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*m.

ON THE INTERACTIONS BETWEEN THE INNER AND

OUTER REGION MOTIONS IN TURBULENT BOUNDARY LAYERS

BY

Joseph Charles Klewicki

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1989

ABSTRACT

ON THE INTERACTIONS BETWEEN THE INNER AND

OUTER REGION MOTIONS IN TURBULENT BOUNDARY LAYERS

By

Joseph Charles Klewicki

Time resolved spanwise vorticity measurements were made in very
thick.zero pressure gradient boundary layers over the Reynolds number
range 1.010 S R0 S 4,850. The scale of the flow field relative to the
size of the four-wire spanwise vorticity probe resulted in very good
spatial resolution measurements. Detailed results relating to the

accuracy of the present measurements are presented.

Physical arguments are employed to justify the hypothesis that the
important instantaneous vortical motions in turbulent boundary layers
organize into essentially two geometries. Near the solid surface it is
proposed that the vorticity distributions have a two dimensional sheet-
like character. Farther away from the wall it is proposed that the
important vortical motions organize in the form of closed loops. From
these deductions, it is hypothesized that the essential features of the
so-called inner/outer interaction in turbulent boundary layers may be

described in terms of the interaction of these loop-like and sheet-like

vortical motions.

The single probe results suggest that in the near-wall region the

spanwise vorticity statistics are universal under inner variable scaling
However, inner variable normalizations of the Reynolds stress and
velocity vorticity correlations related to the transport of the stress
exhibit significant R0 variations. The present higher order spanwise
vorticity statistic profiles are shown to be consistent with the process
of lrigh.intensity hairpin vortex-like and/or lifted shear layer-like
motions arising from the sublayer. Interpretation of <uwz> data reveals a

plausible initiating mechanism for the formation of these lifting

sublayer motions .

Two point spanwise vorticity correlations at a constant spanwise
'probe separation decreased in magnitude as the distance from the wall
increased. Analysis of the correlations with probe separations normal to
the wall revealed the frequent occurrence of organized counter-rotating
spanwise vorticity interactions in the wall region. In general, the
correlations did not provide evidence supporting the existence of large
scale motions endowed with significant spanwise vorticity. Preliminary

results from two-view flow visualization movies support the present

hypotheses.

 

 

.. one should also consider the existence of small-scale convected
vortices to explain the observed sublayer structure . . ." W. W. Willmarth
(1975)

Copyright by
Joseph Charles Klewicki

1989

ACKNOWLEDGMENTS

Although this dissertation has only one author, its completion

depended on the help and support of a number of people.

First and foremost I would like to acknowledge the support of my
family. Over the course of my graduate studies I have benefited from the
patience, love and understanding of my wife Cindy, our two children Brian
and Maureen, my parents, my brother and my sisters. Also I would like to

specifically thank my sister Casey who inspired me to become a mechanical

engineer .

I would like to express gratitude to my advisor, Robert Falco, for

his guidance and assistance, and for allowing me the freedom in the

laboratory to explore interesting issues.

Concerning the work presented here I would like to acknowledge the

Support of the Air Force Office of Scientific Research, and specifically

to Our grant monitor Dr. J. McMichael.

Concerning my work in the Turbulence Structure Laboratory, thanks

are due to all of my co-workers, but especially to Daniel Chu, Frank

Cummings and Chuck Gendrich.

Finally, I would like to thank my committee members, Steve Shaw,
John Foss, Manooch Koochesfahani, and T. Y. Li for their useful guidance
throughout my Ph.D. program. Special thanks go to Steve for his

frifindship, and to John for his detailed reading and remarks concerning

this manuscript .

vi

B F CONTENTS

page
Ll§I_QE_IABL§§ ............................................ x
IS 0 S ........................................... xi
LIST OE SYMBOLS ........................................... xix
935mg
1. TE E EV EW B C VES OF IS Y .... l
1.1 INTRODUCTION ...................................... l
1.2 DETAILS PERTAINING TO INNER/OUTER INTERACTIONS . . . . 5
1.2.1 Average Structure ........................... 5
1.2.2 Inner Region Motions ........................ 7
1.2.3 Outer Region Motions ........................ 12
1.3 OBJECTIVES AND METHODOLOGY ........................ 14
1.3.1 Objectives .................................. 14
1.3.2 A Bias Toward Vorticity Measurements ........ 15
1.3.3 The Inner/Outer Interaction in Terms of
Vorticity ................................... 16

1.4 PREVIOUS MEASUREMENTS OF TIME RESOLVED VORTICITY
AND VELOCITY GRADIENTS, SMALL SCALE STRUCTURE, AND
REYNOLDS NUMBER DEPENDENCE IN WALL TURBULENCE

STATISTICS ........................................ 17
1.4.1 Previous Measurements of Time Resolved
Vorticity and Velocity Gradients ............ 17
1.4.2 Evidence of Small Scale Structure ........... 22
1.4.3 Evidence of Reynolds Number Dependence ...... 24
0 ;. 113.1 3.5.1; . ..; 0 .‘1 W
C C OUND RS .... 27
2.1 ON RESOLVING NEAR-WALL MOTIONS .................... 27
2.2 EXPERIMENTAL FACILITIES AND CONDITIONS ............ 30
2.3 THE SPANWISE VORTICITY PROBE; ITS CHARACTERISTICS
AND CALIBRATION ................................... 33
2.3.1 Physical Characteristics .................... 33
2.3.2 Calibration ................................. 34

2.4 EXPERIMENTAL PROCEDURE AND FACTORS AFFECTING

DATA QUALITY ...................................... 36
2.4.1 Sampling Rate and Cut-Off Frequency ......... 36
2.4.2 Digitizer Resolution ........................ 37
2.4.3 Total Integration Times ..................... 38
2.4.4 On Taylor's Hypothesis ...................... 38
2.4.5 Data Reduction .............................. 39
2.4.6 Free Stream Uncertainty ..................... 40

2.4.6.1 Short Term Low Frequency
Variation in Uco ..................... 41

2.4.6.2 Long Term Variations in
Experimental Conditions ............. 42

2.5 SPATIAL RESOLUTION AND STATISTICAL

CONVERGENCE RESULTS ............................... 44
2.5.1 Spatial Resolution .......................... 45
2.5.1.1 Wire Length Effects ................. 45
2.5.1.2 Wire Spacing Effects ................ 50
2.5.1.3 The Resolution of wz ................ 57
2.5.2 Convergence of Turbulence Statistics ........ 58
2.5.2.1 Methodology ......................... 59
2. 5.2. 2 Velocities and Reynolds Stress ...... 61
2.5.2.3 Velocity Gradients .................. 63
2. 5. 2. 4w wz ................................. 65
2. 5. 2. 5C onvergence Summary ................. 65

2.6 DISCUSSION AND CONCLUSIONS - RELATIVE TO CHAPTER 2 66

O 8 TS O D
NUMBEB_DE£ENDENQE ..................................... 72
3.1 INSTANTANEOUS MOTIONS, AVERAGE STRUCTURE AND

TOWNSEND'S DECOMPOSITION .......................... 72
3.2 REYNOLDS NUMBER DEPENDENCE OF BOUNDARY LAYER

STATISTICS ........................................ 76

3.2.1 Mean Flow Characteristics ................... 76

3.2.2 Fluctuating Velocities and Reynolds Stress .. 77

3.2.2.1 Mean and RMS Quantities ............. 77
3.2.2.2 Higher Order Statistics ............. 84

3.2.3 Spanwise Vorticity Statistics ............... 92

3.2.4 Summary of u, v, uv and w Results .......... 99

3.2.5 Conclusions Pertaining to u, v, uv and ”2

Results ..................................... 102
3.3 THE ACTIVE AND INACTIVE MOTIONS ................... 103
3.3.1 The Inactive Motions ........................ 104
3 3.2 The Active Motions .......................... 106
3.3.2.1 <wwz> Data .......................... 106
3. 3. 2. 2 (vwz )' Data ......................... 108
L 2.2.3 vwz zSpectra ......................... 109
3. 2. 2. 4 The Dependence of <Nwz > on
Probe Scale ......................... 111

viii

3.2.2.5 The Vortex Force: A Plausible
Mechanism For Lifting Vortical
Sublayer Motions .................... 113

3.4 DISCUSSION AND CONCLUSIONS - RELATIVE TO CHAPTER 3 118

4. N S I UC PANWISE VORTICITY IN
E O D ER ............................. 122
4.1 ISSUES PERTINENT TO AN INNER/OUTER INTERACTION .... 122
4.2 EXPERIMENTAL PROCEDURES AND CONDITIONS ............ 124
4.2.1 Additional Equipment and Procedures ......... 125
4.2.2 Experimental Conditions ..................... 129
4.2.2.1 Correlation Experiment Overview ..... 129

4.2.2.2 Matching the Reynolds Numbers of
the Single Point Experiments ........ 129

4.2.3 On Interpreting Spanwise Vorticity

Correlations ................................ 131
4.3 TWO POINT SPANWISE VORTICITY CORRELATIONS ......... 132
4.3.1 Probe Separations Normal to the Wall ........ 132
4.3.2 Spanwise Probe Separations .................. 138

4.4 DISCUSSION AND CONCLUSIONS - RELATIVE TO CHAPTER 4 140

5. ONCLU IONS D ROS CTUS ............................ 146
5.1 CONCLUSIONS ....................................... 146
5.1.1 Hypothesis #1 ............................... 147

5.1.2 Hypothesis #2 ............................... 150

5.1.3 Hypothesis #3 ............................... 152

5.2 PROSPECTUS ........................................ 154
TABLES .................................................... 159
FIGURES ................................................... 167
AP EN I ................................................ 250
APPENDIX 2.1 ON COMPUTING STREAMWISE DERIVATIVES ..... 250
APPENDIX 2.2 DATA SMOOTHING PROCEDURE ................ 254

APPENDIX 2.3 ON MEASURING THE MEAN VELOCITY GRADIENT . 257

APPENDIX 3.1 VELOCITY-VORTICITY CORRELATIONS RELATED
TO THE GRADIENTS OF THE REYNOLDS STRESSES IN PARALLEL

TURBULENT WALL FLOWS .................................. 265
APPENDIX 3.2 OUTER VARIABLE NORMALIZATIONS ........... 278
L;§I_QE_B§EEREN§E§ ........................................ 292

ix

Table

Table

Table

Table

Table

Table

Table

Table

Table

A2.2.

LIST OF TABLES

page

Summary information pertaining to the spatial
resolution of wall-flow investigations cited
in Section 1.4 .................................. 159

Principal characteristics of the zero pressure
gradient boundary layers of the present study;
fc - low-pass cut-off frequency; fs - sampling
frequency ....................................... 160

Summary information concerning the wall-flow
investigation data presented in Figure 2.8a,b.

Note that R9 indicates boundary layer flow,

and Rd 2 indicates channel flow. Also note

that t e probe scale given is computed using

the largest dimension necessary to measure

(compute) v ..................................... 161

Normalized spanwise vorticity probe dimensions;

0+ - nuT/u; "min and n refer to minimum and
maximum measured Kolmogoroff scales found in

each R distribution (a estimated by using c

from equation 2.5). The individual wire length

and x-array spacing are also equal to Ay ........ 162

Summary information concerning other wall-flow
investigation data presented in Chapter 2 (except
Figure 2.8a,b). Note that R0 indicates boundary

layer flow, and Rd 2 indicates channel flow. See
Table 2.3 for information pertaining to the

present probe dimensions ........................ 163

General criteria to ensure the convergence of
statistics up to 4th order moments in boundary
layer flows for the range 1,000 < R0 < 5,000 .... 164

Summary information concerning other wall-flow
investigation data presented and referenced in
Chapter 3. Note that R9 indicates boundary

layer flow, and Rd 2 indicates channel flow.

See Table 2.3 for Information pertaining to

the present probe dimensions .................... 165

Summary of two point wz-wz correlation

experiments ..................................... 166

1 Sample output from the velocity data smoothing
subroutine ...................................... 256

L OF GURES

page

Frontispiece Flow visualization of an apparent ring-like
motion interacting with the sublayer (see
Section 5.2)

Figure 2.1 Schematic of the 17 meter low speed tunnel ....... 167
Figure 2.2 Logarithmic mean velocity profiles ............... 168
Figure 2.3 The spanwise vorticity probe ..................... 169

Figure 2.4 Measured w 'u/ur2 values at different
locations In the R0 - 1,010 boundary layer
as a function of experiment time.
Solid lines represent values from the
R0 - 1,010 distribution of Figure 2.14
at the y+ values indicated ....................... 170

Figure 2.5 The maximum non-dimensional axial intensity
as a function of sensor length in viscous
units. The curve-fit of the Ligrani and Bradshaw,
and Johansson and Alfredsson data is given by,
2.859 - 0.01752+ + 0.0000175£+2, 3.3 3 1+ 5 34.
Note that a curve fit of the Johansson and
Alfredsson data alone (not shown) is given by,
3.002 - 0.02532+ + 0.000282+2, a s 2* s 32.
Details pertaining to the other researcher's
data are given in Table 2.4 ...................... 171

Figure 2.6 The maximum non-dimensional axial intensity
as a function of R . Symbols same as in
Figure 2.5, and the solid line represents
a linear least squares curve fit given by
(u'/ur)max - 2.616 + (0.000092)R0 ................ 172

Figure 2.7 Near wall distributions of the skewness of
au/at. For probe scale information and Reynolds
numbers of other researchers see Table 2.4 ....... 173

Figure 2.8 Near wall distributions of the skewness of
the v component velocity fluctuations
(also see Table 2.2).

a) Distributions which are positive in the range
5 s y+ s 30 ..................................... 174

b) Distributions which are negative in the range
5 s y+ s 30 ...................................... 175

xi

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure
Figure
Figure
Figure
Figure
Figure

Figure

.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

.20

.21

.22

.23

Present turbulent dissipation rate estimates
as given by equation 2. 5 and normalized by
inner variables. The solid line - l/Icy+ ,

where n - 0.41 .................................

Attenuation of gradient intensity ratios as
a function of the non-dimensional outer
array wire separation (see insert) in the
- l, 010 boundary layer. Solid symbols
give results at y+ 3 38. Open symbols give

results at y+ 5 53 .............................

Frequency spectra of au/ay at y+ E 53 in
the R0 - 1,010 boundary layer for different
Ay spacings. The horizontal lines through
the au/ay spectra are the electronic noise
levels as predicted by ¢(6u/6y) - e u2/Ay2 ,
where e 2 is the energy level of the

electric noise in C(u) .........................

Approximate cut-off frequencies of ¢(6u/6y)
due to the spatial filtering effect of
finite wire separation (point of divergence
with the Ay/n - 0.94 curve) as a function

of non-dimensional wire spacing ................

Correlation coefficients of u, v, and av/ax
as a function of spanwise separation at

y+ a 140 in the R0 - 1, 010 boundary layer ......

Inner variable non- -dimensionalization of the
rms spanwise vorticity distribution.
The error bars at y+ - 6.2.13.4 and 100.5

are derived from the data of Figure 2.4 ........

The rms spanwise vorticity distribution
normalized by the function y/u and plotted

versus y/0 .....................................

Convergence data for the u velocity

statistics .....................................

Convergence data for the uv statistics .........

Skewness of uv distributions vs y+

Rurtosis of uv distributions vs y+

Convergence data for the au/at statistics ......

Convergence data for the wz

Skewness of ”z distributions vs y+

Convergence data for the vwz

xii

OOOOOOOOOOOOO

OOOOOOOOOOOOO

statistics .........

statistics ........

.. 177

.. 179

.. 180

.. 181

.. 182

.. 187

Figure 3

Figure 3

Figure 3.

Figure 3.

Figure 3.

Figure 3.

Figure 3.

Figure 3.
Figure 3.

Figure 3.

Figure 3.
Figure 3.

Figure 3.

Figure 3

Figure 3

.1

.2

a)
b)

3

4

5
6
a)
b)
7
a)
b)
8
9

10

11
12
13
a)
b)
e)

b)

.14

.15

Inner variable normalized streamwise intensity
profiles ......................................... 191

Inner variable normalized normal intensity
profiles

Present data ..................................... 192
Data of Wei (1987) ............................... 193
Inner variable normalized Reynolds stress

profiles ......................................... 194
Reynolds stress correlation coefficient

profiles versus y+ ............................... 195
Velocity variance ratio profiles versus y+ ....... 196

Streamwise velocity skewness profiles versus y+

Present data ..................................... 197
Present data plus those of others ................ 198
Streamwise velocity kurtosis profiles versus y+

Present data ..................................... 199
Present data plus those of others ................ 200
Normal velocity skewness profiles versus y+ ...... 201
Normal velocity kurtosis profiles versus y+ ...... 202

Inner variable normalized Reynolds stress

intensity profiles ............................... 203
Reynolds stress skewness profiles versus y+ ...... 204
Reynolds stress kurtosis profiles versus y+ ...... 205

Spanwise vorticity intensity profiles

Normalized by 0 and U6 versus y/0 ................ 206
Normalized by v and u, versus y+ ................. 207
Normalized by the function y/uf versus y+ ........ 208
Normalized by 6, u and u, versus y/6 ............. 209
Spanwise vorticity skewness profiles versus y+ ... 210
Spanwise vorticity kurtosis profiles versus y+ ... 211

xiii

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.16

.17

.18

.19

.20

.21

.22

.23

.24

.25

a)
b)

.26

a)
b)

.27

a)

.28

.29

.30

.31

.32

Profiles of the average streamwise gradient
of the streamwise velocity variances

under an inner variable normalization ............ 212
Profiles of the average streamwise gradient

of the normal velocity variances

under an inner variable normalization ............ 213
Profiles of the rms streamwise gradient

of the streamwise velocity variances

under an inner variable normalization ............ 214
Profiles of the rms streamwise gradient

of the normal velocity variances

under an inner variable normalization ............ 215
Inner variable normalized <sz> profiles ......... 216
Normalization of the <sz> profiles using

the function y/ur2 ............................... 217
Inner variable normalized (vwz)' profiles ........ 218
Normalization of the (vwz)' profiles using

the function u/<uv>3/2 ........................... 219
Normalization of the (vwz)' profiles using

the function y/uf2 ............................... 220
Spectral functions of vwz at y+ 5 20 ............. 221
Inner normalization (see eq. 3.5)

Outer normalization (see eq. 3.6)

Spectral functions of vw at y+ 3 120 ............ 222

Inner normalization (see eq. 3.5)
Outer normalization (see eq. 3.6)

Reynolds number dependence of the vortex

ring-like motions as measured by Falco

(unpublished) .................................... 223
Inner normalization

Outer normalization

Correlation coefficient profiles of vw

versus y+ ........................................ 224
Vortex force directions for the given

velocity vorticity products ...................... 225
Inner variable normalized <Uwz> profiles ......... 226
Normalization of the <nwz> profiles using

the function y/u'2 ............................... 227
Inner variable normalized (uwz)' profiles ........ 228

xiv

Figure 3.33 Normalization of the (uwz)' profiles using
the function y/ur2 ............................... 229

Figure 4.1 Traverse mechanism used in the A2 probe
separation correlation experiments ............... 230

Figure 4.2 Mean velocity measurements from the moving
probe of the Ay correlation experiments as
normalized by the friction velocities in Table
2.2 and overlaid on the single probe profiles
of Figure 2.2 .................................... 231

Figure 4.3 Spanwise vorticity intensity measurements
from the moving probe of the Ay correlation
experiments as normalized by the friction
velocities in Table 2.1 and overlaid on the
single probe profiles of Figure 2.14 ............. 232

Figure 4.4 Spanwise vorticity intensity measurements
from the R0 g 1,010 Az probe separation
experiments at y+ - 6.7 as normalized by the
friction velocity in Table 2.1 and compared
with the relevant values of Figure 2.14, open
symbols -- stationary probe, solid symbols --
moving probe ..................................... 233

Figure 4.5 Spanwise vorticity intensity measurements
from the R0 5 1,010 Az probe separation
experiments at y+ - 15.3 as normalized by the
friction velocity in Table 2.1 and compared
with the relevant values of Figure 2.14, open
symbols -- stationary probe, solid symbols --
moving probe ..................................... 234

Figure 4.6 Spanwise vorticity intensity measurements
from the R 5 1,010 Az probe separation
experiments at y+ — 33.5 as normalized by the
friction velocity in Table 2.1 and compared
with the relevant values of Figure 2.14, open
symbols -- stationary probe, solid symbols --
moving probe ..................................... 235

Figure 4.7 Spanwise vorticity intensity measurements
from the R 5 2,870 Az probe separation
experiments at y+ - 20.4 as normalized by the
friction velocity in Table 2.1 and compared
with the relevant values of Figure 2.14, open
symbols -- stationary probe, solid symbols --
moving probe ..................................... 236

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.10

.11

.12

.13

.14

.15

.16

.17

.1a

.1b

Spanwise vorticity intensity measurements

from the R0 3 2, 870 A2 probe separation

experiments at y+ - 33. 6 as normalized by the
friction velocity in Table 2.1 and compared

with the relevant values of Figure 2.14, open

symbols -- stationary probe, solid symbols --

moving probe ..................................... 237

Ratio of the present spanwise vorticity

intensity profile at R - 1,010 to the mean

vorticity profile, 0 3 -8U/3y, of Van Driest

(1956) as given in Figure A2.3.l ................. 238

Results of the w -w correlations for positive Ay
separations and the stationary probe positioned
at y+ as 100 ...................................... 239

Results of the w -w correlations for positive Ay
separations and the stationary probe positioned
at y+ 3 30 ....................................... 240

Results of the w -w correlations for positive Ay
separations and the stationary probe positioned
at y+ 3 15 ....................................... 241

Results of the w -w correlations for positive Ay
separations and the stationary probe positioned
at y+ - 7.5 (R0 - l, 010 only) .................... 242

Two dimensional probability contour map as
derived from the fluctuating spanwise
vorticity signals contributing to the data
point of closest separation in Figure 4.13 .......
Results of the w -wz correlations for Az
separations and the probes positioned

at y+ 3 30 ....................................... 244

Results of the w -wz correlations for Az

separations and the probes positioned

at y+ - 15. 3 at R9 - 1, 010 and y+ - 20. 4

at R0 - 2, 870 .................................... 245

Results of the w 'wz correlations for A2
separations and the probes positioned
at y+ - 6. 7 (R0 - 1, 010 only) .................... 246

Example of the more probable high angle
small scale lifting sublayer motion
(early in its evolution) ......................... 247

Example of the more probable high angle
small scale lifting sublayer motion
(later in its evolution)

xvi

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

A2.1.

A2.1.

A2.3.

A2.3.

A2.3.

A2.3

A3.1.

A3.1.

A3.1.

A3.l

.4 Approximate high Reynolds number u<Mw

Example of the less probable low angle
large scale lifting sublayer motion .............. 249

1 Comparison of "(Bu/8x)?" as estimated by

equation A2.1.2 with as estimated by

equation A2.1.3 as a function of Ax and for

a given noise level .............................. 252

2 Comparison of "(6u/6x)2" as estimated by

equation A2.1.2 with as estimated by

equation A2.1.3 as a function of noise level

for a given Ax ................................... 253

1 Near-wall mean velocity gradient profiles

as computed by finite differencing the mean
velocities measured from the parallel-array

element contained within the spanwise vorticity

probe ............................................ 261

2 Product of the relative systematic error of

the gradient, E, and the sensor separation, Ay,

as a function of probe body and probe tapered

head diameter Reynolds numbers (Rp and Rh
respectively) .................................... 262

3 Near-wall mean velocity gradient profiles

for the two lower Reynolds numbers shown in

Figure A2.3.l as "corrected” under the

assumption of a constant additive error .......... 263

.4 The product of the relative systematic error of

the gradient, E, and the sensor separation, Ay,

as a function of probe body and probe tapered

head diameter Reynolds numbers (R and R

respectively) for the ”corrected” data 0

Figure A2.3.3 .................................... 264

l u<ww >/uf8 profiles from the experimentally
deriveé data base of Klewicki and Falco‘;
R0 - 1,010; Re " 2,870; R0 “4,850 ......... 273

2 v<uw >/u,3 profiles from the experimentally
derived data base of Klewicki and Falco“;
symbols same as in figure 1 ...................... 274

3 Deduced u<ww /ufs profiles using the 6<uv>/8y
data of Klewic i and Falco“, the data of figure 1,
and, Eq. (2); symbols same as in figure 1 ........ 275

>/u 3
profiles deduced using the formulas of Phillips7,
the R{ -l,010 data of figure 2, and Eq. (10);

s

symbo same as in figure 1 ...................... 276

xvii

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

A3.

A3.

A3.

A3.

A3.

A3.

A3.

A3.

A3.

A3.

A3.

A3.

A3.

