of all scales on the large scales. As the duty cycle decreases the curvature

increases. This can be explained by the limiting cases given in the illustrative example earlier;

as the duty cycle is decreased the system tends towards the limit of a quiescent fluid with

low energy and small standard deviation of velocity being combined with a high energy, large


-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3

3.5

<(Δur)2|Σu

┴

<(Δur)2>

>

Σu┴

√<u2>

Figure 4.9: Conditional structure function vs. large scale velocity showing one separation distance, r/η = 10.67

to 21.33 (similar to figure 4.2). The grid oscillation frequency was modulated between 3 and 0 Hz, where the

motor was in the high state for different percentages of the period, also called duty cycles. Each curve represents

the following duty cycles: + = 100%, ◦ = 75% Hz, ∗ = 50%, × = 25%


0

0.1

0.2

0.3

0.4

0.5

0.6

101

102

r/η

b



function of r/η. This is plotted in the same way as figure 4.3 Each curve represents the fittings for the following

grid duty cycles: + = 100%, ◦ = 75% Hz, ∗ = 50%, × = 25%


standard deviation of velocity fluid. The disparity between these two modes increases as the

duty cycle decreases. For small duty cycle cases the energy in the active mode has enough time

to dissipate thoroughly, and dictates the statistics near low large scale velocity. When the motor

is on, the standard deviation of velocity increases and the large magnitude large scale velocity

is represented and has a large structure function. This creates the higher curvature, and thus

higher dependence on the large scales we observe.

Varying the Period

As we will see here the simple picture of two discrete and separate states of high or low energy

that was described in the illustrative example earlier was a hypothetical limit of the system,

and here we will experimentally explore how the flow transitions between these two states. We

will achieve this by varying the period of the frequency modulation. The grid frequency was

modulated from 3 to 0 Hz with a a duty cycle of 50%, but the duration of the period was varied

from 3,6,12,24,48, and 384 seconds.

To illustrate the two regimes and the transition between them figure 4.11 shows the energy vs.

cycle phase for various periods. Energy is simply u2x + u2

y + u2z. The motor was turned off at

2π and turned on at π. The long period cases (symbols � and ⋄) show the energy saturating

regime. Their maximum is at about 95 (cm/s)2 and they reach a lower limit of less than 20

(cm/s)2. The shortest period case (symbol +) does not come close to the energy saturation

regime. It does not saturate in the maximum nor the minimum energy limit. This is because

the turbulence takes a finite time to decay and rise, approximately on the time scale of the eddy

turnover time. The shortest period case modulates the driving at about the eddy turnover time

scale, thus not allowing for much decay or rise. Intermediate cases (symbols ◦, ∗,and ×) shows

the transition between the two regimes.

Predicting the results of altering the period on the observed dependence is not as straightforward

as the previous cases, it may not be obvious which case would have the highest intermittency.

The base case (with continuous driving) may no longer have the least influence from large scale

intermittency. Guidance on this topic was provided by von der Heydt et al. [27], which showed

the response to very fast modulations was minimal. This means that in the limit of very short

periods the frequency modulation was just like oscillating at a single, constant frequency which

was approximately an average between the two driving frequencies.


0 1.57 3.14 4.71 6.280

20

40

60

80

100

120

motor phase

ener

gy (

cm/s

)2

Figure 4.11: The phase average energy shown as a function of cycle phase. The motor was halted at 2pi and

turned on at pi. Each curve represents the following cycle periods r/η + = 1.5/3”, ◦ = 3/6”, ∗ = 6/12”, × =

12/24” � = 24/48” ⋄ = 192/384”


-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

3

Σu⊥√

〈u2〉

〈(∆ur)2|Σu⊥〉

〈(∆ur)2〉

Figure 4.12: Conditional structure function vs. large scale velocity showing one separation distance, r/η =

10.67 to 21.33 (similar to figure 4.2). The grid oscillation frequency was modulated between 3 and 0 Hz and

the period was varied. Each curve represents the following time the motor was on/total period (seconds): + =

1.5/3”, ◦ = 3/6”, ∗ = 6/12”, × = 12/24”, � = 24/48”, ⋄ = 192/384” Note all have a duty cycle of 50%.

Whereas, for longer periods the turbulence had time to fully reach equilibrium and the fluid

could reach two distinct states. When the motor is off the energy decays rapidly, and has reached

a low energy state. When the motor is on the energy increases rapidly and has time to plateau

at a high energy limit dictated by the 3 Hz motor driving.

Figures 4.12 and 4.13 are plotted in the same manner as figures 4.2 and 4.3, These results are

similar to those shown in figures 4.2 and 4.3, in that an increase in large scale intermittency

results in an increase in the dependence of all scales on the large scales.

Figure 4.13 shows how great the disparity in curvatures can be from varying the modulation

period. The two fastest modulation periods (symbols + and ◦) have very similar and relatively

low curvatures. this suggests we have approached the lower limit in cycle period where we are

no longer increasing the large scale intermittency even though the driving frequency is being


101

1020

0.5

1

1.5

2

r/η

b



function of r/η. This is plotted in the same way as figure 4.3 Each curve represents the fittings for the following

cycle periods: + = 1.5/3”, ◦ = 3/6”, ∗ = 6/12”, × = 12/24” � = 24/48” ⋄ = 192/384” Note all have a duty

cycle of 50%


modulated. This contrasts to the longer modulation periods, as the period grows the curvature

grows significantly greater. This is again, due to the disparity in energy in the fluid in the two

modes. The greater the disparity in energy between the fluid with the motor on and off, the

greater the disparity between the higher points at the large Σu⊥ and lower point in the center,

and thus higher curvature. This is particularly striking for symbol ⋄ (the longest period, 384

seconds) where the statistics within ± 1 standard deviation of Σu⊥ are clearly dominated by

the lower energy fluid. As mentioned earlier, for the longest period the disparity in the amount

of samples between the high and low modes is seen clearly in Fig. 4.6 at all length scales.

Table 4.1: Experimental parameters and resulting statistics for large scale intermittency tests

Frequency (Hz) Duty Cycle Duration On (s) U (cm/s) L (cm) τ (s) ε (cm2/s3) Reλ

3 100% n.a. 5.46 7.69 1.41 21.2 250

3-2 50% 15 4.72 7.48 1.58 14.1 230

3-1 50% 15 4.23 7.13 1.69 11 213

3-0 50% n.a. 4.21 7.16 1.7 10.4 212

4 50% 15 7.14 6.87 0.96 52.9 271

4-2.66 50% 15 6.04 7.23 1.2 30.5 256

4-1.33 50% 15 5.41 7.22 1.33 21.9 242

4-0 50% 15 5.38 7.61 1.41 20.5 247

3-0 50% 1.5 4.44 8.28 1.86 10.6 235

3-0 50% 3 4.71 8.51 1.81 12.3 245

3-0 50% 6 4.54 8.44 1.86 11.1 240

3-0 50% 12 4.42 7.87 1.78 11 228

3-0 50% 24 4.07 6.48 1.59 10.4 198

3-0 50% 192 4.06 5.76 1.42 11.6 187

3-0 75% 22.5 4.92 7.58 1.54 15.7 236

3-0 25% 7.5 3.27 6.86 2.1 5.1 183

5 100% n.a. 7.71 5.95 0.77 77 262

2 100% n.a. 4.05 9.3 2.3 7.15 237

1 100% n.a. 1.96 6.59 3.36 1.15 139

Table 4.1 shows the experimental parameters and resulting statistics for the large scale inter-


mittency tests.

4.2 Discussion

Although rarely addressed, large scale intermittency is almost certainly present in a large fraction

of real world and experimental flows. This experiment tries to shed light on this prevalent, yet

poorly understood topic. It is experimentally very difficult to lower the large scale intermittency

in our flow; however, by modulating the energy injection in a methodical way we can effectively

add large scale intermittency to our flow and systematically observe the results. By doing so we

demonstrate that large scale intermittency affects the structure functions at all scales, including

the small scales, significantly.

It is a concern that the addition of large scale intermittency into our flow is not a realistic

comparison to the large scale intermittency that exists in real world and experimental flows.

Addressing this concern without an accurate and accepted method for measuring large scale

intermittency is difficult. However, there are important flows where the driving mechanism

is significantly modulated, such as the active grid wind tunnel [28] and LEM [29], and it is

reasonable to expect that these flows have considerable large scale intermittency. In addition

to these flows where the driving mechanism is actively modulated, there is reason to suspect

that most steadily driven flows, and almost any low mean velocity flow (usually designed for

Lagrangian experiments) has some large scale intermittency, which will be discussed in the

next chapter. This experiment can grant some experimental backing to concerns that actively

modulated, and low mean velocity flows bring up. As an example for comparison, consider the

most modest case we measured, where the motor frequency was modulated between 3 and 2 Hz,

two turbulent states that are distinct, yet less pronounced than the 40% variation in propeller

driving frequency seen in the LEM. Even in this case the dependence on large scales increased

by approximately a factor of 2 at all length scales. Because this effect is large it is reasonable to

suspect that the majority of real world and experimental flows would also be affected by such a

sensitive dependence on large scale intermittency.

As mentioned above, an accurate and accepted method for measuring large scale intermittency is

an open problem. Work has been done to quantify intermittency in the transition from laminar

to turbulent flow in simulations, however it is unclear if this work can be translated to measuring


transitions between turbulent fluid and turbulent fluid with a different energy level.

4.3 Conclusion

We have previously established that the small scales in turbulence are not completely indepen-

dent of the current state of the large scales. This chapter has provided a comprehensive set of

experiments that has quantified the affect of one of large scale property, large scale intermit-

tency, has on all scales. This was achieved by adding large scale intermittency in a controlled

way and measuring how the structure functions dependence on large scales changed. We found

that large scale intermittency can play a significant role in the dependence structure functions

show on the large scales. We therefore conclude large scale intermittency should be included as

a large factor when considering the interactions between large and small scales.

Chapter 5Signatures of Non-Universal Large Scales

in Conditional Structure Functions

Compared in Various Turbulent Flows

This chapter1 contains a systematic comparison of conditional structure functions in eight turbu-

lent flows. The flows studied include direct numerical simulation of turbulence in a periodic box,

passive grid wind tunnel turbulence, active grid wind tunnel turbulence (in both synchronous

and random driving modes), the flow between counter-rotating disks, oscillating grid turbu-

lence, and the flow in a Lagrangian exploration module (in both constant and random driving

modes). We compare longitudinal Eulerian second-order structure functions conditioned on the

instantaneous large-scale velocity in each flow (for a description of this method see section 3.6).

