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MSU I. An Affirm“ Action/Equal Opportunity Inflation
cw:

A MULTIPLE-VIEW VISUAL STUDY
OF THE INNER/OUTER REGION INTERACTIONS
OF COHERENT MOTIONS IN A TURBULENT BOUNDARY LAYER

By

Kue Pan

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1992

ABSTRACT

A MULTIPLE-VIEW VISUAL STUDY
OF THE IN NER/OUTER REGION INTERACTIONS
OF COHERENT MOTIONS IN A TURBULENT BOUNDARY LAYER

By

Kue Pan

A time resolved multiple-view laser sheet and multiple location, multicolor dye
injection flow visualization technique has been used to observe the interactions between
sublayer motions and the coherent motions above the wall in a turbulent flat-plate flow at
R9~805. Pairs of low-speed streaks were observed to form and evolve underneath
passing typical eddies in the log region and lower portion of the outer region. The pairs
of streaks were also found to be associated with the formation of a pocket between the
streaks near their downstream end. Statistical characteristics of the observed coherent
features and their interactions are presented. The streak pair spacing was found to be on
average l+~75. Observations in laser sheets perpendicular to the flow direction
consistently showed counter-rotating vortex pairs. Because of the highly resolved
temporal record and the complementary spatial information, it was determined that the
streamwise extent of these pairs was similar to the length characteristic of the typical
eddies. Thus, they were not long axial vortices, or stretched hairpin legs. Instead they
were identified as typical eddies. This was confirmed both by the dye with which they

were colored, which indicated they were not connected to the wall, and by visual

identification in the side view laser sheet of the typical eddy pattern. Correlations of the
concurrent occurrence of the typical eddies and the sublayer features, pockets and streak
pairs, below them were examined. Results show that average sizes of these coherent
motions are similar, and that the dimensions of both sublayer features increase as the
spanwise diameter of the typical eddy increases. The sampling of low-speed streaks using
a classical counting method was also performed to confirm the universal mean streak
spacing. An argument utilizing the statistical results of the typical eddy spanwise
distribution spacing and the streak pair spacing, enables one to recover the result that the
overall Streaky structure has a spacing of l+~100 at this Reynolds number. An overall
picture of the inner/outer region interactions and the pinch-off phenomena of hairpin

vortices at low Reynolds numbers is also presented.

Copyright by
KUE PAN
1992

To

My Parents

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my wife, Linda, whose moral
support and patience have carried me through difficult time when I was discouraged;

without her this work could never have been completed.

I wish to thank Professor Robert E. Falco for his motivation, guidance, and
advice for this experimental investigation. I also appreciate Drs. J. Foss, D. Yen, and M.
Koochesfahani who reviewed the manuscript and provided many valuable comments.
Additionally, I gratefully acknowledge my colleagues and friends S.-C. Yoo, J. Klewicki,
S. Buchner, B. Slider, C. Olbrich, M. DeFilippis, M. Gieselmann, A. Folz, F. Cummings,
T. Stuecken, and C. Gendrich for assistance and friendship they have provided. Finally, I
must also thank the financial support of U.S. Air Force Office of Scientific Research
under contract 87-0047.

vi

TABLE OF CONTENTS

LIST OF TABLES ..............................................................................................

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

LIST OF SYMBOLS ...........................................................................................

CHAPTER

1 INTRODUCTION ..............................................................................

1.1 Random Turbulence to Organized Turbulence ..............................
1.2 Important Coherent Structural Motions Recognized in the Wall
Region of Turbulent Flows .................................................................
1.3 The Production of Turbulence in the Near-Wall Region ................

1.4 Proposed Models of Turbulence Production in the Wall Region

1.5 The Overall Production Module (0PM) ........................................
1.6 The Motivation .............................................................................

1.7 Subject of the Current Study .........................................................

2 EXPERIMENTAL TECHNIQUES ....................................................
2.1 The Water Tunnel Facility ............................................................
2.2 Mean Velocity Acquisition ...........................................................
2.3 Flow Visualization Setup ..............................................................
2.4 Data Recording .............................................................................
2.5 Data Reduction and Principles of Conditional Sampling ...............

vii

Page

xi
xiii

XX

12
14
15

17
17
20
23
26
28

Page

 

RESULTS .......................................................................................... 35
3.1 Mean Velocity Profile and Flow Parameters ................................. 35
3.2 Statistical Results ......................................................................... 36
WW ....................................... 36
We .......................................................................... 37
3.2.2.1 Magnitude ...................................................................... 38
3.2.2.2 Correlation of Diameter and Distance From the Wall 41
3.2.2.3 Comparison with Outer Layer Data ................................ 43
3.2.2.4 Orientation ..................................................................... 44
3.2.2.5 Period of Occurrence ..................................................... 47
3.2.2.6 Center-to-Center Spanwise Spacing ................................ 48
121mm ............................................................... 49
3.2.3.1 Spacing of Neighboring Long Streaks (Unconditionally
Sampled) .................................................................................... 50
3.2.3.2 Spacing of Conditionally Sampled Long Streak Pairs ..... 51
W ........................................................................................ 53
3.2.4.1 Length Scale ................................................................... 53
3.2.4.2 Period of Occurrence ..................................................... 54
3.2.5 Length Seale Cemelan'ens .......................................................... 55
3.2.5.1 Typical Eddy and Sublayer Streak Pair .......................... 55
3.2.5.2 Typical Eddy and Pocket ................................................ 58
3.2.5.3 Pocket and Sublayer Streak Pair .................................... 59
3.2.6 {Iihe Effects Qf {Typical Eddy Dismee E'mm the Wall ................ 61
3.2.6.1 On the Overall Length Scale Correlation ........................ 61
3.2.6.2 On the Ability of Initiating an Interaction ....................... 63
68
70
3.3 Templates of Flow Visualization ................................................... 71

viii

Wen: ................................ 71
Wagner] .................................... 74
3331] E. l-QEE [LE ”1' . 1E . [II I . l
Eddies ................................................................................................. 76
4 DISCUSSION .................................................................................... 78
4.1 Scaling of Sublayer Streak Spacing ............................................... 78
4.2 The Formation and Evolution of Typical Eddies ........................... 81

4.3 The Estimate for the Mean Vertical Velocity of Typical Eddies

Using the Size-Distance Correlation and the Diffusion Equation ......... 84
4.4 The Mechanisms Responsible for the Formation of the Sublayer
Features .............................................................................................. 86

WW ................................................................. 86
4.4.1.1 Stagnation Flows and the Orientation of Typical Eddies. 86
4.4.1.2 The Average Height Condition for Typical Eddies to

Create Pockets ........................................................................... 88
We .......................................... 90

4.5 Use of the Statistical Correlation of Events to Support the
Interaction Hypothesis ........................................................................ 91
4.6 Implications for Drag Reductions .................................................. 94
5 CONCLUSIONS ................................................................................ 98
TABLES .............................................................................................................. 102
FIGURES ............................................................................................................ 108
APPENDICES .................................................................................................... 162

 

W ............................................................................ 162

ix

Page

B I] D' . [I II I! .11” . [I . lEll
QMWW ................... 164

C 81' II . IS Ell IIl°
Measurements ............................................................................... 166
D W ...................... 169
LIST OF REFERENCES ..................................................................................... 171

TABLE

3.1

3.2

3.3

3.4

3.5
3.6

3.7
3.8

3.9

3.10

3.11

3.12

3.13

3.14

LIST OF TABLES

Page
Summary of principal characteristics of the turbulent boundary layer
flow ......................................................................................................... 102
Characteristics of C; distribution as shown in Figure 3.6 ...................... 39
Characteristics of Cx+ distribution as shown in Figure 3.7 ...................... 40
Characteristics of d+ distribution as shown in Figure 3.9 ......................... 41
Characteristics of d’“ distribution as shown in Figure 3.12 ....................... 42

A comparison of typical-eddy dimensions between outer-layer data and
the present data at R9z805. The data are non—dimensionalized by both

variables 0 and u1/V. The definition sketch of each scale is also
illustrated. Note the difference of CK between his measurement and ours. 103

Characteristics of e distribution as shown in Figure 3.13 ......................... 45
The statistical features of the typical eddies in each characteristic mode of

orientation ............................................................................................... 104
Characteristics of tug" distribution as shown in Figure 3.15 .................... 47
Characteristics of 5"” distribution as shown in Figure 3.16 ....................... 49
Characteristics of W’ distribution as shown in Figure 3.17 ....................... 51
Characteristics of 1+ distribution as shown in Figure 3.18 ........................ 51
Characteristics of w+ distribution as shown in Figure 3.19 ...................... 54
Characteristics of t?+ distribution as shown in Figure 3.21 ...................... 55

xi

3.15

3.16

3.17

4.1

A.1

A.2

Page

The principal properties of the probability density distribution for each

class of interactions based upon the distance (d+) of typical eddy from
the wall ................................................................................................... 105

The principal properties of the probability density distribution for each
class of interactions based upon the normalized distance (d*) of typical
eddy from the wall .................................................................................. 106

The principal properties of the probability density distributions for
various correlated interaction events based upon the period of

occurrences (At"'). The mode for each distribution is determined using
histogram ................................................................................................ 107

The corresponding occurrence of typical eddy in the log region and lower
portion of the outer region based upon the occurrence of pocket and/or

long streak pairs in the wall layer ............................................................ 93
Measurements using surface floats at y=4.375 inches. The length scale (s)

between two measuring stations is 4 feet ................................................. 162
Measurements using pressure probes for y>699=1.85 inches ................... 162

xii

FIGURE
1.1

1.2

2.1
2.2

2.3

2.4
2.5

2.6

LIST OF FIGURES

A conceptual model of the outer flow organization of the Overall
Production Module, showing the spatial phasing of typical eddies and
LSMs as seen by an observer moving with the convection velocity of
the upstream side of the LSMs. This will result in wall-region
interactions as typical eddies are being brought to the proximity of the
wall (figure adapted from Falco et al. 1989b) ........................................

A plane view illustrating four sequences in the evolution of the wall
region subset of the OPM. It begins with (a) the formation of a pair of
streaks, followed by (b) the development of streaks and formation of
pocket, followed by (c) the formation of vortices in the pocket and over
the streaks, followed by (d) the interaction between the stretching pocket
vortex and the secondary hairpin vortices (figure adapted from Falco et
al. 1989b) ..............................................................................................

Schematic of the low-speed water tunnel ...............................................

Schematic diagram illustrating the setup of pressure anemometry used
for the mean velocity measurement. [3:30 degrees .................................

Side and end views of a schematic of the experimental apparatus
employed in the visualization of turbulent boundary flows ....................

Schematic of the double-dye slit ............................................................

An isometric view of our experiment showing the designated area
incorporated into the multi-color dye system and observation
arrangement ...........................................................................................

Examples describe the principles of the conditional sampling of
coherent features. All typical eddies are sampled. Sublayer features are
sampled for conditions 1, 2, 3, and 11. Only a pocket is sampled in
condition 13 ...........................................................................................

xiii

Page

108

109

110

111

112

113

114

115

3.1

3.2
3.3
3.4

3.5

3.6

3.7

3.8

3.9

3.10

Dimensional mean velocity profile obtained from pressure
measurements. The velocity of the free surface was measured by floats.

The determination of skin friction coefficient utilizing Clauser plot .......
Logarithmic mean velocity profile .........................................................

The average length scale of the visualized cores of typical eddies, in the
intermittent region of turbulent boundary layers, as a function of R9

when non-dimensionalized by the boundary layer thickness 599. The

curve fit is Cy/899=30.5R9'0-758 (Falco 1991). The definition sketch of
Cy is also illustrated ..............................................................................

The average dimensions of typical eddies in the outer intermittent region
as a function of Reynolds number. The data were non-dimensionalized

by both variables, 0 and utlv, in the respective figures. The definition
sketches of Cx and Cy are also illustrated ..............................................

The probability density distribution of the spanwise diameter of typical
eddies. The dashed lines are the normal and log-normal distributions.

The definition sketch shows two probable appearances of typical eddies
in the cross—stream view ........................................................................

The probability density distribution of the streamwise length scale of
typical eddies. Normal and log-normal distributions are also shown in
dashed lines ...........................................................................................

2—D probability contour map and 3-D probability density distribution of
random variables, Cz” and C,{" .............................................................

The probability density distribution for the distance of typical eddies
from the wall. The dashed lines are the normal, log-normal, and
Rayleigh density functions. Definition sketch is also shown. Sample size
is 417 .....................................................................................................

The dependence of the spanwise dimension of typical eddy on its
distance from the wall. Histogram of d""s is also shown. The dot—dashed
lines represent the upper and lower bounds of standard deviation of Of.

The various symbols used stand for the data obtained by binning
processes with various bin widths ..........................................................

xiv

Page

116
117

118

119

120

121

122

123

124

125

3.11

3.12

3.13

3.14

3.15

3.16

3.17

3.18

3.19

3.20

3.21

A possible linear relationship between the spanwise size of typical eddy
and its distance from the wall. The approximation is calculated using the
least square method ...............................................................................

The probability density distribution of the normalized distance of typical
eddy from the wall. Normal, log—normal, as well as Rayleigh
distributions are also shown in dashed lines ...........................................

The probability density distribution of typical eddy orientation in the
cross-stream plane. The curve fit is a normal distribution. Definition
sketch is also illustrated .........................................................................

Definition sketch of typical eddy characteristic modes based upon its
orientation with respect to the wall in the x-y plane. The sequence of the
cross-stream views for each typical eddy mode is also illustrated ...........

The probability density distribution of the period between typical eddy

occurrences for y+<300. Log-normal and Rayleigh density functions are
also shown to fit the distribution ............................................................

The probability density distribution of the center-to—center spacings in
the spanwise direction between neighboring typical eddies. Three
density functions are also shown to approximate the data set .................

The probability density distribution of the spanwise spacings between
neighboring low-speed streaks. Normal and log-normal density
functions are also plotted to fit the data. Sample size is 500 ...................

The probability density distribution of the sublayer streak-pair spacings
in the spanwise direction. Normal and log-normal density functions are
also presented to fit the data set. Definition sketch illustrates that the

measurements were the average distances between the most dense lines.

The probability density distribution of pocket dimensions in the
spanwise direction. The dashed lines are the normal and log-normal
distributions. Definition sketch shows that the measurements were taken
between the most dense lines .................................................................

The comparison of the mean spanwise dimension of pockets between
our data at R9z805 and results of many others in a log-log scale ...........

The probability density distribution of the period between pocket
creations. Log-normal and Rayleigh density functions are also drawn to
represent the distribution .......................................................................

XV

Page

126

127

128

129

130

131

132

133

134

135

136

3.22

3.23

3.24

3.25

3.26

3.27

3.28

3.29

3.30

The comparison of the period of pocket formations between our
measurement at R9==805 and results of many others in a log-log scale

The correlation of the typical eddy spanwise diameter and the streak pair
spacing. Histograms of both scales are also shown. Simulation data are

scaled as l" versus D+ (the diameter of ring vortex as it was being
generated) ..............................................................................................

2-D probability contour map and 3-D probability density distribution of
random variables, Cz+ and 1+ ................................................................

The dependence of the streak-pair spacing on the typical-eddy spanwise
diameter. Histograms of both scales are also shown. The straight line is
the first order least square fit. Various symbols represent that the data
are obtained by binning processes based upon various bin widths.
Sample size is 273 .................................................................................

The correlation of the spanwise dimensions of typical eddies and
pockets. Histograms of both scales are also shown. The linear fit is
obtained by first order least square approximation. Various symbols
stand for that the data are binned with various bin widths. Sample size is
173 ........................................................................................................

2—D probability contour map and 3-D probability density distribution of
random variables, Cz"’ and w"' ..............................................................

The correlation of spanwise dimensions of pockets and sublayer streak
pairs under the condition that a typical eddy is observed to propagate
over the wall. Histograms of both dimensions are also shown. The solid
straight line is obtained using their individual linear relationships with
typical eddy. The dashed straight lines are the first order least square fits
using either scale as the independent variable (the one with greater slope

uses w+ as the independent variable) .....................................................

2-D probability contour map and 3-D probability density distribution of
random variables, l"' and w+ ..................................................................
The correlation of spanwise dimensions of pockets and sublayer streak
pairs in the absence of typical eddy convecting over the wall. The

straight line is same to the solid line shown in the previous figure. Mean
value of the data is marked as a plus (+). Sample size is 18 ...................

xvi

Page

137

138

139

140

141

142

143

144

145

3.31

3.32

3.33

3.34

3.35

3.36

3.37

3.38

3.39

The overall length-scale correlation between typical eddies, pockets, and
sublayer streak pairs. The dependence of pocket size on the distance or
normalized distance of typical eddy from the wall is shown by the data
points. The dispersion of the data is indicated by the upper/lower bounds
of standard deviation as shown in dashed lines. The histogram of pocket
size is also given. The average spanwise dimensions of the
corresponding typical eddies and streak pairs for each data point are
listed near the top of the figure (standard deviation in brackets). The
sample size of data points are 16, 91, 34, and 8 (from left to right). Total
sample size is 149 ..................................................................................

The correlation of pocket size (w+) and the normalized distance (d*) for
typical eddy. The histogram of the pocket size is presented as well.
Sample size is 149. The mean value for the events in which only pocket
was formed is also shown as a star (sample size is 24) ...........................

The probability of initiating an interaction with the sublayer for typical
eddy, based upon its distance or normalized distance from the wall. The

probability functions based upon d* increase monotonically as d*
decreases. However, the probabilities based on d+ are not ascending

functions for d+ less than about 70 wall layer units. The intersections of
the functions and the vertical axis are approximately 218 and 155 wall

layer units for (1+ and d* respectively. Total sample size is 417 .............

The probability density functions for various classes of interactions
based upon the distance of typical eddy from the wall ...........................

The probability density functions for various classes of interactions
based upon the normalized distance of typical eddy from the wall .........

The probability density representations of the period between
occurrences for various classes of interaction events ..............................

Formation of a streak pair and the downstream development of a pocket.
Flow is from left to right (similar picture was also present in Falco,
Klewicki, and Pan 1989) ........................................................................

Pocket evolution resulting in the engulfrnent of secondary hairpins that
have developed on the streaks (similar picture was also present in Falco,
Klewicki, and Pan 1989) ........................................................................

A sequence of simultaneous plan and cross-stream views illustrating a
typical eddy/wall layer interaction that results in the formation of a pair

xvii

Page

146

147

148
149
150

151

152

153 ‘

3.40

3.41

3.42

3.43

4.1

4.2

Page

of long streaks and a pocket near its downstream end. Flow is from right

to left for the plan view and into the paper for the cross-stream view.

The evolution of the sublayer features is indicated by the black arrows,
whereas the appearance of the typical eddy is indicated by the white

arrows ................................................................................................... 154

A sequence of simultaneous plan and side views illustrating the

formation of a pocket by a typical eddy, as well as the subsequent

interaction between the typical eddy and the lifted sublayer fluid near

the downstream end of the pocket. Flow is from left to right. The

evolution of the pocket is indicated by the black arrows, whereas the
appearance of the typical eddy is indicated by the white arrows ............. 155

A sequential side-view pictures showing the formation of a new typical

eddy from the pinch-off of a lifted hairpin (as indicated by the white

arrows). The hairpin is called "primary hairpin" (see Chapter 1) which

initially formed near the downstream end of the pocket in Figure 3.40.

Flow is from left to right ....................................................................... 156

An example illustrating the evolution of a hairpin into a new typical

eddy (as indicated by the white arrows). Flow is from right to left. Note

that, in frames 5 to 9, a upstream typical eddy can also be seen to move

into the streamwise laser sheet ............................................................... 157

Another example showing the formation of a new typical eddy emerging
from the wall layer fluid through a pinch-off of lifted hairpin (as
indicated by the white arrows). Flow is from right to left ....................... 158

An illustration of plan and cross—stream views showing the spanwise

phase relationship of typical eddies and the corresponding sublayer

streak pairs. Mean value (in wall layer units) is used to represent each
characteristic length ............................................................................... 159

A hypothesis of the formation and evolution of Taylor micro-scale

eddies in the log and the lower portion of the outer regions, suggesting

that a typical eddy is created out of a lifted hairpin vortex through

pinch-off mechanism. The subsequent evolution of the typical eddy is
governed by the mean shear layer, instantaneous flow field of LSMs,

and diffusion process. The isometric view shows the mechanism

responsible for pocket formation is a stagnation point (for an observer

moving with the speed of pocket center) created by a typical eddy in the
vicinity of the wall (in side view, the streamwise distance between the
stagnation point and typical eddy is exaggerated) .................................. 160

xviii

4.3

Page

The estimate vertical convection velocity of typical eddy as a function

of distance from the wall. The rms of y-component velocity fluctuation

at a comparable flow is also provided for comparison (see Klewicki

1989). The dashed line is an approximation of rms data in its maximum

region .................................................................................................... 161

xix

O
N

any

’1']

mo:

'62 5"

"U
m

LIST OF SYMBOLS

Quantity normalized by wall layer variables, v/ut or v/uzc2

Mean quantity

Streamwise length scale of typical eddy

Diameter of vortex core of typical eddy in y direction

Diameter (maximum dimension) of typical eddy in the y-z plane
Skin friction coefficient, 2 V(du/ 07y)...” / U00:

Diameter of ring vortex in the physical simulations

Distance of typical eddy center from the wall

Normalized distance, d+ - 0.5Cz+

Flatness (fourth central moment, i.e Kurtosis) coefficient, Wk"

Gravity, 32.2 ft/sec2
Boundary layer shape factor, 8d/6

Pressure differential normal to the horizontal, Pt - P8 = t Sin [3

Number of samples

Probability density

Static pressure

Total pressure

Reynolds number based on the streamwise distance from the tripping, Umx/v
Reynolds number based on the distance from the wall, Uooy/v

Reynolds number based on momentum thickness, Uooelv
Radius of typical eddy, equivalent to Cz/2

XX

X,y,Z

WPNN-e-CD

<

an

Initial radius of typical eddy as it is formed

Skewness (third central moment) coefficient, (33/03

Streamwise spacing between two measuring stations in surface float
measurements, which is 4 feet

Temperature

Time

Period of occurrences of pockets

Period of occurrences of typical eddies

Period of occurrences of correlated interaction events

Free stream (surface) velocity

Velocity components in x, y, and 2 directions respectively

 

Friction velocity, \I Wall / 8),)mll

Vertical convection velocity of typical eddy

Spanwise dimension of pocket-like depression

Spatial coordinates

Angle between pressure tubes and the horizontal, which is 30 degrees
Momentum thickness

Angle between positive y axis and typical eddy axis in the y-z plane
Streak-pair spacing

Pressure differential along the direction of pressure tubes

Streak spacing

Center-to—center spacing between two neighboring typical eddies in spanwise
direction

Coefficient of variation, 0/?)