.1 Mean velocity profiles normalized by uf and

plotted versus y/fl ............................... 279

.2 Streamwise fluctuating velocity variance

profiles normalized by Uco and plotted
versus y/fl ....................................... 280

.3 Normal fluctuating velocity variance profiles

normalized by U6 and plotted versus y/0 .......... 281

.4 Reynolds stress profiles normalized by Ug

and plotted versus y/0 ........................... 282

.5 Reynolds stress rms profiles normalized

U6 and plotted versus y/0 ........................ 283

.6 <ww > profiles normalized by 0 and U5

and p otted versus y/0 ........................... 284

.7 <Nw > profiles normalized by the function y/Um2

and pIotted versus y/0 ........................... 285

.8 (vw )'profiles normalized by 0 and UL

and pIotted versus y/0 ........................... 286

.9 (vw )' profiles normalized by the function y/Uoo2

and p otted versus y/0 ........................... 287

.10 <uwz> profiles normalized by 0 and U;

and plotted versus y/0 ........................... 288

.11 <nwz> profiles normalized by the function y/U'ao2

and plotted versus y/9 ........................... 289

.12 (uwz)'profiles normalized by 0 and U6

and plotted versus y/0 ........................... 290

.13 (uwz)' profiles normalized by the function y/Um2

and plotted versus y/0 ........................... 291

xviii

LI OF SYMBOLS

Remap

A Additive constant in King's law calibration equation

B Multiplicative constant in King's law calibration equation
C Additive ”law of the wall" constant; Calibration correction
Cf Skin friction coefficient - rw/(l/ZpUmZ)

Cp Pressure coefficient - dp/pUao2

d Channel flow width

E Hot-wire voltage

f Frequency

fc Cut-off frequency

fk Kolmogoroff frequency — U1/2nn

fs Sampling frequency

h Wire spacing appropriate to velocity gradient measurements
H Boundary layer shape factor

1 Hot-wire length

K Kurtosis, fourth central statistical moment

n Exponent in King's law calibration equation

p Pressure

rms Root mean square

R Number of A-to-D registers representing Smax

RA Microscale Reynolds number - u'A/u

R9 Boundary layer flow Reynolds number - ugo/u

Rd/2 Channel flow Reynolds number - Ucd/2u

s X-array wire spacing

S Skewness, third central statistical moment

xix

Sma Maximum hot-wire signal level

t Time

T Averaging time

u Streamwise velocity

u' Friction velocity - (rw/p)1/2

U1 Local mean velocity

Uc Mean channel centerline velocity; Convection velocity
Uno Boundary layer free stream velocity

v Normal velocity

V Calibration velocity vector

w Spanwise velocity

x Streamwise Cartesian coordinate

y Normal Cartesian coordinate

z Spanwise Cartesian coordinate

@215

a Angular frequency

6 Hot-wire angle correction

6 Boundary layer thickness also 699

Ad Digitizing increment — Smax/R

At Time increment - l/f

AU+ Deviation of mean wake profile from the logarithmic law
Ax Small displacement in the streamwise direction
Ay Small displacement in the normal direction

Az Small displacement in the spanwise direction

1

2
cu

Dissipation rate; Signal noise level

e'jk Alternating tensor

Streamwise velocity signal noise level

XX

1 Angle of probe axis with the free stream
0 Kolmogoroff microscale - (us/e)1/‘
n von Karman's constant - 0.41

A Taylor microscale - u'/(6u/8x)'

u Kinematic viscosity

H Coles wake parameter

O Frequency spectrum

W Spectral function - ao¢

p Fluid density

0 Momentum deficit thickness

r Shear stress

'w Wall shear stress

w Vorticity, may be fluctuating or total depending on context
wx Streamwise vorticity component

my Normal vorticity component

”2 Spanwise vorticity component

Oz Mean spanwise vorticity component

W

i Refers to an inner variable normalization

o Refers to an outer variable normalization

+ Normalized by v and ur

Subscr t

f Refers to "forward" x-array wire orientation, flow -> \
i Refers to inner parallel-array elements

0 Refers to outer parallel-array elements

r Refers to "rearward” x-array wire orientation, flow -> /

xxi

x Refers to streamwise direction
y Refers to normal direction

2 Refers to spanwise direction

Other

' Indicates an rms quantity

< > Indicates a time averaged quantity, same as an overbar

xxii

CHAPTER 1

LIIERATURE REVIEW AND THE OBJECTIVES OF THIS STUDY

1.1 INTRODUCTION

Studies of the production of turbulence in wall bounded shear flows
have revealed a process generically termed bursting. This process takes
place intermittently within (predominantly) the inner regionl, and is
comprised of two major events, see for example, Kline, Reynolds, Schraub
and Runstadler (1967) or Corino and Brodkey (1967). One of these events
involves the local deceleration of streamwise velocity, and the ejection
away from the wall of low momentum sublayer fluid. The other of these
events is termed a sweep, and is characterized by a larger scale high
speed front entering the near-wall region at a shallow angle. Both of
these events have been identified with significant Reynolds stress
production, see for example, Kim, Kline and Reynolds (1971) and Lu and

Willmarth (1973).

The bursting events are complex and involve a broad spectrum of
length and time scales. Much of the research has therefore attempted to
dissect this process and discern those scales of motion which initiate
bursts from those scales of motion which are a result of the process. In

his review Bushnell (1985) poses the question of understanding turbulence

 

1The inner or wall region may be roughly identified as 0 S y/6 S 0.2, and
then the outer region is thus identified as y/6 > 0.2. The near-wall
region will be used in this study to mean y+ s 50, and the viscous
sublayer is identified as the region y+ S 5.

production in boundary layers as understanding "the origin of ubiquitous
wall streaks, and details of the inter-relationship(s) between the three
(or more) scales involved”. The three scales of motion to which he refers
may be generally labeled as those which are characteristic of the inner
region (i.e. are approximately invariant with Reynolds number when
normalized by u and ur), those which characterize the outer region (i.e.
are approximately invariant with Reynolds number when normalized by 6 and
uf or Um), and those which are intermediate in scale to the inner and

outer region motions.

The large scale outer flow eddies are of order one to three
boundary layer thicknesses, and have been observed to maintain their
identity for more than ten 8 as they convect in the streamwise direction,
see for example, Kovasznay, Kibens and Blackwelder (1970), Falco (1977)
and Brown and Thomas (1977). Furthermore, while the overall scale of the
outer region eddies is relatively insensitive to R0 (in relation to the
total thickness of the flow), Murlis, Tsai and Bradshaw (1982) and
Antonia, Rajagopalan, Subramanian and Chambers (1982) show that the shape
of the turbulent/nonturbulent interface at the outer edge of the boundary
layer is Reynolds number dependent for R0 less than roughly 5,000. The
intermediate scale motions have predominantly been identified by Falco
(1974), (1977), (1983) as vortex ring-like motions of the order of 100
inner variable units in scale. These motions, which can be found both in
the inner and outer regions, have also been shown to exhibit a
significant Reynolds number dependence, see Falco references above.
Within the wall region numerous vortical motions have been observed and
studied. These motions, which will be discussed in the context of an

inner/outer region interaction below and in terms of their viscous unit

scale in the introduction to Chapter 2, are the inner region "flow

modules" that participate in the bursting phenomena.

For the purposes of boundary layer control, one would like to
determine whether the flow conditions initiating the bursting phenomena
arise mainly from inner region dynamics or are dominantly a function of
the scales that characterize the outer region. A popular method by which
this determination has been attempted is to utilize the intermittent
nature of the bursting events and estimate the scaling properties of the
so-called bursting frequency (i.e. the time between bursts). Numerous
studies, using a variety of techniques, have been conducted with this
intent, and have often yielded conflicting results. For example, one can
compare the results in the earlier studies of Rao and Narasimha (1971)
and Blackwelder and Haritonidis (1983) or compare the more recent studies
of Luchik and Tiederman (1987) with Shah and Antonia (1988). These
conflicting results pertaining to the scaling of the bursting frequency
persist even though many of the reasons for the conflicting results, such

as the spatial resolution of the sensor, have been explained.

There does however, seem to be general agreement regarding the
scaling of the duration of sweeps and ejections. The duration of the
bursting events have been shown to exhibit wall layer scaling; see for
example, Lu and Willmarth (1973) and Sabot and Comte-Bellot (1976). This
result appears reasonable. That is, even if the bursting events were
initiated by outer region motions, the duration of these events would
probably remain a function of the response of the inner region motions to
this external perturbation. A similar argument however, cannot be applied
to the scaling of the times between the bursting events. One can easily

envision either inner region and/or outer region physics controlling the

bursting frequency.

Outer variable scaling of the bursting frequency is supported by
the classical spectral energy distribution that associates the production
of the Reynolds stresses to the large scale motions; see for example
Hinze (1975). Inner variable scaling is supported by the apparent
universality of the constants in the scaling law for the logarithmic mean
velocity profile. Evidently however, the dynamics that take part in
sustaining and initiating the bursting events are either too complex,
have no preferred scaling, or this scaling cannot be determined by
current experimental techniques. Most assuredly, the present
understanding of the bursting process is based upon incomplete evidence
and is probably biased by preconceptions about what the process should
entailz. Therefore, it is felt that the most useful models used to
explain the essence of the process should allow for enough variability so
as to embrace those areas of doubt where either evidence is scarce or

conflicting.

The problem of determining a definite scaling behavior for the
bursting frequency leads one to consider a significant interaction
between the inner and outer region motions. Kline (1978), using the
logical method of negative inference, gives convincing reasons why either
an inner region dominated or an outer region dominated physical model of
the production process should be rejected and replaced by a model that
incorporates a significant interaction between the motions of the inner

and outer regions. The data to be discussed in the following section

 

2While it is a certainty that the bursting phenomenon appears to repeat
itself (presumably randomly in space and time), based upon the existing
disagreement pertaining to the frequency of its quasi-periodicity,
perhaps all that should be assumed about the process is that it might be
regenerative.

indicate that a significant level of control of the transport across the
boundary layer may be achieved through modifying this interaction.
Modifications of this kind appear preferable in that they would alter the
underlying mechanisms sustaining the bursting events, rather than only

suppressing the symptoms of the process.

1.2 DETAILS PERTAINING TO INNER/OUTER INTERACTIONS

This section presents a partial review of previous studies that
have addressed issues pertinent to understanding and quantifying the
existence of the interaction between the motions of the inner and outer

regions.
1.2.1 Average Structure

In the boundary layer, the turbulence is maintained via the
continual degradation of streamwise momentum (initially extracted from
the free stream) through the irreversible action of the Reynolds
stresses. As early as the studies of Laufer (1950) and Klebanoff (1954)
it was known that the streamwise intensity, u', and both the production
term, P - -<uv>aU/ay, and the dissipation term, 6 - 2v<sijsij>, in the
equation for the turbulent kinetic energy exhibit distinct peak values
very close to the wall (y+ 3 15). The spatial location of these peak
values indicate that, on average, the boundary layer does not convert the
streamwise momentum extracted from the free stream into three dimensional
turbulent fluctuations until very near the wall. Thus, in an average
sense, the there is a continual transport of (predominantly) streamwise
momentum through the outer region to the inner region; where this

momentum is converted into three dimensional turbulent fluctuations and

eventually lost through dissipation. Sreenivasan (1989), suggests that "a
potentially useful point of view to take in explaining the boundary layer
dynamics is to consider the wall as sink of momentum, and that the inward
flux of momentum governs the boundary layer dynamics to a first

approximation".

Evidence for the existence of an interaction between the inner and
outer region motions may in found by examining the effects of changing
the inner and outer boundary conditions. Concerning the boundary
condition at the outer edge of the flow, there is a substantial body of
evidence suggesting that in general the profiles of many statistical
measures of wall bounded turbulent flows are (approximately) universal
under an inner variable scaling for some region near the wall regardless
of outer flow boundary condition. (The extent of this scaling region
depends on the particular statistic.) This indicates that to some degree
the dynamics near the wall are independent from the source of energy for
the flow (i.e. the irrotational free stream for flat plate boundary

layers or the mean pressure gradient for channels and pipes).

Concerning the boundary condition at the wall, it is useful to
consider the effect of a step change in surface roughness as studied by
Antonia and Luxton (1971) and discussed by Kline (1978). Immediately
downstream of the roughness and near the surface the axial intensity
exhibits a distinct rise, but the outer region profile is almost
identical to the smooth wall case. As one moves downstream of the step
change in roughness, deviations from the smooth wall case become
increasingly apparent until the entire intensity profile is significantly
different from the smooth wall case. This example clearly illustrates

that the outer region structure is not independent of the boundary

condition at the wall.

1.2.2 Inner Region Motions

That fact that many statistics in the inner region exhibit
approximate inner variable scaling has provided useful criteria in
constructing physical models of this region of the flow. One of the most
popular inner region models was first proposed by Townsend (1956).
Townsend's model (which originated through the interpretation of two
point velocity correlation data, and which continues to gain experimental
support, see for example Guezennec (1985)) features counter-rotating
vortex pairs oriented predominantly in the streamwise direction that
scale with distance from the wall, and thus in this sense are ”attached"
to the wall. In agreement with inner region data, these motions are
independent of the outer flow boundary condition and scale with distance
from the wall. Furthermore, there is substantial experimental evidence
supporting the existence of streamwise vorticity containing motions in
the near-wall region, see for example, Bakewell and Lumley (1967) and
Blackwelder and Eckelmann (1979). These motions have been associated with
producing the flow fields responsible for the low speed streaks, as well
as ejections -- since they are proposed to "pump" low speed sublayer

fluid away from the wall.

However, given the intermittent nature of the bursting phenomenon
in addition to the highly unstable environment of the buffer region, it
is doubtful that a sustained double-roller eddy interpretation accurately
represents the instantaneous physical reality. It seems more likely that
the average character of the near-wall region comes from the continual

interaction, creation and destruction of motions rather than the long

term properties of any sustained resident motion. Thus in order to retain
the above mechanism for the ejection of sublayer fluid one must account
for the continual regeneration of counter-rotating streamwise vortices

near the wall.

The presence of organized motions in the near-wall region that
contain streamwise vorticity have been explained by many models. Typical
examples are given by Willmarth and Tu (1967), and Offen and Kline (1975)
(also see for example, Head and Bandyopadhyay (1981), Wallace (1982) and
Smith and Lu (1988)). These models feature hairpin vortex-like motions
which are formed predominantly by the organized reorientation and lifting
of sublayer vorticity filaments originally oriented such that their
vorticity vector points in the spanwise direction. The legs of the
hairpin vortices are then seen to account for the observed counter-
rotating vortex pairs -- thus preserving the above mentioned mechanism
for the ejection of sublayer fluid. In general, these models propose that
the regeneration and subsequent breakdown of the hairpin vortex-like

motions are intimately related to the observed bursting phenomena.

The Offen and Kline model suggests that the cyclic occurrence of
bursts is due to convected vortical motions organized during an upstream
burst promoting the instability of the motions associated with the
sublayer streaks further downstream. The proposed instability mechanism
comes in the form of locally adverse pressure gradients, observed by Kim
et a1. (1971), which serve to lift up the streaks in a process they view
as a local separation of the sublayer. The initiating pressure
disturbances are proposed to be a result of wallward moving motions
originating in the inner region. Offen and Kline suggest that the large

scale outer flow eddies are maintained through a pairing process

involving newly organized vortical motions resultant from bursts. Thus,
models such as theirs propose that the interaction between the motions of

the inner and outer regions is triggered predominantly by inner region

dynamics.

Willmarth (1975), along with Kibens, Winkel and Christians,
performed conditional sampling on wall pressure and Reynolds stress
signals. These results are in agreement with Offen and Kline's in that
they give evidence for the presence of locally adverse pressure gradients
associated with the occurrence of bursts. However, additional to the
results of Offen and Kline, Willmarth attributed the initiation of the
bursting phenomena to a ”massaging" action on sublayer by the pressure
field due to the large scale outer flow eddies. This pressure field is
seen as responsible for the occurrence of the locally adverse pressure
gradients, and the formation of unstable high-shear layers associated
with the creation of hairpin vortices. Thus in models such as that this,
the interaction between the motions of the inner and outer regions is

initiated through the outer region pressure field.

In contrast to Willmarth's results however, Thomas and Bull (1983)
have performed experiments to test the role of the outer region pressure
field on initiating the bursting events. From their conditional sampling
study using wall pressure transducers and hot-wire anemometry, they
concluded that although there is an identifiable pressure pattern closely
associated with the passage of large scale outer region motions, this
characteristic pressure pattern is not of sufficient magnitude (as
compared with local inertial forces in an order of magnitude analysis) to
initiate the bursting events. Their results also indicate that at the

beginning of the burst sequence the near-wall flow field experiences a

10

favorable pressure gradient. Furthermore, they concluded that while
adverse pressure gradients may be associated with stages of the bursting

process, this pressure pattern is not responsible for its initiation.

A more recent model that features lifted sublayer motions other
than hairpin vortices has also been proposed by Jimenez, Moin, Moser and
Keefe (1988). In their interrogation of the computational results of Kim,
Main and Moser (1987) they found large scale (relative to experimentally
observed hairpin-type motions) vortex sheet or shear layer-type motions
extending from the wall at shallow angles. Furthermore, they tentatively
associated the creation of these nearly two dimensional sheet-like
motions to near-wall instability mechanisms observed during transition.
Thus this physical model (plausibly) associates the occurrence of
ejections with the instability of motions almost exclusively resident
within the near-wall region. At high Reynolds numbers the validity of
this type of generation process in supporting the outer region turbulence
becomes questionable. This is because as the Reynolds number increases
the ratio of the near—wall region to the boundary layer thickness becomes

extremely small.

A different model for the initiation of the bursting phenomena near
the wall that apparently embraces the existence of a significant
interaction between the inner and outer region motions (as well as the
conflicting results pertaining to establishing the scaling behavior of
the bursting frequency) has been proposed by Falco (1977), (1983),
(1987). In these studies, Falco has identified the above mentioned
intermediate scale vortex-ring like motions as important contributors to
the Reynolds stress in both the inner and outer regions. Furthermore, the

convection of these motions toward the wall and their subsequent

ll

interaction with the vorticity distributions of the sublayer has been
identified as an initiating mechanism for significant turbulent stress
production. In a related study, Chu (1988) has simulated the interaction
of these vortex ring-like motions with the sublayer in experiments with
laminar vortex rings impinging on the flow field produced by a suddenly
accelerated plate. In these physical simulations it was shown that many
motions associated with the near-wall region (such as hairpin vortices,

pockets, streaks and lifted shear layers) may be produced.

The model of Falco includes a significant interaction between the
intermediate scale vortex ring-like motions and the large scale outer
region motions (which are plausibly seen to influence the frequency in
which the vortex ring-like motions interact with the sublayer). Thus a
Reynolds number dependence in the interaction between the vortex ring-
like and outer region motions would appear to account for the ongoing
lack of agreement concerning the scaling of the time between bursts. This
model is also apparently supported by the wall pressure data of Emmerling
(1973) indicating that the most energetic motions perturbing the sublayer
are of small to intermediate scale. In connection with this, Willmarth
(1975) has suggested ”that one should also consider the existence of
small-scale convected vortices to explain the observed sublayer
structure”. Given the existence of an interaction between the inner and
outer region motions, Willmarth's suggestion appears to be the emerging
picture; with the large scale outer region motions playing a possibly
major role in the convection of highly vortical small scale motions above

and toward the wall.

12

1.2.3 Outer Region Motions

Probably due to their spatial extent, the large scale motions in
turbulent boundary layers are not as easily characterized as the motions
near the wall. Furthermore, the existence of organized large scale
motions remains in doubt. For example, Coles (1987) states that "the
large coherent structure in a boundary layer, if it exists, is concealed
in a tremendous clutter of noise". However, while the body of evidence
supporting the existence of a dominant and well defined outer region
motion is small, some information is known concerning specific features

of the outer region flow.

Brown and Thomas (1977) and Falco (1977) present an approximately
equivalent picture of the generic outer region motion. This large scale
motion, termed a bulge, is tilted in the streamwise direction at an
average acute angle with the wall reported to be between 18 and 48
degrees. These bulges, which convect at about 0.8Um, have also been
observed to exhibit a low frequency overturning motion in their interior
consistent with the sign of the mean vorticity. Furthermore, the above
investigators have also revealed the existence of deep valleys of
essentially non-turbulent fluid between the large scale bulges (sometimes
extending well within the wall region). This feature has also been
confirmed in the temperature contamination study of Chen and Blackwelder
(1978). In connection with this, Wallace (1982) states that the results
of Chen and Blackwelder "provides direct evidence that a relation between
the large scale outer structure and the events occurring near the wall
exists", and he further interprets this as evidence for hairpin vortex-
like motions extending well into the outer region, see also Head and

Bandyopadhyay (1981).

13

The upstream portion, or back, of a large scale bulge is exposed to
the irrotational stream, and in this region a stagnation point-type flow
(in a convected reference frame) has been observed to exist. Brown and
Thomas (1977) and Falco (1977) show that this stagnation-point type flow
redirects irrotational fluid around the bulge such that a wallward motion
occurs in the irrotational valleys between the bulges. The existence of
strong wallward velocities in these nonturbulent valleys was also clearly
exhibited in the earlier study of Kovasznay, Kibens and Blackwelder
(1970). Falco has further identified the backs of the large scale bulges
as a region in which one is likely to observe the intermediate scale

vortex ring-like motions.

Relevant to an interaction between the motions of the inner and
outer regions, both the studies of Brown and Thomas and Falco as well as
the study of Praturi and Brodkey (1978) associate the region near the
wall, in the proximity of the non-turbulent valleys, with highly
turbulent activity. In the Brown and Thomas study this region was
associated with wallward sweep-type motions. Consistent with this
observation, Falco (1983) has noted the frequent observation of the
vortex ring-like motions convecting wallward in this region. Thus it
appears that the introduction of essentially non-turbulent fluid near the
wall as a result of the shape of the large scale bulges plays a role in

initiating the bursting events.

Other large scale features of the boundary layer have more recently
been educed in the studies of Guezennec (1985) and Wark (1988). Both of
these studies used multi-point multi-sensor detection and mapping
techniques in an effort to uncover the details of the outer region

motions, and their influence on the observed wall region events. From his

14

study Guezennec concluded that large scale counter-rotating streamwise
vortex-like motions encompass much of the outer region. (Note he did not
actually measure streamwise vorticity.) Furthermore, he linked these
motions with the occurrence of sweeps or ejections near the wall
depending on their sense of rotation and the presence of inflow or
outflow between them. Thus these motions may either directly effect wall
region dynamics, or play a role in the convection of smaller scale highly
vortical motions consistent with the suggestion of Willmarth (1975), see
previous section. In general, the study of Wark further reinforced the
results of Guezennec, but tended to support a strong inner/outer
interaction through a "hierarchy" of scales of motion. Furthermore, her
study gives evidence for alternating sweeps and ejections to exist in the

spanwise direction.

1.3 OBJECTIVES AND METHODOLOGY

1.3.1 Objectives

The intent of the present study is to uncover essential details
pertaining to the interaction between the inner and outer region motions
in turbulent boundary layers. Mechanisms relating to the transport of the
turbulent stresses are identified and investigated. Furthermore, when
possible, links are made between average structure and observed
instantaneous motions. In particular, plausible initiating mechanisms
responsible for the lifting of vortical sublayer motions are identified
and discussed. This entails examining data relevant to the spatial
structure of the vorticity field in the inner region. Pertinent issues

relating to possible Reynolds number dependencies in boundary layer

15

structure are also addressed.
1.3.2 A Bias Toward Vorticity Measurements

Blackwelder (1983) and Hussain (1986) give the following

definitions of coherent motionsa.

Blackwelder; "A coherent eddy structure consists of a parcel of
vortical fluid occupying a confined spatial region such that a
distinct phase relationship is maintained between the flow
variables associated with the structure's constituent components as

the structure evolves in space and time."

Hussain: ”A coherent structure is a connected turbulent fluid mass

with instantaneously phase-correlated vorticity over its spatial

extent."
Note that both of these definitions attribute the intrinsic vorticity of
a coherent motion to be an essential property. Based upon this feature of
coherent motions, it is a hypothesis of this study that the optimal

under ta d n of turbulence ma be a ned throu h th stud

gfi_§hg_yg;;1g1£y_£igld‘ In practice, this hypothesis translates into the
hope that statistical descriptions in terms of one or more of the
vorticity components will give a more unique indication of the important
instantaneous turbulent motions than that provided through the

interpretation of velocity statistics.

 

3Within this dissertation the suggestion of Kovasznay (1979) is followed
in that instantaneous properties will be referred to as "motions” and the
permanence implied by the word "structure” will be reserved for time
averaged properties.

16

1.3.3 The Inner/Outer Interaction in Terms of Vorticity

The solenoidal condition on the vorticity field,

Vow-=50,

holds at every instant throughout a turbulent wall flow. As a
consequence, the vorticity filaments comprising the flow must either end
at isolated points on the solid surface, close upon themselves or extend
to infinity.

Very near the wall the dominance of viscous diffusion and the
geometric constraints imposed on the flow through the no-slip condition
results in the vorticity distributions of this region to be predominantly
oriented in the spanwise direction, and to (presumably) have a two
dimensional vortex sheet-like character. Further away from the wall, the
three dimensional effects of stretching and reorientation dominate, and
the important vorticity distributions (i.e. coherent motions) develop a
more limited spatial extent in the x and 2 directions. Attaching this
physical observation to the constraints imposed by the solenoidal

condition on the vorticity field leads to the hypothesis that a; some

w e ta on v
a e r nantl no on e connected t t e eet-like
d c a a te i of the subla er bu n tea take the fo
of closed loops‘; the simplest of which comes in the form of a vortex

ring. This hypothesis finds support in the outer region by the fact that
the velocity intermittency profiles of free shear flows and boundary
layers are virtually identical, see for example Hinze (1975). Note also

that the above notions concerning the limited spatial extent of coherent

 

‘In this dissertation these vortical motions will be referred to as
"reconnected”.

l7

motions lessens the possible significance of vorticity distributions that
extend to infinity -- especially outside the sublayer. In connection with

this hypothesis, it is further hypothesized that SEE essential faatures

 

9f Ehfi iatagaagioa beggeeg the inne; and outer region motions can ba
t o te act n between the reconnected and
laaallyf thraa gimaaaianal vorticigy aiatributions characteristic of the
u n the o a two d e sional disturbed sheet- ike
aiatriaagioas that ariae out of the sublayer.