This is used to assess the ways in which the large scales affect the small scales in a variety

of turbulent flows. Structure functions are shown to have larger values when the large-scale

velocity significantly deviates from the mean in all flows except the passive grid wind tunnel,

suggesting that dependence on the large scales is typical in many turbulent flows. The effects

of the large-scale velocity on the structure functions can be quite dramatic, with the structure

1Note this chapter is largely adapted from work which I lead and organized that has been submitted to be

published in the Journal of Turbulence as Blum et al. “Signatures of non-universal large scales in conditional

structure functions from various turbulent flows”

73

Chapter 5 - Conditional Structure Functions Compared in Various Turbulent Flows 74

function varying by up to a factor of 2 when the large-scale velocity deviates from the mean by

± 2 standard deviations. In general, the effects of the large-scale velocity are similar at all the

length scales we measured in the conditional structure functions. This indicates the large-scale

effects are scale independent.

5.1 Experiment

Parameters for each of the eight flows are summarized in Table 5.1. We define a characteristic

velocity by U = (〈uiui〉/3)1/2 where U represents the fluctuating velocity, and a characteris-

tic length scale by L = U3/ε, where ε is the dissipation rate of turbulent kinetic energy per

unit mass. The Taylor Reynolds number reported is Reλ = (15uL/ν)1/2, where ν is the kine-

matic viscosity. We briefly describe each system with further details available in the referenced

work.

The direct numerical simulation data set analyzed has a Reλ of approximately 650, with 20483

grid points and periodic boundary conditions. The turbulence is maintained stationary by

stochastic forcing at large scales [30]. The domain size is 2.1L. Statistics are calculated from

15 snapshots of the simulation. The pseudospectral algorithm of Rogallo was used. The time

stepping is second-order Runge-Kutta. Aliasing errors were controlled by a combination of

truncation and phase shifting techniques.

One wind tunnel was used for passive and active grid turbulence measurements. The wind

tunnel is 0.914× 0.914 m2 and 9.1 m long, with a horizontal, open circuit design. In the passive

grid configuration it produced a flow with Reλ=66. The grid mesh was M=2.54 cm and mean

flow was 10.3 m/s. Measurements were taken using hot wire anemometry at x/M=60 which is

x/L = 74 [31].

In the active grid configuration, the grid mesh was M=11.4 cm and a mean flow of 3.3 m/s was

used for the synchronous grid experiments and 7.0 m/s for the random grid experiments. It

produced a flow with Reλ=140 for the synchronous mode and Reλ=582 for the random mode of

operation. Measurements were taken using hot wire anemometry at x/M=62 which is x/L = 64

in synchronous mode and x/L = 16 in random mode [28].

The flow between counter-rotating disks is inside a 48.3 cm diameter cylindrical water tank.


The two counter-rotating disks are separated by 43.9 cm. At a propeller frequency of 3.5 Hz,

the flow reaches a high Reynolds number of Reλ=690. Nearly neutrally-buoyant polystyrene

particles 25 µm in diameter and 1.06g/cm3 in density were seeded as tracer particles. Their

trajectories were recorded using 3 cameras at a frame rate of 27,000 images per second. Two

pulsed frequency-doubled Nd:YAG lasers (wavelength 532 nm) with a combined power of roughly

150 W illuminated a 5 cm3 observation volume in the center of the tank. Particle velocities were

measured through 3D particle tracking techniques [7].

The oscillating grids are inside a 1 × 1 × 1.5 m3 octangular prism tank. The two grids were

spaced 56.2 cm apart with an 8 cm mesh spacing. A typical grid frequency was 3 Hz with a 12

cm peak to peak stroke. They produced a flow with Reλ = 260. Polystyrene tracer particles

of diameter 136 µm = 0.94η were recorded using 4 high-speed digital cameras recording 450

frames per second. The video data was filtered through real time image compression circuits,

allowing for continuous data recording. An approximately (5 cm)3 cubic detection volume was

illuminated in the center of the tank using a pulsed 50 W Nd:YAG laser (wavelength 532 nm).

Particle velocities were determined using 3D particle tracking techniques [20].

The Lagrangian Exploration Module (LEM) is a regular icosahedron with an edge length of 40

cm and a volume of 140 liters. A propeller is installed on each of the 12 vertices. The propellers

are driven by 12 independently controlled DC motors. Here we report data corresponding to a

motor frequency of 1.67 Hz. When the motor speeds were maintained constant, they generated a

water flow with Reλ=195 in the center of the apparatus. When the motor speeds were randomly

adjusted within ±40% of the nominal speed of 1.67 Hz, the Reynolds number of the flow was

Reλ = 210. Hollow glass spheres with diameter 60-70 µm and average density between 1 − 1.2

g/cm3 were used as tracer particles. A 15×10×10 cm3 detection volume was illuminated with a

30 W Nd:YAG laser (wavelength 532 nm). Tracer particle motions in the detection volume were

recorded with three high-speed cameras at a frame rate of 1,000 frames per second. Particle

velocities were measured from 3D particle tracking techniques [29].

5.2 Data

We focus on the second-order longitudinal Eulerian structure functions, labeled 〈(∆ur)2〉. From

the fluctuating velocity measured at two points separated by a distance r, the component of the

Chapter

5-Conditio

nalStru

cture

Functio

nsCompared

inVario

usTurbulen

tFlow

s76

Flow Reλ ε η L W/L Ref

Direct Numerical Simulation † 650 1.33 0.00141 3.08 12.8 [30]

Passive Grid Wind Tunnel 66 0.448 m2/s3 295 µm 0.0207 m 44.2 [31]

Active Grid Wind Tunnel (Synchronous Driving) 140 0.042 m2/s3 550 µm 0.11 m 8.3 [28]

Active Grid Wind Tunnel (Random Driving) 582 0.94 m2/s3 260 µm 0.43 m 2.1 [28]

Counter Rotating Disks 690 1.15 m2/s3 30 µm 0.071 m 6.2 [7]

Oscillating Grids 260 2.31 ×10−3 m2/s3 144 µm 0.079 m 7.1 [20]

Lagrangian Exploration Module (Constant Driving) 195 2.10 ×10−4 m2/s3 260 µm 0.094 m 6.4 [32]

Lagrangian Exploration Module (Random Driving) 210 2.70 ×10−4 m2/s3 250 µm 0.10 m 6.0 [32]

Table 5.1: Flow parameters in the various apparatus. W represents the flow width, which is the smallest system dimension containing the flow.

† In DNS the choice of units is arbitrary.


100 101 102 103 1040

0.5

1

1.5

2

2.5

r/η

<(Δur)2>

<(εr)2/3>

Direct Numerical Simulation

a

100 101 102 103 1040

0.5

1

1.5

2

2.5

r/η

<(Δur)2>

<(εr)2/3>

Passive Grid Wind Tunnel

b

100 101 102 103 1040

0.5

1

1.5

2

2.5

r/η

<(Δur)2>

<(εr)2/3>

Active Grid Wind Tunnel (Random Driving)

d

100 101 102 103 1040

0.5

1

1.5

2

2.5

r/η

<(Δur)2>

<(εr)2/3>

Active Grid Wind Tunnel (Synchronous Driving)

c

100 101 102 103 1040

0.5

1

1.5

2

2.5

r/η

<(Δur)2>

<(εr)2/3>

Oscillating Grids

f

100 101 102 103 1040

0.5

1

1.5

2

2.5

r/η

<(Δur)2>

<(εr)2/3>

Reλ = 650

Reλ = 140

Reλ = 260

Reλ = 582

Reλ = 66

Counter Rotating Disks

eRe

λ = 690

100 101 102 103 1040

0.5

1

1.5

2

2.5

r/η

<(Δur)2>

<(εr)2/3>

LEM (Constant Driving)

gRe

λ = 195

100 101 102 103 1040

0.5

1

1.5

2

2.5

r/η

<(Δur)2>

<(εr)2/3>

LEM (Random Driving)

h Reλ = 210

Figure 5.1: Eulerian second order longitudinal velocity structure functions compensated by (εr)2/3. a) Direct

Numerical Simulation b)Passive GridWind Tunnel c) Active Grid (Synchronous Driving) d) Active Grid (Random

Driving) e) Counter Rotating Disks f) Oscillating Grids g) Lagrangian Explorer Module (Constant Driving) h)

Lagrangian Exploration Module (Random Driving).

velocity difference parallel to the separation vector is extracted and the second moment gives

the longitudinal structure function [4]. Note that different experimental techniques access this

quantity in different ways. Hot-wire measurements in the wind tunnels use fixed probes, so


-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3Direct Numerical Simulation

a

-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3Passive Grid Wind Tunnel

b

-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3Active Grid Wind Tunnel (Random Driving)

d

-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3Active Grid Wind Tunnel (Synchronous Driving)

c

-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3Oscillating Grids

f

-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3Counter Rotating Disks

e

-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3LEM (Constant Driving)

g

-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3LEM (Random Driving)

h

Reλ = 650

Reλ = 140

Reλ = 210

Reλ = 260

Reλ = 582

Reλ = 66

Reλ = 195

Reλ = 690

Figure 5.2: (Color Online) The Eulerian second order longitudinal structure functions are conditioned on the

transverse velocity sum (Σu⊥), and plotted versus Σu⊥. Symbols represents the following separation distances:

r/η:+ = 4, ◦ = 8, ∗ = 16, × = 32, � = 64, ♦ = 128, △ = 256, ▽ = 512 ⊲ = 1024. a) Direct Numerical

Simulation b)Passive Grid Wind Tunnel c) Active Grid (Synchronous Driving) d) Active Grid (Random Driving)

e) Counter Rotating Disks f) Oscillating Grids g) Lagrangian Explorer Module (Constant Driving) h) Lagrangian

Exploration Module (Random Driving).

the separation vector has a fixed direction. Particle tracking measurements sample randomly

positioned tracer particles and average over separations with all directions. Figure 5.1 shows


0.5

1

1.5

2

2.5

3

<(Δur)2|Σu

z>

<(Δur)2>

-3 -2 -1 0 1 2 3Σu

z

√<u2>

Counter Rotating Disks

a

Oscillating Grids

-3 -2 -1 0 1 2 3Σu

z

√<u2>

0.5

1

1.5

2

2.5

3

<(Δur)2|Σu

z>

<(Δur)2>

b

Figure 5.3: (Color Online) Structure functions conditioned on the vertical component of the velocity sum, Σuz.

Symbols represents the following separation distances r/η:+ = 4, ◦ = 8, ∗ = 16, × = 32, � = 64, ♦ = 128, △

= 256, ▽ = 512 ⊲ = 1024. a) Counter Rotating Disks b) Oscillating Grids .

the second-order longitudinal structure function for each of the flows we consider. The structure

functions are compensated by inertial range Kolmogorov (1941) scaling, (εr)2/3, to better com-

pare the various flows. Most of the flows are at modest Reynolds numbers and show, at most, a

very small region of inertial range scaling. A larger separation of scales is clearly visible in the

higher Reynolds number data in Figs. 5.1a, 5.1d, and 5.1e.