Kinematic viscosity

Density

Standard deviation

xxi

599 Boundary layer thickness based upon 99% of free stream velocity

5a Displacement thickness

xxii

CHAPTER 1
INTRODUCTION

1.1 Random Turbulence to Organized Turbulence

For many years a concept of turbulence had been accepted as random motions
with no organized structural feature and therefore led to the exploration of its behavior
predominantly by means of statistical approaches. During this period, most statistical
properties of turbulence were obtained by using hot wires or other single-point
alternatives, which were not capable of revealing the existence of organized flow motions
owing to the high degree of variation in size, orientation, and occurrence of these three-
dimensional motions. Over the past three decades, the aspects of turbulence have been
reconsidered since experimental investigators such as Kline, Reynolds, Schraub, and
Runstadler (1967), and Kovasznay, Kibens, and Blackwelder (1970) documented the
existence of wall region Streaky structures and large scale coherent motions (bulges) in
turbulent boundary layers respectively. Although these investigations may have not been
necessarily the first to discover the respective features, they are surely the ones who
ignited the numerous subsequent studies concerning the structure of turbulent boundary
layers. Since then, the objective of many research efforts have been focused upon finding
the recognizable outer layer motions, wall region flow structures, as well as the
correlations between outer layer and wall layer structural features. As a result, both large
scale motions and inner region have been extensively studied. Various types of small to

intermediate scale coherent structural elements have been observed or deduced.

2

However, the three-dimensional, unsteady, strongly varying convection velocity has

made understanding the inner/outer region interaction a most difficult problem.

It is necessary to mention that the intent of this chapter is not to present a survey
of contributions to the understanding of turbulent boundary layer structure. Nevertheless,
the pieces of knowledge gained from an extensive review of the past literature, which
provide an excellent background as well as motivation for the current study, will be
briefly described, with particular emphasis on the wall region events and the motions
associated with their occurrence. The phrase "wall region" used here is referred to as
being the zone adjacent to the wall comprising viscous sublayer and the buffer region
(i.e. y+<15). The "inner region" (y+<70) is roughly defined as the combination of the
wall region and lower portion of the logarithmic region, whereas the "outer region"
(70$y+<400) is the region beyond the inner region and bounded by the mean edge of the
boundary layer thickness.

It is also important to note here that turbulent boundary layers at low Reynolds
numbers can be computed by direct numerical simulation (i.e. DNS). Kim, Moin, and
Moser (1987) simulated a turbulent channel flow using unsteady Navier-Stokes equations
and compared the computed results with experimental results at comparably low
Reynolds numbers. It was found that the general characteristics of the computed
turbulence statistics were in good agreement with the experimental results. However the
detailed comparison revealed consistent discrepancies, particularly in the wall region.
Computations by Moser and Moin (1984) and Spalart (1985) also resulted in similar
problems. Zang (1991) pointed out that although current DNS of turbulent flows have
been proven to be able to provide reliable long-time statistics, they may not adequately
capture the instantaneous shapes and dynamics of turbulent structures even at low
Reynolds number flows (also see Robinson, Kline, and Spalart 1989). Zang, Krist, and

Hussaini (1989) presented evidence suggesting that reliable computational simulation of

3

the near-wall turbulent structures required a considerably higher resolution level than
those presently used for the high-fidelity DNS. A consequence of the present state of
uncertainty is that experimental investigations are needed to provide both new structural

information as well as a data base to check the accuracy of computed turbulent structures.

1.2 Important Coherent Structural Motions Recognized in the Wall Region of

Turbulent Flows

Kline et a1. (1967) observed the redistribution of a series of arrays of hydrogen-
bubble markers into elongated streamwise oriented streaks, moving roughly one half of
the local mean velocity, in the wall region of turbulent boundary flows. They proposed
that these low-speed streaks were the signature feature of wall-bounded turbulent flows
and that their spanwise characteristic spacing constituted an universal feature in the inner
layer (also see Kim, Kline, and Reynolds 1971). Later, extended studies by Smith and
Metzler (1983) characterized the low-speed streaks for a Reynolds—number range of
740SR9<5830. They concluded that the probability density function of the spanwise
spacing was approximately a log-normal distribution (i.e. the logarithm of the streak
spacing is normally distributed) with mean value of about 100 wall layer units and
invariant with Reynolds number over the range studied. It was also noted that the
streamwise extent of the streaks varied over a vast range, from 50 to greater than 2,000

wall layer units.

In a visual investigation of smoke contaminant turbulent boundary layers, Falco
(1974) found an identifiable family of motions which occurred over both inner and outer
regions. This family of motions was recognizable by their vortex-ring-like shapes and
similar evolutions. He called them the "typical eddies" (also referenced as "compact
vortical flow structures" or "Falco eddies" by some others). The diameter of the typical

eddies was found to be of the order of Taylor micro—scale, typically between 50 and 150

4

viscous units. They have also been shown to contribute significantly to the Reynolds
stress in the outer part of the turbulent boundary layer, and exhibit a considerable

Reynolds number dependence (see Falco 1977 and 1983).

Experiments conducted by Falco (1978a, 1979) using combined simultaneous
visual/hot-wire anemometry revealed that, other than the well-known low-speed Streaky
structure, a much more energetic flow module also existed in the viscous sublayer of wall
bounded turbulent flows. The flow module was created as one result of the frequent
rearrangement of existing wall region vorticity, by outer region interactions. It was
characterized by the initial movement of marker away from a local region, leaving a
scoured area of low marker concentration. Because of its nature, appearance, and
intermediate scale, he called the flow pattern a "pocket". More detailed observation
showed that the pocket flow module was characterized by having spanwise widths
typically between 50 and 150 wall layer units and lengths about 30% greater than its
lateral dimension. Analogous to the low-speed streaks, the presence of pocket flow
motions is arguably a sufficient condition to distinguish a given boundary-layer flow as
laminar or turbulent transitional, since pockets were observed to form in the transitional
flows, in the upstream edge of the arrowhead-shaped turbulent spots as well as under
spots (see for example, figure 111 of Van Dyke 1982), and in all turbulent regions
downstream. Some investigators felt that the pockets were merely visual artifacts with no
dynamical importance. Falco (1980) showed that it is necessary to have both the correct
visual technique and the quantifiable results to investigate pockets. The numerical
calculation of a turbulent channel flow using DNS by Kim et a1. (1987) revealed that
different flow patterns were emphasized by different visualization techniques. Their
results illustrated that the formation of low-speed streaks was well presented using
hydrogen-bubble time lines parallel to the wall, whereas the pockets were clearly shown

by the illumination of smoke-filled boundary layer. The simultaneous visual/hot-wire

5

results of Falco (1980) and Lovett (1982) documented the kinematic signature of the
pocket flow module.

Using inclined laser planes to study flow characteristics of turbulent boundary
layers at modest Reynolds numbers, Head and Bandyopadhyay (1981) proposed that
turbulent boundary layer was filled almost exclusively with vortex loops or hairpin—like
vortices of a variety of scales originating in the wall region and extending outwards at a
characteristic angle of about 45 degrees to the flow direction. Their assertion led to the
controversy concerning the coherent shape of the important vortical motions existing
across the outer region of the boundary layer. Are the typical eddies best characterized as
ring-like or hairpin-like motions? Using flow visualization recorded in two mutually
orthogonal laser sheets by Falco (1979) indicated that the coherent feature was indeed
ring-like vortical motion (also see Falco and Signor 1982). Stanislas and Hoyez (1990)
also visualized mushroom-shaped coherent motions existing close to the wall in oil-
smoke filled turbulent boundary layers at modest Reynolds numbers. They pointed out
that these motions seemed to be produced near the wall and needed to be taken into
account in the wall-region dynamics. Falco (1991) noted that the disagreement is
primarily in the outer portion of boundary layer. He postulated that hairpin vortices are
dynamically important only in the near wall region, and are the dominant flow feature
solely in transitional flows and downstream of tripping devices around which the flow is
separated This physical picture was given further support by the studies of Lu and Smith
(1988), who indicated that hairpin heads were observed most probably in the range of
28<y+<32. In addition, computational data base of Jimenez, Moin, Moser, and Keefe
(1988) suggested that the tip portion of the elevated sheet-like shear layers tended to

level-off at y+~35.

The origin of wall region hairpin vortices has also been investigated by a number

of researchers. Falco (1982, 1983) observed the sublayer vorticity re-organizing into

6

hairpins in the downstream end of pocket flows. Acarlar and Smith (1984, l987a,b)
produced evidence showing that the shear layer instability of long low-speed streaks was
responsible for the formation of hairpin vortices. Experiments involving the interaction
of a laminar ring vortex with a laminar Stokes' layer by Chu (1987, also see Chu and
Falco 1988) clearly showed that both of the mechanisms described above could create
hairpins near the wall. In addition, Falco, Klewicki, and Pan (1989a) also witnessed the
evolution of hairpin flows with their legs straddling the long streaks in a low Reynolds

number turbulent boundary layer.

The existence of other important structural features such as longitudinal
streamwise vortices has also been addressed (see, for example, Bakewell and Lumley
1967, Lee, Eckelman, and Hanratty 1974). Many proposed that pairs of long counter-
rotating axial vortices existing in the wall region provided a natural cause for the
formation of long low-speed streaks. The high-speed streaks, which were occasionally
observed to form for short extent very close to the hydrogen bubble wires, would be the
result of high momentum fluid being induced towards the wall between these vortex
pairs. However, this conjecture has not been proven since the streamwise vortices in the
wall region found were all of short extent (see, for example, Praturi and Brodkey 1978,
Falco 1979, Smith 1982, and Nishino, Kasagi, and Hirata 1987). In contrast to the
hypothesis regarding the existence of counter-rotating vortex pairs, there are also
investigators reporting that solitary vortices are more prominent than vortex pairs in the

vicinity of the wall (see, for example, Moser and Moin 1984, and Nishino et al. 1987).

The kinematic signatures of the various structural features observed have been
studied by means of visualizations, quantitative measurements with conditional sampling
process, or computational simulations by a number of investigators. The dynamical
significance of these motions have been put forth in many differing inferences. Although

many results have suggested that these characteristic features were, more or less,

7

associated with the production of Reynolds stress, an overall relationship existing
between these motions, which leads to a chain of events that can be labelled the

turbulence production process, has yet been fully established.

1.3 The Production of Turbulence in the Near-Wall Region

Since Prandtl proposed the boundary layer theory in the early years of the
twentieth century, attention has been drawn to the thin layer of fluid adjacent to the wall
which has been found to be strongly influenced via viscous effects. In turbulent boundary
layers the nature of the wall region is not completely understood, but the importance of
this thin layer regarding the generation, transport, and maintenance of turbulence has
been sufficiently documented. Laufer (1954) presented turbulence data from a fully
developed flow in a smooth pipe, indicating that the peak production of turbulence
occurs just outside the viscous sublayer, in the buffer region. If we integrate the energy
production outwards from the wall, it can be shown that nearly 80% (qualitative
estimate) of the net production takes place within the first 50 wall layer units, well inside
the lower portion of the log region. Therefore, flow motions in the inner and log region

play an extremely important role in the production of turbulence near a wall.

The question could be raised : What is the importance of understanding the
turbulence production process? The answer is simple, since energy consumption has been
- and continues to be - a crucial factor in the world economy, the reduction of drag and
the associated savings in energy consumption in the innumerable devices of our modern
technologically based world has been the subject of intensive and numerous studies for
many engineers. Much work has been done (see, for example, Wilkinson and
Balasubramanian 1985, Falco 1989, Moin, Kim, and Choi 1989) in seeking the
mechanisms which are responsible for the bursts of turbulence production in boundary

layers, in the hope that substantial drag reduction could be achieved through

8

modifications of the processes. The philosophy has been that learning the underlying
process provides the best way to understand how to interact with the process so as to
affect a control. Thus, understanding the production process is essential to drag reduction

studies.

1.4 Proposed Models of Turbulence Production in the Wall Region

The search for mechanisms responsible for the turbulence production process in
the near wall region has resulted in a number of hypotheses. Attention has been focused
upon the wall region breakup described by Kline et a1. (1967), who observed that the
sublayer Streaky motions lifted up, oscillated, and eventually broke down. A similar
observation was reported by Corino and Brodkey (1969), who used suspended solid
particles of colloidal size in a pipe flow to create physical pictures very near the wall.
This sequence was later collectively called the 'bursting' phenomena (Kim et al. 1971).
They also reported that all the net turbulence production within the first 100 wall layer
units of the flat-plate flow occurred during times that involved lifting of the low-speed
streaks and ejection processes. Many investigators, who believe the Streaky structure in
the near wall region is dynamically significant, have proposed different models to
describe the event as though its formation and the subsequent instability were the
essential cause of the turbulence production near the wall. However, there are also many
others that feel the streaks are passive and thus not a factor of the near-wall-region
dynamics. Kline (1988) clarified the view by pointing out that the low-speed streaks,
when lying on the wall, appeared to be passive, but they were certainly involved in much
of the turbulence production in the layer above the wall region as they were lifted away
from the wall interacting with the outer part of the flow. The important question of why

they lifted up was not answered.

9

Early attempts were made by Lee et a1. (1974), Townsend (1976), Blackwelder
and Eckelmann (1979), and many others to explicitly model the longer time scale events,
which were pairs of streamwise oriented vortices that were attached to the wall. The
models were primarily deduced from statistical correlation measurements. Their
conjecture suggested that the ensuing rapid breakdown of the wall region flow was a
result of the evolution of these motions which instantaneously inflected the mean
velocity profile. Some investigators attributed the origin of these streamwise vortices to a
Taylor-Gdrtler type instability in which the centrifugal force was supplied by the motion
of large scale structure in the outer boundary layer. The suggestions that conditions for
onset of the turbulent boundary instability remains questionable because they are based
upon a homogeneous steady-state stability analysis of Taylor and Gortler, which may not

be applicable to the unsteady flows in the wall region.

A model proposed by Offen and Kline (1974, 1975) suggested that the rapid
breakdown of the sublayer flow was a result of local adverse pressure gradients that
occurred near the wall due to the presence of a transverse vortex with rotation of the
same sign as that of the mean vorticity in the boundary layer. Laufer (1975) and Oldaker
and Tiederman (1977) also proposed models to interpret the events. Oldaker and
Tiederman suggested that pairs of sublayer streaks were created as a result of the
response to a sequence of local high speed outer region eddies convecting over the wall.
Laufer attributed to a relatively larger scale sweep containing regions of small scale

vorticity that had the size of streak formation events.

Smith and Metzler (1983) suggested that the low-speed streak merging process
often observed in the region of buffer layer (10<y+<20) could be closely related to the
turbulence production. They also postulated that the small stretched and lifted hairpin
vortices near the wall were responsible for the merging behavior and the streak

persistence. Head and Bandyopadhyay (1981) suggested that streak was formed between

10

the legs of a lifted hairpin. However, the ability of the proposed mechanism to provide
the persisting time for a streak of length in the order of x+=1,000 is suspicious. This
problem was later resolved by the modification of Smith and his co—workers (1984,
1987b), who described that the formation of long sublayer streaks was due to a series of
lifted hairpins forming in roughly a longitudinal chain, building up the necessary
persistence time for long streaks to develop. Falco (1979) observed that the long streaks
could be formed as a result of the joining of a staggered array of the streaks that bound
pockets, which provided an explanation of the persistence time of long streaks. Although
there is some rationale for each physical assertion mentioned above, the apparent
mechanism for long streak formation and its subsequent bursting is not completely

understood because it has not yet been confirmed by an objective statistical analysis.

Other models such as vortex induced ejection of wall region fluid (see, for
example, Willrnarth and Tu 1967, Nychas, Hershey, and Brodkey 1973), as well as high-
speed inflows and continuity (see, for example, Eckelman, Nychas, Brodkey, and
Wallace 1977, Praturi and Brodkey 1978) have also been put forth. Despite many
detailed differences in these proposed models, it was overwhelmingly clear that they
support the hypothesis that the sublayer instability occurred as a result of the interaction
between the outer and the inner region motions. These differed from older models in

which the inflows relied solely upon the law of continuity.

Research focused on the formation and evolution of pocket flow modules has
been led by Falco and co-workers (1978-1980). He presented experimental evidence
showing that the most intense ejection of sublayer fluid occurred at the downstream end
of the pocket in which a hairpin-like vortex filament could be readily observed. He also
classified the evolution of pockets, with average lifetime in the order of 30 viscous units,
into five stages of development. The physical processes, that were phased to the

evolution stages were closely associated with the sweeping event and bursting sequence.

11

In addition, be advanced a hypothesis for the cause of the pocket creations. He proposed
that they were the footprints of vortical sweeps whose vorticity was organized in ring
configurations as suggested by the scale of pockets and the existence of counter-rotating
vortices. This constructed a model which coupled both the inner and the outer region
known coherent features through the interactions of typical eddies and the existing wall
region vorticity distribution. However, the relative importance regarding the contribution
to the Reynolds stress in the near wall region, of pocket related bursting seen by Falco
and the long sneaky structure related bursting observed by Kline and his co-workers

apparently exists.

In a visualization experiment, Falco (1983) found that the wall region vortex
filaments could evolve into typical eddies. Therefore, he hypothesized a cyclic turbulence
production process in which ejected fluid evolved into typical eddies and many of them
returned to the proximity of the wall as sweeps to create wall region disturbances as well
as new ejections. Nevertheless, he pointed out that most of the ejected fluid from the
downstream end of pockets was not directly observed to form new typical eddies.
Although a computational work using the Biot-Savart law by Moin, Leonard, and Kim
(1985) also clearly showed that the tip region of an elongated hairpin vortex evolved into
a vortex ring in turbulent shear flows, additional evidence is needed to support the model

which suggested a closed cycle.

In order to examine the typical eddy/wall layer interaction hypothesis, physical
simulations of the interaction between a laminar ring vortex and Stokes layer were
performed to model the process (Chu 1987, Chu and Falco 1988). The simulations
enabled an uninterrupted environment in which important structural features associated
with the production process could be distinguished. The adjustable parameters in the
physical simulations were optimized to match the observations in fully turbulent wall

flows. As a result, the simulations successfully exhibited all of the known structural

12

features and their evolutions in turbulent boundary layers. With a Galilean transformation
on the velocity field, the simulation clearly indicated that a pair of long streaks was
created, normally accompanied by a pocket formed near its downstream end, as a ring-

like vortex impinged the wall layer with a shallow incident angle.

Motivated by these findings, subsequent experiments by Falco, Klewicki, and Pan
(1989a) presented data of long time averaged two-point spanwise vorticity correlations,
with a fixed probe at the edge of the viscous sublayer and an upper probe at various
positions between 15 and 125 Ay+ above the fixed probe, in order to statistically
characterize the spatial interactions of inner/outer layer distinct vortical motions (also see
Klewicki 1989). Results showed that the negative vorticity fluctuation correlation was
consistent with the presence of typical eddy at the upper probe (which would result in
positive total vorticity during its passage), and a strong negative vorticity fluctuation at
the fixed probe (which could be either the pocket vortex or the hairpin vortex at its
downstream end). Note that the mean shear had vorticity of negative sign. The joint
probability density function also showed the counter-rotating contributions to the
correlation occurred only when total vorticity measured at the upper probe was positive.
This statistical result reinforced the hypothesized relationships of the known inner and
outer region coherent motions. An Overall Production Module (OPM), which integrated
the details of the production model developed through years (Falco 1977-1983), was put
forth accordingly (Falco et al. 1989a, Falco 1991).

1.5 The Overall Production Module (OPM)

'Ihe OPM is composed of two major subsets. These two subsets complimentarily
account for all the dynamically significant motions found in both the inner and outer

regions of turbulent boundary layers. They are briefly described as follows.

l3

Log/Outer Region Subset : motions initiating the near wall region response

An essential feature of the OPM dynamics is the initiation of near wall region
response through the interaction with highly vortical microscale typical eddies
propagating over the wall. Figure 1.1 shows a sketch illustrating the spatial relationships
between typical eddies and the large scale motions (LSMs). 'Ihis phasing results in the
typical eddies being brought to the close proximity of the wall. The intrusion of these
outer region perturbations is proposed to be responsible for the onset of the near wall
region subset of the OPM. An important feature behind this inner/outer region interaction
is the LSMs, which provide or induce the instantaneous wallward momentum carrying

the typical eddies towards the wall to trigger the wall region responses.

Wall Region Subset .° evolution and transport in the near wall region

Being excited through the outer region perturbation, the existing near wall region
vorticity undergoes rearrangements as well as specific spatial/temporal interactions,
which comprise the wall region subset of the OPM. Figure 1.2 illustrates a conceptual
model showing four sequential stages of the wall-region evolution in the plan view. The
evolution proceeds as follows. First, a pair of long sublayer streaks is formed. Then, a
pocket is created at the downstream end of the streak pair, in between the streaks. The
deformed upstream sublayer vortices, which actually form the upstream boundary of the
pocket observed in visualization, are called the pocket vortices. In the downstream side
of the pocket, the elevated vortex loop is called the primary hairpin vortex. In the
evolution of the streak pair, a certain time period is necessary for the growth of an
instability that often results in the formation of hairpin vortices over the streaks. 'Ihese

vortices are called the secondary hairpin vortices.

During the evolution of the various coherent features, each feature contributes a

portion to the total turbulence created by the production module. However, the proposed

14

OPM encompasses a complete spatial/temporal interaction, which is seldom seen in
highly perturbed environment of turbulent flows. A significant number of the observed
turbulent wall-flow evolutions appeared to consist of only fractions of the wall region

subset.

1.6 The Motivation

Based upon our survey of investigations of the wall-region dynamics, results
suggest that two aspects are most important (There are, of course, some grey cases, but
on the basis, the distinction is clear). The first one emphasizes the formation and the
subsequent bursting of the wall-region Streaky structures that were first observed by
Kline and his co-workers as the mechanism of turbulence production, whereas the second
attributes turbulence production mainly to the typical eddy/wall layer interactions leading
to the pocket evolutions, interactions with the streaks, and breakdown (Falco and his co-
workers). It is obvious that the former has so far drawn much of the attention of
researchers. In spite of the fact that there are about as many differing inferences as the
researchers involved in the investigations of this topic, the input of views are fruitful and
often unassailably rational. It seems that more research focused on the evolution of the
individual coherent motions is not what is primarily needed at this moment. Research
focused on interactions that can help explain the initiation of the events as well as their
evolution and phasing is of primary importance. The Overall Production Module
proposed by Falco et al. (1989a) and Falco (1991) rationally links essentially all
structural features found in the turbulent boundary layer, including typical eddies, long
sublayer streaks, hairpins, pockets, LSMs, and streamwise vortices. It provides a frame
work and overall interaction model. On the surface, it appears to contradict a number of
view points held by other investigators. Thus it appears worth doing further research to

statistically investigate the precepts of the model.

15

1.7 Subject of the Current Study

On the basis of mounting experimental evidence, it is increasingly apparent that
the interactions between Taylor micro-scale typical eddies and the existing sublayer
vorticity distribution is a crucial process concerning the production of turbulence in wall
bounded shear flows. On the other hand, when it comes to understanding how various
structural elements relate to and/or interact with each other, and what the parameters are
governing this important event, our knowledge is still very limited. Hence, searching for
reliable data characterizing the important spatial/temporal inter-relationships associated
with the occurrence, formation and alignment of these coherent features, that participate

in the near-wall production process, is very desirable.