A general goal of this study is to assess the usefulness and
validity of the above hypotheses. Furthermore, in making the last
hypothesis it is fully realized that complications affecting its validity
probably exist; such as the effect of possible large scale vortical
motions in the outer region, and the influence of the irrotational
valleys associated with the large scale bulges as discussed in Section

1.2.1.

1.4 PREVIOUS MEASUREMENTS OF TIME RESOLVED VORTICITY AND VELOCITY
GRADIENTS, SMALL SCALE STRUCTURE, AND REYNOLDS NUMBER DEPENDENCE IN WALL
TURBULENCE STATISTICS

1.4.1 Previous Measurements of Time Resolved Vorticity and Velocity
Gradients
In order to better understand the significance of the present
spanwise vorticity measurements, it is important to have a knowledge of
previous studies that have made time resolved velocity gradient and/or

vorticity measurements. As implied by the above coherent motion

 

5In this hypothesis the term "locally" is implied to mean relative to the
scale of the interacting sheet-like sublayer vorticity distributions and
the reconnected vorticity distributions hypothesized to occur away from
the wall.

18

definitions of Blackwelder and Hussain, it is generally believed that
coherent motions embody organized vorticity, or that they result from the
action of organized vorticity. This belief has lead to an increased
emphasis on the measurement of velocity gradient quantities and their

associated statistical properties.

As a rule of thumb it is generally acknowledged that to measure the
smallest scales in a turbulent flow the probe must be capable of
resolving motions of approximately the Kolmogoroff scale, a (2
(us/e)‘/‘). In the near-wall region of a fully turbulent boundary layer
this corresponds to less than 3v/ur. Few flow facilities or computations
allow for studies that have the capability to achieve this. Since multi-
wire probes are needed to directly obtain spatial gradients, both the
length of the wires and the spacing between wires should, in principle,
satisfy this criterion. The following partial survey gives an indication
of resolutions and integration times that have been used in studies of
wall-bounded turbulent shear flows that have made time resolved velocity

gradient and/or vorticity measurements.

Early boundary layer studies used probes which were relatively
large compared to the Kolmogoroff scale. Corrsin and Kistler (1954)
studied the intermittent region of a rough wall turbulent boundary layer
at R0 3 7,900 using a streamwise vorticity probe designed by Kovasznay
(1950). This probe's wire length, 2+ 2 Eur/v, was approximately 100 wall
units, and the average wire spacing was 3 70 wall units. Using an
estimate of n (Tennekes and Lumley 1972, p. 159), at y/6 - 0.9 yields 0+
3 5.6. Kovasznay, Komoda and Vasudeva (1962) used a parallel array with
spacing y+ 3 8 and wires of length 2+ 3 24 to study the later stages of

the transition process. The wire spacing in this study was about one

l9

tenth of the total boundary layer thickness. Kovasznay, Kibens and
Blackwelder (1970) used a pair of single wire probes configured to
measure au/ay to detect the vorticity fronts and backs of the large scale
motions in a fully turbulent boundary layer at R0 3 3,100. Their wires
had a length of 25 wall units and a spacing that varied from 40 to 60
wall units (or about 10 to 16 Kolmogoroff scales). Using the result of
Wyngaard (1969), Blackwelder and Kovasznay (1970) estimate that this

spacing results in resolving only 702 of the true rms gradient.

Later works involved more complicated multi-wire probes that
enabled one or more vorticity components to be measured. Eckelmann,
Nychas, Brodkey and Wallace (1977) studied oz and my as close as y+ - 15
in a channel flow at Rd/Z - 4,000. Their five film probe had films with
an average spacing and length of 1.75 wall units. An interesting
additional point is that they found that averaging times, 2TUc/d, of
greater than 3 5,000 were needed to obtain stable ensemble averages.
Falco (1980) using the same type of probe as in the present study,
measured wz in a fully turbulent boundary layer at y+ - 16 for R0 -
1,068. The spatial scale of this probe was approximately 3.6Ay+ and
11.5Az+, and the wire length, 2+, was 3.6. In this combined
visualization/hot-wire study, approximately 210 boundary layer
thicknesses were observed. Kastrinakis (1977) studied a turbulent channel
flow using Kovasznay (1950) type ”x probes (2+ 3 h+ 3 5 at Rd/2 3 6,250).
He measured both ”x and two point ”x correlations as close as y+ 3 9.
Further study of this probe by Kastrinakis, Eckelmann, and Willmarth
(1979), indicated that since the four wires were not independently
operated the probe output was significantly sensitive to all three

velocity components. Later Kastrinakis and Eckelmann (1983) measured

20

streamwise vorticity with a 4 wire (independently operated) probe in a
fully developed channel flow as close as y+ 3 19 for Rd/Z - 12,000. The
scales of this probe were h+ - 11.5, 2+ - 9.1, and their signal record
lengths were 2TUc/d - 2,670. Subramanian, Kandola and Bradshaw (1985)
studied the low wave number aspects of the outer part of a fully
developed turbulent boundary layer at R9 - 14,500. Their probe had a wire
spacing h+ 3 375, a wire length of 2+ 3 75, and they averaged for TUg/6 3
7,700. Balint, Vukoslavcevic, and Wallace (1987a) in a study of a fully
turbulent boundary layer at R9 - 2,100 measured for the first time all
three components of vorticity from y+ - 14.5 to y/6 - 0.95. Their nine
wire probe had an average wire spacing appropriate for computing
gradients of 8.9 wall units (Wallace, private communication), with wire
lengths 2+ - 2.3. Their averaging time was TUg/S 3 3,100. More recently
(Balint et al. (1987b)) they have used a probe with an improved signal to
noise ratio that had an average wire spacing of 10.4 wall units, with 2+
- 2.3 at R0 — 2,850. Klewicki and Falco (1986, 1987) using the same
apparatus described herein (Chapters 2 and 4), have measured spanwise
vorticity distributions and two point wz correlations across turbulent

boundary layers in the range 1,010 5 R0 3 4,850.

Using an array of two probes in a 'v' configuration that is flush-
mounted in the wall, it is possible to obtain vorticity measurements in
the immediate vicinity of the wall. Hogenes and Hanratty (1982) in a
conditional averaging study of turbulent pipe flow at Rd/2 - 18,040 used
electrochemical probes with an 2+ 3 8.5 and a Az+ 3 3.8, and 2TUc/d -
540. Blackwelder and Eckelmann (1979) studied the wall region vorticity
associated with the bursting phenomena in a turbulent channel flow at

Rd/2 - 3,850. The elements of this v-array were positioned at 45 degrees

21

to the mean flow direction. The length of these films was 2+ - 1.3, and
they were spaced Az+ - 5.3 apart. The averaging time used was 2TUC/d -
3,440. Kreplin and Eckelmann (1979), also using the Gottingen oil
channel, used v-array probes to obtain long time averaged statistics.
Using the same probe at the same Reynolds number as Blackwelder and
Eckelmann, they determined that an averaging time 2TUC/d - 2,870 was
required to obtain "adequate" convergence for velocity statistics up to

the fourth moment.

Recently Alfredsson, Johansson, Haritonidis, and Eckelmann (1988)
have attempted to clarify sources of measurement error using wall shear
stress sensors, and hot-wire/film probes in the sublayer. With respect to
probe resolution effects, they found that for streamwise velocity
measurements in the sublayer, probes of length 2+ — 2 and 10 resulted in
no differences within experimental error. (Measurements made with a probe
of 1+ - 8 in air resulted in discrepancies that were attributed to wall
heat transfer effects.) These experiments covered a range of Reynolds

numbers up to approximately R9 - 2,800.

Resolution problems are also inherent in computational studies of
turbulent wall flows. The accuracy of a solution of the discretized
Navier-Stokes equations is affected by spatial resolution constraints in
a way different from that of laboratory measurements. In a laboratory
experiment the job of the measurement device is to resolve the physical
flow field. In contrast, the job of the computation is to create the
physical flow field. Therefore to computationally create a flow field to
a given accuracy, grid spacings smaller than the probe scale necessary to
measure an equivalent physical flow field to the same accuracy are

probably required.

22

To date the two highest resolution computations of fully turbulent
wall flows have been performed by Kim Moin and Moser (1987) in a channel
flow, and Spalart (1988) in boundary layers. Kim et al. used a grid
spacing of AxT 3 12, Az+ 3 7, and 0.05 < Ay+ < 4.4 at the wall and
channel center line respectively (a 192 x 129 x 160 grid), for Rd/Z 3
3,300. Spalart in a study of turbulent boundary layers at R9 - 300, 670
and 1,410 varied the number of grid points to maintain a grid spacing of
Ax? 3 20, Az+ 3 6.7, and a non-uniform Ay+ such that there were 10 grid
points in the first 9 wall units (a maximum grid of 432 x 80 x 320).
Spalart's computational duration was only about TUé/5 - 40. However, as
stated in Kim et al., the computations allow averaging over planes
parallel to the wall. For example, using the computations of Spalart, if
averages were computed from data in a given plane separated by 7
displacement thicknesses (to maintain statistical independence), then the

total averaging time is TUg/S 3 140.

Investigators using computational techniques thus have used grid
spacings which are comparable to the probe scales used by
experimentalists. However, since it is currently unknown what scales are
important with respect to the formation of turbulent motions, it is not
clear that this is sufficient resolution to create the correct

instantaneous representation of the flow.
1.4.2 Evidence of Small Scale Structure

What measurement resolution is necessary to resolve the origin of
coherent events in the wall region? Emmerling (1973) using an array of
wall pressure transducers (of scale 3 54 viscous units) in a turbulent

boundary layer at R9 - 2,000 found examples of strong pressure

23

disturbances that were at least as small as 1/10 the transducer size.
Schewe (1979) working in the same tunnel at R0 3 1,400 showed that the
wall pressure intensity, skewness, kurtosis and the frequency of the
occurrence of pressure peaks continue to change with decreasing
transducer size. Even for his smallest diameter transducer, d+ - 19,
there was no indication that these trends had leveled off. Willmarth and
Bogar (1977) working in a boundary layer at R0 - 11,700 with hot-wires
that were 2+ - 2.5 and 3+ - 2.5 (s - the spacing between a pair of x-
wires) estimated that velocity gradients "in the small scale structure
near the wall will only become small over a distance that is less than
approximately 1/20 of the Kolmogoroff length”. Later Willmarth and Sharma
(1984) using probes of length less than one viscous length (at R0 - 6,840
and 9,840) give direct evidence for the existence of near-wall ”shear
layer fluctuations whose scale is of the order of the viscous length". On
the other hand, Johnson and Eckelmann (1983) working in the Gottingen oil
channel at the Rd/2 - 3,800 and using x-films with 2+ 3 s+ 3 1.7 did not
find evidence of these ultra—small scale motions. However, as they noted,
the Reynolds number of this study was approximately thirty times lower

than that of Willmarth and Bogar.

Experiments at widely varying Reynolds numbers using probes of
equivalent 2+ may not be comparable. This is based upon the hypothesis
that important changes in the small scale physics of the flow may occur
with Reynolds number. This is suggested by the disagreement between the
results of Johnson and Eckelmann and Willmarth-Bogar, Willmarth-Sharma
discussed above. Table 1.1 presents a summary of the spatial resolutions

of the velocity gradient and vorticity studies discussed above.

24

1.4.3 Evidence of Reynolds Number Dependence

The Reynolds number dependence of near wall turbulence statistics
is currently unclear. This is due to excessive data scatter and existing
contradictory results. Laufer (1950) in fully developed turbulent channel
flows (12,300 g Rd/2 3 61,600, 12.3 g 2* s 61.7) found that the maximum
measured value of u'/uf decreased as the Reynolds number increased.
Comte-Bellot (1963) also investigating turbulent channel flows (57,000 5
Rd/Z S 230,000, 26.0 S 1+ S 90.6) found a similar trend. Huffman and
Bradshaw (1972) in a analysis of wall bounded flow data, showed that at
low Reynolds number (say R9 < 5,000) the logarithmic mean velocity
profile showed a dependence in the intercept but not the slope. This
dependence was shown to be proportional to the gradient in shear stress
in the direction normal to the wall, ar+/ay+. Purtell, Klebanoff and
Buckley (1981) support this conclusion in their study of turbulent
boundary layers in which they found that the logarithmic law was Reynolds
number independent for R0 as low 465. However, their highest resolution
measurements (1+ 3 8) of turbulence intensity, u'/uf, show a Reynolds
number dependence opposite that of Laufer and Comte-Ballot. Murlis, Tsai
and Bradshaw (1982) using an x-array (1+ 3 30, 3+ 3 25) studied the outer
region of turbulent boundary layers (y/6 > 0.2) over the range R0 - 791
to 4,750. They found that the anisotropy parameter, "<u2>/<v2> increases
monotonically with Reynolds number while the shear correlation
coefficient, <uv>/u'v', reaches a weak maximum at R9 - 2,000 and then
decreases to an asymptotic value at high Reynolds number". They also
found that the triple products normalized by the local shear stress
showed a weak increase with Reynolds number. Andreopoulos, Durst, Zaric'

and Jovanovic (1984) using probes with 20 3 2+ S 100 measured the

25

statistics up to the fourth moment of the streamwise fluctuating
velocity. Their intensity profiles showed the same trend with Reynolds
number as Laufer and Comte-Bellot. Furthermore, their skewness and
kurtosis profiles exhibited experimentally significant differences over a
Reynolds number range 3,624 5 R0 5 15,406. Erm, Smits and Joubert (1985)
also show a Reynolds number independence of the von Karman constant, but
found an increase in u'/uf, v'/ur and <uv>/u1,2 with Reynolds number
across the inner region for y+ > 15. The length of their wires were in
the range 24 g 2* g 51 for 617 3 R0 3 5,010. Wei (1987) using laser
doppler anemometry in a channel flow, found a stronger but qualitatively
the same Reynolds number dependence in u'/ur and <uv>/uT2 for y+ 2 15,
and in v'/ur for y+ 2 4. Furthermore, Wei's higher order statistics and
spectra of the above quantities also show a significant Reynolds number
dependence. His probe dimension in viscous lengths ranged from 0.66 to
6.43 over a Reynolds number range 2,970 5 Rd/2 S 39,580. To date, little
information is available on the effect of Reynolds number on the

vorticity field in turbulent wall flows.

Reynolds number dependence however, has been observed to affect the
results of coherent motion studies of turbulent wall flows. Blackwelder
and Haritonidis (1983) found that the results obtained using the VITA
turbulence detection technique were strongly influenced by the probe
length. Over a range of R0 3 10,000, they observed that the bursting
frequency scaled on inner layer variables when they used wires of length
smaller than 20 viscous lengths. From this they concluded that earlier
results were erroneous due to spatial resolution problems. Falco
(1974,1977) using combined visualization/anemometry techniques observed

that important vortex ring-like motions found throughout the boundary

26

layer undergo a rapid decrease in scale (relative to 6) as R9 increases
for R0 S 10,000. This implies that quantifying these motions in the most
objective way would require decreasingly smaller probes as R0 increases.
In the outer part of the boundary layer both Antonia et al. (1982a) and
Murlis et a1. (1982) have found significant changes in the structure of
the turbulent/non-turbulent interface as R0 increases; especially at low
R9. Antonia et al., using temperature as a marker, found that for R0 less
than about 3,000 the average time between coherent temperature fronts was
strongly R9 dependent. Also using temperature as a marker, Murlis et al.
found similar results, and hypothesized that as R0 increases (for R9 3
5,000) the motions determining the shape of the turbulent/non-turbulent
interface shift from small to large scale. In a study nearer the wall,
Wei (1987) has attributed the non-universality of wall layer scaling of
u, v, and uv statistics and spectra to a relative increase in the
production of streamwise vorticity. He has conjectured that this increase
is due to the legs of the resident hairpin vortex-like motions becoming
more intense and smaller in wall unit scale as the Reynolds number
increases. Note that this is consistent with the conflicting results of
Johnson and Eckelmann and Willmarth and Bogar discussed above. Issues
pertinent to Reynolds number dependence are further addressed in Chapter

3 of this study.

CHAPTER 2

ON ACCURATEL! MEASURING STATISTICS ASSOCIATED WITH SMALL SCALE STRUCTURE
IN TURBULENT BOUNDARY LAYERS

2.1 ON RESOLVING NEAR-WALL MOTIONS

It is becoming increasingly apparent that experiments intending to
educe boundary layer physics (especially near the wall) must take in to
account the spatial scale of the measurement device. Within the near wall
region, various small to intermediate scale motions have been observed or
deduced. These motions include: a) low speed sublayer streaks with
lengths varying from 50 to greater than 1,000 wall units, with widths
between 5 and 30 wall units, and an average spanwise spacing of
approximately 100 wall units (Kline, et a1. (1967), (also see Smith and
Metzler 1983)), b) longitudinal counter-rotating streamwise vortices,
(see for example, Blackwelder and Eckelmann (1979), Lee, Eckelman and
Hanratty (1974), Bakewell and Lumley (1968)) which have an average
transverse spacing of 20-150 wall units, c) intermediate scale and
Reynolds number dependent vortex ring-like motions, with scales between
50 and 150 wall units, observed by Falco (1977, 1983) to be present in
the regions both near and away from the wall, d) intense wall pressure
pulses, of diameter 30 to 70 wall units, characterized in both the visual
studies of Falco (1982), and the interferometric pressure transducer
study of Emmerling (1973), and, e) hairpin or horseshoe shaped vortices
with scales ranging from 5 to 70 wall units which have their legs

connected to the vorticity distribution at the wall, reviewed in detail

27

28

by Willmarth (1975) and Wallace (1982). Probes that are large compared to

the above motions do not accurately resolve their kinematic signatures.

Long time averaged statistics depend upon the instantaneous nature
of the flow. Therefore, in order to obtain accurate long time statistics
one must resolve the above mentioned instantaneous motions. Reliable
experimental guidelines related to obtaining an accurate statistical
representation of wall turbulence are incomplete. Examination of the
scales of motion reviewed above indicates that measuring their vortical
signatures with sufficient resolution requires compact probes and/or
relatively large scale flow fields. Furthermore, little is known about
the convergence of statistics derived from time resolved velocity
gradient measurements. The problem of the convergence of statistics, in
general, further limits the accuracy of both experimental and

computational studies.

Numerous problems have hindered progress in understanding the
physical processes and motions occurring near a wall, but perhaps the
most important is related to the spatial resolution of probes used for
measurements. Uberoi and Kovasznay (1953) were the first to show
analytically that for isotropic turbulence the output of a hot-wire is
attenuated as the length of the wire is increased. In their channel flow
study of imperfect spatial resolution of single wire probes Johansson and
Alfredsson (1983) found significant attenuation in the measured value of
u' due to spatial averaging caused by a finite wire length. They also
concluded that the dependencies of u velocity statistics on the flow
Reynolds number are small compared to finite probe scale effects. Similar
results have more recently been found in boundary layers by Ligrani and

Bradshaw (1987). They concluded that for studies outside of the sublayer

29

adequate resolution for single wire probes may be obtained if 1+ 5 20,
and the wire length to diameter ratio is I/d > 200. However, they also
showed that at y+ - 17 significant changes in the high frequency end of
the u spectrum occur for variations in 2+ from 14.0 to 3.3. Mestayer
(1982), using Wyngaard's (1968) correction, showed significant
attenuation occurs in the high wave number spectra of u and especially v
for s/n and l/n both equal to 4.5. Therefore, since vorticity is
concentrated in the higher wave numbers, it is expected that the accurate
measurement of derivative quantities requires even better resolution.
Furthermore, the accuracy of derivative measurements as obtained by two
wires is degraded by problems additional to finite wire length. Antonia
Browne and Chambers (1985) discuss the most important of these to be due
to unequal time constants of the wires, mismatch in the wire

calibrations, and the effects of finite wire separation.

Most analytical studies of the effect of finite wire spacing in
multi-wire arrays, such as Wyngaard (1968), Wyngaard (1969), and Roberts
(1973), use isotropic assumptions. A recent compilation by Browne,
Antonia and Shah (1987) shows that most turbulent shear flows exhibit
significant levels of anisotropy. Thus, the results of the above
analytical studies may be best thought of as instructive but not

definitive under anisotropic conditions.

As indicated above, both wire length and wire spacing effects
hinder the accurate measurement of turbulent statistics. As efforts are
made to make probes smaller, presumably a more accurate representation of
a given probability distribution can be obtained. This increased
information comes as the result of resolving the fine scale motions.

Nearer wall however, the scales of the important motions decrease with

30

respect to the size of any given probe. Thus, it is unclear how the
enhanced resolution of wall region motions affects the convergence of

statistics.

The purpose of this chapter is to document the experimental
equipment and procedures of the present study, and to examine some of the
factors concerning the accurate measurement of various turbulence
statistics. Results were obtained using a four wire spanwise vorticity
probe in a very thick turbulent boundary layer flow that resulted in very
good resolution measurements. The experiments were carried out in fully
developed zero pressure gradient turbulent boundary layers at the three
Reynolds numbers R9 - 1,010, 2,870, and 4,850. A description of the flow
facility and the experimental conditions is given in Section 2.2. Section
2.3 describes the probe, its physical characteristics, and its
calibration. The experimental procedure and the accuracy of data are
considered in Section 2.4. Results in Section 2.5.1 include information
concerning the measurement of gradients in an anisotropic flow, as well
as, an evaluation of the effect of the probe's asymmetry on its
measurement accuracy. Section 2.5.2 presents information pertaining to
statistical convergence as a function of Reynolds number and position in
the boundary layer. Finally, Section 2.6 includes a discussion and

conclusions concerning the results of this chapter.

2.2 EXPERIMENTAL FACILITIES AND CONDITIONS

The experiments were performed in the 17 meter low Speed wind
tunnel in the Turbulence Structure Laboratory at Michigan State

university. A schematic of this tunnel is shown in Figure 2.1. The test

31

section of this suction tunnel is 17.1 m long, 1.21 m wide and nominally
0.61 m high. The top and one side wall of the tunnel are made of
plexiglass to allow for flow visualization, and the other two sides are
made of plywood. The tunnel is positioned in the center of a 18.3 m x
30.5 m x 6.1 m pressure and temperature controlled laboratory which acts
as the return circuit when the tunnel is used in the closed return mode.
Suction for the tunnel is provided by a low-noise axial fan, and is kept
at constant speed by an eddy current speed controller. The fan assembly
is mounted on vibration absorbers and is isolated from the test section
via flexible joints. A carefully adjusted set of screens and honeycombs
developed via an iterative procedure, and based upon the work of Loerke
and Nagib (1977) and deBray (1967), make up the tunnel inlet. This inlet
configuration was constructed to avoid the formation of Taylor-Gortler
vortices associated with tunnel inlet contractions. The resulting free
stream turbulence intensities at the speeds of the present experiments
are less than 0.2%. The tunnel exit consists of a 2:1 axial diffuser
followed by a 2:1 radial diffuser. For the present experiments the
adjustable top wall of the tunnel was set at a divergence of 0.25 degrees
over its entire length; resulting in a differential pressure coefficient,

de/dx (where C — dp/png), of less than 20.002. This value is well

P
within the tolerance of 0.02 deemed negligible by Murlis, Tsai and
Bradshaw (1982). The spanwise uniformity of the flow at the present
measurement location is z 2.3% peak-to-peak across the center 0.46 m as
determined by the Preston tube surveys of Rashidnia (1985). For more

details concerning the flow facility and its qualification the reader is

referred to Rashidnia (1985).

The data acquisition, signal conditioning, and probe positioning

32

apparatus consisted of the vorticity probe (described in Section 2.3.1),
4 DISA 55M01 constant temperature anemometers, a custom built analog
signal amplifier, a MKS baratron model 398 differential pressure
transducer, 2 Krohn Hite model 3323 analog filters, a Data Translation
DT3368/DT3369 simultaneous sample and hold A/D subsystem contained within
a PDP 11/23 computer, a cathetometer capable of measuring vertical
distances to within 0.01 mm (: 0.001mm) used to locate the probe center
from the wall, and a vertical traverse mechanism capable of positioning
to within 0.0254 mm (2 0.00254 mm). Detailed discussion of the use of

this equipment is given in Sections 2.3 and 2.4.

The zero pressure gradient boundary layers studied develop along
the lower wall of the flow facility. A 6.35 mm threaded rod is used to
trip the flow approximately 0.5 m downstream of the tunnel inlet.
Tripping the flow was not necessary to obtain a fully developed state at
the measurement station but did serve to localize transition and ensure
spanwise uniformity. The measurement station was approximately 15.25 m
downstream of the tunnel inlet. This extreme flow development length
resulted in enhanced spatial resolution of the vorticity probe even for
highest Reynolds number considered. A summary of the principal
characteristics of the boundary layers at the three Reynolds numbers
considered is given in Table 2.1. Logarithmic mean velocity profiles as
measured by the spanwise vorticity probe's x-array are presented in
Figure 2.2. Determination of the friction velocity was made using the
Clauser plot technique, in conjunction with Coles (1968) law of the wall.
As can be seen in the R0 - 1,010 data of Figure 2.2 this method of
obtaining u, is in very good agreement with the slope of the sublayer

profile.