To measure the effects of the large scales on each flow, we condition the structure functions

on a quantity that is representative of the large scales. We use a velocity sum, Σu, where

Σu = (u(x) + u(x + r))/2 because it is a convenient and measurable quantity that primarily

reflects the instantaneous state of the large scales. This velocity sum also can be decomposed

into longitudinal (Σu‖) transverse (Σu⊥) or directional (Σui) components.

One possible concern is that the longitudinal velocity difference is kinematically correlated with

the longitudinal component of the velocity sum. Since the same measurements are used in

their calculation it is conceivable that the two values are not independent, either because of a

straightforward statistical correlation between velocity differences and velocity sums, or because

of a correlation intrinsic in Navier-Stokes turbulence. As we discuss in section 3.8, kinematic


0.5

1

1.5

2

2.5

3

-3 -2 -1 0 1 2 3

Direct Numerical Simulationa

0.5

1

1.5

2

2.5

3

-3 -2 -1 0 1 2 3

Passive Grid Wind Tunnel

b

0.5

1

1.5

2

2.5

3

-3 -2 -1 0 1 2 3

Active Grid Wind Tunnel (Random Driving)

d

0.5

1

1.5

2

2.5

3

-3 -2 -1 0 1 2 3

Active Grid Wind Tunnel (Synchronous Driving)

c

0.5

1

1.5

2

2.5

3

-3 -2 -1 0 1 2 3

Counter Rotating Diskse

0.5

1

1.5

2

2.5

3

-3 -2 -1 0 1 2 3

LEM (Random Driving)

h

0.5

1

1.5

2

2.5

3

-3 -2 -1 0 1 2 3

LEM (Constant Driving)g

0.5

1

1.5

2

2.5

3

-3 -2 -1 0 1 2 3

Oscillating Grids

f

Figure 5.4: (Color Online) The Eulerian second order longitudinal structure functions are conditioned on the

longitudinal velocity sum (Σu‖), and plotted versus Σu‖. Symbols represents the following separation distances

r/η:+ = 4, ◦ = 8, ∗ = 16, × = 32, � = 64, ♦ = 128, △ = 256, ▽ = 512 ⊲ = 1024. a) Direct Numerical

Simulation b)Passive Grid Wind Tunnel c) Active Grid (Synchronous Driving) d) Active Grid (Random Driving)

e) Counter Rotating Disks f) Oscillating Grids g) Lagrangian Explorer Module (Constant Driving) h) Lagrangian

Exploration Module (Random Driving).

correlations would not contribute in Gaussian random fields since the second moment of the

velocity difference is uncorrelated with the velocity sum, and we confirm this later in this chapter.


-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3

Gaussian Random Field

Figure 5.5: Conditional structure functions for Gaussian random fields. The Eulerian second order longitudinal

structure function is conditioned on the longitudinal velocity sum. Symbols represents the following separation

distances r/η:+ = 4, ◦ = 8, ∗ = 16, × = 32, � = 64, ♦ = 128, △ = 256, ▽ = 512 ⊲ = 1024.

However, Hosokawa [25] has shown that the Navier-Stokes equations require certain correlations

between longitudinal velocity sums and differences. There is no known relationship between the

second moment of the velocity difference and the even moments of the velocity sum, which

are the correlations that could dominate the data we present here. To minimize any possible

effects of kinematic correlations we use the longitudinal structure functions conditioned on the

transverse component of the velocity sum as our primary indicator of the state of the large scales.

The large differences between the flows in our results argues that the conditional dependence

cannot be explained by a universal kinematic correlation.

Figure 5.2 shows longitudinal structure functions for each flow conditioned on the transverse

component of the velocity sum. Figure 5.2b from the passive grid is nearly flat for all length

scales, indicating that there is almost no dependence on the large scales in the structure functions

at any r. Another flow with very weak dependence of the structure functions on the large-scale

velocity is counter-rotating disks in Fig. 5.2e. The other six flows all show significant dependence

on the large scales. The upward curvature indicates that large velocity differences preferentially

occur when the magnitude of the velocity sum is large.


Figure 5.2a shows that the DNS has a significant dependence of small scales on the large scales.

For velocities different from the mean by less than 2 standard deviations, the dependence is

very weak, but it becomes much larger for velocities 3 standard deviations from the mean. The

smallest scales show the strongest dependence on the large-scale velocity. In contrast, data

from the passive grid wind tunnel in Fig. 5.2b at much lower Reynolds number is essentially

flat. This DNS data differs from other published results [1] in which both DNS and passive

grid wind tunnel data show no dependence on the large-scale velocity. The reason for the

difference is still not clear, although we have considered several possible causes. It does not

seem to be Reynolds number since a different simulation at Reλ = 140 shows similar results

to Fig. 5.2a. The difference does not seem to be due to lack of statistical convergence since

individual snap-shots each give a similar result. Similar results also appear in data from a

simulation of decaying turbulence, indicating that the forcing mechanism is likely not the cause.

It appears that the small scales in these simulations are indeed correlated with the instantaneous

state of the large scales, and that this correlation is much smaller in turbulence from a passive

grid wind tunnel.

Figures 5.2c and 5.2d illustrate the difference in the effects of the large scales when the active

grid wind tunnel is driven in synchronous and random modes. In random mode, the curvature

is significantly larger even though the Reynolds number is much larger. A possible explanation

is that L is much larger in the random mode (See Table 5.1), rendering the flow less homoge-

neous and therefore more sensitive to large-scale effects. This explanation is consistent with the

differences between the data from the active grid wind tunnel and the passive grid wind tunnel

in Fig. 5.2b. Again, despite the lower Reynolds number, the flow with greater homogeneity has

less curvature.

A surprise in the data presented here is that data from counter rotating disks shown in Fig. 5.2e

show very little dependence on the large scales. This can be compared to the much stronger

dependence in the oscillating grid flow shown in Fig. 5.2f, even though the oscillating grid flow

is more homogeneous and isotropic. To investigate this further, we conditioned not on the

directionless transverse velocity sum but on different components of the velocity sum that are

fixed in the lab frame. Figure 5.3 reveals that the disparity between these two flows disappears

when the second order structure functions for both flows are conditioned on the component of the

velocity sum in the axial direction, Σuz which is vertical in the lab frame. Here the dependence


on the large-scale velocity for the counter rotating disk flow in Fig. 5.3a is approximately the

same as the oscillating grid data in Fig. 5.3b. Further evidence about effects of the direction

of the large-scale velocity comes from conditioning structure functions from each of these flows

on directions perpendicular to the axial direction (data not shown). Here the counter rotating

disks flow shows almost no dependence and the oscillating grids flow shows dependence similar

to the axial dependence. It appears that the direction of the large-scale conditioning can play a

significant role for systems where large-scale anisotropy is important.

Figures 5.2g and 5.2h show that the Lagrangian exploration module in constant and random

driving modes has structure functions with strong dependence on the large-scale velocity. Unlike

the different driving modes of the active grid, the different driving modes of the Lagrangian

exploration module show similar curvatures as well as similar Reynolds numbers.

A major point which is seen throughout Fig. 5.2 is that separation of scales alone is not sufficient

to produces small scales that are independent of the large scales. For most of the flows, there

is very little dependence of the conditional structure functions on the length scale, r, so the

dependence on the large scales does not diminish with decreasing scale. The DNS shows the

most dependence on the length scale, with the structure function at the smallest scales curving

upward more strongly than larger scales, exactly the opposite of what would be expected if

limited separation of scales was the cause. From this observation, we infer that the trend of the

conditional structure functions depending strongly on the large-scale velocity is likely to persist

to yet higher Reynolds numbers despite the wider range of scales that would be present.

Figure 5.4 shows longitudinal structure functions again, but this time conditioned on the longi-

tudinal component of the velocity sum. This data is largely similar to Fig. 5.2 conditioned on

the transverse sum, but conditioning on the longitudinal sum reveals a few new insights. The

similarity between Figs. 5.2 and 5.4 is evidence that kinematic correlation is not a large factor

in the observed conditional dependence. Data from the wind tunnels (b-d in Figs. 5.2 and 5.4)

shows the most differences between the conditions on the transverse and longitudinal velocity

sums. Note that the structure function minimum has been shifted towards negative large-scale

velocity by over one standard deviation. A major factor here is the orientation of the hot-wire

probes used for the wind tunnel measurements. They are fixed in space so that the positive

longitudinal direction is always downstream in the wind tunnel. The other data is taken from

particle tracking or simulations and the data is averaged over many different random longitudi-


nal directions. The wind tunnel data shows a correlation between larger downstream velocity

and larger structure functions. For the passive grid, this dependence is weak and independent

of scale, but for the active grid the structure functions conditioned on the downstream velocity

are somewhat different at different length scales.

We calculated conditional structure functions for Gaussian random fields which yielded some

additional interesting insights. Figure 5.5 shows the second order structure function conditioned

on the longitudinal velocity sum for Gaussian random fields. The Gaussian random fields are

constructed as vector fields where each component of the velocity vector is Gaussian, indepen-

dent and identically distributed. The velocity components are also subject to the condition of

being divergence free and having the same energy spectrum as the DNS velocity field at a given

Reynolds number. The result is almost perfectly flat showing no influence of the large-scale

velocity on any length scale [33]. This provides a baseline for evaluating conditional struc-

ture functions and confirms the argument presented in chapter 3 that the observed conditional

dependence is not a simple kinematic correlation between sums and differences of the same

measurements.

We also studied the effect of conditioning on the velocity at the midpoint between the two

points rather than the velocity sum. The midpoint velocity is inaccessible to particle tracking

experiments, but can be studied in the numerical simulations. Conditioning the DNS data on

the midpoint makes very little difference for small r as expected since here the two velocities are

nearly the same. At larger r there are some small quantitative differences between conditioning

on the velocity sum and the midpoint velocity, but the main features of the conditional structure

functions in Fig. 5.2a are unchanged. We also considered how the conditional dependence

changes if the conditioning point is selected away from the midpoint. For Gaussian random fields,

this introduces a kinematic correlation that is avoided by conditioning on the midpoint.

5.3 Discussion

We provide here a discussion of some possible interpretations of this data. The data clearly reveal

that the small scales have a different dependence on the large scale velocity in the different flows.

However, the causes of the differences are difficult to conclusively identify.