The subject of my dissertation is to investigate a turbulent boundary layer flow,
with particular emphasis on the fluid motions near a smooth wall, and to gain insights
into the physical picture of the coherent structure interactions that are embodied in the
OPM. In so doing, a conditional sampling procedure has been employed with the aim
that it will most directly extract statistically significant features of the correlated coherent
motions and their interactions. The intent is to provide statistically reliable quantitative
data. In addition, we have coupled this with qualitative long time boundary layer
observations to get a better understanding of the interaction process between the
inner/outer region distinct vortical motions. Special emphasis has been placed on
uncovering the underlying mechanism that is responsible for the formation of long low-
speed streaks and pockets. The visual data base for the inner/outer region interactions has
been constructed in both two and three dimensions. This enables us to observe the
coherent motions in all complementary views. Furthermore, the description of the
interactions in terms of the probability for the simultaneous occurrence of the correlated

structural elements will be made. The results would serve as a measure of the importance

16

of the interaction in the production of turbulence near walls. We will also present visual

evidence that supports the conceptual description of the OPM.

CHAPTER 2
EXPERIMENTAL TECHNIQUES

2.1 The Water Tunnel Facility

Experiments were performed in a recirculating water tunnel in the Turbulence
Structure Laboratory at Michigan State University. The tunnel, which was made of
plexiglass, was 15.2 cm high by 91.4 cm wide by 6.4 meter long. It was part of the
closed flow system. Figure 2.1 shows the top view and side view of the flow facility
which basically consisted of the tunnel, a driving motor with shaft, two reservoirs, a
honeycomb screen, a contraction, and a 15.2 cm radius back flow pipe which was
installed underneath the tunnel. Fresh water could be filled in the flow system from the
top of the main reservoir as indicated in the figure. The maximum capacity of the flow
facility was approximately 4500 liters. The main reservoir contained a sheet metal elbow
which was attached to two fixed plywood boards to ensure that the circulating flow
stream would have a fairly smooth 180 degree route to invert its direction. The corner
effects in the main reservoir such as the secondary vortices were able to be isolated from
the main stream since the reservoir was divided into three chambers by the presence of

the two plywood boards.

Care was also taken in the design of the inlet and outlet configurations of the
tunnel to avoid any possible influence upon the flow in the observation section due to the
return circuit arrangement. A honeycomb screen was inserted in the conjunction of the
main reservoir and the contraction (also in between the downstream end of the back flow

pipe and the main reservoir). This fine honeycomb screen was designed to diminish any

17

18

vorticity that might have existed in the flow before entering the tunnel, especially for the
large scale vortical flow structures created by the redirection of the flow in the main
reservoir. After the water had passed through the honeycomb screen, it would be
gradually increased in speed with the aid of the 1.14 meter long contraction, whose area
ratio was 5.2 to 1, before flowing into the constant cross-section tunnel. At a point about
28 cm downstream of the tunnel inlet, 6.4 mm diameter threaded rods were attached to
both the lower and upper surfaces to trip the flow. Tripping the flow was not necessary to
ensure a fully developed turbulent flow at the measuring station, but did serve to localize

transition as well as to achieve spanwise uniformity.

The downstream end of the tunnel and the back flow pipe was connected via a
secondary reservoir. A flat plexiglass plate with varying diameter holes was placed on
the top of this reservoir and aligned with the lower surface of the tunnel. Since the
diameter of the holes was designed to grow with distance downstream, the resistance for
the water to be delivered into the lower secondary reservoir decreased as the distance
downstream increased. This could extend the area for water to fall so that the sudden
hydraulic head loss, which might result in a very stirred stream in the entrance of the
back flow pipe as well as a possible disturbance back to the upstream observation zone,

was prevented

The flow driving assembly was mounted on vibration absorbers and located in the
lower part of the secondary reservoir. At the inlet of the back flow pipe, a shaft with a
propeller blade at one end was installed coaxially with the center of the pipe. The
diameter of the propeller was selected to fit the inside diameter of the back flow pipe.
The other end of the shaft was extended to the outside of the reservoir and driven by a
300 Volt 5.8 ampere Reliance Super 'T' D-C motor with an adjustable-speed power
supply. A V-shaped belt (not shown in the picture) was provided to transmit the power

19

from motor rotor to the axle of the shaft. The water was then pumped back to the main
reservoir through the return pipe to continue the recycling process. Furthermore, a 91.4
cm long wooden cross was inserted in the end of the return pipe in order to nominally
regulate the flow direction before the fluid streamed into the main reservoir. A drain was

located near the bottom of the main reservoir.

It is emphasized that the flow system was originally built for developing a
channel flow in the constant cross-section tunnel. However, in the present work the flow
facility was employed to deliver a turbulent shear flow along the lower surface of the
tunnel with other side of the flow un-sheared (i.e. an open channel flow or flat-plate
flow). No modification of the flow system was required in order to run a flat-plate flow.
Nevertheless, a 76.2 cm by 43.2 cm window was constructed on the top of the
observation area to gain a better access of adjusting the optical setup of the cross-stream
view. The advantage of using a flat-plate flow instead of a two-dimensional dual
boundary layer channel flow in this investigation was that the clarity of the plan view and
the cross-stream view was considerably improved by directly viewing through the water
surface without crossing the upper plexiglass wall. In addition, the laser sheets were able
to illuminate the flow in full strength without being interfered by the presence of the

plexiglass wall.

The observation section was a 10.2 cm wide by 10.2 cm long by 3.6 cm high
volume of space that started at the wall 3.2 meters downstream from the trip wire. A 61
cm wide by 15.2 cm long hatch was fabricated on the lower plexiglass wall for installing

experimental apparatus such as wall slits, Pitot tube and wall taps, or hot film probe.

20
2.2 Mean Velocity Acquisition

In addition to the advantages mentioned above, the employment of an open
channel flow tended to reduce the physical scale of the coherent motions in the turbulent
boundary layer. This is because that, for constant power input, the shear stress at the wall
of an open channel flow is substantially greater than that of a channel flow. Accordingly,
u; is also greater for the open channel flow, which would allow structural features
characterized on wall variables to reduce their physical magnitude. Since the flow
visualization of smaller coherent features would be difficult, compromise must have been
made to set the flow speed as low as possible, but not too low so that the Reynolds
number could be kept in the order of 1000. As a result, a low speed flow has been
employed throughout the present study. The speed dial (scaled from 0 to 4.0) of the
motor was set at 1.5, which provided the power to drive the shaft at approximately 122
RPM. The decision of choosing a low Reynolds number flow was also affected by
another factor, which will be addressed in Section 2.5.

Hot film anemometry was supposed to be the ideal method of obtaining the
velocity profile in a liquid flow. However, the attempt did not succeed in our experiment
because the design of the flow facility made it impossible to keep the circulating water
clean enough. The water contained small particles that stuck on the hot film probe and
caused false voltage readouts. The damage of the hot-film equipment led us to utilize an
alternative approach for the measurement of the mean velocity profile. The pressure

measurements of Pitot tube and wall taps just provided the necessary mean.

Figure 2.2 shows the setup for the pressure measurements. The pressure readings
were taken with one Pitot tube and two wall taps. The nose of the Pitot tube was flattened

to a 4.1 mm wide and 0.8 mm high opening. The elbow was also bent towards the wall

21

approximately 15 degree enabling the probe to reach the wall as close as possible. Note
that the tip portion of the Pitot tube was still kept parallel to the flow direction. The Pitot
tube was then mounted to a vertical traverse mechanism, which was capable of
positioning within 10.025 mm backlash, to allow the probe to move over the entire
boundary layer. The closest point could be measured was 1 mm above the wall. Two 3.2
mm diameter wall taps were located symmetrically at the same cross-stream plane as the
nose of the Pitot tube and 2.54 cm separation at its both sides. The pressure hoses
connected to the wall taps were merged into one hose and hence provided the mean static

pressure at the measuring station.

Since no micromanometer was necessary to display the readouts, special
arrangement was taken to ensure accurate pressure readings. Two glass tubes placed side
by side along with a ruler were fastened to a board to show the distinction of pressure
levels. The board was painted in black to provide a background that allowed the water
surfaces in the tubes more visible. Flexible plastic tubing was used as the intermediate
connection between the pressure probes and the glass tubing. Glass tubing was chosen to
display the pressure readings simply because it was clean, straight, and easy to read. In
addition, the tendency of (electrical) static buildup was low compared to that of plastic

tubing.

One advantage of this velocity measurement setup was its simplicity, which
enabled the velocity to be measured directly without the application of a correction factor
or a calibration process. With these pressure probes, the mean velocity in the streamwise

direction could be computed by the equation,

'6 = J2gh

where, h is the pressure differential (Pt-PS), and g is the gravity.

22

Since the water was flowing at a low speed, the pressure difference, Pl—PS, was

expected to be small. This would make the measurement difficult and allow small
velocity variation undetected. In order to decrease the sensitivity of the measuring
scheme, the anemometer board was rotated to a position in which the angle between the
glass tubing and horizontal was 30 degrees (the analysis of the relative uncertainty and
sensitivity for this mean velocity measuring scheme is described in Appendix C). This re-
arrangement resulted in that the reading scale of the pressure differential along the side of
the glass tubing was doubled as illustrated in the Figure 2.2. The meniscus in the glass
tubing caused by the water surface tension could also create difficulty for pressure
reading. However, the only measurement needed to calculate the velocity was the
pressure difference. To rid of the meniscus problem, we chose the upper edge of the
meniscuses as the reference points and carefully measured the pressure difference based
upon the reference points. To avoid false readouts, care must be taken to ensure no air

bubble existing in any segment of the tubing.

The time needed to complete the velocity measurements was about two hours.
During this time period, the static pressure decreased slowly with time owing to small
leaks in the flow system. In order to minimize the effects of this pressure loss, water was
added to the reservoir from time to time for keeping the pressure head constant. Since
only pressure differential was of interest, this pressure fluctuation should have negligible

influence upon our measurements.

The mean velocity on the free surface was determined approximately by the use
of surface floats. Two measuring stations were chosen for observing and timing the
drifting floats. One was located 61 cm upstream from the position of the Pitot tube, and
the other was 122 cm downstream of the fast measuring station. Styrofoam of about 5

mm diameter was used as the floats. In order to collect a sufficient number of data, we

23

had repeated the experiment 15 times. Since the free-stream surface was free of eddies
and waves, this method should have given a fairly close estimate of the surface velocity.
More importantly, this measurement provided a matching condition for the data obtained
from the pressure probes and hence helped us to evaluate the accuracy of the pressure

measurements.

2.3 Flow Visualization Setup

In the experiments, the boundary layer motions were made visible by using two
(and at times three) orthogonal views of two or more differently colored dyes being
emitted from the wall slits. The setup of the experimental apparatus employed is
schematically sketched in Figure 2.3. A special parallel wall slit was fabricated that
allowed the introduction of two colors of dye through slits separated by 5.1 cm (about
440 wall layer units) in the streamwise direction. Figure 2.4 shows the diagram
illustrating the design of the double-dye slit. The slits had openings of 10.2 cm wide and
one tenth of a millimeter long in the streamwise direction. They fit flush with the wall so
that the dye was emanated tangentially, and hence the flow would not be disturbed. In the
past, flow visualization studies of the wall region were complicated by the "history" of
the flow. The concentration of the markers (dye or oil droplets) was diluted or dissipated
rapidly after several hundred wall layer units downstream of the wall slit, and thus was
unable to provide reliable information in determining the temporal evolutions and/or the
instantaneous spatial correlations for events that had length scales longer than the order
of a hundred wall layer units. In the present set of experiments, the double-dye slit
enabled a "refresh" of the visual information. This allowed the separation of old and new
events, and put undisturbed markers a position downstream of events that redistributed

markers from the upstream slit. We thus minimized the "history" effect as well as gaining

24

the capability of observing some longer-time sequencing of wall region evolutions. For
convenience, hereafter we name the upstream slit of the double-dye slit the "first slit" and
the downstream slit the "second slit". A 3.2 mm diameter dye injection nozzle, which
was covered by a manifold with a 25.4 cm wide and 2.54 mm high opening allowing dye
to be emitted parallel to the flow, was located 22.9 cm (about 2000 wall layer units)

upstream from the first slit.

The illumination of the dyed fluid in the boundary layer was provided by a laser
light which was spread into sheets that could be placed parallel to the flow and
perpendicular to the wall (side view), or perpendicular to the flow and the wall (cross-
stream view or end view), or both. The plan view was illuminated by two 300 watt flood
lights. The laser light was supplied from a Plasma Kinetics Model 451 Copper Vapor
Laser. The 40 watt laser beam was divided, by a 50/50 beam splitter at 45 degree
inclination, into two orthogonal beams that were eventually transformed into laser light
sheets that mutually intersected at 2.42 cm (about 210 wall layer units) downstream from
the second slit. The excess brightness in the plan and side views caused by the laser
sheets was partially blocked using black filter strips allowing the flow motions in the

neighborhood of the bright spots more visible.

The dye suppling system included three reservoirs from which fluids, which
could be dyed, passed through the adjustable valves, whose openings could be varied, to
the inlets of double-dye slit as well as the upstream injection nozzle. The volume rate of
dye emitted depended upon both the height of the dye reservoir and the opening of the
adjustable valve. The dye reservoir connected to the upstream injection nozzle was filled
with a mixture of 20 ppm (part per million) fluorescent Sodium Salt Sigma No. F-6377
green dye and water solution. As the valve opened, the mixture was released from the

upstream injection nozzle by the pressure head. This resulted in that the dye marked fluid

25

rolled away from the wall as it drifted towards downstream, and thus enabled the
fluoresced flow motions in the log and outer regions of the boundary layer to be
delineated by the laser sheets in the observation section. On the other hand, the fluid
emanating from the first slit was a mixture of 20 ppm fluorescent Exciton Kiton No. 620
red dye, red food coloring, and water solution. This allowed the formation/evolution of
the dynamical structures in the sublayer to be visualized via the illumination of the flood
lighting as well as the laser sheets. The second slit emitted a mixture of 20 ppm
fluorescent green dye, green food coloring, and water solution. This was designed to
extend the visual information further downstream. Care was taken to ensure that the mass
and momentum flux of the dyed fluid, injected through the slits in the wall, was small
compared to the flux in the viscous sublayer. It is important that the latter needed to be
essentially undisturbed. It is also worth noting that the dye coloring combination used in
the experiments offered the best visual effects (provided that the stainless steel surface of
the double-dye slit gave a white background) among many combinations that we had
tested preliminarily. Furthermore, the additive of food coloring to the dye solutions used
in the wall slits was of particular value in showing the sublayer structures in the plan

view under the flood lighting.

With the aid of mirrors, three mutually orthogonal views were directed to a
collecting mirror, in which the images could be filmed by a camera, mounted over the
top of the window. A 7.6 cm wide by 6.1 cm high front-silvered cross-stream mirror was
placed on the wall inclining to the flow direction in the middle of the channel (to assure
the symmetry of the flow field). It reflected the images of the cross-stream plane up to
the collecting mirror. The inclination angle of the mirror was optimized at 45 degrees,
which was the result of the compromise between the blockage effects and the reflection

angle. The presence of the cross-stream mirror would, however, have complicated the

26

flow field in the regions of interest if the mirror had been positioned without sufficient
distance from the observation area. Preliminary experiments were conducted at a series
of locations upstream of the mirror in determining its position so that the mirror imparts
would have minimum influence on the flow and virtually no effect on the xisual
(qualitative) information presented in the observation area. As a result, the cross-stream
mirror was located at a point where the distance between the second slit and the leading
edge of the mirror was 19.3 cm (about 1670 wall layer units). This resulted in that the
downstream end of the observation section (i.e. the position of transverse laser plane)

was approximately 1520 wall layer units upstream from the mirror.

2.4 Data Recording

The primary visual data consisted of simultaneous plan view, side view, and
cross-stream view of time resolved images which were recorded using 35 mm and/or 16
mm high speed cameras. In order to acquaint visual information with acceptable clarity
as well as tractable temporal relationships between the orthogonal views, three methods

have been employed.

Method 1 : Dumped all the information in a single film (i.e. one camera with split
field). Theoretically, this should have been the simplest and best way because all the data
stored in one frame was automatically synchronized and the temporal relationships for
events of interest could be easily quantified by manipulating the known framing rate.
Nevertheless, this method required the path lengths (i.e. the length between the object
and the camera) for all three views to be approximately identical. The path lengths for
side and plan views were able to be set equal by simply adjusting the positions of the
mirrors. Since the path length for the cross-stream view was much longer than that of the

side and plan view, a combination of one concave lens with 150 mm focal length and one

27

convex lens with 200 mm focal length was installed along the path as shown in Figure
2.3 to virtually eliminate the excess length. The cross-stream images could be enlarged, if
needed, by increasing the distance between the two lenses. However, cautions must be
taken to do this because it would degrade the clarity of the visual data as well as distort
the images in the comers of the cross-stream view. The method had been carried out
using a Photo-Sonics Series 2000 35 mm-4ML camera with 135 mm telelens as well as
an 1.4 teleconverter. A 122 meter Kodak No. 5294 high speed color film was exposed at
1/500 seconds and 100 frames per second.

Method 2 : Stored the plan and side view information in one film, and the cross-
stream data in another film. In doing so, a clock was required to provide the time
information so that the two data sets could be related for later analysis. To use this
method, the installation of the extra concave and convex lenses became unnecessary.
This enabled the transverse view information to be taken directly by shooting at the
cross-stream mirror. The main advantage of using this method was that the gains of
higher resolution for all three views (especially for cross-stream view) allowed a more
detailed picture of the flow to be studied. In addition, the possible image distortion in the
cross-stream view caused by the extra lenses was avoided. This method had been
employed using the 35 mm-4ML camera with 58 mm lens and two teleconverters (1.4X
& 2X) for the plan and side views, and a Locam Model 500002 16 mm camera with 135
mm telelens for the cross-stream view. The film used in the 16 mm camera was a 61
meter Kodak No. 7250 high speed color reversal film. The film for the 35 mm camera
was the same as used in method I. The exposure times were 1/500 seconds and 1/900
seconds for 35 mm and 16 mm cameras respectively. The framing rate was set at 200

frames per second for both cameras.

28

Method 3 : In order to exclusively examine the wall region evolutions, a third
method that only plan view was filmed has also been employed. Experiments have been
carried out by directly shooting at the wall plane with flood lighting as the illumination.
Laser light and upstream dye injection were not used for these experiments. Utilizing this
method, we lost the correlation information with the structural features above the wall
region, but we gained pictures of much higher clarity. This enabled the wall-region
events of fine scale, particularly for flow motions of Kolmogorov scales (less than 10
viscous units) such as vortex filaments of hairpin vortices, to be possibly seen. The 35
mm camera and films used were the same to those used in method I. The exposure time

and framing rate were set at 1/500 seconds and 50 frames per second respectively.

Experiments utilizing method I were carried out with some degree of success.
However, in order to have flow pictures with sufficient details so that human eyes were
able to easily identify the micro-scale coherent structures, higher resolution for images in
the cross-stream plane was required. Therefore, the visual information collected using
method 2 provided the major data base for our statistical study. The pictures obtained
using method 1 and method 3 provided good illustrations of sequential evolutions that

revealed some important features of the interaction in the wall region.

2.5 Data Reduction and Principles of Conditional Sampling

Since the major purpose of the current study was to statistically characterize the
structural features, which are involved in the wall-region interaction process, and their
spatial/temporal inter-relationships, the extraction of the events of interest from the bulk
raw data via a conditional selection process was necessary. Only the behavior of the
principal flow motions such as typical eddies, pockets, and long sublayer streaks have

been investigated. The characteristics of the hairpin vortices (primary and secondary)

29

were not put in analysis because the resolution of our correlation data was incapable of
showing flow details with sizes in the order of Kolmogorov scale consistently. Forcing
the analysis will simply result in unreliable or false statistics. Additionally, since the
formations of the primary and secondary hairpin vortices are often accompanied by the
formations of pockets and long sublayer streaks respectively, their occurrence was
deducible and thus was not the best interest regarding to the purpose of this study

although they might be important in the contribution to the Reynolds stress near the wall.

The principles of the conditional sampling process were constructed based upon
the objective of quantifying the correlation between major coherent motions that
participated the inner/outer region interactions. In order to fulfill the need to study the
phase relationship of these correlated events, we setup a designated area in the wall plane
of the observation zone as our sampling reference. This designated area started from the
second wall slit and ended at the streamwise location of the cross-stream laser sheet. The
reason for setting up the designated area was also closely coupled with the utilization of

the double dye slit. It is explained as follow.

It is understood that the wall dye injection is capable of detecting pockets near the
injector. However, the formation of long sublayer streaks cannot be observed within the
short duration and space over which the maker remains uniform enough to see the
pockets. In regions further downstream from the injector, the dye will primarily be
collected to form long low-speed streaks which usually have a longer lifetime compared
to that of the pockets. As a result, diluted dye will be no longer able to outline pockets in
these regions. Therefore, it is almost impossible to tell whether or not a pocket will form
near the downstream end, and in between a long streak pairs if only one wall slit is used.
This will severely inhibit the investigation of spatial phasing between pockets and

sublayer streaks, particularly the long streaks. Thus, in order to make the entire version

30

of the interaction and evolution visible, we employed the double dye slit. The basic idea
is that we seep the red dye from the first wall slit enabling the development of long
streaks to be visualized. Then, in the second wall slit, a different color dye is seeped in to
continue (refresh) the visualization of the long streak formation and, more importantly, to
provide an opportunity for the formation of the pockets to be observed in the designated
area while the long streaks are still visible. In the downstream end of the designated area,
a cross-stream laser plane was setup to observe the spanwise geometry of structural
features. We were particularly interested in events in the log region and lower portion of
the outer region, which might be associated with the initiation of the wall region
evolution. The spatial arrangement of the designated area and the cross-stream laser sheet
was set up according to the assumption that the pockets are produced right beneath the
typical eddies impinging the wall as suggested by previous observations. This spatial
arrangement enables the typical eddies, which are responsible for the formation of the
sublayer features in the designated area, to be identified right after the onset of the
interactions, and before they become unidentifiable due to the interactions that lead to
significant distortion. Note that a special effort was also made to see if any other Taylor
micro-scale coherent motions appeared in the cross-stream laser plane when pockets were
forming in the designated area. We did not find other Taylor micro-scale coherent

motions.

Figure 2.5 shows an isometric view of our interaction studies illustrating the
general flow field, as well as the spatial phasing between the designated area, the
observation arrangement and the multi-color dye system. It shows that the typical eddies
delineated by upstream green dye convect downstream. Some of them will interact with
the existing wall region vorticity field in the vicinity of the designated area. The principle

of the sampling is simple, only sublayer features -- pockets and/or long streak pairs --

31

which are associated with the interactions are sampled. All typical eddies identified in the
transverse laser plane within the first 300 wall layer units are also sampled. The
simultaneous occurrence of a typical eddy in the transverse plane and the sublayer
feature(s) in the designated area (they must be approximately aligned vertically) is
labelled as a correlated (interaction) event. Samples containing typical eddies which
occur in the transverse plane without sublayer features formed in the designated area
underneath them are considered as events with no interaction. Based upon the principle,

the rules of the conditional sampling procedure are described in details as follow.