33

2.3 THE SPANWISE VORTICITY PROBE; ITS CHARACTERISTICS AND CALIBRATION

2.3.1 Physical Characteristics

The spanwise vorticity probe used is similar to that described by
Foss, Klewicki, and Disimile (1986a). However, the calibration and
computational scheme used to obtain the vorticity time series was less
comprehensive than theirs. The probe, shown in Figure 2.3, consists of a
parallel-array, and an x-array located at the same streamwise position
but displaced in the spanwise direction. The spacing of the wires in the
parallel-array is nominally 1.0 mm, as is the spacing of the rearward and
forward slant wires comprising the x-array. The spacing between the
centers of the parallel and x-arrays is 3 3.4 mm. The slant wires of the
x-array are nominally at an angle of 45° with respect to probe axis; a
refined determination of their "effective" angles are made during

calibration.

The individual hot-wires of the probe are 5 pm diameter tungsten
wire which are copper plated at the ends. The copper plating allows the
wires to be soft soldered to the ends of the support prongs, and also
aerodynamically isolates the active region from the support prongs. The
overall wire length is 3 mm with a center active region of 1 mm. This
gives a length/diameter ratio of 3 200. For the measurements discussed in
this chapter all of the hot-wires were operated at an overheat ratio of
1.7. According to the study of Champagne, Sleicher, and Wehrman (1967),
under these conditions end heat conduction loss from the wires should be
less than 81 of the convective heat loss. The wires of the parallel and
x-arrays are mounted on 3 20mm long jeweler's broaches. The ratio of the

broach length to the tapered probe head diameter is about 8. This is

34

believed to greatly reduce possible probe body effects on the gradient
measurements, as discussed by Bottcher and Eckelmann (1985). For further
details concerning the probe's physical characteristics one is referred

to Foss, Klewicki, and Disimile (1986a).
2.3.2 Calibration

Calibration of the parallel-array wires was performed by fitting
the data to a King's Law type equation; E2-A+BVn, where the best least
squares fit was chosen from set having n - 0.40 to 0.60. To minimize
errors in au/ay resultant from mismatched calibration of the parallel-
array wires, one wire was calibrated with velocities inferred from the
pressure transducer and the second wire was then calibrated against the
first; a procedure suggested by Foss (private communication). The
procedure used for the x-array was initially devised by Foss and Falco in
the late 1970's, has been used extensively (see Falco 1980, Signor 1982,
Lovett 1982), and is a variation of the "effective angle" technique
assessed in the recent study of Browne, Antonia and Chua (1989). In this
procedure, equations of the type first used by King (1914) are derived
for the x-array wires oriented to measure u and v, and with the probe

body axis at 1 — 0° to the flow.

Er2 - Ar + BrVrnr (2.1a)

Ef2 - Af + vafnf (2.1b)

Under the assumptions that the slant wire angles (subscripts refer to
rearward and forward orientations) with the probe axis are close to :45°
(i.e., - 145° + 66) and that the A's, B's, and n's in equations 2.1 are

insensitive to small angle changes one can arrive at the following

35

relations for the u and v velocity components.

v - (Vf - Vr)/(Cf + Cr) (2.2a)

The angle correction parameters, C and Cf, in equations 2.2 are related

r
to their respective slant wire angle deviations from :45° by

spr - (cr - 1)/(1 + or) rad (2.3a)

53f - (1 - cf)/(1 + cf) rad (2.3b)

By gathering data with the probe at small yaw angles, 1 3 :10°, to
the known flow direction, and computing the relative u and v velocities

via the relations

u - Vcosy

v - Vsiny,

one can then use equations 2.3 to determine Cr and Cf. For the
experiments of this chapter, the average slant wire angles (averaged over
different flow speeds) were,

fir - 43.l° : 0.7°

sf — -48.6° 1.0°.

l+

These small angle variations presumably arise from changes in the
aerodynamic forces at different flow speeds, as well as, other
uncontrolled effects as discussed by Vukoslavcevic and Wallace (1981).

Repeatable probe alignment with the free stream is accomplished by

36

positioning the probe in line with the cross-hairs of a telescope
positioned about 12 m upstream, and a stationary marker about 2.5 m
downstream. The yaw angles of :5 and :10 degrees used for calibration
were accomplished by a device similar to that described by Bradshaw
(1971). All calibrations were performed using mean flow data (i.e., are
static), and based upon results of Foss et al. (1986b) no corrections are
used to compensate for transverse velocity contamination of the x-array
data. At least one determination of the King's law constants was made

both prior to and following each acquisition session.

2.4 EXPERIMENTAL PROCEDURE AND FACTORS AFFECTING DATA QUALITY

In subsections 2.4.1-2.4.5 information is presented pertaining to
what are considered to be "standard” factors indicative of data quality.
Furthermore, in subsection 2.4.6 possible data inadequacies associated
with maintaining the constant, but relatively low speed free stream

velocities of the present study are examined.
2.4.1 Sampling Rate and Cut-Off Frequency

The frequency response of the wires and anemometers were checked
using the standard square wave test. For the R0 3 1,010 free stream
velocity, this test indicated a response greater than 5 kHz. The
fluctuating signals from the four hot-wires were filtered1 via 4th order
Butterworth filters (24 db/octave) at the cutoff frequencies indicated in
Table 2.1. The signals were then digitized at the Nyquist criterion based

upon the free stream velocity and the minimum Kolmogoroff length scale

 

‘ Using a known input, the channel-to-channel phase shift caused by the
filters was measured. This was found to be about t 2ps.

37

found across the layer (using equation 2.6 and the preliminary
measurements of Klewicki and Falco 1986). This conservative criterion
resulted in digitizing anywhere from approximately 3 to 6 times the given
local Kolmogoroff frequency, fK - Ul/2nn, (U1 is the local mean velocity)
found in the layer. This relatively high sampling rate (fs - l/At) was
used to ensure that the Ax between consecutive data points under Taylor's
hypothesis (Ax - UcAt) was less than 1.0 mm. In the study by Antonia et
al. (1982b) in a circular jet flow it was found that the optimal low-pass
cut-off frequency is approximately 1.75fK for the purposes of measuring
higher order statistics of both velocities and velocity derivatives. The
low-pass cut-off frequency in the present study was set at 0.5fs. This

resulted in fc being anywhere from 1.5 to 3.0 times the given local fK.
2.4.2 Digitizer Resolution

Digitizer resolution is an important consideration concerning the
accurate computation of the even-order moments, as well as, the closure
of the tails of a probability distribution. Tennekes and wyngaard (1972),
describe the digitizer resolution problem as a trade-off between having
enough digital registers to faithfully reproduce large fluctuations, and
not so many that the signal noise causes registers to shift. In their
study they developed an approximate criteria for laboratory scale flows

based upon the maximum signal level, 8 the number of registers being

max'
flipped by the largest excursion, R, the digitizing increment, Ad (8
smax/R)’ and the rms noise level of the digitized signal, 5 (not to be

confused with the dissipation rate). The criteria:
26 3 Ad S 0.1,

states that the A/D register width should be at least twice the signal

38

noise level, and less than one tenth the largest excursion. The hot-wire
voltages in the current study were amplified prior to filtering in order
to increase the 12 bit (plus sign bit) A/D resolution (3 2.5 mv per
register). As estimated in Figure 2.11 (see Section 2.5) the total
electronic noise on the digitized signals was always less than a
millivolt -- thus satisfying the first inequality in the above criteria.
Considering the problem of resolving large excursions, it was desired to
obtain a worst case estimate of the digitizer resolution. To do this, an
estimate of the maximum v fluctuation was found at y+ 3 4.5 in the R9 3
1,010 boundary layer. This estimate for a typical large excursion was
obtained using the kurtosis of v and the rms of v at that location. It
was found for this case that Ad 3 0.050; thus satisfying the second
inequality above.

2.4.3 Total Integration Times

The near-wall data files for the three Reynolds number single-wz
probe considered contained 4x106 points (1x106 points in each channel),
whereas away from the wall the data file size was 2.4x106 points (6x105
points in each channel). Acquisition times ranged between 8.3 and 33.3
minutes depending on the sampling rate and the sample size. This
corresponded to sampling between about 3,600 and 8,600 integral scales as
defined by TUg/6. Following acquisition, each file was transferred to a

PDP 11/73 computer, and then written to tape. A detailed discussion on

the convergence of various statistics is given in Section 2.5.2.
2.4.4 On Taylor's Hypothesis

Numerous studies have been devoted to assessing the validity of the

Taylor's hypothesis method of measuring streamwise derivatives (i.e.,

39

[6()/6t a -Uca()/6x)]. Included are the four component LDA study of Lang
(1985) in a two dimensional turbulent mixing layer, and the spatial
temperature derivative study of Browne, Antonia and Rajagopalan (1983) in
a plane turbulent jet. Lang concluded that the use of Taylor's hypothesis
produces, at best, "fair" results. Browne et a1. (1983) found the
hypothesis to produce satisfactory results regardless of the choice of UC
(either the average or instantaneous velocity), and deemed corrections to
be unwarranted. Given the current undecided state within the turbulence
community, and for lack of a better method2’3, Taylor’s hypothesis was
used in the present study with the equivalent Ax S 2.5" and Uc defined to
be a short time average local velocity averaged over time intervals

ranging from Aturz/u — 10.6 to 42.3.
2.4.5 Data Reduction

The data reduction was performed on a PDP 11/73 computer. This
consisted of demultiplexing the hot-wire voltages, converting from
voltages to velocities, and then computing derivatives, spanwise
vorticity, etc., and their statistics. The components of ”z (2 av/ax -
au/ay) were resolved by computing au/ay via a 2 point finite difference
approximation using the fluctuating velocities from the parallel-array,
and av/ax (2 -(l/Uc)6v/6t) by evaluating the derivative of a local second
order least squares fit (using 5 points centered about the point of
interest) of the v-component velocities derived from the x-array data.

This method of computing av/ax is preferred to multipoint finite

 

2 Actually, Foss et al. (1986a) have devised a method which takes into

account the direction of Uc.

3 Perhaps the best method, in principle, is to measure all of the other
gradients as do the workers at the University of Maryland (i.e., Balint
et al. 1987a,b), and then check accuracy of the hypothesis through the

continuity equation.

40

difference approximations in that it has been observed to be both less
sensitive to the choice of step size and to signal noise. A simple

illustration of this is presented in Appendix 2.1.

With such large data samples the possibility existed for stray,
non-turbulence related noise spikes, to corrupt the data. A small number
of these points, if sufficiently far from the mean, could significantly
effect higher order statistics; especially quantities dependent on time
derivatives. To avoid this possibility, a point to point check (see
Appendix 2.2 for details) of all the velocities was made during the
voltage to velocity conversion process. This check consists of defining a
criterion based upon the standard deviation of the velocities for a 100
point moving averaged sample, and correcting those points which fail to
be within this criterion. The criterion was optimized such that it best
eliminated isolated discontinuous points. The criterion used was three
standard deviations. This resulted in affecting less than 0.0060! of any
given data file (i.e., s 60 points per million). In all but very few
cases data was altered only during quiescent times when the criterion

became very small.

2.4.6 Free Stream Uncertainty

Due to the relatively low flow speeds and the long integration
times involved in this study it became necessary to quantify the
variations in the free stream velocity. In particular, answers to the
following two questions were desired: 1) Within a given data record
acquisition time, what is the maximum uncertainty associated with a
possible low frequency variation in Um? 2) Was there a quantifiable

change in the wire calibration constants or the free stream conditions

41

over the duration of an entire measurement session? Answering the first
question is particularly important in assessing the level of certainty to
which the convergence results in Section 2.5.2 are meaningful. The answer
to the second question provides a more general statement about the

overall experiment quality.

2.4.6.1 Short Term Low Frequency Variation in Udo

In order to quantify the effects of low frequency variation in Uco
it is felt to be unsatisfactory to simply measure the variation in the
mean velocity over the duration of a data record (say by computing the
variance of an ensemble of short time averages). This is believed because
changes in the mean velocity do not necessarily provide an accurate
measure of this effect on fluctuating quantities. Instead an optimal
measure would be to examine this effect on the fluctuations directly. One
would expect higher order odd moments to be most sensitive to mean
variations. In general however, a skewness converges very slowly, and
thus using the variance of a short time skewness would also include
additional convergence error. Given these considerations it was decided
to quantify the possible effects of low frequency variations by observing
the maximum percent variation in <uv> from its final converged value at
y/6 3 0.37, as seen in Figure 2.17. This variable was chosen since it is
both an odd (i.e. the first) moment of a turbulence quantity and it
converged quite rapidly. The criterion was employed at y/6 3 0.37 since,
in general, statistics in this region of the flow converged most rapidly.
Thus a criterion was established that best isolates the convergence
errors from those that result from a low frequency variation in U”. The
uncertainties as indicated by this criteria for the R9 - 1,010, 2,870,

and 4,850 boundary layers are 1.9%, 1.5%, and 1.2% respectively. These

42

values are interpreted as the "noise" level in the convergence data of

Section 2.5.2.

2.4.6.2 Long Term Variations in Experimental Conditions

To ensure quality control over calibration drift errors at least
one determination of the King's Law constants (but often several
determinations) was performed before and after each experiment. Visual
inspections were made of the calibration data taken both immediately
before and after each experiment. If an identifiable trend could be
discerned between the "before" and "after" calibrations, then the entire
experiment was repeated. If the variation was deemed acceptable, the
individual "before" and "after" calibration equations were averaged; with
the exponent set at a constant (see equation 2.1). It is worth noting
that the probability of having to repeat an experiment was greatly
reduced by allowing the wind tunnel, as well as the electronics, to ”warm
up" until stable conditions were obtained. This was done by placing the
probe in a constant free stream and monitoring (say, every half hour) the
A/D (integer) output until subsequent runs produced the same numbers.
Typically it took between 1 1/2 to 4 hours for the equipment to

stabilize.

Since there was good climate control within the laboratory (:
0.25°C and no detectable variation in barometric pressure), the variation
in Uh over the duration of an experiment could be attributed to the
performance of the fan speed controller. To measure this performance, and
also to get an indicator of experiment reproducibility, an auxiliary
experiment was run. Furthermore, comparisons were made with data derived
from the two-point wz correlation experiments described in Chapter 4. In

these experiments, a single wz probe was positioned at a given y+ value

 

 

43

for the duration of the experiment. The auxiliary experiment consisted of
positioning the wz probe at y+ 3 6.2 in the R0 - 1,010 boundary layer and
collecting 5 data files at equal time intervals over the course of

approximately 8 hours. The size of each data file was 1/5 the size of the

data file represented at that location in Figure 2.14. The results of

1+

2 , as well as the data from the

this experiment, in terms of w
correlation experiments (with probes fixed at y+ 3 13.4 and 100.5) are
given as a function of experiment time in Figure 2.4. The horizontal
lines represent data taken from the distribution of Figure 2.14. A least
squares fit of each of the three data sets in Figure 2.4 was made using
the model wz' - Bo + Blt (where t is the experiment time). Given the
assumptions of additive, zero mean, constant variance, uncorrelated, and
normally distributed errors associated with the measurement of wz' as a
function of experiment time, and that the parameters Bo and B1 are non-
random, an F-test of these results indicated that to a level of
significance of 0.99 one cannot reject the hypothesis that B1 - 0. Thus,
it was concluded that over extended periods of time U; did remain
constant. Note also that the auxiliary experiment results tend to show a
greater variation about the data from Figure 2.14 than the correlation
experiment data, which had much longer averaging times. Due to the
relatively small sample size of the auxiliary experiment, a good part of
this point-to-point variation is probably attributable to convergence

error. This conclusion is further supported by the convergence results in

Section 2.5.2.

To obtain an error bar on wz', standard deviations were developed
from the data of Figure 2.4. The error bars in Figure 2.14 are

represented by :1 standard deviation. The maximum standard deviation (at

44

y+ - 6.2) is 4.6% of the measured value. Two significant points

pertaining to these error bars are:

a) the agreement between the mean values of the data of Figure 2.4
and the corresponding points in the distribution of Figure 2.14 are
within :21; indicating, in general, an ability to accurately
reproduce the free stream condition and,
b) the largest error bar represents a maximum error estimate for
all of the distributions since it may be conservatively estimated
that 1/3 of this error bar's length is attributable to a lack of
convergence and, as shown above, the R0 - 1,010 boundary layer had
the largest percentage of experimental uncertainty associated with
low frequency drift in Um.
As a final comment on data quality, it should be noted that no wire
breakages occurred during the course of acquiring the three R0
distributions described in Chapters 2 and 3. Thus this eliminates the
consideration of probe related uncertainties in comparing the different

Reynolds number results.

2.5 SPATIAL RESOLUTION AND STATISTICAL CONVERGENCE RESULTS

In this section spatial resolution and convergence results from the
four wire wz probe measurements are presented, and compared with other
well flow studies. Section 2.5.1 presents results which are used to infer
conclusions about the spatial resolution needed to accurately resolve
near-wall physics. Also included in this section is information
specifically pertaining to the validity and accuracy of the present
measurements. Section 2.5.2 presents results concerning the averaging

times necessary to obtain statistical convergence.

45

2.5.1 Spatial Resolution

Assessing the overall effect of finite spatial resolution on multi-
component probe measurements is difficult due to the numerous
contributing factors involved. In general however, it appears to be
fairly well documented that increasing the spatial scale of a probe will
result in an attenuation of the statistical moments of the probe's
signal. Given a wire construction with a sufficiently large length to
diameter ratio this attenuation can occur by two major effects. Finite
wire length effects (see for example, Johansson and Alfredsson (1983) or
Ligrani and Bradshaw (1987)) tend to average high amplitude small scale
fluctuations with spatially adjacent low amplitude motions over the
length of the sensing element. Finite wire spacing effects (as discussed
by Subramanian et al. (1985)) tend to spatially filter derivative signals
at wave-lengths about equal to the wire spacing. Both of these effects,
primarily by causing a failure to resolve high amplitude fine scale
information represented in the tails of a given probability distribution,
will generally cause an attenuation in the measured values of both the

even and odd moments of that distribution.

2.5.1.1 Wire Length Effects
An apparent indicator of wire length effects on wall flow velocity

measurements is the maximum measured value of u'/u Figure 2.5 presents

f.
a comparison of the maximum u'/uf value versus the non-dimensional wire
length as found using the u-wire closest to the wall in the ”2 probe, as
well as, from other wall flow investigations. Also included is a curve
fit of the data of Johansson and Alfredsson (1983) and Ligrani and
Bradshaw (1987). This curve fit shows that, at a given Reynolds number,

if one increases the length of a sensor the peak value of u'/uT will

46

decrease.

However, the variation in the peak value of u'/ur cannot be
explained in terms of non-dimensional wire length alone. For example, the
present data, that of Wei (1987), and Ueda and Hinze (1975) (all having
1+ < 8) show a consistent decreasing trend as probes of smaller 2+ are
used. Furthermore, the data of Purtell, Klebanoff and Buckley (1981) as
well as that of Andreopoulos et a1. (1984) apparently contradict the
trend shown by the curve fit. In the case of Purtell et a1. a difference
of 152 in the peak value is obtained for probes of nearly identical 2+ 3
8, and then for larger 1+ the attenuation is less severe than either the
Johansson and Alfredsson or Ligrani and Bradshaw data. In the case of

Andreopoulos et al., the data is roughly 152 higher than that predicted

by the curve fit.

To gain an understanding of this apparent scatter, the possibility
of a Reynolds number dependence in the "true" value of u'/urlmax was
examined. To do this, the curve fit of Figure 2.5 was used to correct the
data by removing the attenuation caused by finite probe scale effects.
Note that the use of this curve fit implies that this attenuation is only
a function of the non-dimensional probe scale, 2+. Figure 2.6 presents
the corrected data of Figure 2.5 as a function of Reynolds number. This
figure clearly suggests that the "true" peak u'/ur value is Reynolds
number dependent. Note that only the data of Purtell et al. (for wire
lengths 2+ 3 10.9, 20.4, 29.9), and that of Andreopoulos et a1. (1+ 3
20.9, 33.4) actually required the correction for spatial attenuation in
order to exhibit this trend. This apparent Reynolds number dependence
explains why the low Reynolds number data of Purtell et a1. is 15% lower,

and why the data of Andreopoulos et a1. is 152 higher than that suggested

 

47

by the curve fit in Figure 2.5.

Thus it appears that there are, in practice, competing effects

between an increase in u'/u1| as a result of a Reynolds number

max
dependence in its "true" value, and an attenuation in its measured valued
resultant from finite probe scale effects. Given that many lower
resolution studies in wall flows predict that the maximum in u'/uf
decreases as the Reynolds number is increased, detecting the opposite may
provide a simple measure of the minimum probe spatial resolution required
to study Reynolds number effects. Using the above conclusion, the data of
Purtell et a1. shows the correct Reynolds number dependence for small 2+,
but then as the Reynolds number increased (and thus increasing the non-
dimensional probe scale) an opposite trend is observed. This trend is
presumed to be a consequence of spatial averaging effects. It therefore

appears that the attenuation effect is stronger than the Reynolds number

effect.

These results show that a Reynolds number dependence occurs in the
fine scale structure. Furthermore they support the hypothesis mentioned
in the discussion of Table 1.1 that an even greater Reynolds number
effect may exist. However, this effect may possibly only be demonstrated
by using probe/flow configurations whose Ro/fi+ are increased even

further.

To gain an understanding of the effect of wire length on time
derivative statistics the values of the skewness of au/at in the near
wall region were examined. The present results are derived from the u-
wire closest to the wall contained in the wz probe. Figure 2.7 presents

an adaptation of a plot by Johansson and Alfredsson (1983) with additions

 

e ...—"‘3 -...3. A.

48

from other channel and boundary layer investigations (also see Table
2.4). The generally good agreement between the present data and that of
Wallace et al. (1977) and Ueda and Hinze (1975) for y+ 2 15 probably
reflects the fact that the spatial resolution (either wire length and/or
wire spacing) of the probes in all of these investigations is about
equal. It should be noted that Reynolds number effects apparently do not
manifest themselves in this region of the flow over the range 300 S R9 5
5,000 for this statistic. Apparently the data of Johansson and Alfredsson
shows attenuation due to their relatively large wire lengths. Closer to
the wall this figure also shows that considerable data scatter exists (up
to 30%) for wire lengths in the range 5 < 2+ < 15. In light of the
discussion above concerning u'/u1 this could be due to spatial resolution
and/or Reynolds number effects. Estimates of the S(6u/6t) have also been
obtained from the x-array of the ”2 probe. As the Reynolds number
increased these profiles showed an identifiable decrease in the magnitude
for y+ < 50. This trend is consistent with an attenuation due to the
additional spatial averaging effects resultant from the wire spacing of
the x-array. Thus, it appears that spatial averaging has a greater effect
on skewness of au/at than on those of u'/ur. This result is consistent
with the notion that greater spatial resolution is required to obtain

derivative information with the same accuracy as the variable itself.

It should be noted that comparisons between v statistics (rather
than u statistics) would provide a much more stringent measure of spatial
resolution. This is because v statistics are associated with smaller
scale motions. This objective however, proves to be difficult since there
is significant scatter in the v statistic data. This scatter is probably

due to numerous factors such as Reynolds number effects as indicated by

49

Wei (1987) and the results in Chapter 3, and inherent spatial resolution

effects due to the need for multiple wires.

Figure 2.8 presents measurements of the skewness of v along with a
representative sample of existing profiles from turbulent boundary layers
and channel flows. Comparing Figure 2.8a with 2.8b, the data in the
buffer region shows two distinctly different trends. Figure 2.8a (which
includes all of the present measurements) shows S(v) data which are
positive in the range 5 S y+ S 30. Figure 2.8b shows data which are
negative in this range. Note that in Figure 2.8b one of the data sets is
from the discrete Navier-Stokes simulations of Kim et a1. (1987). Of
particular interest, is the apparent fact that there is no obvious
dependence on either probe spatial resolution or Reynolds number in

either Figures 2.8a or 2.8b.

To gain insight into the potential causes behind the observed data
scatter, Table 2.2 was prepared. In this table the sign and magnitude of
S(v) at y+ 3 20 was chosen to enable more data to be represented, even
though in both cases the maximum deviation from zero occurs at about y+ 3
13. Table 2.2 gives information pertaining to experiments of different
flow types, Reynolds number, probe scales, facility types, and the
presence/absence of a tunnel inlet contraction. There does not appear to
be a clear-cut indicator among these parameters that explains the

observed data scatter.

Looking more carefully however, a correlation possibly exists
between the influence of a tunnel contraction and the sign of S(v) in the
buffer layer. In the three boundary layer studies in which S(v) is

positive in the buffer layer, the flow facilities either have no

50

contraction, or the layers developed along a plate suspended in the
center of the tunnels. Furthermore, for these studies S(v) was measured
using an x-array of wires or films. In the two boundary layer studies in
which S(v) is negative in the buffer layer the flow facilities have inlet
contractions and the layers develop along the tunnel wall. The
possibility of creating streamwise vorticity in a contraction has long
been acknowledged. This vorticity, which is approximately fixed in

spanwise location, could account for the near wall differences in S(v).