A central question is what distinguishes the flow that shows very little dependence of the con-


ditional structure functions on the large scales (passive grid wind tunnel turbulence) from the

other flows. The DNS is an intermediate case with very little dependence on the large scale

velocity when it is small (less than 2 standard deviations) but significant dependence beyond

this. One possible hypothesis is that the flows with weak dependence on the large scales are

homogeneous, not just across the detection volume but also well beyond it. Table 5.1 reports

the number of integral length scales that span the flow in its narrowest dimension, W/L. The

passive grid flow which has the largest W/L ratio also shows the least dependence on the large

scales. For DNS with periodic boundary conditions, W/L is defined using the size of the pe-

riodic box, and it is possible that the limited size of this domain contributes. The other flows

have smaller W/L and stronger dependence on the large scales. In highly homogeneous flows,

any fluid entering the observation volume would likely be statistically similar, and therefore

less likely to show a dependence on the large scales. In less homogeneous flows, fluid could be

swept into the detection volume from regions with different energy, leading to dependence of

the structure functions on the large-scale velocity that is responsible for the sweeping.

A related hypothesis is that structure functions depend on the large scales whenever significant

energy exists at scales much larger than L. In a passive grid wind tunnel, energy is injected at a

fairly well defined scale and then measurements are made before there is time for much energy

to be transferred up to larger scales. But in DNS and in the zero-mean-flow systems designed

for Lagrangian measurement (a and e-h in Figs. 5.1, 5.2 and 5.4), the flow is usually driven

continuously and the flow in the region studied is statistically stationary. In these circumstances

the largest length scales are dictated by the system dimensions which are typically much larger

than the integral scale, and these scales can accumulate energy. The existence of energy in these

large scales can lead to large-scale intermittency either through fluctuations in the energy flux

into and out of these scales or through transport of more energetic turbulence from near the

energy injection mechanism into the detection volume. Such energy fluctuations may result in

the dependence of all scales on the largest length scales shown in Figs. 5.2-5.4.

Another question is why the small scales in some flows show a stronger dependence on the large

scale velocity than larger scales (Fig. 5.2a and 5.2f), while in other flows all scales collapse (Fig.

5.2c, 5.2d and 5.2h). One possible consideration here is that large scale intermittency of the type

described in the previous paragraph will produce temporal variations in the energy dissipation

rate which would cause the Kolmogorov scale to fluctuate. If the Kolmogorov scale is smaller


when the large scale velocity is larger, it would increase the structure functions at small scales

when the large scale velocity is large. This is exactly what is seen in Fig. 5.2a and 5.2f. However,

more work is needed to understand this effect since it is not seen in all data sets.

We also note an intriguing connection between conditional Eulerian and Lagrangian statistics.

Lagrangian accelerations have been conditioned on the instantaneous large-scale velocity and

found to have a significant dependence in both a counter rotating disk experiment and isotropic

DNS [34–37]. The results are quite similar to the smallest scales of the Eulerian structure

functions presented here. However, here we also have data from the passive grid wind tunnel

where the conditional dependence is nearly absent. It would be very interesting to probe the

dependence of accelerations on the large scale velocity in a wider variety of flows to see if similar

flow dependence is observed there.

5.4 Conclusions

We have presented Eulerian structure functions conditioned on the large-scale velocity for eight

different turbulent flows. One flow (passive grid wind tunnel turbulence) shows very little de-

pendence on the large-scale velocity while the rest show a significant dependence that contains

information about how the unique large scales of each flow affect the small scales. This sys-

tematic comparison of conditional structure functions in different turbulent flows provides a

reference that can be used to compare the effects of the large scales in new flows as they are

developed.

Chapter 6Conclusions

The bulk of this work has been the study of a turbulent flow between oscillating grids with

3D particle tracking and a novel real-time image compression system. Using this apparatus we

gathered a large amount of velocity statistics which have been thoroughly analyzed. We have

made significant progress in quantifying the effects of various properties of the non-universal

large scales on the inertial range and small scales.

We measured the mean and variance of the velocity as a function of distance from the grids. The

oscillating grid motion produced a weak mean flow as well as a region near the grid with high

velocity variance that falls off quickly to a very homogeneous, lower velocity variance, region in

the center. This profile has been key in the determination of the role inhomogeneity plays in

small scale statistics.

To determine the role of non-universal large scales, second order Eulerian velocity structure

functions were conditioned on the phase of the grid, an obvious source for periodic large scale

energy input. Results show little dependence of the structure functions in the center region and

surprisingly little even near the grid.

Eulerian and Lagrangian structure functions were also conditioned on the instantaneous large

scale velocity. A large dependence was found in the center of the tank, with the Eulerian

structure function increasing by a factor of 2 or more when the large scale velocity is large. The

dependence of the Lagrangian structure functions is somewhat larger. Conditioned structure

87

Chapter 6 - Conclusions 88

functions show that, in the center of the tank, all length scales are affected in approximately

the same way by the large scale velocity. The region near the grid was also analyzed and

compared with the region in the center. Near the grid, we found a much stronger dependence on

the instantaneous large scale velocity for both the Eulerian and Lagrangian structure functions

than we found in the center. Near the grid, there are clear signatures of the effects of large

scale inhomogeneity on the small scales. Fluid coming up from the energetic region nearer

the bottom grid has large structure functions, while fluid coming down from the relatively

quiescent region in the center has much smaller structure functions. The functional form of the

conditional structure functions are also different indicating the different histories of the different

fluid. These measurements provide a clear picture of the way inhomogeneity affects the small

scales of turbulence.

Although inhomogeneity was shown to be a significant factor in the dependence we observed

of the small scale structure functions on the large scale velocity, it is not the only factor, large

scale intermittency also plays a role. In order to evaluate the role large scale intermittency plays

we modulated the driving frequency and measured the structure functions in the center of the

tank. Adjusting the modulation parameters allowed us to augment the large scale intermittency

already present in the system and explore the extent to which small scales were affected by its

presence. The addition of energy fluctuations on a time scale longer than the eddy turnover

time (the hallmark of large scale intermittency) was found to greatly increase the dependence

of the small scale structure functions on the large scales.

Plotting the conditional structure functions versus the large scale velocity provides a powerful

method for visualizing the effects of the large scales on all length scales in turbulent flows.

This has been done for grid turbulence and homogeneous, isotropic DNS (Ref. [1] which show

almost no dependence of the structure functions on large scale velocity. Our oscillating grid

flow and high Reynolds number atmospheric boundary layer turbulence [1] show very similar

dependence.

To complement this comparison to previous work and further explore the effects of the large

scales in turbulence we have collaborated with other researchers in the field to compare 8 tur-

bulent flows. These flows include forced isotropic turbulence simulated on a periodic domain,

passive grid wind tunnel, active grid wind tunnel (in both synchronous and random driving

modes), counter-rotating disks, oscillating grids, and the Lagrangian exploration module (in

Chapter 6 - Conclusions 89

both constant and random driving modes). We observed that the dependence of the small scale

structure functions on the large scales is prevalent in all but one flow. The turbulent flow behind

a passive grid is unique in that no dependence on the large scales is observed..

By comparing these flows we have identified several possible causes for the dependence. The

causes include the presence of significant energy contained in the largest scales which can result

in energy flux into or out of the largest scales, and can also result in fluid transport from

regions of greatly different energies. Another possible cause is the energy of fluid adjacent to

the observation volume being significantly different from those in the observation volume, a

characteristic of the inhomogeneity of the flow. In less homogeneous flows, fluid could be swept

into the observation volume from adjacent regions with different energy, leading to dependence of

the structure functions on the large-scale velocity that is responsible for the sweeping. We found

this was best quantified by the ratio of the narrowest system dimensions to the integral length

scale W/L. In addition to identifying possible causes, the systematic comparison of conditional

structure functions in different flows provides a valuable reference that can be used to compare

the effects of the large scales in new flows as they are developed.

It is my hope that this work continues, tackling issues like the role anisotropy plays in the

dependence of the small scale structure functions on the large scales, as well as research into

finding statistics that are robust and universal, even in systems which currently show dependence

on the non-universal large scales.

Appendix AMeasurement Error Considerations

An important consideration in any experimental study is the measurement error. In experimen-

tal turbulence research a formal analysis of error propagation is generally not performed for the

higher level statistics such as the conditioned structure functions. This appendix will add on to

the measurement error discussed in the text and briefly touch on some of the error introduced

by our particle tracking algorithm, and examine statistical convergence.

A thorough study of our particle tracking algorithm was performed by undergraduate researcher

Nicholas Rotile. He compared our algorithm to the one used by the Luthi research group at

Eidgenossische Technische Hochschule (ETH) in Zurich Switzerland. Two dimensional images

of particles in a flow were processed using the algorithm at ETH and here at Wesleyan (Wes).

Details of the comparison of these two algorithms can be found in Nicholas’ report; I will provide

here only the summary. In short, he found that our algorithm is sufficient, and under certain

circumstances more accurate then the algorithm used at ETH. The Wes algorithm does have

some distortion of the x and y axes, the distortion is minute and the velocity measurements

from both groups are very similar. The difference in particle position was most pronounced in

the x direction, and varied between the two groups by tens of microns. The difference in the

velocity measurements was also quite small, averaging only about 200 µm/s. Although this is

just the difference between two algorithms, and not the difference between our algorithm and

the true particle position, it does give us confidence that our measurements are so close to those

of an independent research group. It should also be noted that the position measurement error

90

is on the order of the known ray matching error, and is smaller than a typical Kolmogorov

length.

Another concern is for the statistical convergence in our data. In a typical calculation we take the

second moment of the velocity difference distribution as a function of r and Σu⊥, a calculation

that requires a large amount of data. With this in mind we limited the number of r bins across

the range of interest to 8 and Σu⊥ bins to 16. Smaller bins would have lead to fewer samples

in each bin, and greater statistical error. To gain understanding of the statistical error in our

measurements we used a standard convergence determination. The entire data set was randomly

divided into 8 subsets. The conditioned structure functions were then calculated separately in

those subsets, and the standard deviation was found between the 8 subsets for each point in the

calculation. The standard deviation at each point was then normalized by√8 to account for

the error in the mean, and then plotted as error bars seen in figure A.1.

Figure A.1 shows the statistical error in a typical conditional structure function for various

separation distances at continuous 3Hz driving. As expected, the error is very small for all but

the largest Σu⊥ and the smallest r. We are able to measure r that is very small, but it seems

that at this particle seeding density that is relatively rare. However, this shows that in the range

of r and Σu⊥ where we are interested there is a high confidence level in the statistics.

-3 -2 -1 0 1 2 30.5

1

1.5

20.5

1

1.5

20.5

1

1.5

20.5

1

1.5

2

Figure A.1: Eulerian second order conditional structure function versus large scale velocity, plotted in a similar

manner to figure to 4.1a, with error bars representing the statistical error. The following separation distances

are shown in separate plots for clarity: r/η: ◦ = 5.33 to 10.67, × = 21.33 to 42.67, ⋄ = 85.33 to 170.67, ▽ =

341.33 to 682.67.