1. Pocket Sampling : A pocket was recorded and measured only if it was observed
to form in the designated area of the plan view. Note that, if any portion of a pocket was
located inside the designated area while it was still evolving, the pocket would be
sampled.

2. Typical Eddy Sampling : A typical eddy was sampled if it was identified and
located between y+=300 and the wall in the cross-stream and/or side view. If sampled
from the side view, it must be identifiable when crossing the transverse laser plane. In
order to avoid mistakenly recognizing some strong hairpin vortices as typical eddies
(since the visual impression of their spanwise geometry was, at times, similar to that of
typical eddies), only images that were dyed with green dye, that was introduced
upstream, were chosen. Since hairpins would be dyed red, because they were lifted wall
layer fluid from the immediate upstream wall region. This extra degree of discrimination

is gained because of the multi-color dye system.

3. Long Sublayer Streak Pair Sampling : A long low-speed streak was identified
as a well-defined streamwise concentration of dyed fluid marking an extended region of

low-speed motion. Its streamwise extent must have been greater than 100 wall layer

32

units. A pair of the long streaks was recorded and measured only if at least one of the
following conditions was met:

(i) In the plan view, a streak pair was sampled when a pocket in the designated
area was formed between the streaks, and in the neighborhood of the downstream end of
the streak pair.

(ii) In the cross-stream view, a streak pair was sampled when the center of an
identified typical eddy was approximately aligned over the center of the streak pair
whose downstream end was also in the vicinity of the line intersected by the transverse
laser sheet and the wall. Under this condition, only the streak pairs whose downstream
ends located inside the designated area or did not exceed 100 viscous units downstream

of the intersection were sampled

4. Criteria for the Correlated Interaction Events : As mentioned earlier, a streak
pair and/or a pocket were considered to be correlated with a typical eddy if the sublayer
feature(s) were observed to form underneath the passing typical eddy. In a couple of
occasions, the words "approximately aligned" were used in this section to describe the
spatial phasing between the typical eddies and the sublayer features. Here, a criterion is
constructed to quantify the definition of the vertical alignment of coherent features. That
is, the spanwise spacing between the centers of the typical eddy and the sublayer feature
must have been less than a quarter of the spanwise dimension of the sublayer feature in
order that they were accepted as a correlated interaction event. If both the sublayer
features were formed, the comparison was made with the pocket width (because the
streak pair was usually well aligned with the pocket). It is also important to note that no
correlation between a typical eddy and a sublayer feature was considered to exist if the

difference of their spanwise dimension was two times greater than either dimension. This

33

constraint would have automatically rejected the cases, in which irrational size

differences of coherent features occurred, as correlated interaction events.

With regard to the sampling of typical eddies from the transverse view, a question
to be answered is : Were all the observed vortex pairs dyed with green dye simply
connected vortex loops? It is believed that our samples do not contain vortex loops that
had "legs" attached to the wall because of our spatial arrangement of dye emission. The
distance between the first slit and cross-stream laser plane was approximately 7.6 cm
(:650 x+). This constraint could probably eliminate the possibility that the hairpin
vortices, created in the wall region between the upstream injector and the first slit, were
able to elongate in the streamwise direction for the necessary distance to appear in the
transverse laser plane. It is more convincing if one realizes that the distance between the
origination of hairpins at the upstream dye injector and the transverse laser plane was
about 2650 viscous units. It is very hard to believe how hairpin legs could stretch this far
without severe re-orientation of their vorticity field. This inference is reinforced by the
findings of Head and Bandyopadhyay (1981). They reported that the hairpin vortices
were very much less elongated in the streamwise direction at low Reynolds numbers
(R95800) and were better described as vortex loops or horseshoe vortices. Accordingly,
at current flow condition (R9z800), the identified typical eddies dyed with green dye in
the cross-stream laser plane are more likely to be vortex loops than hairpins with legs
that connected to the wall. This was the another reason why we chose a low Reynolds

number flow to conduct the investigation.

Figure 2.6 illustrates the on/off conditions for the sampling process employed in
the present study. Fourteen conditions are shown to represent the most often seen
combinations between plan and cross-stream views. Based upon the rules described

above, all the observed typical eddies were sampled, and the chosen samples of the

34

sublayer structures correspond to conditions 1, 2, 3, 11 and 13. Note that, for condition

13, only a pocket was chosen.

The visual data were analyzed on a Nac Model PH-3SOB Film Motion Analyzer
with a X-Y calibrated scale for 35 mm films, and/or a Nac Model DF-16C Analysis
Projector for 16 mm films. The images of latter were displayed on a ground glass screen
and read from the reverse side of the screen to eliminate the parallax errors associated
with using a standard movie screen and viewing from an angle. The visual analysis of the
movies consisted of observing the individual events, identifying the events in a number
of runs, making a thorough study of each of the events and their relationships with other
simultaneous events, and quantifying, if possible, the characteristic features of statistical
importance. The reduced data were statistically manipulated and displayed by using a

Sun 4/260 computer workstation and its peripheral devices.

CHAPTER 3
RESULTS

3.1 Mean Velocity Profile and Flow Parameters

A velocity traverse was carried out at about 3.3 meters downstream from the trip
(i.e. Rex a 5.72x105), which was exactly the streamwise location that the first slit was
placed in the visualization experiments. The measured velocity profile was used to
indicate the internal consistency of the measurements as well as to obtain the flow
parameters at the measuring station. Figure 3.1 shows the dimensional velocity profile
plotted as the mean streamwise velocity versus the distance from the wall. The
momentum thickness was computed using trapezoidal rule of integration. Since obtaining
data very close to the wall was constrained by the physical size of the probe, the shear
stress at the wall was unable to be estimated reliably from the slope of the sublayer
profile. Therefore, determination of the friction velocity was made using the Clauser plot
technique (1954), in conjunction with Coles (1968) law of the wall. This technique is
good to 15% based upon the comparison of drag balance measurements and the Clauser
plot.

Examination of Figure 3.1 indicates that two measurements near the intermittent
region of the boundary layer appeared not to be in good agreement with the overall
profile. The cause of this inconsistency is, however, not well understood. Since the
positions of these two data points are well beyond the log region, they definitely have
very little effects in the decision of the skin fi'iction coefficient from the Clauser plot.

Figure 3.2 shows the utilization of Clauser plot in determining the skin friction

35

36

coefficient. Using friction velocity based upon the chosen skin friction coefficient, the
universal logarithmic mean velocity profile is presented in Figure 3.3. It shows that the
flow at the measuring station satisfactorily complied with the behavior of a normal
turbulent boundary layer flow over a smooth wall. A summary of the principal
characteristics of the flow at the measuring station is given in Table 3.1. As we can see,
the shape factor of our boundary layer flow (H5127) was very close to the theoretical
value (H=1.3) for turbulent flat-plate flows at low Reynolds numbers as suggested by

White (1986). Note that the thickness of the viscous sublayer was approximately 0.7 mm.

The accuracy of the measurements was checked using the matching condition at
the free surface. By comparing the mean value of the measurements using surface floats
and the mean value of the measurements for y>899 using pressure probes, we found that
the relative difference between them was about 5.1% of the average free stream velocity

based upon the pressure measurements. Calculations are described in Appendix A.

3.2 Statistical Results

32”... [Ill] [1 E [2 I1]

Before the statistical results are presented, it is important to know what
information our visual data are able to provide. The experiment is designed to measure
the following :

--- physical size and spatial relationships of coherent motions.

--- velocity of a coherent motion as a whole.

--- frequency of occurrence and temporal relationships between coherent features.
Because a passive dye technique was employed, we will not be able to obtain quantitative

information about the :

37

--- vorticity or pressure field (however, in some cases their relative intensity can
be qualitatively deduced).

--— fluctuations in any direction.

In addition to the time-space measurements that can directly reveal important
physics, there are few hypotheses (inferences or suggestions) that will be made, mainly
presented in next chapter, based upon piecing together the visual evidence and statistical
results. Note that a statistics concerning the creation and the evolution of hairpin vortices
has not been compiled, partially because the insufficient resolution of our visual data is
incapable of consistently showing small hairpins with vortex filament of Kolmogorov
scale. More importantly, its formation is not the major interest of this study as mentioned

earlier.

3221 'lElT

The statistical characteristics of typical eddies will be presented in this section.
Regarding the observation of the typical eddies, one should bear in mind that the
formation and evolution of the typical eddies were occurring in a highly perturbed
environment, thus it is unrealistic to expect that typical eddy with idealized shape, such
as an undistorted laminar vortex ring, could constantly be seen. Particularly in the cross-
stream view (y-z plane) study, not only are typical eddies with unperturbed shapes rare,
but also are they not easy to be distinguished as they are in the side view (x-y plane). In
order that we could correctly identify typical eddies from the cross-stream view,
preliminary exercise had been made. That is, we had repeatedly practiced viewing the
typical eddies in the simultaneous laser sheets parallel and normal to the mean flow

direction as the eddies passed the intersection of both laser sheets.

38

According to the rules of conditional sampling described in Section 2.5, the
typical eddies were essentially sampled unconditionally (i.e. a typical eddy was sampled
as long as it was visually identified) except that the eddy must have occurred withimhe

W.

3.2.2.1 Magnitude

Dimensions of typical eddies in the outer intermittent region have been measured
over a wide range of Reynolds numbers by Falco (1991). Figure 3.4 shows the average
length scale of the diameter of the vortex core in dimensionless form (see the definition
sketch in the figure) as a function of Reynolds number. Figure 3.5 shows two similar
plots as the mean dimensions of typical eddies are normalized by the momentum
thickness and wall variables. Observation of both figures indicates that the magnitude of
typical eddies decreases as Reynolds number increases when non—dimensionalized by
boundary-layer thickness or momentum thickness, and increases with Reynolds number
when scaled on wall variables. However, no one has ever revealed the length
characteristic of typical eddy at its cross-stream section (i.e. y-z plane). Finding the
characteristics of this dimension is very desirable because important sublayer features
such as pockets and low-speed streaks are normally characterized by their spanwise

dimensions, thus, for consistency, to describe their relationships with the typical eddy

using its spanwise dimension is more adequate than using Cy or Cx.

For the current work, a significant number of samples have been gathered to
compute the typical-eddy diameter (maximum dimension) in the cross-stream section at
Ree—805. For convenience, we will denote this length scale as Cz hereafter. In the
experiments, the most common coherent motions with size of Taylor micro scale

observed in the cross-stream laser plane were counter-rotating vortex pairs and crescent-

39

shaped features in the dye. The sampled vortex pairs and crescent-shaped features are
considered as ring-like vortices (i.e. typical eddies). An explanation for this will be made
later when the streamwise extent (Cx) of the samples is examined. Figure 3.6 shows the
definition sketch as well as the probability density distribution of the spanwise diameter
of the typical eddies in wall layer units. The probability density functions of the log-
normal and the normal distributions based upon the statistical parameters of the data set
are also shown. In the present study, the probability density functions of various
distributions for discrete random variables are given in Appendix D. The statistically
significant properties of CZ+ distribution, such as the sample size, mean, mode, standard
deviation, variation coefficient, as well as skewness and flatness coefficients are
tabulated in Table 3.2 as shown below. It appears that the log-normal distribution, with a
typical dense range from 60 to 170 wall layer units, represents the overall data better than
the normal distribution.

Table 3.2 Characteristics of CZ+ distribution as shown in Figure 3.6

 

Characteristic N Mean Mode 0 qt S F

 

 

 

 

 

 

 

 

 

Value 417 101.3 87.5 24.9 0.25 1.32 6.56

 

The streamwise length scale (Cx) of the typical eddy is also estimated by

counting the number of frames that the typical eddy had appeared consecutively in the
cross-stream view. Based upon preliminary side-view observation, the typical-eddy
convection velocity was found to be, on average, close to the local mean velocity.
Therefore, the approximate convection velocity of a typical eddy can be obtained by

using the known distance of the typical eddy from the wall to find the corresponding

 

40

mean velocity in Figure 3.1. The streamwise length scale of the typical eddies is

computed by the formula

Cx -_—. (local mean velocity) x (number of frames) x (framing rate)‘1

Figure 3.7 shows the definition sketch and the probability density distribution of
CK in wall layer units. Table 3.3 lists the statistical parameters of the data. In a manner

analogous to the behavior of Cf, the overall data are well represented by a log-normal

density function. This set of data is very important although they represent estimated
values. It confirms that our samples were not long axial vortex pairs, but conforms well
with the expected compact vortical shape with which typical eddies are characterized.
Additionally, since the sampled vortex pairs were dyed florescent green, they were not
hairpin vortices that had long legs attached to the wall. Based upon these findings and the
fact that the vortex lines must have closed on themselves, the simplest hyporhesis for the
coherent shape of the sampled vortex pairs is that they were ring-like vortices. The joint
probability density of random variables Cz+ and Cx+ is graphically presented in both a
two dimensional contour map as well as a 3-D probability density distribution as shown
in Figure 3.8. The peak occurrence is clearly located at the neighborhood of Cz+=92 and
Cx+=73.

Table 3.3 Characteristics of Cx+ distribution as shown in Figure 3.7

 

Characteristic N Mean Mode 0' ‘1’ S F

 

 

 

 

 

 

 

 

 

 

Value 417 76.9 72.5 30.2 0.39 0.73 3.28

 

41
3.2.2.2 Correlation of Diameter and Distance From the Wall

The distance of a typical eddy from the wall ((1) is defined as the distance
between the center of the typical eddy and the wall. Figure 3.9 gives the definition sketch
in the y-z plane and the probability density distribution of the distance (d) in wall layer
units. Table 3.4 tabulates the statistical characteristics of the samples. The various dashed
lines represent the normal, log-normal, as well as the Rayleigh probability density
functions. It appears that the length scale (1+ is close to be a random variable of a log-
normal distribution with a mean of approximately 118 wall layer units. It is also found
that the most probable distance of the typical eddies from the wall were slightly under

y+=100 when propagating over the near wall region.

Table 3.4 Characteristics of d+ distribution as shown in Fi 3.9

 

Characteristic I N Mean Mode 0 I}! S F

 

 

 

 

 

 

 

 

Value I 417 117.9 97.5 42.26 0.36 0.71 3.45

 

In the scaling analysis, the relationship between the spanwise diameter (C2) of a

typical eddy and its distance from the wall has also been explored. Figure 3.10 shows the
dependence of Cz+ on d+. The data are obtained by performing several binning

processes, with various bin widths, on both variables. Sample size is 417. Along with the
data, the histogram of (1+ and upper/lower standard deviation boundaries of Cz+ are also
presented to show the degree of data dispersion. The results indicate that the spanwise
length scale of the typical eddies gradually increases as their distance from the wall
increases. This is expected if the formation process of the typical eddies and the diffusion
of their vortex cores are the primary causes of this scaling dependence (more comments

will be made in Chapter 4). The findings also serve as an evidence against the conjecture

 

42

that the pairing process of typical eddies is important in the outer part of turbulent
boundary layers. Furthermore, a reasonable linear relationship between these two length
scales is found to exist over the range of 55<d+<280 as shown in Figure 3.11. The linear

approximation is written as follow.

cz+ - 0.097d+ - 88.6 = o

A normalized distance (d*), defined as d"’-0.5Cz+, is also statistically

characterized here although its use is not immediately revealed at this point. This scale
provides an approximate measure that stands for the distance between the lower
boundary of the vorticity concentration of the typical eddy and the wall. The accuracy of
this estimated scale mainly depends on the orientation of the typical eddy with respect to
the wall in the x-y plane. Figure 3.12 presents the probability density distribution of d*
and the normal, log-normal, as well as the Rayleigh probability density functions. The
statistically significant properties of the data set is tabulated in Table 3.5. It is found that,
among these tluee probability density functions, the Rayleigh distribution is obviously
the best curve fit for the data set. Utilization of this length scale will be shown in Section

3.2.6.

Table 3.5 Characteristics of d* distribution as shown in Fi mre 3.12

 

 

Characteristic N Mean Mode 0 \y S F

 

Value 417 67.47 43.75 41.36 0.61 0.74 3.29

 

 

 

 

 

 

 

 

43

3.2.2.3 Comparison with Outer Layer Data

Table 3.6 shows the comparison of the present Cz/2 and CK measurements with

Falco's data, which were measured in the outer intermittent region by side-view
observations, at a comparable Reynolds number. The data are compared in dimensionless
forms as normalized by variables u1/v and 0. The definition sketch for each scale is also
provided. It shows that the current data are in good agreement with Falco's measurements
although there is minor difference in scale definition (see the definition sketches for
Falco's Cx and Cy). Qualitative comparison of these data demonstrates that the selected
samples using cross-stream observations were typical eddies. Furthermore, because both
side view and cross-stream views have the configuration drawn, we also confirm
observations that typical eddies are indeed ring-like. Their toroidal conformations led to
the appearance of counter-rotating vortices as they passed through the transverse laser
sheet. Quantitatively, this comparison shows that the data are consistent with the trend

that the size of typical eddies increases with distance from the wall. This is because that
although CZ/2, the half spanwise diameter of the typical eddy, is comparable to Cy, the

diameter of the vortex core in the x-y plane, CZ/Z should have been greater than Cy if

measures were taken out of the same typical eddy. However, the results show a nearly

identical mean value of Cz/2 and Cy for typical eddies in the two distinct regions of the

turbulent boundary layer. The indication is clear that the length scales of the outer-layer
typical eddies are, on average, larger than that of typical eddies existing close to the wall.

The interpretation may also be made by the following mathematical expressions.

Since Cz/2 2 Cy , for same typical eddy (T .E).

the comparison shows (Cylouter 115 E (Cz/2)inner T.E

44

result in (Cy)outer TE 2 (Cy)inner T.E

By similar interpretation, the comparison of our estimated Cx and Falco's measurements

will simply result in the same conclusion. Additionally, the flow-visualization
measurements of Hoyez (1990) also confirmed that statistically the diameter of typical
eddies in the x-y plane increases with distance from the wall. Similar results are drawn
although her samples were taken in the upper portion of the outer region at moderately

high Reynolds numbers.

Examination of Table 3.6 suggests that, for Reynolds numbers in the order of the
present experiment (01000), the typical eddy normally has an axisymmetric donut-like
configuration with diameter of the Taylor micro-scale (~100 wall layer units) and

streamwise span about 75% of the diameter.

3.2.2.4 Orientation

In the laser sheet cross-stream observations, we have also measured the typical-
eddy orientation, which was characterized by the angle (4)), ranging from -1t to 1:,
between positive y axis and the typical-eddy axis. The measurements were taken when
the maximum dimension of the typical eddies appeared in the laser plane. The definition
sketch and probability density distribution are shown in Figure 3.13. The dashed line
stands for the probability density function of a normal distribution. Table 3.7 lists the
statistical properties of the data set as shown below. In this measurement, the error bar is
32.5 degrees since the data were recorded by units of 5 degrees. As we can see, the
normal distribution provides a fairly good curve fitting for the samples. This is also
confirmed by the small skewness coefficient and relatively high flatness coefficient

shown in the table. Results indicate that, within the first 300 wall layer units of turbulent

45

boundary flows, the most probable orientation of typical eddies in the y-z plane is almost
identical to their average orientation, which is approximately zero degrees. Incidentally,
the symmetrical distribution demonstrates the excellent quality of the water tunnel flow
which kept the mean lateral flow fluctuations negligible (i.e. WEI—:0) in the

measuring station throughout the experiment.

Table 3.7 Characteristics of 4; distribution as shown in Figure 3.13

 

Characteristic N Mean Mode 0 S F

 

 

 

 

 

 

 

 

 

Value 417 - l .650 0° 32.960 0.27 6.60

 

By examining the sequential cross-stream views of a typical eddy, its orientation
with respect to the wall in the x-y plane could also be deduced. Based upon their
inclinations with the wall in x-y plane, typical eddies are divided into six category of
orientations. Each category of orientation will be called a "mode". Figure 3.14 illustrates
four of the six different modes of typical eddies in both side and cross-stream views as
they are propagating past a transverse laser sheet. The sketched sequencing of the cross-
stream views associated with each mode of the typical eddies shows how we managed to
obtain their modes. It is noted that the figure illustrates an idealized scenario, and the real
case is unlikely to be as regular and clear-cut as the sketch might indicate. We must
anticipate the real pictures to be complicated because of the distortion by highly
perturbed environment and complex typical eddy interaction, as well as the smearing
effect of diffusion. Typical eddy of Mode 5 is defined as the transition stage from Mode
1 to Mode 2. In other words, the change of the typical eddy elevation was very small,
essentially not measurable in the cross-stream view sequence (i.e. the self-induced

velocity vector was pointing outwards and normal to the wall in the side view when

46

crossing the transverse laser plane). Similarly, typical eddy of Mode 6 is defined as the

transition stage from Mode 3 to Mode 4.

The statistical characteristics of typical eddies based upon their modes are
presented in Table 3.8. Total sample size is 417. The first three columns of the table give
the distribution of typical eddies in different mode categories. The length-scale properties
associated with each mode are also tabulated in columns 4 and 5 (discussion concerning
the rest of the table will be made later in this chapter). Examination of the table suggests
that not only did Mode 1 represent the most probable geometrical orientation in the x-y
plane, but also were typical eddies of this mode, on average, the closest ones to the wall.
On the other hand, the occurrence of typical eddies of Mode 4 or 6 was almost
nonexistent (less than 1%). In general, the majority of the typical eddies were oriented so
that they induced themselves away from the wall (i.e. 97.9% of the typical eddies were
Modes 1, 2, and 5).

Earlier, we have measured the mean spanwise diameter (Cz) and estimated mean

streamwise length scale (Cx) of the typical eddies. Therefore, the average inclination

angle with the wall of Mode-1 typical eddies in the x-y plane can be calculated by the
term, Cos’1(76.9/101.3)=40.6 degrees, assuming that the typical eddies on average had
axisymmetrical forms. The denominator and numerator of the operand stand for the
average diameter and the mean streamwise length of Mode-1 typical eddies respectively.
By comparing this to the characteristic angle (about 45 degrees) of hairpin vortices as
suggested by Head and Bandyopadhyay (1981), it is not difficult to realize how easily
these two structural features could be misinterpreted in the x-y plane, and how closely
the relationship they might have with regard to the evolution of wall region hairpins and
the formation of typical eddies. Further discussion concerning a possible link between

hairpin vortices and typical eddies will be made later.