For the channel flow data, the results of Wei appear to indicate a
clear Reynolds number dependence -- with S(v) changing sign from negative
to positive as the Reynolds number increased. However, this observation
is in direct contrast with the results of Kreplin and Eckelmann (1979)
who found S(v) to be positive at about the same low Reynolds number. It
is also interesting to note that only the low speed LDA measurements of
Wei (1987) and Barlow and Johnston (1985) (and the nine wire probe of
Balint et al. (1987a)) have reported negative values of S(v) for 5 < y+
30. As noted above the Kim, Moin and Moser computation also gives

negative values.

2.5.1.2 Wire Spacing Effects
According to the study by Wyngaard (1969), to resolve fluctuating
velocity gradients in an isotropic turbulent flow the wire separation, h,

should satisfy the inequality
1.0 < h/n s 3.33 (2,2,)

where, n - (vs/e)1/‘. In the present study 6 was estimated using four of

the twelve terms.

51

e/v - <(au/6x)2 + (av/6x)2 + 3.5(8u/dz)2 + 2.5(6u/8y)2> (2.5)

This relation is derived by making the following assumptions:
<(aw/ay)2> - <(au/8y)2>, <(av/az)2> - <(au/az)2>,
<(8v/ay)2> - 1/2<(au/ay)2>, <(aw/az)2> - 1/2<(6u/6y)2>,

and <(aw/6x)2> - <(au/8z)2>,

as well as assuming that the sum of the three cross derivative terms is
negligible. These assumptions are similar to those given by Klebanoff
(1954).

While describing probe dimensions in viscous units allows objective
comparisons to be made between different studies, often times one would
also like to know the probe scale in terms of n. Many wall flow studies
do not have the capability to accurately estimate 6, and thus it would be
advantageous to have a simple compact formula for 6. An inner variable
non-dimensionalization of the turbulence energy equation under the
assumptions of production equals dissipation, <uv>/u’,2 - 1, and that the

law of the wall is valid allows e+ (3 ev/ur‘) to be expressed as:

5+ = l/ny+ (2.6a)
or
e - ufs/ny. (2.6b)

Figure 2.9 presents estimates of 5+ obtained using equation 2.5, as
well as the estimate derived by assuming isotropy. For R0 2 3,000
equation 2.6b provides a good agreement with the given dissipation rate
estimate for y+ > 10. As the Reynolds number decreases the agreement
becomes progressively worse in the inner region. Note also that by using
the isotropic estimate (6 s 15u<(6u/ax)2>) one obtains differences with

the estimate of equation 2.5 (or 2.6) of over 300% in e (or 401 in n),

52

and thus the estimate given by equation 2.6 is preferred.

Table 2.3 presents a summary of the minimum and maximum non-
dimensional wire separations encountered in the three boundary layers of
the present study. In this table Ay represents the wire spacing in the
parallel-array and A2 represents the separation between the centers of
the parallel and x-arrays. It should be noted that the resolution of ”2
does not depend on 6()/dz gradients. However, the given probe arrangement
does assume that av/ax at z - av/ax at z+Az. Thus, it was necessary to
evaluate the validity of this assumption, as well as, the validity of
equation 2.4 under anisotropic conditions. Furthermore, since many multi-
wire probes use spatial averages of temporal and streamwise derivatives
to approximate measurements at the given probe center, the effect of wire

separation on this process is also examined.

Evaluation of wire spacing effects on the measured value of the
associated spatial gradient was done by performing an auxiliary
experiment. The geometry of this experiment consisted of fixing a
parallel-array about a given point in the R0 - 1,010 boundary layer, and
then positioning at equal Ay spacings above and below this array a pair
of y-traversable single wire probes. The arrangement is shown as an
insert in Figure 2.10. In this figure Ayi is the wire spacing between the
parallel-array elements, and Ayo is the spacing between the two single
wire probes. Simultaneous data from all four wires was taken for various
Ayo; keeping the center position of Ayi and Ayo the same. The matching
calibration procedure described in Section 2.3.2 was performed using the
lower wire of the parallel-array as the reference wire. The experiment

was performed at y+ 3 53 and at y+ 3 38.

Results pertaining to the attenuation in the measured rms of

53

(8u/6y)° (3 (8u/8y)'o) as Ayo was increased are given in Figure 2.10. The
data in this figure are presented as the ratio of (8u/6y)’o over
(6u/8y)'1. Since data from all four wires was taken simultaneously for
each Ayo, data presentation in this way greatly eliminates from the
results the effect of the observed :32 variation in (6u/8y)’ from run to
run. It is important to note that for the given boundary layer Ayi+ 3
1.84, and Ayi/n 3 1.04 and 0.94 at y+ 3 38 and 53 respectively. Thus the
data in Figure 2.10 are (in principle) equal to 1.0 at these values and
not at the origin. One thing to notice about both the y+ 3 38 and y+ 3 53
data is that for Ay/n S 3 very little attenuation (3 32) occurs, and that
changing the level of mean shear has little effect. This suggests that
Wyngaard's estimate (equation 2.4) is valid to a good approximation, even
under anisotropic conditions. Another thing to note is the rapid
attenuation (~ 151) for wire spacings between Ay/n - 3 to 6; which agrees
well with the results to be shown in Figure 2.15. These results are
seemingly in contrast with those of Bottcher and Eckelmann (1985) which
predict that the inner array would give smaller rms values than the outer
array for small Ayo. Significant differences, however, such as flow type
(laminar vs. turbulent), probe type (hot-wire vs. hot film), probe
construction, and calibration method exist between the two studies.
Further reference is made to these factors in the discussion of Figure
2.14. Also, data pertinent to assessing the accuracy of the parallel-
array measurements of the mean gradient, 8U/6y, are presented in Appendix

2.3.

In order to quantify possible spatial filtering effects of finite
wire separation, spectra of the individual (flu/6y)i and (flu/6y)o time

series were computed. Examples of the frequency spectra for Ay/n - 0.94,

54

4.83, and 9.01 at y+ 3 53 are presented in Figure 2.11. By using the wire

spacing to construct an approximate frequency, f - Ul/(2xAy), it can be

AY
clearly seen that the effect of wire spacing is to attenuate the gradient
intensity at higher frequencies. Furthermore, creating separate spectral
plots from each pair of simultaneously acquired (Bu/6y)i and (Bu/6y)o
time records allowed for the determination of effective low-pass cut-off
frequencies (i.e. the frequencies at which the (Bu/6y)o spectra diverged
from the (au/ay)1 spectra). These results, as derived from the y+ 3 53
data, are presented in Figure 2.12 as a function of outer array
separation. In this figure the ordinate is the ratio of the approximate
cut-off frequency over fK a Ul/(Zun). The error bar to the left of the

spectral curves in Figure 2.11 represents the data scatter at lhz, and

was used in determining the cut-off frequencies in Figure 2.12.

Interesting results concerning how noise in the velocity signals
enters into the au/ay signals are also described in Figure 2.11.
Heuristically, one would expect that at the highest measured frequencies
the energy in the u spectra is due entirely to electronic noise. This
noise level, labeled tug in Figure 2.11, is associated with the white
noise portion of the spectrum at high frequencies. For the u spectra in

this figure 6 2 is about e'15'75 (ftz/s), or equivalently, less than 0.3

u
mv of electronic noise. Assuming that the cu” for each wire is
uncorrelated, then in the noise dominated part of the spectrum one may
approximate ¢(au/6y) by tug/Ayz. Using this equation and the above cited
value for an2 allowed for estimates of the associated noise levels in
¢(6u/3y) for the various Ay. The predicted noise levels are given as the

horizontal lines through the au/ay spectral curves. As one can see, the

predicted values agree very well with the actual noise levels in the

55

au/ay spectra. These results suggest that the noise level in a single
velocity signal can be used to determine the minimum acceptable Ay

spacing used to measure au/ay.

Another problem associated with multi-wire array measurements is
that often it is necessary to average the same quantity obtained at two
spatial locations in order to approximate point-wise measurements at the
probe center (Balint and Wallace private communication). The data of the
above described experiment provided a means by which to measure the
effect of this averaging procedure. For each wire a au/at time series was
computed. Then at each instant a spatial average of au/at was computed
using the inner two wires and the outer two wires respectively. From
these (8u/6t)1 and (Bu/6t)o time series' rms values were computed. As
with the au/ay data, the ratio of rms (au/at)o (I (au/at)'°) over
(du/dt)'i is presented in Figure 2.10. The attenuation shown by the data
is very rapid for wire spacings AY/fl S 15, and then continues to
attenuate, only more slowly, for greater Ay values. Clearly, for a given
non-dimensional wire spacing the effect of increasing the mean shear
(moving closer to the wall) is to cause further attenuation due to this
averaging procedure. Note that this effect is not readily apparent in the
au/ay data. It is also interesting to note that the attenuation of
(6u/8t)’ (due to spatial averaging) for small Ay is greater than that of
(au/ay)'. These results suggest that significant changes in the temporal
gradients exist across Kolmogoroff scale eddies. This spatial sensitivity
in the temporal gradients is also suggested by the comparison that was
made between S(du/at) as derived from the x-array and the single wire of

the parallel array in the discussion of Figure 2.7.

It should be noted that the utility of the results in Figures 2.10,

56

2.11, and 2.12 are dependent on the absolute accuracy of the inner array
results. Of course the absolute values of (6u/8y)’ will never be easily
verified. However, for the au/at data no statistically significant
difference was found between (6u/6t)’ from the inner array (Ay/n 3 l) and

the (6n/8t)’ values derived from the individual wires of the inner array.

An assessment of the validity of the assumption:

av/ax at (z) - av/ax at (z + Az),

(as required to compute wz from the present probe) was made by spatially
separating two x-arrays in the spanwise direction and computing the av/dx
correlation coefficient for various Az separations. The results of this
experiment are shown in Figure 2.13 as a function of the non-dimensional
spanwise spacing at y+ 3 140 in the R9 - 1,010 boundary layer. Two runs
were made per Az spacing to increase confidence in the results. At this
point in the boundary layer (6v/8x)'/(6u/6y)' is about equal to 0.38;
indicating significant anisotropy. Also included in this figure (for
reference) are the u and v correlation coefficients.

The results in Figure 2.13 show, for example, that at the given y+
location an average error of approximately 201 can be expected in the
instantaneous av/ax values for A2 separations of approximately 7
Kolmogoroff scales (which is 3 14 viscous scales at the given position in
the boundary layer). Examination of Table 2.3 shows that in the R9 3
1,010 boundary layer the separation between the parallel and the x-array
center is always less than n - 7. Most of the R0 - 2,870 data, and about
half of the R0 - 4,850 data satisfy this condition. However, these
results should be viewed with some reservation since in general integral
scales decrease as the wall is approached. Furthermore, a significant

Reynolds number dependence may exist for this correlation coefficient. In

57

any case, these results suggest that av/ax (and probably all v gradients)
are largely comprised of small scale contributions, and attempts to
resolve v gradients over wire spacings much greater than a few
Kolmogoroff scales will result in serious errors. Foss, Ali, and Haw
(1986b) support these results in that in a similar experiment done in a
free shear flow they show that the angle of the velocity vector changes
significantly for small spanwise separations. On the plus side however,
the contribution of this error to the spanwise vorticity is diminished in
the near-wall region since in this region au/ay dominates av/ax (see

Figure 2.14).

2.5.1.3 The Resolution of wz

To investigate probe scale effects on w ' the distributions of two

Z

different non-dimensional functions containing wz'

were examined. Figure
2.14 shows an inner variable normalization versus y+. As can been seen,
the wall variable normalization results in a very good scaling of the
data. This is further shown by the fact that wz' for the present data
varies by a factor of ten over the given Reynolds number range. The
results of this figure suggest that spatial resolution effects not only
attenuate the present data, but also the data of Balint et al. (1987a).
In the wall region, a small but identifiable attenuation with increasing
Re can be observed. This attenuation is consistent with a decrease in
spatial resolution resultant from an increase non-dimensional probe scale
caused by an increase in the Reynolds number. Note that if one has
confidence in the apparent scaling of wz' by u/uf2 (at least in this
region of the flow) then the under-estimation of wz' for small wire

spacings (via the under-estimation of au/ay) as predicted by Bottcher and

Eckelmann (1985) is not apparent in the present results. An explanation

58

of the error bars presented in this figure was given in Section 2.4.6.
Additionally, it should be noted that the computations of Kim, Main and

Moser agree well with this scaling except as one nears the sublayer.

To further examine the spatial resolution effects suggested in
Figure 2.14 an alternate non-dimensional function containing wz' was
examined. Figure 2.15 shows distributions of ywz'/ur versus y/fl. (The
data set of Kim et al. was left off this figure because of the large
differences in the wake structure of boundary layers and channel flows.)
This function was chosen since it shows a distinct Reynolds number trend
when plotted versus y/0. One can see in the present data a clear Reynolds
number trend in the opposite direction as that predicted by spatial
resolution effects. Given this Reynolds number dependence (and assuming
that at each R0 this function is represented by a single curve), the
distribution of Balint et al., (R0 - 2,100, h+ 3 8.9), should lie between
the two lower Reynolds number distributions of the present study. The
fact that it does not is probably due to the relatively larger wire
separation (in viscous units) of their probe. This figure indicates that
their w ' measurements show an attenuation between 10% and 15%. This

2

result is in remarkable agreement with the findings in Figure 2.10.
2.5.2 Convergence of Turbulence Statistics

Numerous results concerning the convergence of statistics in
turbulent flows may be found throughout the literature. In fact, most
studies presenting higher order statistics (especially those
investigating the dissipation scales) report, to varying extent,
information pertaining to statistical convergence. Typically however,

since these studies are investigating small scale structure, little

59

attention is paid to more global effects such as the level of mean shear
and the flow Reynolds number, R0. Wall-bounded shear flows present
perhaps the most demanding conditions for obtaining statistical
convergence. In boundary layers one has both the turbulent/non-turbulent
interface at the outer edge, and the highly intermittent production
process near the wall. Furthermore, wall flows (versus free shear flows)
generally have relatively more energy concentrated at higher wave numbers
due to the presence of the wall. This section presents a fairly
comprehensive look at the statistical convergence of a variety of
variables in four regions of the boundary layer in which the physics is

different, and for the R0 range given in Table 2.1.

A short review of popular analytical/empirical relations concerning
the estimation of convergence times necessary to ensure a given accuracy
is given by Antonia et al. (1982b), and therefore will not be repeated
here. Rather, the goal of this section is to both document some of the
factors influencing statistical convergence, and to provide a purely

empirical basis by which to design experiments.

2.5.2.1 Methodology
Cumulative estimates of the statistics up to the fourth moment were
computed for u, v, uv, au/at, av/ax, au/ay, ”z and vwz. (In the following

and vw are shown.)

presentation only the results of u, uv, au/at, wz 2

These estimates were output at regular intervals of 200,000 points (the
corresponding integration times depend on the sampling rate). The
relatively long times between the intermediate averages were chosen to
facilitate statistically meaningful convergence results. The convergence
data for the above variables were compiled for the three Reynolds numbers

given in Table 2.1, and for four regions of the boundary layer. These

60

regions are denoted by:

1 near-wall y+ - 12 - 18
II strong-shear y+ - 40 - 50
III weak-shear y/6 - 0.36 - 0.40
IV intermittent y/6 - 0.75 - 0.80

Convergence data was compiled for each variable and arranged in the
form of Figure 2.16. In these figures, which are comprised of 12
component plots, each row shows data from a different region, whereas
each column shows different Reynolds number data. The ordinates of the
individual component plots are in units of absolute percent convergence
as given by:
(x1 - xf)

xf

where X1 is the intermediate value of the moment, and Xf is the final

x 100%

value found using the entire data record. Thus, presenting data in this
way assumes that Xf has converged. This assumption proved to be valid in
the vast majority of cases. The abscissae of the component plots are in
non-dimensional time units of Tum/6, where T is the averaging time. These
units were chosen to represent approximate non-dimensional integral time
scales. Data presentation as described above allows for the
identification of Reynolds number and/or boundary layer regional trends.
To facilitate the identification of trends, the convergence data were
curve fit by second order polynomials (note that in Figures 2.16, 2.17,
2.20, 2.21 and 2.23 points above the ordinate range in the component
plots are not shown).

Before the convergence data were examined they were divided into

three groups. These groups are velocity and Reynolds stress, velocity

61

gradients, and velocity-velocity gradient products. Within each one of
these categories convergence rate comparisons are first made between the
variables within a group. Then, general trends in convergence among the
four boundary layer regions are examined. Finally, Reynolds number trends

in convergence are explored.

2.5.2.2 Velocities and Reynolds Stress

Comparisons between the convergence data for u, v and uv were made.
The convergence data for u are in Figure 2.16 and that for uv are in
Figure 2.17 (note that the ordinate scales of the component plots may be
different for different figures). Somewhat surprisingly, the present
results indicate that in general the uv statistics converged at least as
fast as the u statistics. Furthermore, the skewness of uv shows a
distinctly more rapid convergence than that of u. For all four regions,
Reynolds numbers, and statistical moments, the v statistics showed the
slowest convergence rates. Except for perhaps the intermittent region of
the R0 - 1,010 flow (see Figure 2.16), the skewness of both u and v
showed the slowest convergence rate. This is not surprising since odd
order moments tend to converge more slowly than even order moments.
Antonia et al. (1982b) showed this to be true for both u and au/at
statistics at the centerline of turbulent jets. Oddly enough however, the
kurtosis of the Reynolds stress converged just as slowly, and in some
cases distinctly more slowly, than the skewness of the Reynolds stress.
Figures 2.18 and 2.19 show the present distributions of S(uv) and K(uv).
Note, that the K(uv) distributions show much greater change across the
layer than the S(uv) distributions. A possible cause for the slower
convergence of K(uv) may be related to the greater variation in K(uv)

itself. Similar results are suggested by the data of Wei (1987) which has

62

been reproduced in Figures 2.18 and 2.19. It should be noted that the

S(uv) and K(uv) profiles shown are central moments.

An examination is now made of the effects of mean shear, and the
near-wall and outer region intermittency on the convergence of u, v, and
uv statistics. As one can see in the left hand columns of Figures 2.16
and 2.17, for the lowest Reynolds number layer there is little variation
from region to region across the layer. Comparisons among the four
regions at the two higher Reynolds numbers indicate distinct trends;
showing the need for increased signal length nearer the wall. These
trends are about the same for the u, v, and uv statistics. In almost all
cases (and, generally, for all of the variables observed) the weak-shear
region showed the most rapid convergence. Both the intermittent region
and the weak-shear region show about the same convergence rate for the

two higher Reynolds number flows.

The intermittent region at the lowest R0 showed slightly slower
convergence rates than the same region at the higher Re. For the other
regions, however, the velocities and Reynolds stress show poorer
statistical convergence as the Reynolds number is increased. Given that
all other factors are equal, this finding is consistent with the
hypothesis that as Rx (RA - u'A/v, where A is the Taylor microscale)
increases, so does the time to convergence. This however, should not be
viewed as a general rule -- as suggested above by the possible reverse
trend in the intermittent region. This is also shown more clearly in the

following subsection.

63

2.5.2.3 Velocity Gradients

The convergence results for au/at, av/ax, au/ay and ”2 were quite
different from those of the velocities and Reynolds stress. While the
overall convergence times necessary to achieve a given accuracy were
about equal to or slightly less than those found for the velocities,
significant changes across the boundary layer and with Reynolds number
were observed. Of all of the gradient quantities examined, the au/at
statistics exhibited by far the most rapid convergence, whereas the av/ax
statistics were clearly the slowest to converge‘. The au/ay and “2
statistics converged at almost exactly the same rates; presumably due to
the dominance of au/ay over av/ax over most of the boundary layer. The
au/at and ”z convergence data are shown in Figures 2.20 and 2.21
respectively. Almost without exception, and for all R9 and positions in
the boundary layer, the skewness of the gradient quantities examined
demonstrated the slowest convergence rates. It is worth noting that in
the weak-shear region the skewness data of Figure 2.20 are in good
agreement with the results of Mestayer (1982). He showed that it takes
between 250 and 1,400 Tug/6 to achieve :51 accuracy in the 3rd order
moments of fluctuating temperature and fluctuating temperature

differences at y/6 3 0.33.

The gradient quantities generally showed less region to region
variation than did the velocities and Reynolds stress. Perhaps the most
striking feature was the relatively long convergence times for the
gradient quantities in the intermittent region (shown in both Figures
2.20 and 2.21). This makes apparent physical sense in that gradient

quantities should be highly sensitive to rapid changes in the flow

 

‘Since Taylor's hypothesis is used, it is unclear whether the relatively
slow convergence of av/ax is a consequence of this.

64

associated with the turbulent/non-turbulent interface. The time to
convergence of the gradient quantities, in general, did not show the same
increase in the presence of high mean shear as did the velocities and
Reynolds stress. In the near-wall region for the two lower Reynolds
numbers the skewness of wz and av/ax took relatively long times to
converge. Figure 2.22 shows the present skewness profiles of wz. S(wz) is
negative over the entire layer, with a distinct peak at y+ 3 40.
Reasonable scaling on wall variables is suggested for y+ < 40. (Further
discussion and implications of these features are given in Chapter 3.)
The smoothness of these curves suggests that the values obtained using
the entire record have little uncertainty associated with a lack of
convergence. Also, this data shows good agreement with that of Balint et
al. (1987a). Their integration time was TUm/6 3 3,100, which according to

Figure 2.21 should give approximately :52 uncertainty.

Reynolds number trends for the gradient quantities were in general
also less significant than for the velocities and Reynolds stress. In
more than one instance however, there appeared to be faster convergence
within a given region as the Reynolds number increased. This is shown in
the weak-shear region in both Figures 2.20 and 2.21. For reference the
microscale Reynolds number, RA' is given in these component plots. The
results here are in direct contrast to the aforementioned hypothesis that
as RA increases so does the integration time. This suggests that basing
comparisons of fine scale structure experiments on matching RA may not be
sufficient, and that other features such as the level of mean shear

should also be considered.

65

2.5.2.4 vwz

As the need to understand the finer details of turbulence structure
increases, the experimental measures required will also increase in
complexity. This subsection presents convergence data as it relates to
the fluctuating quantity vwz. This quantity is known to be related to the
Reynolds stress gradient (e.g. Tennekes and Lumley (1972) p. 78, also see
the present Chapter 3). As might be expected, the convergence rates of
the vw statistics were much slower than the other variables examined.

2

The convergence data for vw are presented in Figure 2.23. Note that for

z
the highest Reynolds number and in the near-wall region the skewness of
vwz never converged. This is due to the fact that for different record
lengths S(vwz) kept reversing sign, thus making it difficult to compute a
meaningful percent convergence. As can be seen, in the other regions of
the layer the skewness shows the slowest convergence rate. The trends
across the boundary layer are very similar to those exhibited by the
velocities in that as the level of mean shear is increased so does the
time to convergence. A fairly consistent Reynolds number trend can also
be seen for all of the regions. Apparently, convergence times for vwz

statistics are adversely effected by increasing R9.

2.5.2.5 Convergence Summary

Table 2.5 presents a compilation of general convergence criteria
based upon the above described data. Note that these criteria are based
upon the assumption that one wishes to obtain profiles across the entire
boundary layer, and thus are conservative for, say, someone wishing to
take data only in the weak-shear region. It is hoped that these criteria

will facilitate better experiment design in wall-bounded shear flows.

66

2.6 DISCUSSION AND CONCLUSIONS - RELATIVE TO CHAPTER 2

Aside from documenting the present experimental arrangement and
procedures, this chapter has examined some features of turbulent boundary
layer hot-wire measurements that are sensitive to small scale structure.
These features may be grouped into two (not necessarily independent)
categories relating to the probe and its spatial and temporal resolution,
and the type of information one wishes to extract -- mainly pertaining to
the practical constraints imposed by the integration times necessary to
achieve statistical convergence. In this section a discussion,'
conclusions, and some general recommendations based upon the results
herein will be given. Furthermore, a critical evaluation of the overall

quality of the present four-wire ”2 probe experiments is presented.

Based upon the data of Figure 2.6 it appears that the maximum value
in u'/uf increases with increasing R0. Imperfect spatial resolution can
hide this dependence because the attenuation (due to increasing wire
length in viscous units as a consequence of increasing R9) is apparently
greater than the increase due to Reynolds number. It is therefore
submitted that detecting this Reynolds number trend provides an indicator
of good probe resolution, and constitutes a practical necessary condition
for studies investigating Reynolds number dependencies. It is probably
true that as a wire of given length becomes greater than about Bur/u (as
R, is increased) spatial averaging effects will begin to wipe-out this
trend. The data of Purtell et a1. (1981) in Figure 2.5 clearly
demonstrate this. The competing effects between spatial resolution and
Reynolds number have apparently hidden the true flow physics, and have

led many to the belief that the maximum value of u'/uf is indeed a

67

constant. In fact, if one were to ignore both of these effects then the
data of Figure 2.5 would appear to support this result in that most of
the data are located in a narrow band about u'/ur 3 2.8. Furthermore, in
regards to previous studies investigating wire length effects, the
present results suggest that Reynolds number effects are smaller but not
small compared to spatial averaging effects (as concluded by Johansson
and Alfredsson (1983)), and that the general criterion set down by
Ligrani and Bradshaw (1987) is somewhat generous since identifiable
attenuation can clearly be seen in Figure 2.5 for 5 < y+ < 20. Concerning
the spatial averaging effects due to finite wire length on velocity
derivative fluctuations, the data of Figure 2.7 support the notion that a

greater resolution is required than for the velocity fluctuations.