Appendix BTank Preparation

In this and the following appendices I will describe the tank apparatus used for turbulence

experiments and how to prepare it for experiments. This is not meant to replace a thorough

tutorial from an experienced researcher who has used the apparatus, and is comfortable with its

operation; this should be used as a supplement to such a tutorial. If a future researcher tries to

run the experiment based on these appendices alone he or she will almost certainly find them

inadequate, and I hasten to say it, blame the author. This and the following appendices are

meant to partially record the procedures for performing experiments and are of limited scientific

interest, instead they are intended for future researchers who will use the apparatus.

The tank was built at Wesleyan University in 2003-2004, primarily designed and by Greg Voth,

James Johnson, and Dave Boule, and constructed in the Wesleyan machine shop. Detailed

blueprints should be located in lab 039 above the desk by the optical table, with copies in the

machine shop. The technicians in the machine shop have been extremely helpful during my

time working on the tank, and I recommend using them as a resource, they are very skilled and

friendly.

First some obvious words of caution. There are several dangerous aspects to the operation of

the tank. When the motor is running the rods and rod supports oscillate with tremendous force.

Do not put any body part underneath the tank when the tank is running. As a safety feature

there is a shear pin in one of the cams which is a built in weak link that will hopefully break

and halt the oscillating movement before the oscillating parts cause too much damage. This

93

have never been tested, and hopefully never will. Another safety device is the mesh fencing that

can hang on the box beams to keep any flying debris inclosed in case of a failure. It is good

practice to keep two of these fences on when the motor is running (the other two cannot be put

on because they would block the laser beam. The tank is sturdily built and it is often necessary

to stand on the ledges, this is safe if one uses caution. The ledges are small, and tools tend to

accumulate on them. One researcher has fallen, although no injury resulted; be sure to wear

appropriate shoes when working on the tank.

There are several dangers that could harm the equipment as well. It is often necessary to

winch the structure that holds the rods (called the spider). Be sure to remove the winch before

powering on the motor, there is a note on the circuit breaker which reminds ’check spider’ for this

purpose. If the winch is fastened to the spider when the motor is running it may be destroyed.

A second point of caution is when filling the tank with water (for details on filling see appendix

D.1. Be sure the tank is open to atmosphere by opening a valve at the top of the tank. If the

tank is quickly filled while it is not opened to atmosphere the air pressure will build and the

tank may rupture. A third point of caution, there are a number of access points on the top of

tank, be sure they are all closed when filling with water. If one is left open water will spill from

the top of the tank and may damage the cameras or other equipment. An additional point of

caution when fastening the windows on to the tank. Be sure to use screws of the proper length.

If screws that are too long are used they could crack the Plexiglas. If a window screw socket

is too tight for a screw leave it without a screw. There is adequate redundancy, the windows

will not leak with one screw missing. Lastly a cautionary story, early in the development of the

tank the Plexiglas was split at a seam and all of the water was spilled. This occurred because

the tank was filled with cold water and the all-thread rods were tightened. The tank was then

filled with warmer water and thermal expansion caused the Plexiglas to split. The Plexiglas was

reglued and springs were put on the all-thread rods so the tank could thermally expand.

It is important to have a clean tank. Any debris in the tank could appear in the observation

volume and be tracked as a tracer particle even though it does not follow the flow. In addition,

dirty water can absorb light and lower the signal to noise ratio. Keeping the tank clean has been

a challenge that I have not satisfactorily met. There are several causes of debris in the tank.

Biological growth in the fill lines can enter into the tank after the filters. Hairs and dirt can

enter from open windows. Paper fibers can enter if paper towels are used to clean the tank. The

grids are corroding and aluminum oxide is contaminating the water. The seals contain metal

that can rust, and rust particles can enter the tank. The rods and linear bearing slowly wear

out and can add metal particles to the water. A good solution to remove the debris would be

to filter the water continuously for a long time before the particles are added. Unfortunately

in the current set up of pumps this adds air bubbles to the water. They may be created by

cavitation, or due to air entering the fill lines. Air bubbles should also be avoided because they

too could be mistaken as tracer particles when recording data. A future project to rearrange the

pumps to avoid the addition of bubbles could be fruitful. A program has already been written

that can occasionally stir the water for effective constant filtering (WashCycle.vi). Before the

experiment is run it is helpful to clean the tank. This can be done by vacuuming with the

shop vac, Kimwipes, or a sponge. It is up to the researcher to find the method they find most

effective. One note of caution, avoid using soap. It will be very difficult to remove all the soap,

especially from the fill lines, and it will cause bubbles which will be hard to get rid of. After

the experiment has been run it is good practice to clean the tank again. Keep a window open

to allow the water to dry quickly and avoid hard water stains on the windows, and vacuum up

the small amount of water that remains on the bottom of the tank. Any hard water stains can

be removed with vinegar, but are generally harmless. Be sure to vacuum the water from the 4

rod wells, stagnant water there maybe overlooked at biological growth may occur.

Appendix CCalibration

The purpose of camera calibration is to set up a 3 dimensional coordinate system and locate

the cameras in that coordinate system. See section 2.3 for a brief outline.

C.1 Acquiring Calibration Images

The first step is to place the calibration mask in the tank at the location of the desired observation

volume. There are interlocking aluminum cylinders stored near the filters for this purpose. They

build off of a divot in the center of the bottom grid and hold the mask in the desired position.

Here is an example of how I set up the observation volume in the center of the tank: I place

several interlocking cylinders on the disk which fits into the divot in the center of the bottom

grid. I then locate the center of the tank by marking 75 cm from the bottom of the tank on a

post-it note stuck to the outside of the tank. I make the same 75 cm mark for the opposite side

of the tank. I then raise or lower the bottom grid via the winch attached to the spider until the

center screw of the mask is level with both of the 75 cm marks. I look through the tank and

try to make my eye level with the two 75 cm marks, and when the mask screw and the two 75

cm marks are all inline the mask is in the center of the tank. As an added check I will put the

meter stick in the tank and try to line it up next to the mask to make sure the mask screw is at

75 cm. I then look from the side of the tank and try to rotate the mask until it is parallel with

the bars of the grid, and therefore at 45 degree angles from the windows. The rear window is

96

then reinstalled and water is added into the tank. It is not necessary to fill the tank all of the

way, just to above the windows so that the refraction of the water is taken into account when

the calibration pictures are taken. Do not calibrate without water in the tank.

When the mask is in position and water is in the tank place a theater lamp behind the mask and

illuminate the mask. Turn on and adjust the camera positions and focus. This can be done by

moving the camera shelves as well as the position of the cameras on the shelves. It is important

to insure all of the cameras are focused on the same volume in the tank. To maximize this,

I open the XCAP (video5-C:\XCAP\program\xcapwxx.exe program and display a vertical line in

the middle of the screen (at 640 pixels) I align this vertical line with the middle of the mask. I

found it useful to align the vertical line on the camera with the vertical line in the ”D” printed

on the bottom of the mask. I followed a similar procedure with a horizontal line aligned with

the mask screw which I used to as a reference for half of the mask height. Once all the cameras

were pointing at the same volume in the tank I adjusted the focus of each of the cameras. I did

this by first increasing the aperture to its maximum (fstop of 2.8 this decreases the depth of

field), then rotating the focus ring until the center of the in-focus region was centered around the

vertical line displayed in XCAP. I repeated this for all of the cameras and return the aperture

to fstop 8 or 11 (note, all cameras do not necessarily need to have the same aperture).

With the cameras now in their proper positions and with proper focus take 10 .tiff still calibration

pictures in XCAP from each camera. One picture would probably be sufficient, but 10 are

normally taken so they can be averaged and the noise can be lowered. One helpful hint for

taking the calibration pictures, the theater lamp can be moved for each camera, there is no

reason why the lighting needs to be the same for calibration images of each camera. What is

important is that the mask be sufficiently bright and have even illumination across the mask

with few air bubbles on the mask.

C.2 Running the Calibration Programs

Now that the calibration images have been taken, the mask can be removed and the exper-

iment can be run (remember to take some data with a low particle density for the dynamic

calibration which will come later). I will here describe the procedure for running the cali-

bration programs which are used to determine the real space position of the cameras. First

Figure C.1: An example calibration image that is an average of 10 calibration images taken from camera 1.

the simple Matlab program ext_hd1-G:\Lab1backup\Matlab_local_lib\dbblum\Calibration\db_

calib_image_averaging_no_background_imgs.m is run, which simply averages all the calibration

images to reduce the noise. An example of such a picture is shown in figure C.1.

The averaged calibration images now need to be edited so they can be input into the calibration

program. The calibration program identifies the dots on the mask and finds their position. It

is important that no air bubbles or extraneous marks are present in the images. In Photoshop

I manually remove all parts of the images that are not dots. An example of such an edited

picture is shown in figure C.2. Note the dots that remain should be present in every calibration

image. It is possible, depending on the field of the view, that a large number of dots will be seen

from every camera. This is desirable, and can create an initial calibration that is as accurate as

possible, however it is also possible that having a large number of dots will cause convergence

problems in the calibration program. If this occurs I have edited the calibration images to lower

the number of dots into an array of 10 × 10.

Once the edited calibration images have been prepared for each camera they can be input into

the calibration program db_PTVsetupPrep_with_end_params.m. This program has a user friendly

interface and was created by Haitao Xu. Note, when running the program it is possible to

submit parameters for the dot recognition that are too lax. This will create a huge number of

Figure C.2: An example of a calibration image that has been edited so all that remains is the dots which are

viewed by each camera

recognized dots and may cause the program to stall, if this happens, restart Matlab.

It is difficult to understand the calibration algorithm; to understand the basics and the output

of the program I recommend first running and understanding gv_tsai_calibration_sandbox.m.

The basic premise is the position of the cameras are given as three angles and the distance from

the origin. The sandbox program demonstrates this and allows the user to convert from the

angles to Eulerian coordinates. When the calibration program is successfully completed the file

camParaCalib.mat will be created which contains all of the calibration parameters. The data is

now ready to be stereomatched, which will be described in appendix E.

One additional note, the original calibration program output a coordinate system which was left

handed. To convert to a right handed coordinate system the program db_left2right_coordsys2.m

was created.

C.3 Dynamic Calibration

Once initial stereomatching has successfully found particles on several thousand frames the

calibration can be adjusted using the ray intersection error found in the initial stereomatches.

Essentially dynamic calibration slightly adjusts the position of the cameras in the lab coordinates

so that the ray intersection error of the particles which have been stereomatched with the initial

calibration is minimized.