47
3.2.2.5 Period of Occurrence

The time between typical eddy occurrences is one of the important features
associamd with the turbulent boundary layer dynamics, especially for these sampled
typical eddies which were close to the wall (for y+<300) having direct impact on the
wall-region turbulence production. The period of occurrence (rm) was able to be
measured by viewing the cross-stream view and recording the time at which typical
eddies passed a pre-chosen measuring station. The station was an imaginary line which
extended from the wall to y+=300. It is noted that a typical eddy was counted if any
portion of it crossed the imaginary line. Since a typical eddy usually required more than
one frame to cross the measuring station, we regulated our time recording scheme so that
the time was marked when the center of a qualified typical eddy appeared in the
transverse laser plane (i.e. time was recorded when its spanwise dimension was
maximum). If more than one typical eddy were crossing the imaginary line
simultaneously, only one sample was counted. In order to have a large enough sample
size, two measuring stations within the spanwise laser sheet had been selected to perform
the investigation. They were separated by 3002+ to ensure the independence of the
selected samples. Figure 3.15 shows the probability density distribution of tm's in viscous
time units, as well as the log-normal and Rayleigh probability density functions. The
principal properties of this set of samples are listed in Table 3.9. It is found that the
highly skewed probability density distribution of tm'l' is better represented by a log-

normal density function with a mean of nearly 26t+.

Table 3.9 Characteristics of tm‘l' distribution as shown in Figure 3.15

 

Characteristic N Mean Mode 6 3g 8 F

 

 

 

————r =
Value 157 26.3 1 1.3 24.2 0.92 2.27 9.34

 

 

 

 

 

 

 

 

 

48

The average period of the typical eddy occurrence can be normalized by the
boundary layer thickness and the average convection velocity of the sampled typical
eddies, Egg/U3”. Note that the value of Uavg is calculated using the local mean velocity
(Figure 3.1) based upon the mean d+ of the typical eddies (Table 3.4). The resultant
value is about 1.1 which suggests that, on average, one typical eddy in the log region or
lower portion of the outer region could be seen by a stationary observer about every
boundary layer thicknesses at low Reynolds numbers. Again, we approximate the mean
convection velocity of a typical eddy to be the local mean velocity about its center of

mass.

3.2.2.6 Center-to-Center Spanwise Spacing

In many visual studies of sublayer features, it has been noticed that the chances
for witnessing the formation of a row of pockets along a spanwise direction in the
immediate downstream of the wall slit are high. It is felt that the cause of this
phenomenon should have been related to the lateral distribution of the excitation coherent
motions in the log region and/or lower portion of the outer region. Thus, an important
feature of the typical eddies which needs to be quantified is the characteristic dimension
of their spanwise spacings. The data base for this investigation has been constructed by
selecting the samples in which more than one typical eddy appeared simultaneously in
the cross—stream view. If the centers of two typical eddies happened to be aligned
vertically in a chosen sample, they were counted as one typical eddy in order to eliminate
the possibility of zero spacing, which is meaningless for the purpose of this
measurement. In fact, examination of the chosen samples indicates that the minimum
distance between two typical eddies was about 56 2"“. Figure 3.16 shows the probability

density distribution of center-to-center spacings (£1) between neighboring typical eddies

49

in the spanwise direction. The normal, log-normal, and the Rayleigh probability densities
are also sketched to represent the distribution. Although none of these distributions
represents the data perfectly, it appears that the log-normal density function fits the
overall distribution better than the other two. The principal properties of the data set are
listed in Table 3.10. It indicates that the typical eddies were, on average, spaced in the
spanwise direction by roughly 0.6 boundary layer thicknesses. Note that, although the
sample size is 84, there were 156 typical eddies involved in the analysis because each
sample usually consisted of two or three typical eddies.

Table 3.10 Characteristics of §+ distribution as shown in Figure 3.16

 

Characteristic N Mean Mode 6 \y S F

 

 

 

 

 

 

 

 

 

 

Value 84 244 128 132 0.54 0.76 2.86

 

Using this spanwise spacing characteristic of the typical eddies, an argument will
be presented later to reconcile the difference between the mean spacing of the sublayer
streak pairs found in the present interaction study and the overall spacing of the low-

speed streaks in turbulent boundary layers.

W

The low-speed, long streaky structure is an important feature in the wall region of
wall bounded turbulent flows. It is well known that the average spacing (1+) of these
streaky structures is about 100 when non-dimensionalized by wall variables. It was also
documented that sublayer streaks with longitudinal length greater than 1000 wall layer
units exist. Furthermore, once formed the streaks were observed to persist for very long

time. Although the mechanism responsible for the formation of the long sublayer streaks

50

has not been fully uncovered at this stage, previous observations suggested that the long

sublayer streaks are formed in pairs as results of inner/outer region interactions.

3.2.3.1 Spacing of Neighboring Long Streaks (Unconditionally Sampled)

The characteristic length scale of long sublayer streaks is represented by its
spanwise spacing (2.) between two neighboring streaks. In the current study, the method
of measuring this length scale was to measure the spanwise distance between the centers
of two neighboring streaks at a transverse measuring station located approximately 50x+
downstream of the second wall slit. It was required that the streaks had well defined
outline (i.e. clear distinction of dye concentration) and streamwise extent of at least 100
wall layer units when sampled. If two streaks appeared to merge within 100x+
downstream from the measuring station, they were counted as one streak; otherwise,
there was no preset minimum spacing. The film that contains only plan view was used to
perform the streak counting. Each frame in the film was spaced by 1.84t+. We evaluated
the streak spacings in still pictures that were evenly separated by 130t+ (i.e. 70 frames
apart) in order to obtain statistical information of independent streaks. The total duration

of investigation was approximately 7280t+.

Figure 3.17 shows the probability density distribution of it's in wall layer units as
well as the normal and log-normal density functions. Table 3.11 tabulates the statistical
parameters of the samples. Results show that the overall data is very well represented by
a log-normal density function with most streak spacings occuning between 40 and 170
wall layer units. Our data confirm the overall streak spacing measured previously by
Kline et al. (1967) and many other investigators. The statistical parameters of our streak
spacing distribution are also very similar to those determined by Smith and Metzler

(1983; 0.75<S<1.0, 0.34<\y<0.4) using hydrogen bubble technique and Oldaker and

51

Tiederman (1977; w=0.36) using dye visualization. The most probable value of the streak
spacing (the Mode) is also consistent with the findings of Smith and Metzler (75~802+)
and the value (about 20% less than the mean) suggested by Kline et a1. (1967).

Table 3.11 Characteristics of 1+ distribution as shown in Figure 3.17

 

 

Characteristic N Mean Mode 0' \y S F

 

Value 500 96.0 78.5 34.9 0.36 0.91 4.44

 

 

 

 

 

 

 

3.2.3.2 Spacing of Conditionally Sampled Long Streak Pairs

In the section, we will present the spanwise length characteristic of the long
sublayer streak pairs which were selected according to the rules of the conditional
sampling (Section 2.5). The characteristic length of a streak pair is represented by its
spanwise spacing (1) between the streaks. In the present study, the method of measuring
this length scale was to measure the ayemge spanwise distance between the centers of the
two streaks, whose widths normally ranged from one to four Kolmogorov micro-scales.
Figure 3.18 shows the measuring scheme and the probability density distribution of 1's in
wall layer units. The normal and log-normal density functions are also sketched. Table
3.12 gives the statistically important properties of the samples. It is found that the overall
data is well fit by the log-normal density function with relatively small skewness and
variation coefficients. Most of the samples ranged between 40 and 120 wall layer units.

The mode is only about 6% less than the mean.

Table 3.12 Characteristics of 1"“ distribution as shown in Figure 3.18

 

 

Characteristic N Mean Mode 0 w S F

 

 

 

 

 

 

 

 

Value 291 74.7 70 15.7 0.21 0.86 4.78

 

 

 

52

The conditional sampling of the streak pairs leads to the apparent disagreement in
the wall-region mean streak spacing between our result (about 75 wall layer units) and
the universal length scale (about 100 wall layer units). However, the results are different
simply because they are two sets of data which were taken under different measuring
rules. In the present experiment, the samples of the long sublayer streak pairs were
conditionally selected based upon the occurrence of pockets in the sublayer or typical
eddies in the log region and lower portion of the outer region. It is important to
emphasize that the classical scheme used to deduce an average spanwise streak spacing
was to count, from the motion pictures, the number of streaks in a number of statistically
independent frames. The classical method of streak counting was not conditioned by the
occurrence of other structural features on and/or above the wall. It is interesting to note
that the streak spacing measurements using the classical method (Smith and Metzler
1983, as well as our application of the classical method, i.e. our unconditionally sampled
data) resulted in a log-normal distribution having a made nearly identical to the mean of

our conditionally sampled streak pair spacings.

It is worth noting that when we use the term "streak pair", we are not necessarily
implying that the length in streamwise direction of one streak must equal that of the
other. In many visual studies concerning wall region features, single streaks could be
observed occasionally because one of the streaks was longer than the other. Another
possibility is that the confined dye-marking region prevented one of the streaks from
being observed The length that a streak was able to extend might depend on many
factors such as the incident angle and the orientation of the excitation eddy, as well as the
existing local conditions in the wall region. Hereafter, the phrase "streak-pair spacing" (l)
is used to denote the spanwise spacing of a streak pair which is identified to be associated

with the occurrence of a typical eddy or a pocket. On the other hand, the words "streak

53

spacing" (A) is still used to denote the lateral spacing between neighboring streaks.
Further discussion on the wall-region streak spacing, involving the information obtained

in Section 3.2.2.6, are made in Chapter 4.

1242mm

The pocket flow module is an important feature of the wall region structure that is
clearly present close to the dye slit. It is necessary to have sufficient concentration of
markers covering an area over the wall to visually detect them. Pockets were difficult to
observe further downstream (> 400 x+) from the wall slit because most of the dye would
have been diluted and/or convected away from the wall. Falco (1980) documented the
formation and evolution of pockets in detail. He also showed that, as the Reynolds
number increases, the spanwise dimension of pockets increases slowly when non-
dimensionalized with wall parameters. In this section, we will describe the length
characteristic and the occurrence frequency of the pockets based upon the present set of

samples.

3.2.4.1 Length Scale

The length scale of a pocket is characterized by its maximum width (w) as the
fully developed stage has reached. In the present study, the method adopted for
measuring this characteristic length was to measure the spanwise distance between the
most densely marked boundaries of the pocket. Figure 3.19 shows the definition of
pocket measurements, the probability density distribution of w+'s, as well as two
probability density functions. The statistical parameters of the samples are tabulated in
Table 3.13. Apparently, the behavior of the pocket width conforms closely to a random

variable of a log-normal distribution having a mean of about 82 wall layer units. Most of

54

the samples are found to range between 50 and 130 wall layer units. Further examination
of the table indicates that the skewness, flatness, and the variation coefficients are very
similar to those of the streak pair spacing as shown in Table 3.12. Figure 3.20 presents
the comparison of the average pocket width between the current data and the outcomes of
some previous experiments in a log-log scale. It shows that our data is well in the range

of the measurements of many others.

Table 3.13 Characteristics of w+ distribution as shown in Figure 3.19

 

 

Characteristic N Mean Mode 6 \y S ’ F

 

Value 195 81.8 75 17.6 0.22 0.93 4.08

 

 

 

 

 

 

 

 

3.2.4.2 Period of Occurrence

In experiments designed to examine only the wall region (i.e. using the film
containing only plan view), the measurement of the time (tp) between pocket creations
has been carried out. The counting scheme employed to determine the frequency of
pocket occurrence is similar to that of counting typical eddies as described in Section
3.2.2.5. The pockets were counted when they crossed the measuring station which was a
pre-chosen imaginary point located at about 120 x+ downstream of the second wall slit.
In order to collect a large enough size of samples, three measurement positions have been
adopted. They were about 275 wall layer units apart in the spanwise direction, so that the
pockets detected at each of these measuring stations provided independent data for the

ensemble average.

The probability density distribution of tp's, in dimensionless form, and two

density functions are shown in Figure 3.21. It is found that the distribution of tp+'s is

55

better represented by a log-normal density function with a mean of approximately 30.
Table 3.14 lists the principal characteristics of the samples, indicating that the data
exhibit a manner analogous to that of the occurrence period for typical eddies which
propagated over the near wall region (see Table 3.9). It appears that both the t1,+ and tm+
have distributions conforming to log-normal behavior, which are highly skewed to the
region of greater value than the mean. The slight difference of the expectation values is
due to the fact that a portion of the typical eddies did not have strong interaction with the
wall layer leading to pocket formations. Figure 3.22 presents the comparison of the mean
t,;" between the current data and the results of some previous experiments in a log-log
scale. It shows a good agreement with the recent data measured over a smooth wall at R9

close to 800 (Yoo 1990).

Table 3.14 Characteristics of t1,+ distribution as shown in Figure 3.21

 

 

 

 

 

 

 

 

 

 

 

Characteristic N Mean Mode 0' \y S F
Value 183 29.8 13.5 24.3 0.82 1.91 6.97
Warts

3.2.5.1 Typical Eddy and Sublayer Streak Pair

Samples in which both a typical eddy and a sublayer streak pair occurred
simultaneously are gathered to examine their length-scale correlation. Figure 3.23 shows
the correlation of the streak-pair spacing and the spanwise diameter of the typical eddy,
plotted as l+ versus Cf, as well as the histograms of both scales. Sample size is 273.
The data indicate that, in addition to the correspondence of streak pair spacing and

typical eddy mean scales (the typical eddy is slightly larger than the streak pair it

 

56

creates), statistically the spacing of sublayer streak pairs increases as the physical size of

the corresponding typical eddies increases.

For comparison, subset of the data obtained from the ring vortex/moving wall
simulations (Chu and Falco 1988) for 3 degree incident angle and Stokes layers of
thicknesses ranging between 32 and 47 viscous units are also presented in the figure.
Note that the simulation data in which both the diameter (D) of ring vortex and the
streak-pair spacing (1*) that were greater than 200 viscous units are not shown, simply
because the nature seldom or never creates those situations for the Reynolds number on
the present experiment. Since the simulation results provided data of 1+ against D+, it is
important to display them in the current coordinates. In the present set of samples, Cz of
typical eddy was measured after the onset of the interaction (recall that the transverse
laser plane was set downstream of the designated area). On the other hand, the diameter
of the ring vortex (D) was recorded at the time when it was generated as an axisymmetric
form. It is rational to believe that the ring vortex would have not been as symmetrical as
it had been when it impinged the wall and stretched the lower part of the vortex laterally.
Additionally, if the diffusion effect PM) is also taken into account, although it might
be small in this case, the diameter of the ring vortex would have been increased.
Accordingly, one would expect that a corrected comparison would involve shifting the

simulation data to the right (i.e. towards large value of Cz+) to some degree when fitting

into the coordinates of Figure 3.23.

However, examination of both correlations suggests that they were similar events
(in the sense of interaction mechanism) in spite of that the details between the turbulent
boundary layer and the laminar simulation were appearing significantly different. The
trend of our correlation conforms well with that of the simulation data in the

neighborhood of the peak occurrence and the region below it. Nevertheless, in the region

57

of greater typical eddy size (for instance, Cz+>120), the rate of growth for the

corresponding streak-pair spacings falls short when compared to the simulation data. An
important reason contributing to this is that those typical eddies of greater size were also,
on average, located with greater distance from the wall in the turbulent boundary layer
experiment. Therefore, the local wall areas underneath them being influenced were
correspondingly smaller than the dimensions of the typical eddies. In the physical
simulations, ring vortices of any size were, however, able to be forced to convect over
regions adjacent to the wall. Furthermore, our experiments have been carried out in a
much more perturbed environment than that of the physical simulations. The higher
strain rate and lateral disturbance could have predominantly governed the shape and
evolution of the sublayer features. For example, the spanwise 'stretch' of a developing
streak pair, resulting from a large typical eddy convecting over the wall, could have been
constrained if another typical eddy was also creating sublayer features to the side of it but
in the same neighborhood. Contrarily, in the physical simulations, the wall-layer features
were able to form freely in the absence of the side perturbations. Intuitively, it is believed
that the constraints on the spanwise spacings of streak pairs due to the lateral
disturbances will be more severe on the events of greater size than on the smaller ones.

Coincidentally, our results also show this trend.

Figure 3.24 shows the 2-D probability density contour map as well as the 3-D
probability density distribution of random variables, l+ and Cz+. It appears that the peak

occurrence is located at Cz+=92 and l+z72. Figure 3.25 shows the dependence of the

streak-pair spacing on the spanwise diameter of the typical eddy at R9==805. The reduced

data are obtained by grouping the samples based upon the binning, with various bin

widths, of Cz's. The figure also shows that a linear approximation reasonably fits,

 

 

 

stra
(lat;

ftla

58

particularly in the high occurrence region, between 1* and C241 The linear relationship is

written as follow.

I+ = 0.29 C; + 45 , for 50 < C; < 200

3.2.5.2 Typical Eddy and Pocket

Analogous to the previous analysis, samples in which both a typical eddy and a
pocket occurred simultaneously were collected to examine their length-scale correlation.

Figure 3.26 shows the dependence of the pocket size (w+) on the spanwise diameter
(Cf) of the typical eddy, as well as the histograms of both scales. Sample size is 173.

The data were smoothed by performing several binning processes based upon C21”.

Observation of the data indicates an apparent correlation existing between w+ and C21“. It

shows that statistically the pocket size increases as the physical size of the corresponding
typical eddy increases. Moreover, it demonstrates a trend similar to that of the typical
eddy/streak pair correlation depicted in Figure 3.25, although the growth rate of the
pocket size is slightly greater than that of the streak-pair spacing for the larger typical
eddies. The argument presented in previous section to explain why the size of the
sublayer streak pairs did not grow as much as the size of the typical eddies in the large
Cz region is also applied here for the pocket formations. At this point, we can see that
both the streak pairs and the pockets are strongly correlated with the typical eddies. The
straight line in Figure 3.26 represents the first order least square approximation of all raw

data (un-binned data), which gives a reasonable fit between w+ and Cf. The linear

relationship is written as follow.

w"' = 0.355 CZ+ + 46.5 , for 50 < Cz+ < 200

59

Figure 3.27 shows the 2—D probability density contour map as well as the 3-D probability

density distribution of random variables, w'*' and C211 The peak occurrence is located at

Cz+~89 and w+~69.

3.2.5 .3 Pocket and Sublayer Streak Pair

Samples in which both a pocket and a sublayer streak pair occurred
simultaneously, with the condition that a typical eddy was simultaneously propagating
over the wall above them, were gathered to examine the length-scale relationship
between the pocket and the sublayer streak pair. Figure 3.28 shows the correlation data,
plotted as w"’ versus 1"”, and the histograms of both scales. Sample size is 149. A fairly
strong grouping is apparent. It also indicates that the majority of the data participate in a
trend except two samples that are located in the region of l"’/w+ ratio close to a factor of
2. Figure 3.29 shows the 2-D probability density contour map as well as the 3-D
probability density distribution of random variables, w‘*' and 1*. It is found that the peak

occurrence is located at I+=71 and w+==75.

It is frequently helpful in analysis to search for an expression in describing a
given set of data. Nevertheless, care must be taken to find such an expression since the
present set of data is not a monotonically ascending or descending function based upon
either scale, thus the binning or curve fitting based upon either scale as the independent
variable will result in very different outcomes. Referring to Figure 3.28, two straight
dashed lines stand for the first order least squares fits of the data using either scale as the
independent variable (1* for the line with lower slope and w"' for the other). It is found
that both of the linear fits are biased, failing to represent the overall trend of the data set
satisfactorily. This is because that use of only (““3 and w""s results in mathematical

representations which are totally blinded to a third feature that governs the two scales. In

60

this case, the proper way to accomplish the correlation is to utilize the individual length-
scale correlations of each feature (the pockets and the long streak pairs) with the typical
eddy, which is responsible for the formation of these sublayer features. Referring to
Sections 3.2.5.1 and 3.2.5.2, the expressions of 1+ and w+ as functions of Cz+ have been
established Therefore, the correlation of the pocket size and the streak-pair spacing,
under the condition that a typical eddy was observed to convect over the wall above

them, is readily derived as follows:

w+ = 1.2251"' - 8.6 , for 30 <l+< 150; 30 < w+< 140

It is very important to emphasize that this expression can not be obtained by
solely using the information of w+ and 1+. The above linear representation is also plotted,
as a solid straight line, in Figure 3.28. It appears to be the best fit among these tluee
linearities with regard to the overall trend of the scattered data. This evidently shows that
the information about the typical eddy is required when characterizing the relationship
between the sublayer streak pairs and the pockets. Examination of the linear
representation suggests that the lateral dimension of both sublayer features are almost
identical in the vicinity of the peak occurrence and the region below it. In the region
beyond the peak occurrence, the pocket size gradually becomes greater than the streak-

pair spacing.

Figure 3.30 shows the same linear representation and a data set that comprises
samples in which both the sublayer features were formed in the absence of a typical eddy
occurrence. Sample size is 18. It is interesting to note that this data set appears to
conform well with the trend of its complementary data. This indicates that these sublayer

events might have also related to typical eddies. A hypothesis suggesting why we were

61

unable to identify the initiating motions is that they lost their characteristic structures or

distorted significantly due to the interactions before the observation station.

325T] BEE [I . 1E1! D. E l 1!!“
3.2.6.1 On the Overall Length Scale Correlation

It is understood that the distance of a typical eddy from the wall is also capable of
affecting the length scale of the sublayer feature(s) that it creates. In this section, samples
which contain all the three coherent features (i.e. simultaneous occurrence of a typical
eddy, as well as a pocket and an accompanying streak pair) are analyzed to uncover their
overall scaling correlation, including the distances of typical eddies from the wall. Figure
3.31 shows the dependence of the pocket size (w+) on both the distance (d+) and the
normalized distance (d*) of the typical eddy from the wall. The histogram of w+ as well
as the upper/lower standard deviation boundaries for (1+ and d* are also provided to show
the degree of the data dispersion. In addition, the average spanwise scales of the
corresponding typical eddies and the sublayer streak pairs, C; and 1+, for each data
point are listed near the top of the figure (standard deviation in brackets). Total sample

size is 149.

Examination of the data based upon d+, one could be misled by the fact that the
pockets and the sublayer streak pairs (see l+'s near the top of the figure) of greater
dimension were, on average, associated with typical eddies positioned further away from
the wall. It is necessary to recall that these typical eddies of greater distance from the
wall also, on average, possessed greater physical size. This, however, raises the question:
What is the criterion of the maximum distance (d+) in which a typical eddy is able to

create biggest sublayer feature(s), or is simply too far away from the wall to make any

62

disturbance at all? The ambiguity between the distance of the center of a typical eddy to
the wall and the distance from the "bottom" of the typical eddy to the wall for different
size eddies can be resolved by introducing a normalized parameter d* (i.e. d+-0.5cz+)
which takes the magnitude of the typical eddy into account. In the same figure, the data
points based upon d* reveal a striking feature showing that essentially all the typical
eddies responsible for the formation of the pockets and the sublayer streak pairs were
equally close to the wall, with d*=50. The invariant d* suggests an average height
condition that typical eddies must comply with in order to initiate a disturbance on the
wall layer. Figure 3.32 shows the details of the correlation by refining the binning
process with various bin widths. Results preserve the apparent constancy of d*. The data
representing the mean value of d* for the events that only pocket was formed is also
marked, which is slightly less than y"'=50. Further discussion of this feature will be made

in next Chapter.