In principle, examining the influence of spatial averaging effects
due to finite sensor size on v velocity fluctuations (as compared to u
fluctuations) would provide more stringent criteria. In practice however,
this is currently not possible because existing excessive data scatter
results in a lack of reliable reference values necessary for meaningful
comparisons. The primary cause of this scatter can no doubt be attributed
to the increased difficulties associated with measuring the v
fluctuations (relative to u). Evidence was presented suggesting that
other factors such as the tunnel and measuring device characteristics may
also be significant. Furthermore, it is not yet clear that the skewness
of v (or other v statistics) are the same in boundary layers and

channels, and thus this issue also need to be clarified.

Perhaps the main result concerning wire spacing effects on the
computed value of spatial velocity gradients is that Wyngaard's (1969)

criterion, equation 2.4, is to a very good approximation valid, even

68

under anisotropic conditions. Furthermore, changing the level of mean

shear seems to have little effect on the criterion's validity.

Conversely however, the au/at results indicate that instantaneously
spatially averaging velocity gradients (in order to, for instance,
approximate pointwise measurements at a probe center) both causes
dramatic attenuation in the resulting rms, and is highly sensitive to the
level of mean shear. This result suggests that for probes with even small
off-center measuring sites (say greater than 30) one is probably better
off accepting non-centered measurements rather than instantaneously
spatial averaging. For example, for the given probe arrangement one would
probably not improve the accuracy of the estimated value of av/ax by
adding another x-array at -Az from the parallel-array center and then
spatially averaging over the 2A2 separation between the two x-arrays.
These results also suggest that there are differences between the
temporal and spatial structure of the fine scale motions in the wall
region. Further experiments into these differences (especially as a
function of Reynolds number) may lead to new insights concerning the
conflicting results (at very different Reynolds numbers) of Willmarth and
Bogar (1977) and Johnson and Eckelmann (1983) as discussed in the Section

1.4.

The results in Figure 2.11 give a good indication of how electronic
noise in the velocity signals enters into the au/ay signals. It was
demonstrated that to a very good approximation ¢(6u/ay) 3 euz/Ay2 in the
noise dominated part of the spectrum. This relation allows one to
optimize the choice of Ay for a given system noise level by simply
performing a single wire measurement. The strategy here would be to make

Ay as small as possible in order to resolve the motions within the flow,

69

but not so small such that the noise level becomes intolerable. Also, it
was demonstrated that the main effect of finite wire separation is to
remove energy from frequencies higher than an approximate cut-off value

of fc - Ul/(ZNAY)-

The assumption that the instantaneous values of 6v/6x(z) - 8v/ax(z
+ Az) specific to the current ”2 probe proved to be on average only about
802 valid for Az/n i 7 in the logarithmic region of the R0 -l,OlO
boundary layer. Thus for the present study its validity is in question
for a good part of the highest R9 distribution. Given the very good
spatial resolution of the parallel array for all of the Reynolds numbers
considered, the recent elimination of this error by the new compact probe

design of Mitchell (1987) is certainly warranted.

In regards to the present experimental procedure it should be
stated that the existing guidelines as put forth mainly by Tennekes and
Wyngaard (1972) and Antonia et al. (1982b) were adequate; excepting the
convergence criteria in the case of Antonia et a1. (see below).
Furthermore, it was documented that the low speed nature of the present
experiments caused little overall uncertainty in the results. Also the
matching calibration procedure used for the parallel array elements
proved to greatly eliminate any detectable correlated errors in (8u/6y)’.
This is shown indirectly in that virtually all of the noise in ¢(6u/ay)
in Figure 2.11 may be attributed to electronic noise. Perhaps however,
the lowest quality experimental technique was the calibration procedure
used for the x-array elements. Evaluation of the relative errors in using
a variation of the present technique, as compared to more sophisticated
methods (such as that used by Foss et al. (1986a) or Lueptow, Breuer, and

Haritonidis (1988)) has, however, been performed by Browne et al. (1989).

70

They found that, in general, a technique similar to the one of this study
performed quite satisfactorily. Also dynamic calibration techniques (cf.
Perry and Abell (1975)) would provide greater accuracy, but they put
multi-sensor probes at a greater risk to wire breakage. On the data
reduction end, results are given in Appendix 2.1 that suggest the present
least squares method of computing streamwise derivatives can be far

superior to a conventional finite difference technique.

There were many interesting results pertaining to statistical
convergence. Generally it can be concluded that, due to the factors
involved such as the level of mean shear, the intermittency level, and
Reynolds number effects, existing convergence formulas should be used
with caution in wall flows. Also, one should not apply criteria developed
in other flow fields to wall flow studies. For example, at the centerline
of a turbulent jet flow with a centerline velocity Uc and an approximate
transverse scale of L0, Antonia et al. (1982b) show that it takes a time
of approximately lZLo/Uc to obtain :52 accuracy in the skewness of
(Bu/6t). This criteria, applied to boundary layer flows (using 6 and Ug)
is orders of magnitude smaller than that cited in the present study or in

that of Mestayer (1982).

Distinct boundary layer regional and Reynolds number trends in
convergence were observed for the statistical moments of the u, v and uv
fluctuations. These trends showed that increasing either R0 or decreasing
the distance from the wall generally increased the time to convergence.
Both of these trends may possibly be lumped into one which simply states
that for these variables increasing the absolute level of mean shear
increases the time to convergence. Also comparisons between the

velocities and Reynolds stress showed that the uv statistics converged at

71

least as fast as either of the measured velocity components.

The convergence of the velocity gradient statistics showed much
smaller regional or R0 variations than the u, v, or uv statistics. Their
convergence did however, appear to be adversely affected in the
intermittent regions both very near and very far from the wall. In terms
of absolute time to convergence, the gradient quantities appeared to
converge slightly faster than the velocities or Reynolds stress. However,
since S/Ug is probably an integral time scale most appropriate for the
velocities, the equivalent number of integral time scales appropriate to
the gradient quantities necessary for convergence is probably greater
than that found in terms of 6 and Um. The results also suggest that
spatial gradients converge much more slowly than temporal gradients.
Again this may be connected to the differences between the fine scale
spatial and temporal structure. As was expected, the vwz statistics took

a very long time to converge.

Finally, the overall convergence criteria developed indicated the
necessity for considerably longer averaging times than expected. It is
felt that there are three plausible explanations for this. The first
being that this subject has never been (to this author's knowledge)
thoroughly investigated, and thus the expectations were based upon
limited evidence. The second is that because of the very good resolution
of the probe, more of the high intensity fine scale information in the
tails of the respective probability distributions was represented than in
earlier studies. And third is that statistical convergence may be
profoundly adversely effected by even small levels of "noise" (as
described in Section 2.4.6) due the inconstancy of the free stream

velocity.

CflAETER 3

REXNOLD§ STRESS TRANSPORT AND ITS REYNOLDS NUMBER DEPENDENCE

3.1 INSTANTANEOUS MOTIONS, AVERAGE STRUCTURE AND TOWNSEND'S DECOMPOSITION

All statistical characteristics of a turbulent flow field are a
consequence of the instantaneous dynamics associated with its complicated
spatial-temporal evolution. This complicated spatial-temporal behavior
makes it difficult to educe the true dynamics that produce the
kinematical information acquired from a small number of stationary
probes. Furthermore, interpretations based upon statistical information
derived from a small number of stationary probes tend to endow the
proposed turbulent motions with an unjustified permanence of form
associated with their spatial extent. Therefore, in physically
interpreting the long-time statistical information presented in this

chapter, the bias toward this tendency was recognized and avoided.

Based on the information reviewed in Chapter 1, it is clear that
the complicated interactions of the bursting events occur over a
significant spatial extent and involve motions covering a range of
scales. Given the discussion at the end of Section 1.2.2, the spatial
phase relationships between the different scales of motion would seem to
be an essential feature associated with transport across the boundary
layer. This chapter examines data relevant to the transport of the

Reynolds stresses normal to the wall.

As it pertains to boundary layers, the reality of the opening

72

73

sentence to this chapter leads one to assess the implications of the
active/inactive motions decomposition first put forth by Townsend (1961)
from the point of view of the instantaneous physics. In this highly
plausible as well as physically appealing decomposition Townsend
hypothesized that at any given point the motion consists of "an active
component responsible for turbulent transfer and determined by the stress
distribution and an inactive component which does not transfer momentum
or interact with the universal component". Given the present knowledge of
the existence and importance of the various coherent motions that have
been shown to reside in the wall region of turbulent boundary layers,
concerning turbulent transport Townsend's decomposition appears highly
justified. That is, there is a large body of evidence suggesting that
these coherent motions are associated with the bulk of turbulent
transport in the wall region. However, a model that demands that the
local shear stress, 1, and the distance from the wall, y, solely
determine the statistical properties of a universal active component may
not incorporate enough of the essential physical mechanisms by which

turbulent stresses are created and transported.

Townsend's original justification for the active/inactive motions
decomposition was based upon the fact that the velocity intensities in
the wall region can vary dramatically between different flows having the
same stress -- even though the logarithmic mean velocity distribution for
these flows is universal. Since Townsend's article, the universality of

the constants x and C in the log law profile,
U/ur - (l/x)lny+ + c, (3.1)

has been investigated in numerous studies; Simpson (1970), Huffman and

74

Bradshaw (1972), White (1981), Purtell, Klebanoff and Buckley (1981),
Murlis, Tsai and Bradshaw (1982), Andreopoulos, Durst, Zaric' and
Jovanovic (1984), and Erm, Smits and Joubert (1985). The general findings
of these investigations (excluding Simpson's) is that (except possibly at
Reynolds numbers approaching reverse transition and over the range of
Reynolds numbers examined) x is constant, and that at low Reynolds number
C shows a slight variation. Thus, under Townsend's original description
the universal active motions are determined by r and y and are also
responsible for the universality of the mean velocity profile. Bradshaw
(1967) further supports this hypothesis, and points out that "indeed, the
only other course would be to believe that the scales of the mean motion
were universal but that the scales of the shear-stress-producing
turbulence were not -- a sentiment open to doubt". The data of this
chapter suggest that the spanwise vorticity field in the near-wall region
is universal, but not the average ability of the spanwise vorticity
containing motions to create and/or transport the stress.

Regardless of the validity of Townsend's hypothesis however, it
remains plausible that there exist other time averaged properties of the
flow field that remain invariant under an inner variable normalization
over approximately the same y+ range as does the mean profile. These
properties, along with the logarithmic mean velocity profile, could then
be interpreted as the ”defining" features of the turbulence in the inner
region in that self-similarity is maintained with respect to what
Barenblatt (1979) refers to as the local and global Reynolds numbers,
yuf/v and, say, 0uf/u. Given that perhaps the most definitive feature of
a turbulent flow is the presence of a three dimensional fluctuating
vorticity field and that the logarithmic law is essentially derived from

a statement about the mean vorticity (i.e. aU/ay), one might expect that

75

if the above pr0posed invariant non-dimensional functions exist they

would be related to the vorticity field.

Following Hinze (1975, p.680), one may introduce an approximate
decomposition that separates the mean Reynolds stress gradient of a two

dimensional boundary layer flow into rotational and irrotational parts,
6<uv>/8y - <wwy> - <vwz> + 8<v2 + w2>/6x - 6<u2>/ax. (3.2)

Under this decomposition, Hinze identified the first two terms on the
right as an active rotational part, and the last two terms as an inactive
and virtually irrotational part. One of the major goals of this chapter
is to examine the motions contributing to 6<uv>/8y, and the gradients of
other components of the Reynolds stress tensor. Furthermore, contained
within the present results there are certain functions which, when
normalized by inner variables, apparently remain invariant over all or a
portion of the boundary layer (over the given R0 range). The physical
relevance of the plausible invariance of these functions is discussed.
The experimental conditions and procedures for the data presented
in this chapter were documented in Chapter 2. The following section gives
a general documentation of the Reynolds number dependence/invariance of
pertinent velocity, Reynolds stress, and spanwise vorticity statistics.
In presenting the data, many comparisons and contrasts are made between
the current results and previous investigations. A detailed look at the
contributions to the Reynolds stress transport is given in Section 3.3.
Finally, in Section 3.4 a discussion and general summary of the present
results in the context of an interaction between the motions of the inner

and outer regions is presented.

76

3.2 REYNOLDS NUMBER DEPENDENCE OF BOUNDARY LAYER STATISTICS

In this section the observed Reynolds number dependence of
velocity, Reynolds stress, and spanwise vorticity statistics are examined
and discussed. Since the main emphasis of this section will concentrate
on inner region phenomena, almost exclusive attention will be paid to
inner variable normalizations. (One is referred to Appendix 3.2 for outer
variable normalizations.) In this section the u statistics presented are
derived from the lower wire of the parallel array contained in the ”2
probe (to use the information closest to the wall). The v and uv data are
derived from the wz probe's x-array. In the comparisons with previously
reported results care was taken to select studies using probes of

comparable spatial resolution, and in those cases where this is not

possible the effects of finite spatial resolution are discussed.
3.2.1 Mean Flow Characteristics

Perhaps the most recognizable structural change that occurs in
boundary layer flows at low Reynolds number is observed in the wake
region of the mean profile. Coles (1956), (1962), (1968) was the first to
thoroughly document and describe these changes. In a more recent study
and following the hypothesis of Huffman and Bradshaw (1972), Murlis et
al. (1982) give evidence suggesting that the disappearance of the
strength of the wake component, AU/ur, is associated with the direct
effects of viscosity outside the sublayer, and is due to a relative
increase in the dynamical significance of the viscous superlayer for R0
less than about 5,000. Antonia, Rajagopalan, Subramanian and Chambers
(1982) give further details relating to this hypothesis, and also its

relevance in terms of observed outer region coherent motions. Concerning

77

the instantaneous features of the wake region, both of these studies tend
to support the conclusion that the motions determining the shape of the
turbulent/non-turbulent interface change from small to large scale as R0
increases. In these studies it has also been shown that for R0 > 3,000
the average structure of the wake region changes very little. We will
return to this point throughout this section and especially in the
discussion of the spanwise vorticity intensity profile. Pertinent to
Coles wake parameter, H, the present values of AU+ (1.7, 3.0, 3.2 at R0 -
1,010, 2,870, 4,850 respectively) agree reasonably well with the values
given by Coles and Hirst (1968), 1.9, 2.8 and 3.0 at R9 3 1,000, 3,000

and 5,000 respectively.
3.2.2 Fluctuating Velocities and Reynolds Stress

3.2.2.1 Mean and RMS Quantities

The existence of a universal mean velocity profile has prompted
some to hypothesize that a similar "law of the wall" type of analysis is
also appropriate for scaling the velocity intensities. However,
analytical approaches, such as that of Phillips 1987, while providing
useful approximate formulas for the normalized intensities, have limited
relevance to the questions concerning Reynolds number dependence since
they are based upon an infinite Reynolds number assumption. As of yet the
validity of this type of analysis is unclear since the high Reynolds

number data necessary to verify the asymptotic profiles are unavailable.

Furthermore, attempts at experimental verification of a "law of the
wall” type of hypothesis for the intensities are clouded by other
factors. For example, Perry and Abell (1975) give evidence suggesting

that axial velocity intensity profiles in turbulent pipe flow are

78

universal for y/R S 0.1 under an inner variable scaling. However, the
higher resolution LDV data of Wei (1987) in a channel, and the boundary
layer data of Ueda and Hinze (1975), Purtell et al. (1981), and Erm et
al. (1985) all show a clear R0 trend in u'/ur for y+ 2 10. This trend is
also shown in the present profiles of Figure 3.1. Included in Figure 3.1
are the profiles of Purtell et al. (1981), Ligrani and Bradshaw (1987)
and Alfredsson et al. (1988). As one can see, the present data agree very

well with the results of the other studies.

Concerning the detection of the trend shown in Figure 3.1 for y+ 2
10, evidence was presented in Chapter 2 suggesting that spatial averaging
effects due to finite probe scale can begin to obscure this R0 dependence
for probe scales 2+ 2 8 (depending on the Reynolds number). Thus the data
of Perry and Abell, which if anything show a slightly opposite R0 trend,
may be affected more by spatial averaging effects than they suspected1

(see Table 3.1).

The data of all of the above cited references, do however, tend to
suggest that for y+ S 10 the u'/uf profile is universal for all R9.
Furthermore, data of the recent three-facility study of Alfredsson,
Johansson, Haritonidis, and Eckelmann (1988) support this result as well
as the hypothesis that in the region y+ S 10 boundary layer and channel
flow data fall on the same curve. Wei (1987) makes the conjecture that
the R0 trend for y+ 2 10 in a channel flow is primarily due to changes in
the structure of the inner region, and not due to the indirect effect of

non-universal and large scale outer region motions influencing the wall

 

1This conclusion is tentative however, since it is based upon comparisons
between channel, pipe, and boundary layer data. In general however, there
is a fairly large body of experimental evidence suggesting that the inner
regions of these flows are highly similar.

79

region structure.

The v velocity intensities, when normalized by wall variables, show
an even more dramatic R0 trend. This suggests that the mechanisms
responsible for transport across the boundary layer are strongly Reynolds
number dependent. The trend in the present data of Figure 3.2a is in very
good agreement with the trend in the data of Wei (1987) shown in Figure
3.2b, and in it is in good agreement with the findings of Erm et al.
(1985) (not shown). The data of Erm et al. however, exhibit a
significantly smaller Reynolds number trend. Since their non-dimensional
probe scale (see Table 3.1) was significantly greater than in either the
present study or that of Wei, spatial resolution effects may be
responsible for this smaller variation. Also, the different Reynolds
number data of Erm et al. apparently fall on a single curve when
normalized by u, and plotted versus y/6 (not shown). The present profiles

do not exhibit this feature.

It is of interest to note that the observed rise in v' in the
region 4 S y+ S 10 shown in R9 - 1,010 distribution of the present data
can also be seen in Figure 3.2a in the data of Andreopoulos et al. (1984)
at R0 — 3,624. Furthermore, at y+ 5 A.8 Andreopoulos et al. give evidence
that the probability density function of the v velocity fluctuations is
bimodal. This feature however, has not been observed in any of the v
fluctuation probability density functions of the present data. Note that
this increase in v' very near the wall has not been observed (to this
author's knowledge) in any studies of channel flow. Note also that while
the overall shape of the profile of Andreopoulos et al. is in very good
agreement with the present profiles, its relation to the R9 - 2,870 and

4,850 profiles is inconsistent from a Reynolds number dependence point of

80

view. This may be due to the significantly poorer spatial resolution of
their sensor (see Table 3.1) than in either the R9 - 2,870 or 4,850

distributions of the present study.

Note that in both Figures 3.2a and 3.2b the y+ position at which
the peak value is attained increases with increasing R0. This trend has
been previously observed by Sreenivassan (1989). However, by comparing
the data in parts a and b of this figure, one can also see that the
channel flow v'/ur profiles tend to reach their peak at significantly
smaller y+ values, and in general have a distinctly different shape than
the boundary layer profiles -- even in the inner region. In the outer
region this can easily be attributed to the differences in the wake
structure of channels and boundary layers, but in the inner region one
cannot resolve this discrepancy by the same argument without assuming a
strong interaction between the motions of the inner/outer regions. These
differences could also be due to the presence of the opposing wall in the
channel, as referred to as the "geometry effect" by Wei. Even in this
case however, one would still have to assume a strong inner/outer
coupling. Also, both the present data and the data of Wei suggest that
the normal velocity intensity profiles (as non-dimensionalized in Figure
3.2) do not merge for y+ < 20. Wei hypothesizes that this feature is due
to a relative increase in the creation of streamwise vorticity as the

Reynolds number increases.

A feature seen in both Figures 3.1 and 3.2a is that the R0
variation is much greater between the R0 - 1,010 and 2,870 distributions
than between the R0 - 2,870 and 4,850 distributions. This is reminiscent
of what occurs in the wake region of the mean velocity profile, and

leaves open the possibility that these two phenomena are both a symptom

81

of the R9 dependence of the same internal mechanisms. This correspondence
is apparently incomplete however, since as pointed out by Purtell et al.
(1981) the low Reynolds number effect "penetrates much deeper in terms of
the turbulence intensity than it does for the mean velocity". If there is
such a connection however, given Purtell's observation this would also
suggest the existence of a strong interrelation between the physics in
the inner and outer regions. Also, in general, the smaller magnitudes
found in the lower Reynolds number profiles of Figures 3.1 and 3.2 are
consistent with the notion that at low R9 viscous effects serve to damp

the turbulent motions.

The tendencies exhibited by the present Reynolds stress
distributions in Figure 3.3 are similar to those of the streamwise
intensities. For y+ 2 15 a clear R0 dependence is observed, and once
again this dependence appears to be much stronger between the two lower
R0 distributions. Very similar tends have been reported by Wei and to a
lesser degree by Erm et al. The data of Figure 3.3 also suggest that for
y+ S 15 the two lower Reynolds number profiles merge. A universal profile

in this region is also consistent with the findings of Wei.

Examination of the present R0 - 2,870 data in Figure 3.3 and that
of Andreopoulos et al. (1984) (R0 - 3,624) indicates that for y+ values
greater than about 20 the two profiles exhibit very good agreement.
Nearer the wall however, the present data tend to fall below theirs.
Comparison of the R0 - 1,010 profile with the R9 - 1,140 profile of
Barlow and Johnston (1985) (not shown) also shows this same feature. The
reasons for these observed differences are currently unknown. As with the
normal intensity profiles, the channel flow Reynolds stress profiles of

Wei also tend to reach their peak values at lower y+ values.

82

Even better measures of the internal structure of the turbulent
field are the Reynolds stress correlation coefficient, <uv>/u'v', and the
velocity variance ratio, <u2>/<v2>. As can be seen in Figure 3.4,-the
present <uv>/u'v' distributions for y+ 2 15 show a clear R9 trend. These
profiles decrease from a peak value of near 0.49 in the R9 - 1,010
profile to a peak value of about 0.41 in the R9 - 4,850 profile. If one
assumes that for R0 < 1,010 the peak <uv>/u'v' value decreases with
decreasing R0, then the present data are in qualitative agreement with
both the studies of Murlis et al. (1982) and Wei (1987). These studies
show that the peak value (as a function of Reynolds number) occurs at an
R0 value intermediate to the low Reynolds number limit necessary for
fully turbulent flow, and a proposed asymptotic value as the Reynolds
number approaches infinity. The data of Murlis et al. show that this peak
occurs at about R9 — 1,900. However, their <uv>/u'v' profiles also show a
much smaller variation with R9 than the present. This may be a result of
the significantly poorer spatial resolution of their x-array measurements

(see Table 3.1).

As with the streamwise velocity intensity profiles, the present
correlation coefficient distributions indicate that they probably merge
for y+ s 15. A universal <uv>/u'v' profile for y+ less than about 15
suggests that the presence of the wall restricts the organization of the
turbulent shear stress producing motions directly above the wall in such
a way that the mean turbulent shear stress production is directly
proportional to the total (uncorrelated) rms of the these motions (i.e.
u'v') at all R9. The lack of universality for y+ 2 15 however, indicates
that the organization/interaction of the stress producing motions may be

quite different at different R9. Conceptually then, one might envision at

83

different R9 different types of turbulent motions and interactions above
the wall. As one nears y - 0 however, the presence of the wall limits the
manner in which these motions can interact with either themselves or the

instantaneous vorticity distributions in the sublayer.

Murlis et al. (1982) identify the velocity variance ratio,
<u2>/<v2>, as an approximate measure of the efficiency of turbulent
mixing, as well as a measure of the coherence of turbulent eddies. The
present <u2>/<v2> distributions, as derived from the ”z probe's x-array,
are given in Figure 3.5. Included in this figure are some of the data of
Wei (1987) for y+ S 100. Following the reasoning of Murlis et al. (1982),
one can roughly associate a decrease in <u2>/<v2> with an increase in the
efficiency of turbulent mixing. Thus the present data and the data of Wei
in Figure 3.5 imply that predominantly in the near-wall region (and in a
very dramatic way) as the Reynolds number increases the turbulent mixing
process becomes more efficient. These results are obviously a consequence
of the trends in Figures 3.1 and 3.2 which show that (especially in the

inner region) v' exhibits a much greater R0 dependence than does u'.

It is also interesting to note that the data of Wei suggest that
the position of the peak in the <u2>/<v2> profile shifts to larger y+
values as the Reynolds number increases, while the present data suggest
that the position is fixed at y+ about equal to 12. This is a consequence
of the observed significant differences between the channel and boundary
layer v'/ur profiles shown above. It may be indicative of a subtle but
true difference between the inner flow structure of channels and boundary

layers.

84

3.2.2.2 Higher Order Statistics

Since higher order statistics are representative of the information
in the tails of a probability distribution, in general they are a more
sensitive measure of the turbulent structure. The present higher order
velocity and Reynolds stress statistics tend to agree well with
previously reported results, although there are some differences. These

points of agreement and disagreement will now be discussed.