The program used to dynamically calibrate is called db_dynamic_calib2.m. There are several

relevant parameters and I recommend becoming familiar with the program before attempting

to run it (good advice for all programs in general). In order to insure the particles with which

we are dynamically calibrating are good matches they should be seen on each camera and have

a relatively low ray intersection error. Note there are versions of this program that have been

developed to correct for trapezoidal distortions inherent in viewing the particles through the

Plexiglas at an angle. Ultimately we decided not to use the trapezoidal corrections because the

differences were small.

When the dynamic calibration program has been successfully run the ray intersection error

should be reduced significantly. For an example, a typical system could have field of view with

55 µm per pixel and an initial mean intersection error of approximately 80 µm. This could be

reduced to approximately 25 µm for 4 cameras, or 14 µm for 2 cameras.

Appendix DData Acquisition

Data acquisition is a complex task, and I recommend recording test data with an experienced

researcher just to make sure all the procedures are clear. The set up for data acquisition can

take several hours and consists of tank preparation and camera calibration which were covered in

the previous appendices. Here I will describe the data acquisition along with all of the relevant

computer programs.

D.1 Filling the tank

Once the tank has been prepared and the cameras calibrated at least 300 gallons of degassed

water should be in the storage tank. I leave the vacuum pump on over night before the data

collection just to be sure any air bubbles captured in the process of transferring the water from

one tank to the other are removed. I have tried to make the water transfer process user friendly.

I numbered all of the 3-way valves in such a way that when the water is being transferred from

the storage tank to the turbulence tank the numbers are right side up, and when the water is

being transferred from the turbulence tank to the storage tank they are not. The 3-way valves

are also labeled with an ’L’ which shows how the water will flow through the valve. Despite

these attempts to make things user friendly it is still very easy to confuse things, so please trace

the intended path of the water before activating either pump, it is easy to dump water on the

floor, or overpressure a vessel.

101

With these words of caution make sure both tanks are open to atmosphere, all the windows are

in place, and begin pumping the water from the storage tank into the turbulence tank. If both

the coarse and fine water filters are inline (recommended for experiments) the water will enter

the turbulence tank somewhat slowly. It will take over an hour to fill. During this time I have

been able to begin other parts of the setup, but keep an eye on the water level, look for leaks

or when it reaches a few inches from the top. While the tank is filling I often fill the air bubble

suppression reservoirs at the top of the tank. Once the water line is within a few inches of the

top stop pumping and close the valve to the storage tank. The goal is now to fill the turbulence

tank with water and have no air bubbles remain at the top of the tank. Small air bubbles are a

problem because they could be entrained in the flow and be mistaken for tracer particles. Large

air bubbles are not as big of a problem, but should still be avoided if possible. Achieving the

goal of no large air bubbles in the top of the tank is not straightforward. A method I have used

with moderate success is to reduce the flow rate into the turbulence tank by partially closing the

valve near the storage tank and fill the turbulence tank until the fill line is slightly above the top

of the tank. This will cause water to rise through the valve to go to atmosphere and may cause

water to fall into the sink. This is acceptable, but be sure no water splashes on the mirrors or

lenses. If a large air bubble remain at the top of the tank it may be necessary to drain the tank

a few centimeters and try again. Keep in mind some small amount of air trapped at the top of

the tank is common, and can be difficult to remove. Once the experimenter is satisfied close all

the valves, including the one to atmosphere.

Note there is a white overflow tank resting on top of the turbulence tank. This was originally

designed to eliminate air bubbles in the tank, but I have not found it effective and do not fill it

with water.

D.2 Using the laser

I will briefly discuss the laser here. The laser is a dangerous piece of equipment and definitely

should not be run by anyone until they are familiar with it and the safety procedures. I will

give a run down of the steps to operate it here. First the cooling water valve should be turned

on, it is kind of hidden in the wall of the fume hood. Be sure power is being sent to the laser

by checking the circuit breaker. Then the laser key should be turned to ’unlock’ and the power

button pushed. This should cause the laser spring to life. The ’Cooler’ toggle switch should

then be moved to on. If the laser is being used frequently the green ’ok’ light should be on.

However, it is common for there to be a DI water fault. This occurs when the cooling water has

not been circulated through the DI filter for a long time. If this happens keep the laser on, the

cooling water pump should still circulate the water. After approximately 30 minutes shut down

the laser using turning the ’COOLER’ switch off then pressing the ’EMO’ button and try again.

If the problem persist it is possible the cooling water needs to be replaced (there is a stockpile

of steam distilled water by the filters that can be used). It is also prudent to check that there

is a proper amount of cooling water, and that no biological slime has grown on the sides of the

cooling water reservoir. The DI filter should also be replaced periodically according to the user

manual. Once the fault has been cleared and the fault light is green the lamp can be switched

on. After a few seconds the laser shutter can be switched open which will require waiting a few

moments. The display should indicate when the system is ready. At this time the system is

lasing but the light is being blocked by the laser aperture, a final check should be made that the

door to the lab is closed, safety goggles are on, and nothing obstructs the laser path into the

tank (make sure the bags are off the lenses). The ’APERTURE’ switch can be toggled up and

laser light will enter the tank. From here the beam properties can be adjusted, the current, the

pulse rate frequency, the triggering, etc.

A note about the laser, it was not mass produced like other lab equipment and may have design

or construction defects. As of now it has been returned to the manufacturer Quantronix on

Long Island twice. First the ’trans-orbs’on the power shelf malfunctioned. The second time the

whole system was returned, there were several wires that had come loose in the first repair and

biological growth had impeded the flow of cooling water. To inhibit biological growth the laser

should be run for at least an hour every month.

D.3 Using the image compression circuits

The image compression circuits must also be turned on, in a similarly involved process. First

be sure that the image compression circuits are properly cabled. By cable I mean that all

of the cables connecting the cameras to the image compression circuits and from the image

compression circuits to the computers are securely connected to the correct device. Improper

cabling happens surprisingly often. Once the cables have been checked attach the USB Blaster

to the data port on an image compression circuit and turn the image compression circuit on

using the custom made black power switch box. It is theoretically possible to damage the im-

age compression circuit by attaching the USB Blaster after the the image compression circuit

is on, so make sure it is attached beforehand. Upload the program video5-C:\NewICSFiles\

HardwareProgramming\NewImageCompressionVer2\Compression_Migration.qpf to the image com-

pression circuit using Quartus II video5-C:\altera\90\quartus\bin\quartus.exe. This can be

done through the programmer option on the tools menu. Once the program is uploaded press

both reset buttons on the image compression circuit. For the Mikrotron cameras, Impact

(video5-C:\NewICSFiles\Impact3.2\IMPACT3.exe) should now be opened. As a first test, set

the cameras on free run mode and shine a flashlight into the camera lens. Check to make sure

the image comes through (up to the maximum number of bright pixels) and moves smoothly

through the screen. It has happened before where a horizontal band will not respond to light

and the image compression circuit had to be turned off and reprogrammed. In general if some-

thing is awry turning off and reprogramming the image compression circuits is a good first step

for troubleshooting (after checking the cabling of course). Another troubleshooting tip is to

open XCAP with the image compression circuit running. The image will be encoded into what

should be vertical bands of pixels which are changing and unintelligible, but the simple test of

shining a flashlight into the lens will show a clear increase in signal. A similar procedure should

be followed for the Basler cameras, except that the option to have Impact run in free run mode

is absent. Instead the SetupB.fmt file must be swapped with one that was created for free run

mode. This should be more thoroughly described in the work of Dennis Chan.

D.4 Using external trigger

Once the image compression circuits are working in free run mode they can easily be changed over

to external trigger. For Basler cameras simply open Impact with the appropriate SetupB.fmt

file in the containing folder. Figure D.1 shows a diagram for the path of the triggering signal.

Note the pulse selector circuit. The cameras run at 450 Hz, but when the laser is running near

full power (around 13.5 - 14 A) at 450 Hz PRF it becomes unstable and the laser intensity

fluctuates. To remedy this we run the laser at 900 Hz, and the cameras at 450 Hz. To achieve

this the pulse selector circuit divides the signal by two so that the incoming signal is at 900 Hz

Function Generator

Altera Pulse Selector

Sensor DAQ

Video6

Video5

950 Hz

2.5 V

Laser Control

EMO

Video7

Video8

Cam 1

Cam 2

Cam 3

Cam 4

ICC 1

ICC 2

ICC 3

ICC 4

Sync

Figure D.1: A flowchart for the path of the camera trigger.

and the output is 450 Hz. As seen in figure D.1 the signal is output from the function generator

at 900 Hz and is split with one half going directly to the laser and the other half going to the

pulse selector circuit. The pulse selector circuit is nothing but a Stratix FPGA with a simple

frequency halving algorithm. Simply power the circuit and upload the program which is located

at video5-C:\altera\quartus60\Pulse_Selector\Stratix\Pulse_Selector_stratix.qpf. To test

to make sure everything is working I recommend turning off and on the trigger signal and

observing the display stop and start. I also recommend recording a small amount of data

It is important that all 4 cameras and the laser are working synchronously, early experiments

had frame skips which caused incorrect frame numbers and had to be corrected manually. This

problem was fixed by having the frame number input directly from the camera signal. However,

it is still a good idea to check for obvious frame skips. After stereomatching look for file sizes

that are drastically smaller than the others. This may be a sign that the frame numbers are

out of sync. I also recommend closing the laser aperture once or twice during a set of runs so

that a few seconds of recording will be blank. This will help identify any problems by showing

a marker when all the cameras should report zero particles found on the same frames.

For the normal frame rate of 450 Hz the exposure time for the Mikrotron cameras should be set

to 0.7 msec. This allows the cameras to capture one and only one laser pulse in a frame. The

exposure time for the Basler cameras should be set to 0.4 msec in order to capture one laser pulse

per frame. I believe the Basler cameras have the peculiar property that a 0.4 msec exposure

time will be interpreted in the following way: the shutter closed for the first 0.4 msec then open

for 0.4 msec and then closed again for the remainder of the frame period. This hypothesis was

drawn from carefully adjusting the exposure time and observing when the particles could be

seen. It is possible this hypothesis is wrong, but I do believe that these settings will garner one

(and the same) laser flash per frame in each of the cameras.

D.5 Running the motor

Once the tank is completely filled with water and all the valves are closed the motor can

be turned run. After double checking that the spider is free from the winch turn the motor

power on at the circuit breaker and open the motor control program (lab2-C:\ProgramFiles\

NationalInstruments\LabVIEW8.2\user.lib\dbblum\MotorControl.vi) in LabView. This program

controls the grid frequency as well as the acceleration, deceleration, and other parameters of

the motor. Note the frequency you enter differs from the grid frequency in the tank by a factor

of 180. For example entering 180 Hz will result in 1 Hz grid oscillation, 360 Hz will lead to 2

Hz grid oscillation and so on. To be precise the factor is closer to 167, however all previous

measurements have been made using the nominal number 180.