With these findings, the underlying physics that describes the size correlation of
the typical eddies and the sublayer features is more readily identifiable. Since the
bottoms of these typical eddies were, on average, equally close to the wall, some of them
were able to create bigger sublayer feature(s) simply because the larger physical size they
had (see Cz+'s near the top of the figure). This statement, however, can not be seen as
conclusive. Although it describes an important feature necessary to uncover the overall
size correlation, we have not factored in the intensity of the typical eddies, which should
be a crucial factor of the interaction process. The strength of the typical eddies was not
obtainable in this study. In the next section, we will see that the use of d* is more
advantageous than using d+ when the measure of a typical-eddy's ability to initiate an
interaction with the wall is considered. Further observation of Figure 3.31 indicates that a

linearity between w"' and d+ exists. It is found that the slope representing d"'/w+ is about

63

0.33 over a range between 50<w+<l40. In general, Figure 3.31 shows the existence of an
overall length-scale correlation between typical eddies, pockets, and sublayer streak

pairs.

3.2.6.2 On the Ability of Initiating an Interaction

Three subsets of our data base were studied to see how the distance of a typical
eddy from the wall could affect the ability of the typical eddy to initiate an interaction
with the wall layer. These three subsets consist of samples which are selected based upon
the simultaneous occurrence of (i) typical eddy and pocket, (ii) typical eddy and streak
pair, and (iii) typical eddy and either streak pair or pocket, respecrively. Note that subset
(iii) is the union of subsets (i) and (ii). It is important to point out that if a pocket and/or a
long streak pair in the sublayer, and a typical eddy in the log region or the lower portion
of the outer region were visualized to occur simultaneously in accordance with our
sampling criteria, the sublayer feature(s) were assumed to be created as a result of the
interaction between the typical eddy and the wall layer. Discussions concerning the

legitimacy of the interaction hypothesis will be made in next chapter.

Figure 3.33 presents the probability of the occurrence for each of the above
interactions as a function of d* and d+. Total sample size is 417. Generally, it shows that
the probability for a typical eddy to interact with the wall increases as its distance from
the wall decreases. Further examination of the data indicates that, as (1* of a typical eddy
decreases, the possibility of forming a pocket and/or a sublayer streak pair increases
mcnmcnicauy. On the other hand, if using d"' as the measure, the probability of the
interaction reduces when d+ decreases in the region of d+<60. This demonstrates the
apparent advantage of using d'" instead of d+ in this region. The magnitude of d* is a

better direct measure in indicating whether an interaction would take place or not

64

(regardless of the size of the created sublayer features), particularly for the interactions
involving the formation of a pocket (i.e. subset i; as defined in previous paragraph). The
reader is reminded that, according to the results of the previous section, the spanwise
dimensions of the pocket and the sublayer streak pair are found to depend primarily on
the magnitude of Cf. Summarizing of these findings, when considering the interactions
between typical eddies and the existing wall region vorticity distribution, d* of the
typical eddy plays a decisive role in determining whether an initiating interaction with
the wall layer will occur, whereas the diameter (Czl’) of the typical eddy dictates the

lateral dimension of the interaction.

An interesting feature in Figure 3.33 worth mentioning is that the probability of
the typical eddies having an interaction with the wall layer goes statistically to zero when
they were convecting at distances greater than 155 or 218 wall layer units from the wall
based upon d* or d+ respectively. This is consistent with the data of two-point
correlations obtained by Klewicki (1989). He showed that negative spanwise vorticity
fluctuation correlations for positive Ay separations and the stationary probe positioned at
y+=7.5 were weak when the separations were greater than 100 Ay+ (correlation
negligible when Ay+ was greater than about 150) at a comparable Reynolds number (see
his figure 4.13). Furthermore, Figure 3.33 also shows that the probability of creating a
sublayer streak pair was always greater (or, at least, equal) than that of creating a pocket
for a typical eddy at a given distance (or normalized distance) from the wall.

The importance of the typical eddy distance from the wall is now further
discussed. For convenience, the evolution of the typical eddy/wall layer interactions in
turbulent boundary layers is divided into 4 classes based upon the formation of sublayer
features. They are as follows.

(1) formation of a pocket only

65

(2) formation of a streak pair only

(3) formation of both a pocket and a streak pair

(4) neither a pocket or a streak pair was formed (i.e. no interaction)
The samples corresponding to each class of the interactions make up a subset of our data
base. It is noted that the intersection between any two subsets described above is a null
set. Figure 3.34 shows the probability density distributions of the specified interactions
based upon the distance d+. The representing distribution associated with each class of
interactions is the best fit among the normal, log-normal, and the Rayleigh probability
density functions. The sample size, principal properties, and the name of each
representing distributions are tabulated in Table 3.15. Refening to the table, we can see
that about 71% of the typical eddies were able to interact with the wall layer (i.e. union
of classes 1, 2, and 3). Among the interaction classes, the event of the least probability of
occurrence concerned the interactions in which only pockets were formed (i.e. 5.8% for
class 1). Further examination of the table indicates that sublayer streak pairs were formed
by typical eddies which were, on average, propagating at greater distances from the wall
than that of typical eddies which created pockets. The same feature is also reflected in the
peak occurrence as shown in Figure 3.34. This result along with the previous findings
from Figure 3.33 are consistent with the observations of Chu and Falco (1988). In their
physical simulations, the creation of the wall-region long streak pairs occurred even
when the ring vortices were quite distant from the wall, well into the outer part of the log
region. All of these suggest that a typical eddy that convected over the wall close enough
to create an interaction would initially form a pair of streaks. Pockets would have also
been created only if the typical eddies continuously moved towards the wall. It is
interesting to note that the probability density function associated with the interactions
wherein only streak pairs were formed (i.e. class 2) is best represented by a normal

distribution with an amazingly low skewness.

66

The probability density distribution, which describes the events associated with
the class in which only pockets were formed (i.e. class 1), indicates that these typical
eddies were, on average, the closest to the wall, and did not result in the formation of
streak pairs. This may suggest that these typical eddies were brought, by a strong
instantaneous wallward flow field, close to the wall with an incident angle steeper than
the normal for typical eddies, so that the interaction became almost like a quick vertical
impact on the wall by the outer layer high speed fluid. Consequently, both the interaction
time and the region over which the interaction occurred were too short to enable the
formation of a pair of streaks of sufficient length that they were able to be differentiated
from the pocket interaction. Although, as the table indicates, this class of interactions had
the lowest probability of occurrence in turbulent boundary layers, its significance in
physics is not allowed to be overlooked because this class might result in the strongest

interactions of all classes and hence produce high Reynolds stress near the wall.

Figure 3.35 shows the probability density distribution of each class of interactions
based upon the normalized distance d*. The sample size, principal characteristics, and the
name of each representing distribution are tabulated in Table 3.16. It is found that all
distributions are best fitted by Rayleigh density functions except the data associated with
the events that neither a streak pair nor a pocket was formed (i.e. class 4) when a typical
eddy propagated over the wall. Observation of the table shows that the trend of the mean
d*'s for various classes of interactions is similar to that of mean d+'s, thus the physics
drawn from the previous figure/table are also applied when d* is used as the measure of
the distance between typical eddy and the wall. It is also found that the representing
probability density functions and their peak occurrences for interaction classes 1, 2, and 3
shown in Figure 3.35 are unable to be clearly distinguished (i.e. the distributions are very

similar). This shows that the employment of d* provides a measure in determining the

67

possibility of interaction between typical eddy and the wall layer without regard to what
class of interactions might occur. It is believed that the class of an interaction was mainly
decided by not only the incident angle of the typical eddy as mentioned earlier, but also
the intensity of the typical eddy and the local vorticity distribution in the sublayer, which

were not obtainable through our visual investigation.

Comparison of Tables 3.15 and 3.16 indicates that, for each class of interactions,
the difference of F- E is approximately 50.5 wall layer units, which is exactly one
half of the mean Cz+ for all typical eddies. This constant difference suggests a strong
geometric similarity in typical eddies among the different classes of interactions. Further
examination of the present data suggests that the typical eddies which did not create
sublayer streak pairs or pockets (i.e. case 4) were, on average, located way beyond the
distances, or normalized distances, of typical eddies that interacted with the wall layer.
Additionally, both of the probability density functions based upon d+ and d* associated
with this no-interaction class are represented by the ncnnal distributions. The underlying
physics for the distance and the normalized distance of typical eddies from the wall being
the normal distributions is tentatively interpreted as follows. It is known that the
distribution of a random variable is sometimes dependent on the magnitude of the
random variable. For example, the log-normal distribution is formed when the influences
on the random variable are proportional to the magnitude of the variable, whereas the
influences on a normally distributed random variable are independent of the variable
magnitude (see Aitchison and Brown 1957). Usually, the distribution of a random
variable with non-negative value and potentially unconstrained upper limit is non-
symmetrical owing to the constraint of the lower limit (i.e. zero, an impossible
occurrence of magnitude for the random variable) such as the personal incomes which

are log-normally distributed. In the events of no-interaction, the symmetrical feature of

68

the normal distributions for random variables d"' and d* apparently shows that the lower
limit (zero) of the random variables did not influence the distributions. Physically, it
seems to indicate that the presence of the wall (zero) did not affect the distribution of the
typical eddies which did not interact with the wall layer. In other words, these typical
eddies did not sense the existence of the wall region vorticity distribution or the

impermeability condition, so that no interaction could possibly occur.

Wm

In Sections 3.2.2.5 and 3.2.4.2, we have presented the characteristics of the
occurrence periods, tm+ and tp+, for typical eddies and pockets respectively. It is noted
that those results were obtained by performing two independent countings in the cross-
stream view (for typical eddies) and the plan view (for pockets). In this section, we will
utilize the correlation data to study the temporal relationships associated with the
occurrence of the sublayer features and the typical eddies. The procedure adopted for
obtaining the period of occurrence of the correlated events was similar to that described
in Section 3.2.2.5, except that the additional constraint that the occurrence of the sublayer
features was also enforced. In order to collect a large enough size of samples, four
measurement positions have been employed. They were located about 200 wall layer
units apart in the spanwise direction, so that the events detected at each of these
measuring stations provided independent data for the ensemble average. Figure 3.36
shows the probability density function of the period of occurrence (At"‘) for each
specified correlation. The probability density representations of tam+ and t1,+ from
Figures 3.15 and 3.21 are also plotted for comparison. The set number, sample size,
principal properties, and the name of each distribution are tabulated in Table 3.17. Note

that the probability density function which represents each data set is chosen from the

69

best fit among the normal, log-normal, and the Rayleigh distributions. It is found that all
of the data sets are best fitted by log-normal distributions except data set #3. In addition,
the common features of these log-normal distributions are high positive skewness and
variation coefficients (i.e. the mean value and the standard deviation are the same order
of magnitude) as shown in the table. The distribution for data set #3 is represented by a
normal density function having a relatively low standard deviation. However, the small
sample size and the low flatness for this events indicate that the representing distribution
is merely a rough approximation. Note that the intersection of sets #3, 4 and 5 is a null

SCI.

The period of the simultaneous occurrences of a typical eddy and a pocket (no
streak pair) is described by the properties listed in set #3 of the table. It is found that, on
average, about every nine appearances of typical eddies (within the first 300 y“) resulted
in an interaction in which only a pocket was formed in the sublayer (i.e. average
At+ltm+==9.3). However, examination of its probability density representation in Figure
3.36 suggests that interactions of this class took place in a random manner, showing no
clear trend that enables us to predict when it will occur. Refening to the table, the
occurrence of the events that only a streak pair was formed when a typical eddy
convected over the wall is characterized in set #4, showing that statistically the
interactions of this class occurred by a rate nearly twice higher than that of set #3
interactions. Further observation of the table indicates that the interactions which resulted
in both a pocket and a streak pair formations (set #5) took place about every 80 viscous
time units. A very important feature of this distribution is the mode characteristic, which
shows that the most probable period of these correlated interaction events is almost

identical to that of the individual occurrence of Ira-Ts or tp+'s. This suggests that the

majority of the pocket formations were coincident with the typical eddy appearances in

70

the near wall region. It also infers that the typical eddies, which created pockets, usually
formed streak pairs as well. This again supports the strong correlation of the presence of

typical eddies and the formation of sublayer features.

Ilt‘ t'. ‘0 u -. _.U .JI'l.-.!.I.l -' ONO-40!... ’,,|o.' ‘-. .r‘

In Section 3.2.2.4, we have classified the typical eddies into six modes according
to their inclination angle with respect to the wall in the x-y plane. In this section, we will
present the data to show how the interaction could be affected when the orientation of the
typical eddy is taken into consideration. Table 3.8 tabulates the statistical features for
each mode of the typical eddies. The table has been divided into two divisions based
upon the sign of the vorticity in the lower portion of the typical eddy. The first three
rows present the data of Mode 1, 4, and 5 typical eddies which possessed vorticity of the
opposite sign as that of the mean shear in their lower lobes. On the other hand, statistics
for typical eddies of Mode 2, 3, and 6, which were oriented so that the vorticity in their
lower portions had the same sign as that of the mean shear are listed in the last three

rows.

Examination of the table indicates that the majority of the pockets (84.1%) were
created by typical eddies of the first division. Contrarily, only 9 out of 195 pockets
(4.6%) were observed to form as the typical eddies of the second division passed over the
wall. This reveals a strong dependence between pocket formation and the typical eddy
orientation. Note that, in our visual study, 11.3% of the pockets were observed to form in
the condition that no typical eddy was convecting over their tops. Results also show that
the probability of interacting with the wall layer (i.e. creating either pocket or sublayer
streak pair) for typical eddies of the first division was always greater than that for typical

eddies of the second division. However, it is interesting to point out that the probability

71

of creating pockets for typical eddies of the first division was greater than that of the
second division typical eddies with a ratio of nearly 3 to 1 (i.e. 45.7 : 15.5), but the ratio
dropped substantially to approximately 7 to 4 (i.e. 69.6 : 39.7) for the probability of
typical eddies to create sublayer streak pairs as shown in the last column of the table.
This seems to indicate that the dependence of the streak pair formation upon the typical
eddy orientation was considerably weaker than that of the pocket formation. Finally, we
must bear in mind that the distance, or normalized distance, of the typical eddy from the
wall has also been shown to be able to affect the probability of interaction. It is found
that both the distance (d"') and normalized distance (d*) from the wall for typical eddies
of the first division were, on average, less than those for typical eddies of the second
division. This suggests that the strong ability of the first division typical eddies to form
pockets are due to at least two characteristics of the typical eddies, the (normalized)
distance from the wall and the orientation. Further discussions concerning the
mechanisms responsible for the creation of the sublayer features will be made in next

chapter.

3.3 Templates of Flow Visualization
BEIIJIMHE . 5! [II .

In experiments designed to investigate only the wall region, observations
indicated that the sequential evolutions described by the wall region subset of the Overall
Production Module (OPM; see Falco et al. 1989a, and Falco 1991) frequently occurred.
Figure 3.37 shows an example of the observations in the plan view. The flow is from left
to right and the first dye slit is located near the leftmost edge of the pictures. The
production sequence starts with the formation of a pair of low-speed streaks via the

rearrangement of the red dye emanating from the first wall slit. There are, of course,

72

some other dye rearrangements taking place in the pictures due to either parts of similar
events and/or interactions with the current event. The streak pair in Figure 3.37 has an
average spacing along its length of z+s73, which is nearly equal to the mean spacing of
our conditionally sampled streak pairs. It is also found that the pair of long streaks has
developed over the distance between the two wall slits (i.e. #5440) in only the first
three frames. The time needed to accomplish this is in the order of 20 viscous time units.
By the fourth frame, it can be seen that the streak pair is readily marked by the dye
emitting out of the second wall slit and a pocket is forming near the downstream end of
the streak pair. The continuation of the streak-pair formation shown in the pictures by the
two colors of dye infers that the mechanism responsible for this evolution is still active.
By the fifth frame, the pocket is clearly seen. The pocket develops through frames four to
eight (or nine) and becomes a fully formed pocket (stage III or IV) in about 20t+ time
span. Additionally, the lifetime of the pocket (from frame four to the last frame) is found
to be approximately 30t+. These time scales are consistent with the temporal
characteristics of pocket evolution depicted in the paper of Falco (1980). The maximum
spanwise dimension of the fully developed pocket in frame nine is about 802*.
Furthermore, the duration for the entire evolution of the wall-region subset of the
interaction can be estimated from this example to be in the order of 60t+. Note that this
time scale can only be treated as a rough estimate, because how well it represents the
overall duration of the interactions is not yet known. It is interesting to point out that the
flow visualization in the near-wall region using oil droplets and a laser sheet parallel to
the wall at y+=8 by Bessem et al. (1989) has clearly shown the spatial phasing between
the sublayer streak pair and pocket in a stage comparable to the sixth or seventh frame in
Figure 3.37. The picture (figure 3 of their paper) was presented in order to demonstrate
the quality of a visualization technique, which accidentally provided a very illustrating

scenario for the typical spatial phasing of the sublayer features.

73

Figure 3.38 shows another example in describing the sequence in which the flow
development in the vicinity of the pocket is focused. The flow is also from left to right.
This sequence begins with the formation of a streak pair which has been developing in
the region between the first and the second wall slits. Note that the prints have been
enlarged to depict the details of interest so that the first wall slit is not shown in the field
of view. The first frame starts the sequence at the time when a pocket has just formed
near the downstream end of the streak pair. It is also found that secondary hairpin
vortices associated with the pair of streaks appear in the region immediately upstream of
the pocket. The stage of the evolution presented in this frame is almost equivalent to that
of the fifth or sixth frame in the previous example. In the subsequent frames, we can
observe the development of the secondary hairpin vortices. By the second frame, the dye
indicates that the pair of hairpin vortices has moved over the edges of the pocket because
the lifted hairpins has undergone greater convection velocities than the pocket. In the
third, fourth, and fifth frames, the pair of hairpins appears to be rotated towards the
center of the pocket. The last two frames show that the twisted hairpins are being induced
towards the wall into the pocket center. It is important to point out that the entire
sequence of the secondary hairpin rotation takes about only t+=10 (frames third to
seventh). Although the intensity of the pocket vortex is unable to be measured using dye
technique, the rapid evolution clearly suggests that a very strong vorticity exists along the
upstream boundary of the pocket. Note that the average streak-pair spacing and the
pocket size in the third frame are approximately 94 and 118 wall layer units respectively.
Furthermore, the average thickness of the hairpin-vortex filament is found to be of the

order of Kolmogorov scale (about 5 to 10 wall layer units).

74
HZIII'IEIIISH I .

Before examples are shown, it is necessary to note that the visual data to be
presented below are prints selected from split screen images which initially had three
simultaneous views (i.e. using Method 1, see Section 2.4), thus the magnified portions
shown have higher grain than similar views would show in a single view experiment.
Figure 3.39 shows an example illustrating the simultaneous plan and cross-stream
observation of a typical eddy/wall layer interaction. The flow is from right to left for the
plan view and into the paper for the cross-stream view. The scale in these two views are
different due to the additional optical magnification for the cross-stream view (see Figure
2.3). The scale ratio between the plan view and cross-stream view is about 1.5 : 1. Note
that side views are not shown since the interaction was not occuning nearby the
streamwise laser sheet. In frame A, the plan view shows the development of a pair of
streaks (indicated by the black arrows). During this exposure, no organized motion above
the wall can be identified at the corresponding location of the cross-stream view. In
frames B and C, the formation of a pocket, in the designated area, near the downstream
end of the streak pair can be seen in the plan view (also indicated by the black arrows),
whereas there is still no organized motion identifiable in the simultaneous cross-stream
observation. In the next four frames (i.e. frames D to G), we can see that an intermediate
scale vortical motion appears in the cross-stream view (indicated by the white arrow),
whose spanwise location is approximately aligned with that of the sublayer streak pair
and the pocket. In frames E and F, this vortical motion appears as a streamwise counter-
rotating vortex pair. Its spanwise diameter is measured to be about 120 viscous units (i.e.

Cf). In the last frame, the cross-stream view of the vortical motion is no longer seen.

Therefore, the streamwise extent of this vortical motion can be estimated. It is found to

be about 102 viscous units (i.e. Cx+). This indicates that the vortex pair is neither a long

75

axial counter-rotating vortex pair nor the stretched legs of a hairpin. Additionally, the
consecutive cross-stream view of the vortical motion is similar to the sequential cross-
stream geometry of a typical eddy as sketched in Figure 3.14. All of these suggest that
the observed vortical motion is a ring-like typical eddy crossing the transverse laser
plane. The temporal and spatial phasing of the typical eddy and the sublayer features
indicate that they are correlated events as described in the conditional sampling criteria in
Section 2.5. In this example, the pocket size (w"'), streak-pair spacing(l+), and the height
of the typical eddy (d+) are measured to be approximately 80, 68, and 115 respectively.

Figure 3.40 shows another example illustrating the formation of a pocket by a
typical eddy, and the subsequent interaction between the typical eddy and lifted sublayer
fluid near the downstream end of the pocket. The pictures show the simultaneous side
and plan views of the interaction sequence. The corresponding transverse views are not
presented because the interaction took place in the neighborhood of the intersection of
the laser sheets, where the excess brightness makes the visualization of the typical eddy
in the transverse view impossible. The flow is from left to right and the first dye slit is
located at the leftmost edge of the pictures. The sequence starts with the presence of a
typical eddy in the vicinity of the wall (indicated by the white arrow), and no apparent
sublayer feature appears under it (frame one). The typical eddy in this frame might have
just moved into the streamwise laser sheet from the far side as suggested by both the
location of the pocket in the second frame and the poor conformation of the typical eddy
in the first frame. In frame two, the formation of a pocket flow module near the center of
the designated area is clearly observable in the plan view (indicated by the black arrow),
although the image is partially blocked by the brightness of the laser sheet. In the
meantime, the typical eddy is located at the intersection of the laser sheets, which prevent

it from being seen in the side view. Whether or not there is a sublayer streak pair formed

76

upstream of the pocket cannot be determined from the plan view because the presence of
the streamwise laser sheet disrupts the view of streak pair observation. In the third and
the fourth frames, the typical eddy, which moved out of the transverse laser sheet, and
the lifted sublayer fluid, as well as a hairpin vortex (dyed in red) in the downstream end
of the pocket are clearly seen. In the fifth frame, the interaction between the lifted
sublayer fluid and the typical eddy results in the sublayer fluid being ingested into the
typical eddy. As a result, both the typical eddy and the lifted sublayer fluid lose their
coherent appearances (see the last frame). This interaction is similar to the Type III

interactions as classified by Chu and Falco (1988, also see Liang et al. 1983).

It‘ ’dt t-CJ 0 ' ‘.0 12-10; in Hit-.H 0 k'. ' a. _.00.'