The u skewness profiles of Figure 3.6a exhibit reasonable to
excellent agreement with the boundary layer studies of Gupta and Kaplan
(1972), Ueda and Hinze (1975), Andreopoulos et al. (1984), Barlow and
Johnston (1985), Ligrani and Bradshaw (1987) and Alfredsson et al.
(1988), as well as the channel flow studies of Kreplin and Eckelmann
(1979), Johansson and Alfredsson (1983), Kim, Moin and Moser (1987), Wei
(1987) and Alfredsson et al. (1988). The present data show a small but
consistent and discernable difference between the R0 - 1,010 and the R9 -
2,870 and 4,850 flows. In addition to the present data, shown in Figure
3.6b are the channel flow results of Wei, Kim et a1. (a computation),
Kreplin and Eckelmann and Alfredsson et al., and the boundary layer data
of Andreopoulos et al., Barlow and Johnston, Ligrani and Bradshaw and
Alfredsson et al. As one can see, for y+ 2 10 the R0 - 2,870 and 4,850
distributions of the present study and the data of Andreopoulos et al.,
Ligrani and Bradshaw, and Wei exhibit remarkable agreement. (Note that
the boundary layer data of Alfredsson et al. are also in good agreement
but are not well represented in this region of the flow.) It is important
to note that all of the data (except that of Wei, R9 5 1,500) has R9 2

2,600.

On the other hand however, over the same region of the flow the

85

data of Kreplin and Eckelmann, Kim et al. and the channel flow data of
Alfredsson et al. of Figure 3.6b also show good agreement; however, they
define a significantly different curve. Given that this second set of
flows are all at about the same very low R9 (see Table 3.1), and that the
present R0 - 1,010 profile and the data of Barlow and Johnston (R9 -
1,140) are intermediate to the two different (but well defined) curves in
Figure 3.6b, it is probably true that there is a Reynolds number

dependence in the skewness of u for y+ > 10 and R0 < 3,000.

Note that this trend in S(u) has been previously hypothesized by
Andreopoulos et al. (1984) to be a Reynolds number dependence. However,
since the study of Johansson and Alfredsson (1983) showed that this type
of trend may also be produced by spatial averaging effects due to finite
probe scale, the relatively low resolution higher R0 results of
Andreopoulos et al. remained inconclusive. Given that the resolution of
the u component in the present study is very good (see Table 2.3) and
that the R9 - 2,870 and 4,850 flows define a single curve, it is
presently felt that the trend in the higher R0 results of Andreopoulos et
al. predominantly represents spatial resolution effects. Once again, note
that the R9 range over which the proposed trend is shown to be
significant is about the same as that over which Reynolds number effects

are significant in the wake region of the mean velocity profile.

Data comparisons become more difficult very near the wall. This is
because wall heat transfer and/or blockage effects, as for example
studied by Alfredsson et al. (1988), may cause significant errors. Thus
before discussing the data of Figure 3.6b for y+ < 10 some qualifications
need to be made. First of all it is important to note that heat transfer

and/or blockage effects should be negligible in the present results since

86

the lower wire of the parallel array of the wz probe (i.e. the most
susceptible wire) never got closer to the wall than 2.3 mm. Also, based
upon a comparison between the "affected” and "unaffected" u'/U results in
Alfredsson et al., the boundary layer results of that study are probably
only valid down to y+ 3 4. Furthermore, given that the trend in the data
of Andreopoulos et al. for y+ S A has been shown by Alfredsson et al. to
be symptomatic of wall effects, we shall also consider their data valid
down to only y+ 3 4. Also, since it is well known that the mean data
acquisition rate of an LDA system is dependent on the mean velocity (for
a given particle density), the reliability of the measurements of Wei is
in question for y+ < 6 (Wei, private communication). Barlow and Johnston
state that the accuracy of their u and v velocity data is in question for
y+ less than 4 and 7 respectively. Thus for y+ S 4 the only data sets
remaining for consideration on Figure 3.6b are at very low R0 (one of
which is from the computation which may have spanwise and/or streamwise
grid resolution problems in the near-wall region, see S(uv) data below).
This therefore eliminates any discussion of an R0 dependence in this
region of the flow. It does however, point out the need for more
measurements in this region of the flow at higher R9. In the range a S y+
S 10, the present measurements, those of Andreopoulos et al. and a
comparison of the high and low R9 data of Alfredsson et al. suggest that
the R0 dependence detected for y+ > 10 may continue down to at least the

edge of the sublayer, y+ 3 5.

The present u kurtosis profiles of Figure 3.7a show very good
agreement for all three Reynolds numbers. Also for y... > 10 the present
results show good agreement with those of Andreopoulos et al. (1984),

Barlow and Johnston (1985), Ligrani and Bradshaw (1987) and Wei (1987)

87

(see Figure 3.7b). The general trends of the data presented in Figure
3.7b suggest that there may also be a very slight Reynolds number
dependence in the K(u) profiles for y+ 2 10. In the lower Reynolds number

profiles of Figure 3.7b the shallow minimum near y+ 2 12 tends to be a

"2

smaller value, while the shallow maximum near y+ 40 reaches a greater
value. The observed trends in Figure 3.7b are in agreement with the
Reynolds number variations found by AndreOpoulos et al. Although, once
again, it is presently felt that their probe scale may have significantly
affected their higher Reynolds number results. Nearer the wall the K(u)
data in Figure 3.7b are quite scattered, and thus at this time questions
concerning a Reynolds number dependence in this region remain unresolved
-- although the deviation of the boundary layer data of Alfredsson et al.
and Andreopoulos et al. for y+ < 4 can probably be explained in terms of

”wall effects". Once again these results tend to support the notion that

by Re 2 3,000 the effects of Reynolds number become small.

The scatter in the higher order v statistic profiles reported in
the literature is much greater than that of the higher order u
statistics. Much of this variation is undoubtedly associated with the
increased difficulty in resolving the v component. Factors such as the
calibration technique employed and the dynamic loading of heated wires as
examined by Perry and Abell (1975), spatial resolution effects and probe
orientation errors as examined by Vukoslavcevic and Wallace (1981) all
increase the difficulty of resolving the v fluctuations. Furthermore, in
Chapter 2 it was shown that, in general, the v statistics take

significantly longer to converge than the u statisticsq.

88

The present v skewness profiles shown in Figure 3.8 (also in Figure
2.8a) exhibit a discernable positive peak near y+ - 10, and then decrease
toward zero as the sublayer is approached. These results are in
reasonable agreement with the findings of Gupta and Kaplan (1972) and
show very good agreement with the results of Andreopoulos et al. (1984).
Other boundary layer studies however, such as Barlow and Johnston (1985),
give negative values for S(v) near y+ 3 10 and a zero crossing to
positive values in the sublayer and at y+ g 30 (see Chapter 2). Similar
discrepancies in the reported values of S(v) in the near-wall region can

also be found in the channel flow literature.

These discrepancies leave open the possibility that slight
variations in different flow facility and/or measuring instrument
characteristics may cause large differences in the measured value of S(v)
(again see Chapter 2). The channel flow data of Wei also suggest that the
sign of S(v) in the range 5 s y+ s 30 is strongly Reynolds number
dependent; ranging from a value of about -0.65 at y+ 5 12 at his lowest
Reynolds number, to a value of about +0.65 at y+ 3 20 at his highest
Reynolds number. Thus Reynolds number may also play a role in the
variation of the reported value of S(v). The present data show only a
slight (if any) R0 dependence in S(v). At y+ 5 12 a comparison of the R0
- 1,010 and 2,870 profiles suggests that as R0 increases the v component
probability distribution becomes less positively skewed. However,
examination of the R0 - 4,850 profile suggests that the peak value at y+
3 12 probably increases for R0 > 2,870. Thus at this time it is unclear

whether these relatively small differences are indicative of structural

 

2This may be in part due to the problems associated with resolving v.
However, as long as these problems cannot be eliminated the longer
convergence times will remain an experimental reality.

89

changes with Reynolds number, or reflect the difficulty in accurately

measuring S(v).

The v kurtosis profiles of the present study are shown in Figure
3.9. These profiles exhibit the same clear-cut R9 trend of most of the
data presented thus far (i.e. the two higher R9 flows show good
agreement, and are noticeably different from the R0 - 1,010
distribution). The trend in the data is toward an increased peak in K(v)
near y+ - 12 at the higher Reynolds numbers. This suggests an increasing
intermittency in the motions producing large v fluctuations with
increasing Reynolds number. Increasingly intermittent behavior in these
motions is in apparent agreement with the aforementioned hypothesis of
Wei (1987) that proposes the legs of the hairpin vortex-like motions
found near the wall become smaller and disproportionately more intense,
in relation to inner variable scaling, as the Reynolds number increases.
Given this Reynolds number dependence, one would expect it to be
reflected much more dramatically in the v kurtosis than the u kurtosis
since streamwise vorticity has a v-gradient contribution (wx a aw/ay -
av/az). Comparison of Figures 3.7 and 3.9 shows that K(v) undergoes a
much greater R0 variation than K(u). It is interesting to note however,
that the K(v) data of Wei at y+ 5 12 (not shown) exhibit just the

opposite Reynolds number trend of the present data.

The second, third, and fourth central moments of the Reynolds
stress signals are presented in Figures 3.10, 3.11, and 3.12

respectively.

The (uv)'/ur2 distributions show a Reynolds number trend that

resembles the v'/ur distributions above. Like the v'/ur distributions the

90

(uv)'/u,r2 distributions do not suggest that the three data sets merge
anywhere in the inner region. It is interesting to note however, when
(uv)' is used to normalize <uv> the resulting profiles are almost
identical to the <uv>/u'v' profiles shown in Figure 3.4. This indicates
that the correlated, (uv)', and the uncorrelated, u'v', "intensities”
associated with the Reynolds stress are nearly identical in magnitude
over the entire boundary layer, and that they are (apparently) self-
similar with <uv> for y+ S 15. Finally, it is worth noting that the
present (uv)'/ur2 distributions are in very good agreement (in both trend

and magnitude) with those of Wei (not shown).

The uv skewness distributions exhibit good agreement for all three
Reynolds numbers, and in particular across the inner region the two
higher R0 distributions very convincingly define a single curve. That
these distributions show very little R0 variation (even though the
<uv>/u,2 and (uv)'/ur2 do) suggests that the motions/interactions
responsible for the instantaneous uv signal undergo distinct Reynolds
number changes, but that average proportion of :uv content remains nearly

invariant.

The present S(uv) distributions are in excellent agreement with
those of Wei. Both data sets show a zero crossing between 5 < y+ < 10.
Both data sets show remarkable agreement across the inner region at
higher Reynolds numbers (in both cases the data for y+ 2 20 define a
single curve at a nearly constant value of about -l.6). Furthermore, in
both data sets the lowest Reynolds number distributions exhibit a
slightly more negative plateau at about -2.0 across the inner region. The
agreement between the present data and those of Gupta and Kaplan (1972)

is also very good.

91

However, none of the experimental data show good agreement with the
computational profile of Kim et a1. (1987) for y+ < 20. Note also that
this disagreement apparently cannot be attributed to Reynolds number
since the lowest Reynolds number profile of Wei is at a lower Rd/2 than
that of the computation. It is presently believed that this discrepancy
may be related to the inability of this computation (due to the
relatively large spanwise and streamwise grid spacings) to correctly
create the true physics near the walls. Under the assumption that the
characteristics of the uv producing motions that interact with the
sublayer are important element of near-wall physics, the results in
Figure 3.12 suggest that this computation probably has limited
application in uncovering the true details of the physical motions and

mechanisms in the near-wall region.

The K(uv) profiles show more variation than do the S(uv) profiles.
Generally however, the three distributions exhibit reasonable agreement
in the range 20 S y+ s 100. For y+ < 20 the R0 — 1,010 data show a
decrease until about y+ - 7, and then begin to increase rapidly as the
sublayer is approached. This apparent deviation from the two higher R0
distributions means that even though S(uv) is nearly invariant (over R9)
for y+ S 20, the shape of the uv probability distribution for the R9 -
1,010 is different. The present data suggest that the motions associated
with large uv fluctuations are less intermittent for 7 < y+ < 20 at low
R0. Given the u and v data presented above, it is probably true that the
observed variations in K(uv) for y+ S 20 are predominantly due to the

non-universality exhibited by the v component. As with the skewness, the

 

3This notion is further substantiated by the poor agreement between the
u-v quadrant breakdown results of the computation and experiments for y+
< 20, see Kim et al. (1987).

92

present K(uv) distributions (excepting the R0 - 1,010 flow for y+ S 20)
are in very good agreement with both the data of Wei (1987), and Gupta

and Kaplan (1972).
3.2.3 Spanwise Vorticity Statistics

Numerous profiles of non-dimensional functions containing the
spanwise vorticity intensity have been examined. Figures 3.13a, b, c, and
d present outer, inner, and mixed variable non-dimensional functions
containing wz'. In Figure 3.13b the computational channel flow results of
Kim et al. (1987) and the experimentally derived boundary layer data of
Balint et al. (1987a) are also plotted for comparison (same as in Figure
2.14). Under the outer variable normalization of Figure 3.13a the present
data exhibit a small but discernable Reynolds number trend in the outer
part of the boundary layer. This trend becomes increasingly distinct as
the wall is approached. The observed trend is also opposite to that which
would be indicative of probe resolution effects, and thus measurements
with a probe of poorer resolution might give apparent evidence for
universality under this scaling. Also it is worth noting that scaling
with ur and 0 (rather than Uco and 0) makes the trend more pronounced

since ur/Ug decreases with increasing Reynolds number.

The data of Figure 3.13b (a normalization using v and Ur) suggest a
universal scaling of the three data sets for y+ S 80. In this region,
except for the leveling-off of the R9 - 1,010 profile near y+ - 12, the
small differences between the present three profiles and that of Balint
et al. (1987a) were shown in Chapter 2 to probably be a consequence of
the small differences in the respective non-dimensional wire spacing

relevant to the measurement of au/ay. Concerning the deviation in the R9

93

- 1,010 profile near y+ - 12, it is worth pointing out that this feature
has also been observed in the previous measurements of Klewicki and Falco
(1986). Also it is interesting to note that as the wall is approached the
present profile shows a distinctively different shape than the
computational results of Kim et al. (1987), even though the two data sets
apparently have about the same limiting value as y+ approaches zero. This
difference may be indicative of numerous factors such as unaccounted for
measurement errors, a true difference between channel and boundary layer
flows, a Reynolds number dependence, or, as suggested by the S(uv)
profiles above, the inability of this computation (due to its relatively
large spanwise and streamwise grid spacings) to correctly create the true
small scale vortical motions near the wall. The present data appear to
level-off at a value of 3 0.37 near the edge of the sublayer. This value
is in good agreement with the recent findings of Alfredsson et al. (1988)
of 0.38 2 0.02 for the value at the wall. For y+ values greater than
about 80 the R0 - 2,870 and 4,850 data and the data of Balint et al. (R0
3 2,100) show very good agreement, while the R0 - 1,010 distribution
shows considerably higher values. The occurrence of this deviation is
consistent with the observed structural changes in the wake region of the
mean velocity profile at low R0. This deviation also agrees with the
hypothesis of Huffman and Bradshaw (1972) relating to the increased
dynamical significance of the viscous superlayer at low R0 as a result of
the direct influence of viscous effects in the outer region. Note that
the channel flow profile of Kim et al. do not and should not have this
trend (under the Huffman and Bradshaw hypothesis) because of the confined
nature of their flow. Note also that the R0 - 1,010 mean velocity profile
(see Figure 2.2) begins to deviate from the logarithmic law at about y+ -

120, which is in reasonable agreement with the value at which the data of

94

Figure 3.13b begin to deviate from the higher R0 distributions.

In regard to the hypothesis of Townsend concerning the statistical
universality of the active motions, the present wz' distributions have
also been normalized by the variable quantity, y/ur. These distributions
are shown in Figure 3.13c. As one can see, the trends between the
different Reynolds number profiles are very similar to those in Figure
3.13b. Both non-dimensional representations suggest universality for y+ S
80. In the outer region the two higher R9 distributions show little
difference, while the lower R9 distribution gives significantly larger
values. Note also that if y/<uv>1/2 is used rather than y/ur (in
correspondence with Townsend's hypothesis), then examination of the
stress distributions of Figure 3.3 indicates that the differences between
the three profiles of Figure 3.13c would increase. However, given that
there are only small differences between the higher Reynolds number
distributions of both Figures 3.3 and 3.13c, leaves open the possibility
that the appropriate scaling for both the mean and fluctuating spanwise
vorticity fields in the constant stress region uses y and r (or 'w ) as

Ro+ m.

The present result indicating that the data of Figure 3.13b
apparently merge for y+ less than about 80 over the given Reynolds number
range is in disagreement with the findings of Spalart (1988). In his
computational study (300 S R0 3 1,410) Spalart found that in the near-
wall region the vorticity intensity profiles (when scaled by u and ur)
exhibit a significant increase with increasing R0. This is demonstrated

by his uwz'/u72| values which are indicated by the arrows in Figure

w

3.13b. Furthermore, Spalart states that the mixed normalization,

(ufs/su)1/2, for the vorticity intensities when plotted versus y/6,

95

provides for a good scaling of different R9 data. Figure 3.13d shows the
present data under this normalization. Note that over virtually all of
the boundary layer the three distributions exhibit very good agreement.
However, the insert to this figure shows that for y/6 < 0.02 this scaling
apparently fails. Examination of the R0 - 1,010 distribution in Figure
3.13b indicates that the points nearest the wall in Figure 3.13d reach
their peak (a value of about 7) at y/6 3 0.013. 0n the other hand, at the
same y/6 position the R9 - 2,870 and 4,850 distributions are both already
greater than 7. Furthermore, examination of the R0 - 2,870 and 4,850 data
of Figure 3.13b strongly suggests that points nearer to the wall in these
distributions are going to continue to increase. This observation is
directly substantiated by the results of Alfredsson et a1. (1988) that
indicate that the limiting value of wz' is 0.38 :0.02 over the
approximate Reynolds number range 300 < R9 < 2,800. This indicates that
the normalization of Figure 3.13d does not produce a universal profile of
different Reynolds number data for y/6 < 0.013. To explicitly illustrate
this point, if one assumes that the R0 - 4,850 wz'+ profile has a value
of 0.37 at y+ - 4.5, then under the scaling of Figure 3.13d this
distribution would have a value of approximately 16 -- which is more than
twice the peak value of about 7 in the R0 - 1,010 flow. Thus, while the
apparently universal scaling of Figure 3.13b remains to be explicitly
verified at higher R0 for y+ S 10, the above results strongly suggest
that the mixed normalization of Figure 3.13d does not produce a universal

scaling of the data near the wall.

The present w -skewness and kurtosis profiles are shown in Figures

2

3.14 and 3.15 respectively. Included in these figures for comparison are

the nine wire probe data of Balint et al. (1987a). As one can see the

96

agreement between the two studies is very good.

The dominant feature of the three S(wz) distributions is the large
negative peak reached near y+ - 40. This feature is consistent with a
process of high intensity negative spanwise vorticity containing motions
(of the same sign as that of the mean vorticity) lifting from the
sublayer. These motions may plausibly be associated with either the head
portion of hairpin vortices as recently studied in detail by Lu and Smith
(1988), and/or the lifted vortex sheet-like shear layers observed in the
physical simulations of Chu (1988) and in the computational data base
interrogation of Jimenez, Moin, Moser and Keefe (1988). Given these types
of processes, the present skewness data suggest that the proposed lifted
sublayer motions on average reach a maximum relative intensity near y+ -
40, and then decrease in relative strength at greater distances from the
wall‘. This type of physical picture is given further support by both the
template-matching results of Lu and Smith (1988) which indicate that the
most probable region of observing a hairpin head is in the range 28 < y+
< 32, and the computational results of Jimenez et a1. (1988) which show
that the tip of the lifted vortex sheets tend to level-off at about y+ 3
35. This observed feature in the skewness profile could also be in part
resultant from spanwise vorticity interactions involving the intermediate
scale vortex ring-like motions studied by Falco, see Chapter 1. Under
this type of interaction (and given the scale of these motions), as the
positive vorticity in lower lobe of the ring-like motion interacts with
the sublayer, the center of the upper lobe (which has the same sense of

rotation as the mean vorticity) would be located at about y+ - 40.

 

‘Of course the number of proposed motions lifting from the sublayer that
actually reach a given y+ value and the presence of other w -containing

motions, would also influence the value of S(wz). z

97

The S(wz) data of Figure 3.14 exhibit good agreement for y+ < 50,
but farther away from the wall the three distributions are different and
do not exhibit a clear-cut Reynolds number trend. The tendency for the
two higher R9 distributions to agree, and be different from the R9 -
1,010 profile, is however, still observable. That these profiles exhibit
apparent near-wall universality is consistent with the notion that near
the wall the single dominant resident features are the lifted sublayer
motions discussed above. Given that the statistical properties of uv,
discussed in Section 3.2.2.2, tended to exhibit universality under wall
variable scaling for y+ < 15, it is probably true that the active motions
for y+ < 15 are predominantly represented by these lifted sublayer
vorticity distributions and their associated initiating mechanisms.
However, since in the range 15 < y+ < 50 the Reynolds stress statistics
are clearly not universal under an inner variable normalization while the
spanwise vorticity statistics apparently are (up to K(wz)), one cannot
associate the universality of the spanwise vorticity field in this region
with a universal Reynolds stress-producing active motion. Physically
however, one can reconcile this discrepancy under a process in which the
lifted sublayer motions are initiated by mechanisms which are on average
universal in their ”2 content, but are not universal in their ability to
produce Reynolds stress6 (due to, say, relative changes in scale or in

the manner in which these motions interact within the flow).

Features similar to S(wz) are also observed in the K(wz)

 

6Also note that implicit in this argument is that the initiation of the
lifted sublayer motions is predominantly resultant from external
excitation, rather than from instability mechanisms that are local to the
sublayer. This sentiment can be given support by a significant body of
experimental evidence; see, for example, Bradshaw (1967), Willmarth
(1975), Falco (1983) and Wark (1988).

98

distributions of Figure 3.15. However, the higher Reynolds number data of
this plot for y+ > 50 show reasonable agreement. That the K(wz) profiles
increase with increasing y+ in the near-wall region is consistent with
the lifting sublayer motion model discussed above. For example, as an
average vortical motion (say of scale about equal to the sublayer
thickness and of strength characteristic of urQ/u) moves outward from the
sublayer, its relative scale will decrease and its relative circulation
will increase with respect to the average surrounding motions. Thus if
one interprets an increasing kurtosis as an indication of increasingly
intermittent behavior, this process predicts that the K(wz) profile
should initially increase for increasing y values outside of the
sublayer. At some distance far enough from the wall, the probability of
observing only hairpin-type or lifted shear layer-type motions decreases.
At this point K(wz) may cease to increase, and S(wz) will probably cease

to decrease.

Under the above type of physical model the S(wz) data suggest that
the relative intensity of the lifted motions decreases from their maximum
for y+ 2 40. Combining this with the observation that the K(wz) profiles
continue to increase for y+ values significantly greater than 40 (at
least at the higher Reynolds numbers) suggests that intermittent high

intensity positive w containing motions (of sign opposite that of the

2
mean) are also present in the near-wall region. Finally, the apparent
near-wall self-similarity of the S(wz) and K(wz) profiles (when plotted
versus y+) adds further support to the notion suggested by the "law of

the wall" and the present wz'v/ur2 profiles that the structure of wz for

y+ S 50 is universal under inner variable scaling.

99

3.2.4 Summary of u, v, uv and ”2 Results

Before proceeding on to the data presentation concerning the
transport of the Reynolds stresses, it is useful to summarize the results

of this section.

Concerning the velocity intensities for y+ 2 10, an apparent
Reynolds number variation is observed which is consistent with other high
resolution studies. In general, the observed trends are consistent with
an increased damping of the turbulent fluctuations as suggested by a
relative increase in the importance of viscous forces resultant from a
decrease in Reynolds number. Nearer the wall the u intensities appear to
merge, while the R0 variation of the v intensity is much more pronounced
and is maintained over the entire layer. These observed Reynolds number
trends have been hypothesized by Wei (1987) to a result of a near-wall
increase in Dwx/Dt, and are consistent with the results of Willmarth and
Bogar (1977) and Willmarth and Sharma (1984) which indicate that
significant changes in the small scale structure occur with increasing
Ra. Physically, the greater R9 variation in the v intensities suggests
that changing the Reynolds number (within the given range) significantly
changes those features of the flow associated with momentum transport
across the layer. Most of the Reynolds number dependencies observed occur
predominantly for R9 less than about 3,000. This is the same approximate
Reynolds number range (given by Antonia et al. (1982a)) over which
significant changes in the wake region of the mean velocity profile are

most readily observable.

The higher order velocity statistics are shown to be in good

agreement with previously reported results. The R0 - 1,010 S(u) and K(u)

100

profiles differ subtly from the present two higher Reynolds number
distributions; which in both cases exhibit excellent agreement over the
entire inner region. The present S(v) distributions exhibit only a small
near-wall variation with Reynolds number, while the K(v) profiles
demonstrate a clear rise with increasing R9 near the wall. This trend is
apparently consistent with the previously mentioned hypothesis of Wei

(1987).