D.6 Water cooling

As turbulence theory tells us, stirring the water in the turbulence tank will lead to an increase

in temperature. To mitigate this a heat exchanger has been installed at the top of the tank. I

have simply connected a hose from the sink to the heat exchanger and allowed a small amount of

cold water to flow. Since so much water has a large heat capacity thermal changes will be very

slow and so such a method is effective, and really only (hypothetically) necessary for long runs.

Monitoring the temperature can easily be done using the temperature probe which attaches to

the top of the tank.

D.7 Adding particles

The particles are stored in water in a bulk container above the sink as well as some particles in

a smaller container. Be careful with the large bottle, particles are expensive, the large bottle

cost approximately $1,000. To add particles to the tank I have taken a small bottle of particles

to the top of the tank and carefully removed the temperature probe. Be sure not to lose the

small o-ring that fits around the probe and provides a water tight seal, and do not bend the

temperature probe. I then shake the small bottle to mix the particles in the water and use a

pipette to squirt the particles into the tank. The tank is probably a bit over pressurized so it

is common for water to come out of the port when the temperature probe is removed, losing

some water is acceptable but be sure to clean it up and be quick about adding the particles. I

have had the best results by inserting the pipette into the temperature probe port as far as it

will go and then squirting the particles in, and then sucking in some water from the tank and

squirting it out again into the bulk water. This insures fewer particles are left in the pipette.

As a very rough estimate 2 full pipettes corresponds to 50 particles in view, but do not rely

on this estimate. Add particles then stir with the motor and check how many particles are in

view. Keep in mind roughly 40% of particles seen in 2D are stereomatched. Remember to record

some data (several minutes) with a low particle density for the purposes of testing and dynamic

calibration before adding more particles.

D.8 Recording the grid position (optional)

Below one of the rods is a photogate that can record when it is broken as a function of frame

number. This can be used to calculate the position of the grids at any frame. Unfortunately

this system has proven to be needlessly complicated, especially when the grids are not driven

at one continuous frequency. However, if one wishes to record data using this device the Lab-

View program to do so is located at lab2-C:\ProgramFiles\NationalInstruments\LabVIEW8.2\

user.lib\dbblum\GridPhaseMeasurement\GateAndCameraRecorder.vi. The programs used to con-

vert these raw measurements to grid phase are located here ext_hd1-G:\Lab1backup\local_lib\

dbblum\lvm2gdf.

D.9 Recording data

Once everything is in place it is finally time to record data. Begin by setting the motor at

the desired frequency and allowing enough time for the turbulence to equilibrate. Setup each

computer so that Impact is ready to record data when the ’record’ button is pressed. Then stop

the trigger signal on the function generator. This will cause the laser to turn off too. Then press

reset on all of the image compression circuits, this will zero all of the frame numbers. Then

press record in Impact on all of the computers (press record as well on the photogate recorder

if it will be used). Now press the trigger button again on the function generator and the frames

should start recording. Check to see that the frame number is incrementing on all of the Impact

programs as well as the time remaining display. It has been a problem that one of the computers

will hang and Impact will not increment frames. The cause of this is not yet known, and the

experiment must be re-recorded. It is common for the Impact programs to finish at slightly

different times. As long as they all finish this is not a problem. This occurs because each

computer loses a different number of frames, and Impact will not report the recording complete

until the requested number of frames have been recorded. Keep in mind the frame numbers are

correct, so even if the elapsed time for each camera is different, all the cameras have recorded a

given frame synchronously.

To keep the file sizes manageable the maximum run duration I use is 90 minutes, which keeps the

cpv files to approximately 10 Gbyte. A few other miscellaneous tips: plan out a rough outline of

what data you must take and what data you would like to take. Things often take longer than

one would hope, and there must be an end at some point. In addition it has been my experience

that the water will become cloudy after enough run time. Also, taking data can be taxing, it

requires mental concentration in a noisy environment for a long time; try to account for this,

work in teams, and try to make sure the researcher is adequately cared for as well.

Appendix EData Processing

Once the data has been collected the researcher has only compressed video files (cpv files) and

calibration images. The goal is to process this raw data into 3D particles positions then to veloc-

ities then to higher order statistics (and eventually into a greater understanding of turbulence).

The computation time required for this process can be significant and there are many parts.

Here I will describe the largely automated procedure for processing the raw data.

Figure E.1 shows a flowchart which I will use to describe the data processing. We begin with

the two types of files mentioned above, the compressed video files (cpv) which Impact created

and the calibration images which were taken of the mask before the data was collected.

Begin by loading a cpv file into CPVplayer video5-C:\Impact3.3\IMPACT3.exe, and watching

some of the video. Play around some, make sure everything looks as it should, the particles are

a good size and shape, they are bright, they move smoothly, etc. It is also a good idea to check

several places in the file to make sure the data is consistent throughout the run. Try to develop

an intuition for what looks normal.

One of the features of CPVplayer is that it can recognize particles in a 2D image. It finds

the particle centers, total intensity, radii, eccentricity and frame number the particle was found

on, and writes this into an array called featdat.gdf. The user can change the parameters for

finding particles and display which pixels on the screen would be considered particles. Typical

parameters I have used are: a search perimeter of 30, this is how many pixels away from a given

110

run1-4_XX.cpv

Impact.exe

CPVplayer.exe

Xcap.exe

calib_image1-4_X.tif

db_PTVsetupPrep_with_end_params.m

camParaCalib.matcpv2gdf_project_new_8-2007

db_Stereomatch_Function.m db_Stereomatch_RUN_multiframe.m

db_smoothed_velocity_vector_field_creation_ML.pro

db_velocity_vector_field_subtraction_ML.pro

db_dynamic_calib2.m

runjobs1-4_XX.sh

run1-4_XX_featdat_YYY.gdf

parameterlog.mat vel3d2d_YYY.mat pos3d2d_YYY.mat

3d_mean_vel_field_x.gdf

vel3d2d_SMS_YYY.gdf

camParaCalib_DC.mat

db_calib_image_averaging_no_background_imgs.m

OR

parameterlog.mat vel3d2d_YYY.mat

Further Analysis

Shaded represents on

computer cluster

Rounded endges represent programs

Rectangles represent data files

Figure E.1: A data processing flowchart, rounded edges represent programs, rectangles represent data files,

and shaded areas occur on the computer cluster

bright pixel the program should search for other bright pixels. Histogram threshold of 30, this

sets the minimum pixel intensity to be considered as part of a particle. I keep this at the image

compression circuit threshold. Minimum total intensity of 100, the total intensity is the sum

of all the pixel intensities in a particle. Minimum center intensity of 35, one pixel must have

this brightness intensity in order to be considered a particle. Minimum number of pixels of 3,

just as the name implies. Choosing fewer than 3 pixels will lead to the loss of particle finding

accuracy.

The algorithm for calculating particle centers from 2D images was also written in a C++ pro-

gram and uploaded to Wesleyan’s Swallowtail computer cluster where it can be used on many

processors in parallel. Although it is possible to calculate to 2D positions locally in CPVplayer

I recommend processing the cpv files into featdat.gdf files on the computer cluster. Below I will

describe the procedure for doing so, it is not as straightforward as using the CPVplayer, but is

orders of magnitude faster.

First one must get access to the cluster by contacting the administrator (currently Henk Meij).

The particle recognition programmust be set up on their home directory on the cluster. The files

required are located here lab2-C:\DocumentsandSettings\dbblum\MyDocuments\ImportantComputerPrograms\

ParticleRecognitiononCluster. Set up may not be straightforward, and may require assistance,

be needs only to be done once.

To help facilitate the processing on the computer cluster CPVplayer creates a script for use on

the cluster that will instruct the program to run on many nodes in parallel. Create a cluster

script by entering the number of frames to be calculated in a given node (typical is 5,000 frames)

and click the ’create cluster script’ button. This will produce a text document called runjobs.sh

in the local directory of the cpv file. The objective now is to put the cpv files and cluster

scripts on the cluster and run the particle recognition algorithm there. First one will note that,

unfortunately, the current version of CPVplayer does not have a user defined input path, so

the script files need to be altered manually. This means a typical local cpv path would read

C:/dbblum/2010-03-29/run3_45.cpv, but this must be altered to the location of the cpv file on

the cluster, something like /cpv2gdf_project_new_8-2007/home/dbblum/2010-03-29/run3_45.cpv

the exact location will depend on where the cpv file is located on the cluster. Renaming the

path can be done manually with the replace function in Notepad, or I wrote an even more

automated version in Matlab ext_hd1-G:\Lab1backup\Matlab_local_lib\dbblum\utilities\db_

runjobs_path_renamer.m.

Operating on the cluster will be unfamiliar for those who do not have Unix experience. I will

give a brief overview of the steps involved, but I recommend asking someone with experience

for help. To upload the cpv files to the cluster I recommend the SSH Secure File Transfer program

video6-C:\ProgramFiles\SSHCommunicationsSecurity\SSHSecureShell\SshClient.exe for both the

transfer of files and for the terminal to execute commands on the cluster. The cluster’s address

is ’swallowtail.wesleyan.edu’ (users will have to get a password from the administrator). Put the

cpv files in a folder on the home directory and put the runjobs.sh files in the same directory as

the cpv2gdf executable file. The runjobs.sh file can technically be executed, but does so in a way

which is unfair to other cluster users. I therefore strongly recommend a further step of calling on

the runjobs.sh scripts from another script called C:\DocumentsandSettings\dbblum\MyDocuments\

ImportantComputerPrograms\ParticleRecognitiononCluster\simple_edit.txt. This script enters

the processing program on to the cluster nodes properly using queuing. The only parts of the

simple_edit.txt file that need to be altered are the exact runjobs.sh file name and the intended

queue. The options for which queue to choose are ’imw’, ’elw’, ’emw’, ’ehw’,or ’ehwfd’. Choose

one based on the most open nodes as reported here http://petaltail.wesleyan.edu/cgi-bin/

bqueues_web.cgi, ’imw’ or ’elw’ are typically used. Save the simple_edit.txt file and execute

it. It can be executed by typing the command ./simple_edit.txt while the terminal is in the

appropriate directory. It should take a few moments to begin, and when it does the number

of nodes used in the previously mentioned queue status website should increase with your new

jobs.

A few additional notes, do not leave data files on the cluster. It has limited disk space for

everyone to share, bring the files down to lab computers as soon as it is convenient. The

command to check the data used in a folder (including subfolders) is ’du-k’. To check to see

what nodes are being used enter ’bjobs’. The command to stop a program that is running is

’bkill’. Also, the eccentricity calculation does not currently work, and gives meaningless results,

but this is not used in the analysis.