Figure 3.41 shows sequential pictures of the evolution of a hairpin vortex into a
new typical eddy. In these pictures, the development of the hairpin vortex, which initially
formed in the region immediately downstream of the pocket in the last example (i.e.
Figure 3.40), is discussed. The flow is from left to right. Note that only the side view is
shown since the observation of the hairpin evolution is primarily observable in the side
view. The fust frame of this figure is about 37t+ later than the last frame of Figure 3.40.
Through frames two to seven, one can easily observe a new typical eddy emerging from
the wall layer fluid through a pinch-off of the lifted hairpin vortex (indicated by the
white arrow). By frame four, a clearly defined lower lobe can be seen. In the last few
frames, the newly formed typical eddy is observed to evolve from Mode 1 into Mode 5
before it is out of sight. The rapid change of the typical eddy orientation in such a short
time scale was due to the strong shear of the mean velocity in the near wall region. This
example shows that the primary hairpin which is formed as a result of typical

eddy/sublayer interaction may evolve into a new typical eddy. The time needed for this

77

evolution is about 160t+ (i.e. the time between the third frame of Figure 3.40 and the
fourth frame of Figure 3.41).

Figures 3.42 and 3.43 show two other examples of typical eddy formation
process. Only side views are shown. The flow is from right to left and the rightmost edge
of the pictures is the location of the cross-stream laser sheet. In both figures, the process
starts with a hairpin being formed near the wall as indicated by the white arrow (frame
one). In the subsequent frames, we can clearly see the formation of a new typical eddy
from the lifted hairpin. By frame six of Figure 3.42 and frame seven of Figure 3.43, the
clearly defined lower lobes are readily observable. Coincidentally, the evolution time for
the pinch-off phenomena in both examples are nearly identical, which is approximately
133t+. The examples shown in this section indicate the type of visual evidence
supporting the hypothesis that typical eddies are produced in the near wall region via
pinch-off of wall layer fluid, which also provides a link between the typical eddies
observed by Falco and the hairpin vortices observed by others.

CHAPTER 4
DISCUSSION

4.1 Scaling of Sublayer Streak Spacing

In Section 3.2.3, we have pointed out that the conditional sampling of the pairs of
long sublayer streaks results in a mean spanwise spacing that apparently differs from the
overall streak spacing of the unconditionally sampled long sublayer streaks. Here, we are
able to use our statistical results to present an argument addressing that difference; it will
be shown that the mean streak-pair spacing essentially provides complementary
information for understanding the underlying process of the wall-region long streaky

structure spacing in turbulent boundary layers.

Suppose a pair of long low-speed streaks was formed as a result of a typical
eddy/wall layer interaction (the formation of pocket is not the main issue here). Our
statistical results suggest that the sublayer streak-pair spacing and the spanwise
dimension of the typical eddy would be, on average, 75 and 100 wall layer units
respectively. Results also show that the average center-to-center spanwise spacing
between two neighboring typical eddies was about z+=244. The available knowledge for
the length characteristics is summarized by the scenario shown in Figure 4.1, which
illustrates the phase relationship of two typical eddies crossing through the cross-stream
plane simultaneously. The characteristic parameters of the flow structures are assumed to
be random variables with fluctuations about their mean values. Note that only two typical
eddies are sketched although others might generally be found in this view. Since almost

all the typical eddies sampled were close to the wall, it is rational to assume that they

78

79

were capable of initiating interactions with the wall layer in the vicinity of the cross-
stream laser plane. In Figure 4.1, two pairs of long streaks created due to the interactions
are also illustrated in both the plan and cross-stream views. It has been well-known that
once a low-speed streak was formed it often extended to a very long streamwise length.
This fact strengthens the validity of the phase relationship between the typical eddies and
sublayer streak pairs shown in the figure. In the plan view, it appears that three spanwise
spacings were established between these four streaks. Therefore, the estimate of the
average streak spacing in wall layer units could be computed by a simple calculation,
318+3=106. This method is same to the classical streak-spacing measuring scheme as
described in Chapter three, which has been widely employed by many investigators (see
for example, Kline et al. 1967, Oldaker and Tiederman 1977) resulting in a hundred wall

layer units for the mean streak spacing.

The result of our scaling analysis suggests that the low-speed streaks, which are
found to occur randomly in space and time in the near-wall region by many
experimentalists, with mean spacing in the order of 100 wall layer units, could result
from distributions of streak pairs with an average spanwise spacing of about 75 wall
layer units. The coincidence of length scale strongly supports the postulate that, in
turbulent boundary layers, the sublayer streaks are formed in pairs rather than being
generated singly. As we mentioned earlier, the measurements of Smith and Metzler
(1983) clearly indicated that the most probable streak spacing was in the range of 75-80
viscous units (our unconditionally sampled data also verifying the mode is 78). This
length characteristic is nearly identical to our mean spacing of streak pairs. The
implication of this coincidence is that the majority of the low-speed streaks sampled by
Smith and Metzler were essentially paired. More importantly, our results show that these

streak pairs were formed as the result of interactions between typical eddies and the wall

8O

layer. The link between the inner and outer dynamical features, as well as the agreement
on the length characteristics in the wall region, provide statistical support for the typical

eddy/wall layer turbulence production model.

It might be questioned if typical eddies are indeed responsible for the formation
of the long streak pairs in the sublayer as our correlation data suggest. Specifically, why
is the size of a typical eddy Reynolds number dependent whereas the overall spacing of
streaks is not? This brings up the core of the question --- 'Is our assertion concerning the
mechanism for the formation of long streaks still valid when Reynolds number is taken
into account?'. A qualitative answer to this question is made as follows. As we know, at
higher Reynolds numbers, the physical size of typical eddies is, on average, greater than
that at lower Reynolds numbers when scaled with wall variables as shown in Figure 3.5.
On the other hand,'the mean streak spacing has been found to be invariant with Reynolds
numbers up to 5830 (Smith and Metzler 1983). If the trend of the size correlation
between typical eddies and sublayer streak pairs still holds at higher but moderate
Reynolds numbers, the spacings of streak pairs which are created by typical eddies at
higher Reynolds numbers must be greater than those of streak pairs at lower Reynolds
numbers. At R9=805, the average ratio of streak-pair spacing to the spacing between
neighboring streak pairs is about 1 to 2 (see Figure 4.1). At moderate higher Reynolds
numbers, this ratio should increase noticeably towards unity due to both the increase of
average streak pair spacing and the decrease of spacing between neighboring streak pairs.
This will result in an increasing number of samples with 26" about its overall mean value.
By the same token, the probability of observing samples with large 1+ that formed
between two narrow streak pairs should also be reduced Referring to the results of Smith
and Metzler (1983, figure 6), two probability density histograms for Reynolds numbers
differing by a factor of 4 are compared. It apparently confirms the trend that is described

81

above. In their figure, the log-normal probability representation of H at higher Reynolds
number appears to have smaller standard deviation (i.e. greater concentration of samples
about the mean) and skewness (i.e. shorter tail in the large 1+ region) than those at lower
Reynolds number. Although we do not yet know exactly what is the size correlation of
typical eddies and the corresponding streak pairs at higher Reynolds numbers, these
results suggest that our hypothesis would hold at higher Reynolds number.

4.2 The Formation and Evolution of Typical Eddies

In the previous chapter, we have classified the typical eddies into six
characteristic modes based upon their orientation with respect to the wall in the x-y
plane. It was found that the typical eddies existing between y+=300 and the wall were
predominantly oriented as Mode-1 typical eddies. It is also understood that the
probability of the occurrence of Mode-4 and/or Mode-6 typical eddies was nearly zero.
These statistical findings are strongly in favor of the assertion that the formations of
typical eddies are originated from lifted hairpin-shaped vortices through the pinch-off
phenomena (see Falco 1983). This is because the typical eddy that most probably would
have evolved into Mode 1, owing to the sharp mean velocity gradient, was Mode 4 which
has been, however, proven almost nonexistent. Therefore, Mode 1 should represent the
initial orientation of a typical eddy as it was being produced. Coincidentally, ring-like
vortices pinching—off from tip portions of elongated hairpin vortices are also naturally
oriented as Mode 1. Our results (see Section 3.2.2.4) also show that the average
inclination angle of Mode-1 typical eddies is very close to the characteristic angle of
hairpin vortices observed by Head and Bandyopadhyay (1981). In addition, Table 3.8
shows that Mode-1 typical eddies were the ones that were statistically located closest to

wall among the different modes (except Mode 5 which, however, did not occur often),

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implying that their close spatial relationship with the wall layer was analogous to the
nature of the hairpin vortices. More importantly, typical eddies scale on wall parameters
v and wt, and over a few decades their sizes are still close in magnitude to the Taylor
micro-scale. These facts lead to our hypothesis that the coherent typical eddies in
turbulent boundary flows are formed via the wall-region vorticity reconstruction, most
probably through the pinch-off mechanism. In the present study, hairpin-like vortices
have been observed in many occasions. They could be visually distinguished from the
typical eddies since they were dyed with red color instead of green. Some of them having
spanwise dimension in the order of 100 wall layer units were seen to be lifted as high as
150 wall layer units above the wall. In our side-view observations, many hairpins have
been clearly visualized to pinch-off into new typical eddies. The typical examples of this
phenomenon are shown in pictures embodied in Section 3.3.3. The experiments
conducted by Chu and Falco (1988) have also shown that it did occur in the interactions
between the ring vortex and a moving belt at certain initial conditions. Moreover, the
typical eddy pinch-off phenomenon has been demonstrated to take place in the

computations of Moin et al. (1986) as we mentioned in Chapter one.

Examination of our statistical and visual results along with the above supporting
information has resulted in the conceptual model shown in the side view of Figure 4.2. It
represents the deduced picture of a typical eddy evolution in the log region and lower
portion of the outer region. The figure illustrates that a typical eddy of Mode l is being
created out of a lifted hairpin vortex loop via the pinch-off mechanism. It also shows that
the typical eddy will immediately be brought either inwards or outwards by the local
instantaneous flow field At the same time, the typical eddy will, of course, be convected
downstream and possibly evolve into a Mode-2 typical eddy. Referring to Table 3.8, it

indicates that we were rarely able to see a typical eddy which had evolved into Mode 3.

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This might be because the time span required for a typical eddy to evolve from Mode 1
into Mode 3 (through Modes 5 and 2) was usually longer than the typical eddy's life time.
Needless to say this is the reason why Mode-4 and/or Mode-6 typical eddies were almost
non-existent. In the same figure, two possible trajectories (shown as dashed lines) are
sketched illustrating that sublayer feature(s) would be created if the typical eddy is
brought down low enough to interact with the existing wall region vorticity distribution.
In some cases, the typical eddy would lose its physical identity by approaching too close
to the wall. Normally, the typical eddy will start to move away, or maintain a constant
distance, from the wall after having reached the lowest point where its own speed and the
inviscid flow boundary effects have overcome the instantaneous wall-ward flow field
which has brought it down. Although the sketch suggests that the typical eddy starts to
diffuse when oriented as Mode 3, the typical eddy may, in fact, blur or even lose its
characteristic structure due to diffusion in any mode of orientation. It is important to
emphasize that the evolution of the typical eddy proposed here is mainly for those
existing in the log region and lower portion of the outer region where the mean shear
dominates the flow dynamics. Once the typical eddies are displaced beyond the outer
region by, for example, the local instantaneous flow eruptions, they may not follow the
pattern we proposed because the typical eddies are now embedded in a flow field of

higher convection velocity but much less shear.

It is interesting to point out that the hypothesis of the typical eddy origination is
also supported by our statistical results in Section 3.2.2.2. They showed that the mean
scale of typical eddies grew as the distance from the wall increased (i.e. Figures 3.10 and
3.11), which implies that typical eddies were formed in the vicinity of the wall and
continued to grow in size mainly due to diffusion process as they were migrating over the

boundary layer. It also leads to an argument that typical eddies propagating beyond the

84

inner region usually don’t come back to the proximity of the wall because our data clearly
indicate that the typical eddies existing near the wall were, on average, smaller than
typical eddies in the outer region, and it seems unlikely that typical eddies of greater
dimension shrank their physical size as they moved inwards, unless either significant
stretch or interaction with other vortical motions that reinforced their strength occurred
during the time when they were moving towards the wall. Nevertheless, this argument is
by no means to hint that typical eddies propagating in the outer region are unable to
interact with the wall layer (remember that they have advantageous physical size). It does
suggest that the interactions between typical eddies and the wall layer are usually
initiated by newly formed typical eddies which have strong vorticity intensity and

positions close to the wall as their advantages.

4.3 The Estimate for the Mean Vertical Velocity of Typical Eddies Using the

Size-Distance Correlation and the Diffusion Equation

As we know, the mean streamwise velocity for typical eddies is of the order of
the local mean convection velocity since the mean flow dominates the boundary layer
‘ dynamics. We have also learned that typical eddies are normally oriented so that they
induce themselves away from the wall. Results of Hoyez (1990) also showed that typical
eddies moving outwards outnumbered the typical eddies moving inwards (see histogram
in her figure 4-12). It might be interesting to know how the typical eddies migrated
vertically in the turbulent boundary layer. If we accept the hypothesis of the typical eddy
origination in the inner region and also assume that the growth of the spanwise diameter

of typical eddies was purely resulted from diffusion process, then the average vertical
velocity (V113) of typical eddies in the proximity of the wall can be determined. In doing

so, we must use the linear relationship of G; and d+, as well as the governing equation

85

of diffusion process. As a result, the relationship between V113 and (1, when normalized by
the free stream velocity U“, and the boundary layer thickness 899 respectively, is equated

as follows (derivation of this equation is written in Appendix B).

v1.40... = [22.16 + 10.17(d/599)]'1

Figure 4.3 shows the function in the range of 0.14<d/899<0.7 (because the

linearity of Cz+ and d+ used for deriving the function existing only for 55<y+<280).

Examination of the function indicates that the typical eddies on average migrated
outwards with velocity slowly decreasing as the distance from the wall increased. This is
consistent with the physical picture that, as time went by, the typical eddies were losing
their vorticity intensity in the marked vortex core, thus reducing the induced convection
velocity that had an outward normal component. Certainly, part of the typical eddy's
vertical movement was driven by its own induction. Blackwelder and Kovasznay (1970)
used the continuity equation and the measured growth rate of boundary layer thickness
finding out that the maximum mean value of the non-dimensional normal velocity, v/Uoo,
was less than 0.0025, which occurred at the outer boundary. It appears that their number
is negligible when compared to our estimated vm/Uoo. Apparently, the outward
convection of the typical eddies was not provided by the mean normal velocity of the
boundary layer. Figure 4.3 also shows the ms of the y-component velocity fluctuation
for a turbulent boundary layer at a comparable Reynolds number (see Klewicki 1989). It
is found that our estimate is approximately 25% greater than the maximum rms of the
normal velocity fluctuation. However, it is unclear how the inferred mean vertical

velocity of the typical eddies can be related to this quantity.

86

The time needed for a typical eddy to travel over the entire boundary layer can
also be estimated If we use the mean Vrri at d/899=0.5 and assume that typical eddies
were normally produced at d+ between 100 and 150, the time required for typical eddies
to reach the outer intermittent region is found to be in the order of t+=400. The spatial
scale over which this evolution takes place can also be approximated by multiplying the
average convection velocity of typical eddies at d1899=0.5 by the time span, t+M00. This
results in the length scale on the order of 7,000x+. The question is whether the average
life time of typical eddies is greater than 400t+ or not. If not, it means that typical eddies
which appear in the outer intermittent region have lost most of their energy when we
observe them. Thus, the answer seems to be yes, they have a lifetime greater than 400t+
unless there are some other mechanisms that can create typical eddies in the outer region.
If another mechanism of typical eddy generation dominates the outer region flow, these
typical eddies created in the outer region can only contribute to the Reynolds stress in the
outer part of the boundary layer, but have little effect on the wall region turbulence
production. This is because the argument addressed in the end of the previous section
also applies here. It is important to also be reminded that all the discussions made in this
section is based upon the assumption that the diffusion was the only cause for the growth

in size of the typical eddies.

4.4 The Mechanisms Responsible for the Formation of the Sublayer Features
Wm
4.4.1.1 Stagnation Flows and the Orientation of Typical Eddies

Falco (1980) suggested that the outer sweeping eddy, which was oriented so that

the lower portion of the sweep had vorticity of sign opposite to that of the mean shear,

87

was able to create a favorable flow field for the formation of the pocket flow module.
Referring to Table 3.8, our statistical results also showed that the probability of creating
pockets for typical eddies possessing positive vorticity in the lower lobe was nearly three
times greater than that for typical eddies of negative vorticity near the wall (mean shear
had negative vorticity). The findings lead to an interpretation of the mechanism
responsible for the pocket formation in the viscous layer. In Figure 4.2, the side view
illustrates a scenario showing that a pocket would be created in the sublayer if the typical
eddy of Mode 1 or Mode 5 (Mode 4 is nearly non-existent) is brought down low enough
to establish a stagnation point in the very-near—wall region (as seen by an observer
moving with the speed of the pocket center). The stagnation flow is created as a result of
the counter rotational flow field induced by the lower portion of the typical eddy and the
mean shear of the bOundary layer. In the same figure, the sketch of the close-up isometric
view indicates that the presence of the stagnation point locally decelerates the upstream
spanwise vortices of the mean shear and accelerates the downstream spanwise vortices,

thus providing the mechanism for the generation of a pocket flow.

The link between the stagnation flow and the pocket formation suggests that the
orientation of a typical eddy is a crucial factor to determine whether or not a pocket
would be created when the typical eddy is in the position to interact with the wall. Since
a Mode 2 typical eddy possesses vorticity of the same sign as that of the mean shear in
the lower portion of the ring vortex, the establishment of a stagnation point very near the
wall is unlikely even if the typical eddy is in the proximity of the wall. This would
substantially reduce the possibility of producing a pocket. The lack of ability to generate
a favorable flow field for pocket formation by this mode of typical eddies was well
reflected in the statistical outcomes of this study. Table 3.8 shows that merely 6 out of 51

(i.e. 11.7%) Mode-2 typical eddies were able to form pockets. Furthermore, although

88

typical eddies of Mode 3 or 6 have negative vorticity close to the wall, they are capable
of creating stagnation flows in the near wall region, due to the counter rotation of their
own, when moving towards the wall. However, this would have occurred only if the
typical eddies approached very close to the wall. More importantly, the occurrence of
these modes was found to be rare (probability for Mode 3 is 0.014, for Mode 6 is 0.002),
thus their contribution to the near-wall region dynamics is limited. Note that the ability to
form a pocket for Mode-3 typical eddies is also well shown in Table 3.8. It indicates that
3 out of 6 (i.e. 50%) typical eddies of this mode created pockets.

4.4.1.2 The Average Height Condition for Typical Eddies to Create Pockets

The foregoing discussion has addressed the important role of the typical eddy in
inducing a stagnation flow that promotes the pocket formation. In addition to the
orientation of a typical eddy, the distance from the wall for a typical eddy is also crucial
regarding the formation of a stagnation flow very near the wall. Refening to Figure 3.32,
the data based upon d* show a dominant feature; all of the typical eddies responsible for
the formation of both the pockets and the sublayer streak pairs were found, on average, to
be at height of d*=50 when the interactions were occurring or just occurred. We have
suggested that the invariant d* could be a statistical height condition with which typical
eddies must have complied in order to form the sublayer features. Since we have also
learned that the sublayer streak pairs were able to be formed by the typical eddies when
they were further out from the wall, this length characteristic (i.e. d*z50) might well be
the average height condition for the typical eddies to form the pockets. That is, the
"bottoms" of typical eddies on average needed to enter the zone between y+=50 and the

wall in order to create pockets. This is consistent with the findings that the mean d* of

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typical eddies for the events in which only pocket was formed was slightly less than
y+=50 (see Figure 3.32).

Data of two-point correlations concerning spanwise vorticity fluctuation obtained
by Klewicki (1989) showed that the negative correlations appeared to become significant
when the separations of the probes were under about 50 wall layer units (see his figure
4.13). He also presented data concerning the spanwise vorticity skewness and kurtosis
(flatness coefficient) distributions at three Reynolds numbers ranged between 1,010 and
4,850 in his figures 3.14 and 3.15 respectively. The results showed that the skewness
coefficient (always negative) reached the strong negative peak near y+=40, then started
to increase until y+=300. On the other hand, the kurtosis increased from approximately 3
to 8 in the range of 3<y+<40 and stayed high for y+240. Combining these statistical
features shows a good agreement with our physical picture. Interpretation of these

features is as follows.

The trend of skewness suggests an evolution of motions with intense negative
spanwise vorticity (i.e. same sign as that of the mean) that lifted from the sublayer, which
might be associated with either the tip portion of hairpin vortices or pocket vortices
studied by Falco. The data also suggest that the lifted wall region motions reached a
maximum intensity at y+==40 with respect to the mean of the surrounding motions. The
kurtosis profiles showing increase within the first 40 wall layer units confirmed this type
of processes. That is, if kurtosis is an indicator of intermittent behavior, the increase of
kurtosis in the near wall region is consistent with the physical picture that the proposed
motion scaled on wall variables decreased its 1113.111: scale as it moved away from the
wall. In order that these hairpin-type of motions continued to be dominant farther out in
the boundary layer, the skewness probably should have continued to increase negatively.

However, his results showed that the skewness started to decline in magnitude for y+240,

90

whereas the kurtosis still maintained in high levels. This indicates that the importance of
the lifted hairpin vortices began to level off by a more dominant feature containing
motions of strong positive spanwise vorticity for y+240. The high kurtosis in this region
suggested that these vortical motions occurred intermittently. Recapitulating, our results
show that the "bottoms" of the typical eddies were on average located near y+=50 when
forming pockets. All of these observations indicate that the motions containing strong
positive spanwise vorticity might plausibly be associated with the lower lobes (of sign
opposite to that of the mean) of the typical eddies that impinged the wall. In summary,
the statistical results and our visual findings suggest that the typical eddies on average
start to enter the inner-region dynamical picture by themselves at y+=40, after which they
are also able to significantly influence the near wall region vorticity field through strong

vortical interactions.

W

In physical simulations, Chu and Falco (1988) have discovered that long counter-
rotating streamwise vortices were not necessary for long sublayer streaks to be created in
contrast to the suggestion of many authors. Furthermore, the numerical simulations using
full Navier-Stoke equations of a turbulent shear flow by Kim (1986) have shown that the
eddies which have streamwise vortices were not elongated in the flow direction. In
addition, many investigators (see, for example, Praturi and Brodkey 197 8) have
measured the streamwise vorticity in the near-wall region discovering that these vortices
have been of short extent or very weak circulation in the cross-stream plane (also see
Bakewell and Lumley 1967). All of these suggest that the long, low-speed streak pairs
could be formed by a mechanism other than the streamwise vortices (at least that they are

not the primary cause). Referring to our data, it is interesting to note that, by comparing

91

the data of columns 8 and 10 in Table 3.8, the probability for a typical eddy to create a
long sublayer streak pair appeared to be less sensitive to its orientation than the creation
of a pocket. As we know, the pressure field induced by a roundish typical eddy is almost
insensitive to how the typical eddy is oriented. Based upon all the information presented
above, it may be concluded that the long sublayer streaks could be created more by the
influence of the convecting pressure perturbation than the internal vorticity or velocity
field of the incident typical eddy. That is, the directly vortical interactions between
typical eddies and the existing wall region vorticity distribution are not required to form
streak pairs. If we accept the argument, the formation of the long streak pairs in the
sublayer is simply the result of the fluid being pushed away from the center line of the
paths of typical eddies moving over the wall, which then builds up into the streaks. The
streamwise extent of the streaks therefore depends upon both the time and the spatial

length that the typical eddy is able to interact with the wall as it convects downstream.