While the featdat.gdf files are being created on the cluster the calibration can be preformed as

described in appendix C. The PTVsetupPrep calibration program will require the 4 averaged

calibration images as input and the output will be a camParaCalib.mat file which will have 4

structures containing the calibration information for each camera.

E.1 Stereomatching

An initial round of stereomatching should now be performed. The stereomatching program

db_Stereomatch_RUN_multiframe.m is rather complex and is outlined in section 2.4. Here I will

summarize the main functions as shown in figure E.2.

db Stereomatching RUN multiframe

Input: files with 2D particle centers for each camera, parameters for each subprogram

This program should be the only one to actually be executed by the user, it should contain all

the parameters for each subprogram. The algorithm is designed to run by iterating through

the featdat.gdf files with the 2D particle centers listed in ascending frame number. Each file

should be a reasonable length, 5,000 frames is typical. This limit is meant to keep array sizes

manageable, the main array in subsequent programs will be the length of all the particles in all

the frames in a file (approximately 300 particles per frame * 5,000 frames = 3,000,000 rows).

One of the parameters is ’moreinfoflag’, this changes the number of columns and 2d particle

information in the output file. The successful execution of this program will be files which

contain accurate particle velocity data.

Output: vel3d2d.gdf, 3D particle information and 2D particle information from each camera,

with up to 28 columns. The columns are described in figure E.3

db file finder3

Input: path for files with 2D particle center data, and number of cameras.

This small function creates a list of where all the files are, and assures they are in the proper

order

db cam2d loader2

Input: list of files with 2D particle center data

This small function opens the 2D particle center files in a robust way. It is designed to account

for missing frames.

db_Stereomatch_RUN_multiframe

db_file_finder3

db_cam2d_loader2

read_gdf

db_Frame_Number3

db_Angle_Determination5

gv_pixel2unitvector_fast

calibProj_Tsai

db_Candidate_Sort9

db_Candidate_Match6

db_RayIntersection_fast

db_Position3d_Determination_dist_method3

db_Tracking_and_velocity_function

db_track_weeks

dbsp_velocity

Figure E.2: A flowchart for the functions used in stereomatching.

vel3d2d columns: 1 2 3 3D position with respect to the origin (mm): x y z 4 5 ray intersection error(mm): best match, 2nd best match 6 frame number 7 8 9 10 11 12 13 14 2D position(pix):cam1 x y cam2 x y cam3 x y cam4 x y 15 16 17 18 total particle intensity: cam1 cam2 cam3 cam4 19 20 21 22 radius of particle(pix): cam1 cam2 cam3 cam4

23 particle id number

24 25 26

velocity (mm/s): x y z 27 (optional) motor phase 28 (optional) grid phase

Figure E.3: The meaning of each column in the vel3d2d files.

db Frame Number3

Input: number of cameras, 2D data

This function finds the indices of the frames within the 2D particle data files for each camera.

Determining this once in the beginning will save time for many of the upcoming functions which

will use this information.

db Angle Determination5

Input: maxsearchdist, camParaCalib, files for 2d particle centers for each camera

This is the key program, besides this the remainder of the code is essentially bookkeeping. The

idea as described in section 2.4, is to determine what particles seen in 2D are likely candidates

for 3D matches. Doing so greatly reduces the processing time. If a researcher needs to alter

this program please do so with caution, the bookkeeping may seem opaque, and will require a

good deal of effort to understand. It should also be noted that I often refer to the angles of a

camera pointing to a 2D particle. I usually convert this angle to a distance so it can be more

easily compared.

There are several key subfunctions within this function, briefly described here.

gv_pixel2unitvector_fast: Input: 2D pixel coordinates, camera calibration parameters

Output: a 3D unit vector pointing from the camera towards the 2D pixel in the 2D image that

it sees

calibProj_Tsai: Input: camera calibration parameters, 3D particle position coordinate

Output: 2D pixel coordinates of where that 3D point would be on that camera’s image plane.

It is important to note the 3D point can be anywhere, not just in a small box in 3D space in

front of the camera, this function will project the 3D point onto what the camera would see if

it had an infinite image plane.

db Candidate Sort9

Input: maxsearchdist, maxclosematches, angles associated with each 2D particle

The function compares the angles and records rays that are probably pointing to the same

particle, so later these candidates can be submitted to the ray intersection code.

For the inputs, maxclosematches is the maximum number of possible matches for rays pointing

to a particle. If this is a large number there may be combinatoric problems. A typical number is

9 close matches. The maxsearchdist parameter is the distance in pixels that the rays vary when

pointing to the same pixel. If this parameter is too high computation slows down, if it is too

low good matches will be list. An ideal system would have zero error, however a typical number

is .09 pixels, but this can be raised for the initial run, and lowered when dynamic calibration

has been established.

db Candidate Match6

Input: maxintersectionerror, mincamonpart

This function takes all of the combinations of candidate rays generated by db_Candidate_Sort9

and finds their intersection. An important subfunction is db_RayIntersection_fast, which is

the algorithm for determining the 3D point in space where the rays come closest to intersect-

ing.

For the inputs, maxintersectionerror is an important parameter that limits the distance the rays

can intersect and still be considered a single particle. The intersection error is recorded with the

3D particle information, I recommend looking at this statistic carefully and becoming familiar

with it (see how it varies in space, calibration, etc). If it is too small good matches will be lost,

if it is too large incorrect matches will be processed as particles. A typical number is 0.2 mm for

the initial run, and lowered to 0.06 mm after dynamic calibration has been established. However,

the mean for intersection error after dynamic calibration can be as low as 0.025 mm.

The parameter mincamonpart is the minimum number of cameras that need to see a particle in

order for it to be considered. Of course this is between 2 and 4, typically 2. It may be helpful

to set this at 4 for the initial run so that all cameras are properly dynamically calibrated and

the particles found are surely good matches.

db Position3d Determination dist method3

Input: all 3d points found by all camera sets, mingrouponpart, maxgrouperror

The current algorithm in db_Stereomatching_RUN_multiframe works by having the angles of a

main camera compared to the angles determined for each other camera (where ’angles’ refers to

the distance between a reference ray and the ray pointing from the camera to the 2D particle

on the image). The algorithm is somewhat redundant in that it cycles through each camera as

the main camera, but this is useful when one camera does not see a particle. The result is that

3D particles are found for different camera sets, up to 4 camera sets. As an example one camera

set would be cameras 1,2,3,4 where camera 1 is the main camera. A second camera set would

be 2,3,4,1 where 2 is the main camera. Both of these camera sets could find a 3D particle in a

nearly identical place in space. This function was made so that this 3D point is not recorded

twice (once for each camera set). Instead it goes through and finds particles that are nearly

identical and picks one randomly to be recorded.

For the inputs, mingrouponpart is the minimum number of camera sets that a 3D particle needs

to have in order to be considered valid. A typical number is 2 camera sets. The maxgrouperror,

is maximum distance between particles found from different camera sets. It is expected that this

distance is quite small, so a typical number is 0.001 mm, much smaller than the maxintersec-

tionerror. Be sure not to make this number too small, or valid particles will be lost. If it is too

large particles that just happen to be near each other will be recorded as one particle.

E.2 After Stereomatching

Stereomatching is now complete. A list of 3D particles is produced and the processing is nearly

complete. The stereomatching program also includes tracking and velocity functions within

the function db_Tracking_and_velocity_function.m. The tracking program (db_track_weeks.m)

identifies which frames a single particle appears in. Tracking a single particle through frames is

essential for measuring its velocity. As for the inputs for tracking, mem is the number of frames

into the future the program should search for the current particle, a typical number is 4 frames.

The maxdisp parameter is the maximum displacement of the particle between frames, and is the

most important parameter. A typical number is 0.6 mm, but should be judged based on each

experiment because it limits the maximum velocity that can be recorded. The dim is simply

the number of dimensions the particle exists in, of course this is set to 3. The parameter good

is the minimum number of frames a particle needs to be tracked for in order to be recorded, a

typical number is 5 frames. The velocity program dbsp_velocity.m determines the velocity of

each particle. A simple derivative gives unnecessarily noisy results. Instead the algorithm fits

a polynomial to the positions as described in 2.5. The fitorder is the order of the polynomial

that should be used to fit the position data, a typical number is order 2. The window refers to

the points considered within a trajectory for the velocity measurement, the halfwindow is half

of this, and a typical number is 2 frames.

Now that the velocities have been found I recommend running the dynamic calibration as de-

scribed in section C.3. The results of dynamic calibration are more accurate calibration files

which in turn should be used to recreate the 3D particle data more accurately. Usually one

iteration is enough, but I have found it tempting to create more velocities with the dynamic

calibration and then repeat the process.

In addition to running the sterematching code on local machines it can be run on the computer

cluster. This can be very useful if there is a lot of data needs to be processed. Eight nodes

are available for use with a Matlab program, although, I recommend not using every node as

a courtesy to other users. In order to run the stereomatching program on the cluster it must

be executed as a function (see db_Stereomatch_function19.m and related subfunctions as an

example). After the proper files have been uploaded to the computer cluster they can be run by

executing the file MyJob.m. This is done by opening the terminal and accessing the folder with all

of the sterematching function and launching the Matlab program on the cluster (note only one

user can do this at Wesleyan at one time). The command to launch Matlab is simply ’matlab’

once opened simply type the name of the MyJob script to be run. Be sure to type ’exit’ after it

has run to free up the one instance of Matlab that can be running at a time.

This concludes the data processing. It will certainly take some time to become familiar with the

process and develop one’s own best practices. From here I can point future researchers to a few

additional programs I have developed. The program ext_hd1-G:\Lab1backup\local_lib\dbblum\

DataInspection\VelocityField\db_smoothed_velocity_vector_field_creation_ML.pro and ext_hd1-G:

\Lab1backup\local_lib\dbblum\DataInspection\VelocityField\db_velocity_vector_field_subtraction_

ML.pro calculate the mean velocity at each voxel in 3D space and subtract that value from

each particle’s velocity so that what is left is the fluctuating velocity. Structure functions

were calculated using the program G:\Lab1backup\local_lib\dbblum\Collaboration\General\db_

stucture_function_conditional_fewrbins_uncond.pro which can create unconditioned or condi-

tioned structure functions. I have developed quite a few programs for analysis and display. Any

researcher is welcome to use and modify them, just be sure to copy the file into the user’s home

directory to retain a copy of the original.

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Thesis

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large scale velocity

center region

grid region

wesleyan physics department

karen blum

great deal

great joy

great undertaking