4.5 Use of the Statistical Correlation of Events to Support the Interaction
Hypothesis

In our study, if a pocket and/or a long streak pair in the sublayer, and a typical
eddy in the log region and lower portion of the outer region were visualized to occur
concurrently in accordance with our sampling criteria, the sublayer features were
considered to be created as a result of the interaction between the typical eddy and the
existing wall region vorticity distribution. It is understood that if our interaction
hypothesis was correct, the probability for the concurrent occurrence of the sublayer
features and typical eddies should have been high. That is, for example, if the pocket
formations were related to typical eddies as hypothesized, observation of a typical eddy
propagating over the top of an evolving pocket should be highly expected This was what

92

our sampling conditions were constructed for. We sampled the pockets forming in the
designated area and checked to determine if typical eddies or any other Taylor micro-
scale coherent motions were crossing the transverse laser plane which was setup
immediately downstream of the designated area. By doing this, the motions responsible
for the pocket formations could be revealed. As we briefly mentioned before, other than
typical eddies no coherent motion which had size comparable to the size of the pockets

and streak pairs was found to occur with statistical significance.

Referring to Table 3.16, our results show that 41.5% of the typical eddies were
found to propagate over the wall with pockets formed in the sublayer, and 65.4% of the
typical eddies were associated with the formation of long streak pairs in the sublayer.
However, these statistical numbers do not fully reflect the legitimacy of our interaction
hypothesis. This is because some of the observed typical eddies, which were found
unable to interact with the wall layer based upon our observation arrangement, were
either located too far away from the wall to make any disturbance in the wall layer, or
they might be able to create pockets in regions other than the designated area (in
particular, just downstream). As explained above, the best measure for the validation of
the hypothesis is reflected on the ratio, NTE/NP, which stands for the probability of the
typical eddy occurrence when a pocket was visualized to form. Table 4.1 (see below)
shows the corresponding occurrence of typical eddies based upon the occurrence of
pockets and/or long sublayer streak pairs. It is found that 88% of the sublayer events
were observed to take place coincidentally with typical eddies convecting over their tops
at distances varying from 0 to 300 viscous units from the wall. This ratio provides a
strong statistical evidence showing that the typical eddies were indeed closely related to

the creation of pockets and long streak pairs in the viscous layer as hypothesized.

93

Table 4.1 The corresponding occurrence of typical eddy in the log region and lower
portion of the outer region based upon the occurrence of pocket and/or long streak pairs
in the wall layer.

 

 

 

 

 

Coherent Features N % No. of Typical Eddies Simultaneously Nm/N%
Observed in the Sublayer Observed above the
Sublayer Events (NW)
Pocket with or without 195 100 173 88.7
Streak Pair
Pocket with Streak Pair 167 85.6 149 89.2
Pocket only 28 14.4 24 85.7

 

 

 

 

 

 

Examination of the table also indicates that near 90% of the events, which
consisted of both the long streak pair and pocket formations in the sublayer, were found
to be associated with the occurrence of typical eddies. The ratio drops to about 85% for
the events in which only pockets were formed. Although the difference between these
two ratios is not great, it is consistent with the observation that the typical eddies, which
created only pockets in the sublayer, had greater probability to lose their vortex-ring-
shaped structures during the interactions than the typical eddies that created both long
streak pairs and pockets. It is interesting to note that the typical eddies responsible for
events in which only pockets were created had, on average, the lowest d'" among all the
classes of interactions (see Table 3.16). The physics behind this has been partially
discussed in Section 3.2.6.2, suggesting that the typical eddies of this kind were
convected towards the wall with an incident angle steeper than that of the typical eddies
creating both pockets and streak pairs. Consequently, their chances of eluding the wall

were reduced. Furthermore, hot-wire measurements conducted by Klewicki et al. (1990)

94

have shown that vortices with vorticity of sign opposite to that of the mean shear had
essentially zero probability of existing within the first 12 wall layer units. It can be
deduced that a typical eddy with vorticity of opposite sign of the mean shear in its lower
lobe would never maintain its vortex-ring-like structure if the lower portion of the typical
eddy approached to the wall within 12 wall layer units. Referring to Table 3.8, we have
learned that typical eddies with vorticity of opposite sign compared with that of the mean
shear in its lower lobe (for instance, Mode 1) were, however, the main contributors of
creating the pockets. This suggests that some of the typical eddies, which created only
pockets, would have lost their structural identities because their lower portions had
entered into the immediate proximity of the wall in which the vorticity with opposite sign
as that of the mean shear could never be sustained (in an attached flow). The vortex-ring-
shaped structures broke up because the lower portions of these typical eddies strongly
interact with the sublayer vorticity (Type III or IV interaction, Chu and Falco 1988) and
undergo a violent instability. Referring to Table 4.1, this could be the reason why a
statistically smaller number of typical eddies were visualized when this class of

interactions occurred.

4.6 Implications for Drag Reductions

Drag reduction for surfaces exposed to turbulent flows has been a long time
interest of engineering research. Control of turbulent boundary layers has been attempted
using outer layer manipulators such as LEBU's (see, for example, Lemay et al. 1988),
passive wall region modifications such as riblets (see, for example, Walsh 1982), or
combinations of both. Active controls such as air suction and fluid ejection have also
been experimented In the current study, we have learned that many characteristics of the

typical eddy are able to directly affect its ability of interacting with the wall layer.

95

Therefore, modifications of any of these typical eddy characteristics will definitely result
in an alternation of the drag on the wall. Some of these parameters, such as the eddy size
and vorticity intensity, may not be controllable. However modifications of many others
are possible, for instances, the angle of incidence, orientation in the x-y plane, and the

distance from the wall.

Suppose the turbulence production in the near-wall region is indeed undergoing a
cyclic procedure as we proposed, with the new typical eddies produced via the pinch-off
process. As we can see, it is very important for revealing the ordered spatial-temporal
correlations of these coherent features because the uncovered inter-relationships can
provide many places to apply a control mechanism to break the chain and interrupt the
cyclic production process. By doing this, we will be able to alter not only the drag in the
local region but also drag in the downstream. According to the present study, most of the
sublayer feature formations were found to be directly related to the typical eddies
propagating over the region close to the wall. With respect to the drag reduction, the
typical eddy appears to be the key in the production chain that can be modified by outer
layer manipulations. Many passive outer region manipulators have been used to interfere
with the LSMs enabling the sweeping flow field, which could bring the typical eddies to
the proximity of the wall, to be diminished The attempts were somewhat successful as
indicated by the local drag reduction, but the drag downstream was also increased owing
to the disturbances created by the intersection of the wallward side of the wake (formed
behind manipulators) and the wall layer (see Falco and Rashidnia 1987).

Active control of typical eddies in the log and the outer regions is also possible.
This, however, requires an un-intrusive detector which can rapidly monitor the formation
of typical eddies or their current state. A discriminator (with criteria functions) is also

needed in order to determine when and how long a control should apply. If we can, for

96

instance, detect the orientation of typical eddies and apply a control, if necessary, to alter
it, we can easily inhibit the establishment of a favorable flow field for pocket formation.
Although the formation of sublayer streaks can not be prevented by solely altering the
orientation of typical eddies, their contribution to the turbulence production can be
confined by using riblets which will be discussed below. The most probable distance of
the typical eddies from the wall for each class of interactions and the frequencies of the
correlated events are available from our probability density distributions. This
information may provide statistical references needed for a control algorithm to be
developed for interrupting the process of the interactions. Our data also show that the
wall-region interactions caused by the typical eddies did not take place when typical
eddies are on average positioned greater than 218 and 155 wall layer units for d+ and d*,
respectively. Additionally, d*==50 is found to be statistically critical for typical eddies to
form pockets. The implication is that if we could maintain typical eddies above the
critical distance from the wall, the suppression of turbulence would have been substantial
and hence the skin friction. Due to the control capability at this stage, the inability of
accomplishing the foregoing task is understood, but it does provide a subject worth

further studying and possibly be done by a computational work.

Among attempted wall region modifications, riblets are found to be effective on
stabilizing the long sublayer streaks and hence inhibiting their subsequent breakup.
However, the decrease of the drag is limited (5%-7%, Walsh 1982) by solely controlling
the stability of long streaks. This result is foreseeable simply because the production of
turbulence in the near-wall region is dominated by a more important process, that is the
stretching and amplification of the pocket vortices. Owing to the technical difficulty of
achieving active control of typical eddies in the log and the outer regions, the design of

wall region modifications to interfere with the pocket formation and its subsequent

97

evolution seems more feasible for practical implementation. According to our
investigations, the necessity for a pocket formation is a stagnation point in the very—near-
wall region as seen by an observer riding with the mean velocity of the pocket. Finding a
mean to prevent the stagnation flows from occurring and/or to interfere with the
retarding tendency of the pocket vortices inhibiting further defamation of the sublayer

vortices appears to be the main task for the study of drag reduction at this stage.

CHAPTER 5
CONCLUSIONS

The correlations between the major coherent motions existing in the lower
portion of turbulent boundary layers have been studied statistically via visual analysis.
Experiments were carried out in a water channel which provided an incompressible
smooth-surface turbulent flat-plate flow with R9z805 at the observing section. Long
sublayer streak pairs and pocket flow modules were sampled when observed to form and
evolve under passing typical eddies in the log region and lower part of the outer region.
In the cross-stream observation, portions of the typical eddies appeared as counter-
rotating vortex pairs. Efforts have been made through the development of our multi-color
dye system and visualization technique to eliminate the possibility that hairpin legs or
long axial vortex pairs were mistakenly recognized as typical eddies. Results show that
the streamwise extent of the counter-rotating vortex pairs passing through the crosss-
stream laser sheet were on average 77 wall layer units, which conforms well to the length
characteristic of typical eddies that our overall evidence showed they were part of. Our
data showed that the typical eddies exist throughout nearly the entire boundary layer,
which is contradictory to the assertion of Head and Bandyopadhyay (1981).

A statistical relationship between these coherent features has been found. The
probability density distributions of the non-dimensional spanwise length scales for
typical eddies, pockets, and long streak pairs were all well approximated by log-normal

density functions having mean values of about 101, 82, and 75 wall layer units

98

99

respectively. Additionally, the evaluation of the overall spanwise spacing of the long
low-speed streaks using a classical counting method was also performed. The mean
streak spacing was found to be about z+=96, which is consistent with outcomes of many
previous experiments. Correlations of the simultaneous occurrence of the typical eddies
and the sublayer features -- long streak pairs and pockets -- developing underneath them
were formed They show that the dimensions of both sublayer features increased as the
spanwise diameter of the typical eddy increased. It is also found that the correlation of
the pockets and the long sublayer streak pairs was better characterized using the
information of the third coherent motions, the typical eddies. Furthermore, we are able to
use our statistical results of the typical eddy spanwise distribution spacing and the mean
streak-pair spacing to present an argument showing that long sublayer streaks, generated
in pairs, are spaced so as to result in the overall mean streak spacing (h+~100). The
foregoing results strongly support the hypothesis that the coherent typical eddies are the
significant motions that govern the generation of the wall region dynamical features in

the turbulent boundary layer.

Based upon both the statistical findings and the inference from piecing together
the visual evidence, the origination of the typical eddies and the pinch-off phenomena of
wall-region lifted hairpin vortices were linked Accordingly, a cyclic interaction process
was deduced. In addition, the relationship betan the vortex diffusion and the growing
typical eddy magnitude as a function of its distance from the wall provides not only extra
support to the hypothesis of cyclic interaction, but also an argument that the wall region
interactions were mainly initiated by newly formed typical eddies existing adjacent to the
wall. Results have also shown that the pocket formation is associated with (a) the
presence of a typical eddy, and (b) the ability of the typical eddy to form a convected

stagnation flow very near the wall (as seen by an observer moving with the velocity of

100

the pocket center). Statistical results show that the latter is primarily governed by two
overall typical eddy characteristics : the orientation in the x-y plane, and the distance
from the wall. Our data also suggest that the long streak pairs found in the sublayer could
have been formed as a response to the convecting pressure disturbance induced by the
typical eddies propagating over the wall. It also appears that typical eddies are able to
produce pairs of low-speed streaks from a distance greater than from which that they can

create pockets.

A normalized length variable, d* (=d+-0.5Cz+), was introduced as the physical

distance between the "bottom" of the typical eddy and the wall. It was found that this
length characteristic is a potential measure for the ability of a typical eddy to initiate an
interaction with the existing wall region vorticity distribution, particularly for the pocket
formation events in which the bottoms of the corresponding typical eddies were, on
average, located at y+z50 (i.e. d*=50). Furthermore, the probability for the simultaneous
occurrence of the typical eddy and the sublayer feature(s) was found to be remarkably
high. It serves as a measure indicating that the hypothesis of the inner/outer coherent

motions is legitimate.

Qualitative flow visualization was done using a double-dye "refresh" wall slit and
a cross-stream observation arrangement. Results demonstrated that a sequential evolution
existed incorporating all of the important structural features in the viscous sublayer and
the typical eddies in the log region and/or lower portion of the outer region. The entire
sequence, with period of occurrence in the order of 80 viscous units, made up the Overall
Production Module (OPM) depicted by Falco et al. (1989a) and Falco (1991). In the wall
region observation, the rapid induction of the secondary hairpins towards the wall reveals
the predominant streamwise vorticity of the stretching pocket vortex in this event.

Summarizing, both the visual observations and the statistical correlations provide

101

additional support for, and more detail about the inner/outer region interactions
embodied in the Overall Production Module. It also offers information necessary for the
study of drag reduction in wall-bounded turbulent flows.

TABLES

102

 

 

 

 

 

 

 

 

 

 

 

 

Flow characteristic Denotation Value
Reynolds number R3 805
Temperature T 12.5 °C
Kinematic viscosity v 0.0123 cm2/sec
Momentum thickness 0 0.465 cm
Velocity at free surface U00 0.21 m/sec
Boundary layer thickness 599 4.7 cm
Shape factor H 1.271
Density L 1 g/cm3
Skin friction coefficient Cf 0.005
Friction velocity uL 1.065 cm/sec

 

 

 

Table 3.1 Summary of principal characteristics of the turbulent boundary layer flow.

 

103

Falco's Data at R9=740 ! Present Data at R9=805
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Figure 3.37 Formation of a streak pair and the downstream development of a pocet. Flow
is from left to right (similar picture was also present in Falco, Klewicki, and Pan 1989a).

 

153

StreakPair

 

 

t+=0
t+=l.85 ~ V Hailpins Being Lifted
t+=3 .70

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t+=5 .54 The Pocket Center
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Induced Towards The Wall

 

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Figure 3.38 Pocket evolution resulting in the engulfment of secondary hairpinsithatihave
developed over the streaks. Flow is from left to right (similar picture was also present in
Falco, Klewicki, and Pan 1989a).

154
Laser Sheet sum sum EndView ___ PlanView

A

 

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D ;’:- ' _“ ‘. I

Figure 3.39 A sequence of simultaneous plan and cross-stream views illustrating a typi—
cal eddy/wall layer interaction that results in the formation of a pair of long streaks and a
pocket near its downstream end. Flow is from right to left for the plan view and into the
paper for the cross—stream view. The evolution of the sublayer features is indicated by the
black arrows, whereas the appearance of the typical eddy is indicated by the white arrow.

 

Figure 3.40 A sequence of simultaneous plan and side views illustrating the formation of
a pocket by a typical eddy, as well as the subsequent interaction between the typical eddy
and the lifted sublayer fluid near the downstream end of the pocket. Flow is from left to
right. The evolution of the pocket is indicated by the black arrows, whereas the appear—
ance of the typical eddy is indicated by the white arrow.

A, at.

 

 

Figure 3.41 A sequential side-view pictures showing the formation of a new typical
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pin is called "primary hairpin" (see Chapter 1) which initially formed near the down-
stream end of the pocket in Figure 3.40. Flow is from left to right.

 

Figure 3.42 An example illustrating the evolution of a hairpin into a new typical eddy
(as indicated by the white arrows). Flow is from right to left. Note that, in frames 5 to
9, an upstream typical eddy can also be seen to move into the streamwise laser sheet.

 

Figure 3.43 Another example showing the formation of a new typical eddy emerging
from the wall layer fluid through a pinch-off of lifted hairpin (as indicated by the white
arrows). Flow is from right to left.

159

 

 

Cross-stream view

 

 

 

 

 

--------- Laser sheet

Flow

    

Plan view

 

Figure 4.1 An illustration of plan and cross-stream views showing the spanwise
phase relationship of typical eddies and the corresponding sublayer streak pairs.
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APPENDICES

162

6 U... 0 ' It‘ a .-_. o .I‘ it" D it. run. :1}. Mt t‘ t"

The data obtained using surface float technique are listed in Table A.1. The

measurements for positions beyond 599 using pressure probes are tabulated in Table A.2.

They are shown as follows.

Table A.1 Measurements using

   

 

 

 

 

 

 

 

 

surface floats at yfl.375 inches. Table A.2 Measurements using
The length scale (s) between two pressure probes for y>699=1.85
stations is 4 feet. inches.
Data # time sec L y (inch) Velocity (ft/sec)

5.40 1.918 0.67141
5.64 2.097 0.69691
5.43 2.275 0.69498
5.64 2.632 0.69494
5.46 2.990 0.69691
5.56 3.347 0.70266
5.47 Average 0.693

 

 

 

 

5.51

5.64

5.53

5.30
5.25

5.55

5.58

5.49
5.497

 

The mean free stream velocity based on the float measurements can be found as

Um, = s/tavg a 0.728 ft/sec (A1)

163

The relative difference between the result of equation (A1) and the average value of

Table A.2 is defined as follow (based upon the pressure measurements).

Difference (%) = [(Ufloat-Upmsm) / Upressm] x 100% (A2)

which results in ~5. 1% for the difference.

 

164

.3" x” D. :’ Ir '49 .99: 9 r um '19 .. ‘9.9 o u ._ .0.0-' 1311‘;
E . E I D. E I 1!! ll :1:
The function of the linearity between Cz and d, as well as the governing equation

of the diffusion process of a vortex core are

CZ+ = 88.6 + 0.1d"', for 55<y+<280 (131)
r2 = to2 + 4w (B2)

,where r is equivalent to Cz/2, and r0 is the initial radius as typical eddy is formed.

Equation (B 1) can be rewritten in a dimensional form as

r = 0.0168 + 0.05d (133)
If we differentiate equations (B3) and (B2), they become

Ar = 0.05(Ad) (B4)

2r(Ar) = 4v(At) (B5)
Substituting Ar and r by equations (B4) and (B3) respectively, equation (B5) becomes

(0.00168 + 0.005d)(Ad) = 4V(At) (B6)

By rearranging the equation, the relationship between vm and dis equated as follow.
v1.E = Ad/At = 4v/(0.00168 + 0.005d) = v/(0.00042 + 0.00125d) (B7)

Now we can normalize V115 and (1 using the free stream velocity, U00, and boundary layer

thickness, 899, respectively. As a result, equation (B7) becomes

165
vm/U... = [22.16 + 10.17(d/699)]'1 (B8)

Note that the properties v, U00, and 899 of equation (B8) have been substituted by their

nominal values at R9=805 (see Table 3.1). Equation (B8) can also be expanded into a

binomial series as follow (see, for example, Abramowitz and Stegun 1964)

VIE/U” = 0.045/[1 + 0.459(d/699)]'1 = 0.045(1 - x + x2 - x3 + x4 - ...), X<1 (B9)

,where X = 0.459(d/899). Note that function (BS) is shown in Figure 4.3.

166

a" U". '1‘ «9‘ I‘Utt 9” ‘tmnt 0 uni ‘0. U,'2.-1'H‘l

The mean velocity profile in the water tunnel was determined from pressure

measurements and a formula derived from the Bernoulli equation as
E = JZgh (C1)

where
u = velocity in feet per second
g = 32.2 feet per second, and

h = observed velocity head on the Pitot tube, in feet.

We recorded h by subtracting the static pressure from the total pressure. Since the

ruler used to measure the pressure differential was 100 divisions per inch, the uncertainty

(error bar) of h (i.e. Ah) can be estimated as 10.005 inches of H20. The central question

is : "Given the Ah estimate, what is the resulting uncertainty in u?" For the answer, we

recognize that equation (C1) can be differentiated as

dfi I
_ = i ((32)
db 2h

The derivative is also known as the sensitivity coefficient. In our setup, h=lsinB, [3:30

degrees. We now can rewrite equation (C2) as

51—11 _ gSinfi

(ll 21 (C3)

The sensitivity changed due to the rearrangement of the readout device is represented by

the ratio, (C3)/(C2), as

167

 

dfi
3% = ,hsrnfi = Sin fl =% ((34)
38

This means that the sensitivity was reduced by a factor of two.

To evaluate the relative uncertainty of u, we take the natural log on both sides of

equation (C1) and differentiate them with respect to u and h. This results in an expression

as follow.
3 = 3 (C5)
11 2h

The derivative is the relative uncertainty of 11. Similarly, we can rewrite equation (C5) in
terms of I as
91

= 21 (C6)

=1|g1

By simple mathematical manipulations, it is found that the right hand sides of equations
(C5) and (C6) are equal. This means that the relative uncertainty of u was unchanged by
the rearrangement of the readout setup. The maximum and minimum of possible error for
our measurements can be determined by either equation (C5) or (C6). The maximum

error might occur at the measuring point closest to the wall as

 

(3) s 0.5 x 10°05 = 10.1: 110% (C?)
11 max 0.025

Typically, the relative uncertainty is represented as an "1" bar, whose height is the

representative of the numerical value of equation (C7). The minimum of relative

 

 

 

168

uncertainty was possessed by the data measured beyond the boundary layer thickness,
599» as

 

(9:) s 0.5 x *0'005 = i0.028 = i2.8% (C8)

169

EEEEIIDIXD .1] E l 1.]. E . E . [12' E

In the current study, the probability density functions representing the
characteristics of events are formulated from discrete sample sets. Three probability
densities are involved in our data analysis. They are the Normal (Gaussian), Logarithmic

Normal, and the Rayleigh distributions, which are given as follows.

Normal Distribution :
pm — —1-1=.x [Pi—”2] (D1)
427w p 20'2

N
2r,

where f = 341:1— (i.e. mean value),

i0} ' ”2

i-l

N-l

    

(i.e. standard deviation)

In the above equation, P is the probability density, r is a random variable and N is

the sample size.

Logarithmic Normal Distribution :

 

Ian—H?
P(r)= [Zijerxp{ 202 ] , r>0 (D2)

170

_ ilnfii)

where ln(r) = 31T— ,

 

i[1n<r.)-E(T)]’

0.: i-l
N-l

 

Rayleigh Distribution :

P(r) - -’-Ex (1:) r>0
R p 2R ’

N

29
where f = L,
N

“I?!

(D3)

LIST OF REFERENCES

LIST OF REFERENCES

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