The present <uv>/u,2

profiles exhibit an R0 variation very similar
to that of the streamwise intensities (i.e for y+ 2 15 they diverge, and
for y+ S 15 they apparently merge). As with the velocity intensities, the
observed variations in the <uv>/u'v' profiles were more dramatic at lower
R9. These profiles also appear to merge for y+ S 15. The velocity

variance ratio distributions exhibit a definite decrease with increasing

Reynolds number for y+ < 100; implying an increased efficiency in

turbulent mixing in this region as R0 increases.

The (uv)'/uf2 profiles show a high degree of similarity with the v
intensity profiles in that they apparently do not merge over any portion
of the boundary layer. The S(uv) distributions show only a very slight
Reynolds number dependence, and are in excellent agreement with the data
of Wei (1987). This lack of significant Reynolds number dependence is
interpreted to indicate that the average relative :uv content of the
Reynolds stress producing motions is also nearly R0 invariant, and that
the changes in <uv>/u’,2 and (uv)'/uf2 are resultant from a Reynolds
number dependence in the manner in which the stress producing motions
interact within the flow. In general, changes in these interactions may
be tied to the above suggested increased significance of viscous forces

at lower R9, or as inferred in the Section 3.3 changes in scale and/or

101

convection velocity of the motions responsible for the stress transport.
The K(uv) profiles exhibit good agreement for y+ 2 20, but for y+ < 20
the R0 - 1,010 distribution shows a distinct decrease until about y+ - 7,

and then for smaller y+ increases rapidly.

The spanwise vorticity intensities apparently define a universal
profile in the inner region when normalized with v and ur and plotted
versus y+. However, in the outer portion of the boundary layer the lowest
Reynolds number distribution (under inner variable scaling) exhibits
distinctly higher values, while the higher Reynolds number distributions
continue to show excellent agreement and are in very good agreement with
the results of Balint et al. (1987a). The low Reynolds number deviation
is apparently consistent with the structural changes seen to occur in the
wake region of the mean profile. The mixed normalization proposed by
Spalart (1988) produced a very good scaling of the present wz' data for
y/6 > 0.013. However, nearer the wall available data strongly suggest
that this normalization fails to produce a single profile from different

Reynolds number data.

Both the S(wz) and the K(wz) profiles are shown to be consistent
with the physical process of high intensity negative spanwise vorticity
containing motions (i.e having the same sign as the mean vorticity)
lifting from the sublayer. The given shapes of these distributions not
only suggest that, on average, the lifted sublayer motions reach a
maximum relative intensity at about y+ - 40, but also provide indirect
evidence supporting the existence of high intensity positive spanwise
vorticity containing motions (having sign opposite that of the mean

vorticity) in the wall region.

102

3.2.5 Conclusions Pertaining to u, v, uv and ”2 Results

Having reiterated some of the main features of the preceding data
presentation, a brief listing of what are believed to be the main

conclusions relevant to the discussion in the following sections.

. In many ways the present measurements support the notion that the
changes observed in the wake region of the mean velocity profile for R9
less than about 3,000 are symptomatic of structural changes that also
affect many of the u, v and uv statistics throughout most of the boundary

layer for R9 < 3,000.

. The present v'/ur measurements indicate that those features of
the flow associated with transport across the layer are highly sensitive

to Reynolds number (over the given R0 range).

. The Reynolds number trends in the present u'/uT, v’/uf, <uv>/u,2
and (uv)’/uf2 distributions are consistent with the notion that viscous

forces are more prevalent at lower R9.

. The present measurements give strong support to the hypothesis
that for y+ < 50 and under inner variable scaling the statistical
characteristics of the spanwise vorticity field are universal. However,
as it pertains to an active Reynolds stress producing component of the
flow, it was shown that this universality does not imply the same for

<uv>/u,2 for y+ > 15.

. The present spanwise vorticity statistics support a physical

model in which negative w containing motions are lifted from the

Z

sublayer up to the most probable y+ location of about 40. Also,

examination of these statistics further from the wall gives indirect

103

evidence for the existence of intermittent positive w containing motions

Z

in the wall region.

0 Finally (as an aside) comparisons (mainly with the data of Wei)
support the conclusion that there exist subtle but possibly significant
differences between the inner region physics in channel and boundary

layer flows.

3.3 THE ACTIVE AND INACTIVE MOTIONS

For the purposes of boundary layer control one would like to
determine ways in to reduce or alter the Reynolds stress distribution.
One way of accomplishing this would be to alter the mechanisms
responsible for the transport of the stress. In this section, in an
effort to uncover the nature of the motions directly involved in the
transport <uv>, most of the terms of equation 3.2 are examined.
Furthermore, velocity vorticity correlations related to the y gradients
of <u2>, <v2>, and <w2> are also examined. These results are then related

to the findings of the preceding section.

While the equation for the mean transport of <uv> as given by
equation 3.2 is exact, the association of <wwy> - <vwz> with an active
rotational part and 6<v2 + w2>/8x - 6<u2>/6x with an inactive purely
irrotational part is, as pointed out by Hinze (1975), an approximation.
It is clearly true by its composition that <wwy> - <sz> is always
associated with rotation. However, since 6<v2 + w2>/ax - 8<u2>/6x is not
identically zero in a boundary layer (as it is in a channel) it could
make significant contributions to a<uv>/6y. In what follows an assessment

is made regarding the validity of this approximate decomposition. In

104

doing so, the contributions of the motions participating in the mean

transport of the Reynolds stresses will be examined.
3.3.1 The Inactive Motions

Using the wz probe x-array data, 6<u2>/6x and 6<v2>/8x profiles
were computed. An inner variable normalization of these profiles is given
in Figures 3.16 and 3.17. As one can see, these profiles are essentially
zero across most of the boundary layer. Furthermore, comparison of these
profiles with the data in Figure 3.20 and Figure A3.1.3 indicates that
the a<u2>/6x and 8<v2>/6x (and presumably 6<w2>/6x) contributions to
8<uv>/ay are between one and three orders of magnitude smaller than the

<vwz> and <ww > contributions. Thus the active/inactive motions

Y
decomposition proposed by Hinze (1975) in the form of equation 3.2
appears to be consistent in this respect with Townsend's original model.
It is also worth noting that these results are in good agreement with the
results of Bradshaw (1967) in that they show that the mean effect of the
”irrotational component" (i.e the inactive motions) is small.
Furthermore, these results show that to a very good approximation the
present boundary layers may be considered homogeneous in the streamwise

direction, and that virtually all of the mean transport of the Reynolds

stress is due to the "rotational" terms of equation 3.2.

The rms profiles of 6(u2)/6x and 6(v2)/ax have also been obtained.
These are shown under an inner variable normalization in Figures 3.18 and
3.19. A comparison of the data of these figures with the data of Figure
3.22 shows that unlike the mean values, the variance of these quantities
is at least of the same order of magnitude as that of the rotational

contributions to 8<uv>/ay. This is consistent with the comment of

105

Bradshaw (1967) that, "it is only in the mean that we can call the large-

scale motion 'inactive' in the simplest sense of Townsend's hypothesis".

The 6<u2>/ax and 8<v2>/8x profiles of Figures 3.16 and 3.17 tend to
suggest reasonable agreement under an inner variable scaling. However,
given the very small magnitude of these profiles as well as the data
scatter (especially at R0 - 1,010), this issue remains unresolved. The
rms profiles shown in Figures 3.18 and 3.19 exhibit the same feature of
most of the data presented thus far. The two higher Reynolds number
distributions show only small differences, but are significantly
different from the R9 - 1,010 profiles. The rms 8(v2)/8x profiles clearly
do not follow inner variable scaling. However, it is interesting to note
that the y+ position of the peak in each distribution is apparently

Reynolds number invariant.

Under Bradshaw's (1967) interpretation the inactive component is
made up of truly irrotational contributions associated with outer region
pressure fluctuations, as well as large scale vortical motions presumed
to be present in the outer region. To this point in the discussion and
data presentation it has been shown that the streamwise gradient terms of
equation 3.2 have a negligible mean contribution to a<uv>/ay, and thus in
this respect fit the definition of the inactive component. However, it
has been implicitly assumed that these terms are predominantly associated
with the large scale motions in the boundary layer, and that the large
scale motions are essentially irrotational. In Chapter 4 evidence is
given supporting the hypothesis that, on average, large scale spanwise

vorticity containing motions are not present in the inner region.

106

3.3.2 The Active Motions

The relationship between the y gradient of <uv> and the velocity
vorticity correlations given in equation 3.2 may be extracted either from
a more general tensor identity, or may be derived directly from the
equations of motions as shown in Appendix 3.1. The results of the
previous section indicate that to a very good approximation (for the

present flow field) equation 3.2 may be written as

6<uv>/6y - <ww > - <vwz>.

Y

This section will examine the terms on the right hand side of the above
expression. Interpretations of the data in this section will pertain to
issues concerning the scaling of the active motions proposed by Townsend
(1961), and in a context relevant to an interaction between the motions
of the inner and outer regions.
3.3.2.1 <vwz> Data

Figure 3.20 shows the present <vwz> profiles normalized by u and
uf. As can be seen the R0 -l,010 profile reaches a distinct positive peak
near the edge of the sublayer. Closer to the wall the distribution drops
off sharply, as it must, since <vaz>|w - -1/26<uv>/6y|w - 0. Further from
the wall the R0 -l,010 distribution melds smoothly with the R0 - 2,870
distribution, and both cross zero at y+ 3 13. Between y+ - 15 and 20 the
data of Figure 3.20 suggest the three profiles merge under the inner
variable scaling; although the R9 - 4,850 flow is not well represented in
this region. For y+ values greater than 20 however, an apparent Reynolds
number dependence is observed. This dependence indicates that at low

Reynolds numbers the spanwise vorticity fluctuations and the normal

velocity fluctuations are much more negatively correlated in the

107

logarithmic region than at higher R9. Note that the apparent universality
in <vwz>+ for y+ less than about 20 is consistent with the same feature
in the <uv>+ profiles shown in Figure 3.3. This correspondence further
supports the notion that in terms of their ability to create and
transport <uv> the active turbulent motions are self-similar under inner
variable scaling for y+ less than about 20. In connection with the lifted
sublayer motions known to be prevalent in this region, this result
probably indicates that certain average features of these motions (such
as, say, the average amount of sublayer fluid lifted per unit area and

time) scales with the wall shear.

As discussed in Section 3.1, Townsend's notion of the active
component in the flow included that its statistical properties in the
inner region are universal functions of distance from the wall, y, and
the local shear stress, 7. Given that the velocity vorticity correlations
shown in equation 3.2 have the units of [L/T’], one can satisfy
Townsend's proposed scaling by normalizing these variables by v and r (or
possibly uf) and plotting versus y+, or by creating a new non-dimensional
function by using y and r (or possibly ur) and plotting versus y+. As can
be seen in Figure 3.20, the familiar inner normalization fails except for
y+ < 20. Furthermore, examination of Figure 3.3 indicates that an inner-
type non-dimensional function formed by using a velocity scale derived
from the local turbulent shear stress, <uv>1/2, causes the differences in
the distributions of Figure 3.20 (for y... > 20) to become greater. Thus,
while both the <uv>+ and <vwz>+ distributions show a Reynolds number

dependence, these dependencies are not self-similar.

Profiles of the non-dimensional function, ywz'/uf, are shown in

Figure 3.21. As with the profiles of Figure 3.20, these profiles also

108

fail to merge except for y+ less than about 20. Also the use of y and
<uv>1/2, once again, only makes matters worse. With respect to the <wwy>,
examination of the data deduced in Appendix 3.1 indicates that the

profiles of <ww > will exhibit essentially the same behavior as in

y
Figures 3.20 and 3.21. These results indicate that over the given R9
range the individual contributions to <uv> transport normal to the wall

are not universal functions of y and r across the inner region.

3.3.2.2 (vwz)' Data

Further insight into the validity of Townsend's proposed scaling
for the active motions (assuming they are represented by vwz) can be
gained by examining the rms vwz profiles. Figure 3.22 shows an inner
variable normalization of (vwz)'. As can be seen, near the wall all of
the profiles appear to be distinctly different. However, at the two
higher Reynolds numbers and for y+ 2 30 very good agreement is observed.
Note that the region over which this agreement is observed corresponds
very well with the logarithmic region of the respective mean velocity
profiles. Furthermore, normalizing the function (vwz)' with the function
u/<uv>3/2, see Figure 3.23, convincingly removes the near-wall deviations
between the higher Reynolds number profiles, and brings the R9 - 1,010
data into much better agreement. Equally interesting is the normalization
employing the function y/uf shown in Figure 3.24. Under this scaling good
agreement between of all three profiles is observed, and exceptionally
good agreement is seen at the higher Reynolds numbers. Note that the
extent of the agreement between the different R0 profiles corresponds
very well with the extent of the logarithmic mean velocity profile at

each R9.

The above results concerning the rms of vw correspond very well

2

109

with Townsend's proposed scaling for the active motions. However, the
<vwz> results show that the net mean contribution of the vwz motions to
the transport of <uv> is not universal under any inner-type scaling. This
is interpreted to indicate that the organizational features of the flow,
as they pertain to the transport of <uv>, are Reynolds number dependent
over the given R0 range. Information is presented below that indicates
the flow field changes affecting <sz> may be associated with an R0
dependence in the average scale and/or convection velocity of the vw

Z

containing motions.

3.2.2.3 vwz Spectra

In an effort to further understand the observed R9 dependence in
<vwz>, frequency spectra were examined. Following Perry and Abell (1975),
inner and outer variable non-dimensionalizations of the power spectrums,

@, are defined as follows.

Inner normalization:

, u(vwz)' 2
¢1[wz(c)]da1 :-—. [—] (3.3)
u s
o r
where a1 — 2wrf/ur2 is the non-dimensional angular frequency.

Outer normalization:

0(vwz)' 2
<I>°[vwz(t)]da° a [—] (3.4)
U 2
0 co

 

6The power spectrum, Q, of a function x(t) is defined as,
Na) - {Re[X(a)]}’ + {Im[X(a)]}2.

Where a is the angular frequency (5 2xf), and X(a) is the Fourier
transform of x(t).

110

where 0° - 201f/U52. The data to be shown are in the form of a spectral
function, W, defined for the given non-dimensionalizations as

wi - ai-ei, (3.5)
and

w° - a°.¢°. (3.6)

The spectral functions shown in Figures 3.25 and 3.26 were derived
by averaging ensembles of 250 1,024 point digital Fast Fourier Transforms
of vwz for each y+ location and Re. In the computation of the individual
Fourier transforms a l/lO cosine taper function was used to suppress

”side lobe leakage".

Examination of Figures 3.25a and 3.25b indicates that near y+ - 20
neither inner or outer normalizations scales the data; although the high
frequency end of the R0 - 2,870 and 4,850 spectral functions apparently
merge under the inner scaling in Figure 3.25a. The outer normalization of
Figure 3.25b shows no hint of scaling the data over any frequency range.
This conclusion is also reached for the outer normalization at y+ 3 120
in Figure 3.26b. Concerning the inner normalization of Figure 2.26a it is
interesting to note that at low frequencies the spectral functions appear
to merge for all R9, while at higher frequencies there is an apparent

trend. This is just the opposite of what is observed nearer the wall.

An interesting result concerning the scale of the vwz motions can
be deduced from Figures 3.25 and 3.26. As indicated by the arrows, the
peak in the inner variable scaled spectral functions of Figure 3.25a (and
to a lesser degree in Figure 3.26s) show a significant shift toward lower

frequencies with increasing Reynolds number. Conversely, the outer

normalizations of Figures 3.25b and 3.26b show very clearly that the peak

111

in the spectral function moves to higher non-dimensional frequencies with

increasing Reynolds number.

In general, the cause of a shift in a spectral function may be due
to either a change in the scale of the associated motions and/or a change
in the convection velocity of the motions. Given that the Au shift in log
frequency ranges from about 0.15 to 0.3 under the inner normalization,
and from about 0.25 to 0.4 under the outer normalization, it is unlikely
that convection velocity (which is relatively insensitive to R0) is the
sole cause for the observed variations. Therefore, if one associates
large scale motions with low frequencies and small scale motions with
high frequencies, the results in Figures 3.25 and 3.26 indicate that

under an inner normalization the vw containing motions increase in scale

2
with increasing Reynolds number, while under an outer normalization these
motions decrease in scale with increasing Reynolds number. These changes

in scale are probably associated with the Reynolds number dependence in

<vwz> discussed above.

In connection with instantaneously observed motions, it is
interesting to note that the above deduced changes in scale correspond
well with the scale dependence of the vortex ring-like motions observed
by Falco (unpublished). Figure 3.27 shows the observed R0 dependence for
these motions. As can be seen, the scale of these motions increases under
an inner normalization and decreases under an outer normalization with

increasing R0.

3.2.2.4 The Dependence of <vw2> on Probe Scale
Having identified an apparently significant Reynolds number

dependence in the motions contributing to <vwz>, it is now important to

112

assess the significance of the observed trends in the data relative to

measurement inaccuracies.

The largest error source associated with the measurement of vwZ is
readily identified as the Az spacing between the centers of the parallel
and x-array elements contained within the wz probe. In assessing the
errors resultant from the x-array and parallel-array separation it is

useful to expand <vwz>.

<sz> - <vav/ax> + <vau/ay>

Notice that the first term in the expansion is equal to l/2a<v2?/ax,
which is derived solely from the x-array and was shown in Figure 3.17 to
be negligible for all R0. Therefore, the major effect of the A2
separation is associated with the resolution of <vau/6y>.

To get an estimate of the error in this correlation, the data of
Figure 2.13 will be used. A general result that might be concluded from
Figure 2.13 is that velocity gradient correlations decrease more rapidly
than velocity correlations. From this, a logical extension is to assume
that velocity-velocity gradient correlations decrease at a rate somewhere
between velocity and velocity gradient correlations. Based on this
hypothesis, one would expect <v6u/6y>/v'(au/ay)' data to fall somewhere
between the u and av/ax data of Figure 2.13 for a given Az separation. To
be very conservative, let us assume that the <v6u/ay> correlation
decreases like the av/ax data, and that the error due to the A2
separation corresponds to the case for which Az/n - 15.6 (the largest
value given in Table 2.3). Based on these values the predicted

attenuation in the <v6u/3y> correlation coefficient is about 501.

Figure 3.28 shows the vwz correlation coefficient distributions.

113

Assume that the R0 - 1,010 data in this figure are correct. If the
differences between the R0 - 1,010 profile and the R0 - 4,850 profile
were only due to attenuation caused by the finite Az separation, then at
y+ - 140 (the position at which the data of Figure 2.13 were derived) the
above error estimate indicates that the R9 - 4,850 profile should attain
a value of approximately 0.24. The actual value of the R0 - 4,850
distribution at the given location is about 0.07, or less than l/3 the

predicted value.

The above analysis strongly suggests that the observed differences
between the <Nwz> profiles at different R9 cannot be solely attributed to
measurement errors. Examination of the data also suggests this to be the
case. For example, using probe separation error as an explanation, it is
difficult to reconcile the apparent non-zero agreement between the R0 -
1,010 and 2,870 data for y... < 20 in Figures 3.20 and 3.28. All available
studies suggest that probe resolution errors should become more
pronounced in this region of the flow. Similarly, it is difficult to
account for the interesting and physically appealing scaling properties
observed in the (vwz)' profiles if the vwz signals were substantially
affected by finite probe scale effects.
3.2.2.5 The Vortex Force: A Plausible Mechanism For Lifting Vortical
Sublayer Motions

Tennekes and Lumley (1972 p.78) discuss velocity vorticity cross

products in terms of the body force per unit mass that they generate,
F - e u .
1 ijk 3%

This phenomenon was first documented in experiments by Magnus (1853). An
example of this force is given in Batchelor (1967 p. 427), in which it is

shown that a rotating cylinder exposed to a velocity normal to its axis

114

experiences a lift force in the direction of the high flow speed side of
the cylinder'. In the present context the vortex force will be used in
physically interpreting velocity vorticity correlations relevant to
experimental observations concerning vortical motions lifting from the
sublayer.

For reference Figure 3.29 illustrates the lift forces experienced
by vorticity filaments in directions pertinent to the correlations
presented in the previous section and in Appendix 3.1. Concerning uwz
notice that the associated vortex force is in the y direction”. The
generation of this lift force may provide for a simple and quite general

mechanism by which vortical sublayer fluid is transported away from the

wall.

Evidence is presented in Chapter 4 indicating that in an absolute
sense the spanwise vorticity in the viscous sublayer is to an extremely
good approximation always of the same sign as the mean vorticity. In the
coordinate system of the present experiments this sign is negative. To be
consistent, this means that the instantaneous streamwise velocity near
the wall is always positive. Given these signs for the velocity and
vorticity components, Figure 3.29 indicates that instantaneously the
direction of the vortex force acting on spanwise vorticity containing
motions in the sublayer is always in the positive y direction. This

however, does not mean that the sublayer is undergoing continuous

 

7A1though it is clearly realized that vorticity filaments in a flow do
not act like solid bodies, it is true that the velocity and vorticity
vectors, in general, are independent of each other. Thus when vorticity
filaments have a velocity relative to the surrounding motions they will
experience this body force.

8Notice that the wwx force is also in the y direction. This force is, on
average, significantly smaller than the uwz force, see Appendix 3.1. The
negative sign of this correlation in the sublayer is however, consistent
with the flow field associated with Falco's (1980) pocket vortex which
has been observed to induce itself towards the wall.

115

ejections in the sense of the bursting terminology, since instantaneously

this force may be quite weak compared to, say, viscous forces.

Coupled with the experimental observation of sweeps, the
interpretation of the lift force due to uwz provides a plausible causal
link between sweeps and the initial movement of vortical sublayer fluid
away from the wall. Consider first an essentially irrotational high speed
motion of limited spatial extent coming toward the wall at a shallow
angle. If the instantaneous velocity associated with this high speed

front is far enough from the mean, then the w containing motions in the

z
sublayer subjected to this front will experience a significant lift

force.

Consider further this process in the context of convecting vortical
motions interacting with the sublayer as discussed in Chapter 1. Given
that the magnitude of the local lift force acting on the sublayer
spanwise vorticity depends on the local streamwise velocity, it is easy
to verify that vortical motions of rotation opposite that of the mean
vorticity convecting above the sublayer will impose a local positive
velocity perturbation on the sublayer. This will result in a perturbation
vortex force in the positive y direction. Note that if the positive
vorticity of the convecting vortical motion is of sufficient magnitude,
this result is independent of its convection velocity. The region under
this type vortical motion will always be a region of locally accelerated

flow.

The above analysis suggests that a positive streamwise velocity
perturbation imposed at, say, the outer edge of the sublayer will cause

the effected region of the sublayer to experience a lift force. This

116

represents a plausible causal relationship between sweeps and ejections.
Note that this mechanism is consistent with the conclusions of Bull and
Thomas (1983) indicating that "fluid involved in the bursting process is
subjected to a favourable streamwise pressure gradient ... at the time
that the lift-up of low-speed streaks in the wall region begins". In
terms of an interaction between the inner and outer region motions it is
easily seen that the above analysis suggests that a motion with vorticity
of sign opposite that of the mean embedded in a high speed front is the
optimal configuration for providing lift to sublayer fluid through the
uwz vortex force. This result is consistent with the observations of
Falco (1977, 1983) in that the lower lobes of the vortex ring-like

motions (which interact with, and apparently lift sublayer fluid) have a

rotational sense Opposite of that of the mean vorticity.

Figure 3.30 shows the present <uwz> distributions normalized by v
and u'. Under this normalization the data of Figure 3.30 exhibit good
agreement for all R0. This indicates that, on average, the vortex force
experienced by wz containing motions is Reynolds number independent. Note
further that

<uwz> - <u6v/6x> - l/2<au2/6y>.

Given the experience with Figures 3.16 and 3.17, it is probably true that
the first term on the right hand side is negligible. Furthermore, the
recent evidence of Alfredsson et al. (1988) suggests that for y+ < 10
u'/uf is Reynolds number invariant over the range 300 S R0 5 2,800. This
evidence indicates that the <uwz>+ profiles should be universal for y+ <
10. Invariance of profiles of <Zuwz>+ in this region are also consistent
with the notion that the average contributions to the transport of <u2>

from the active motions (in the form of lifting sublayer motions) are

ll7

universal for y+ less than about 15.

Clearly, the most distinctive feature of Figure 3.30 is the large
negative correlation attained near the edge of the sublayer. The sign of
this correlation is consistent with either an average lifting force due
to a positive streamwise velocity perturbation acting on a negative wz
perturbation, or the converse. Since however, S(u) is positive and S(wz)
is negative in this region, the largest contributions to this correlation

are probably from the (+)u - (—)w perturbation combination. When

2
considered in an absolute sense, this combination represents the largest
lift force. Under the inner variable normalization shown the correlation
reaches a value of about -0.63 at y+ 3 5.2. This indicates that, on
average, the correlation between the u and “z perturbations represents a
force that is about 631 of the associated mean viscous forces in this
region. Note that this average force is substantially greater than the

peak average force represented by any of the other correlations shown in

Appendix 3.1.

Further away from the wall <uwz> decreases rapidly, and is
essentially zero for y+ > 25. Note however, that the R0 — 1,010 data show
a small positive trend up to a value of about 0.04 after crossing zero
near y+ - 20. Although in a relative sense this slight positive trend
would seem to be negligible, examination of the <vwz> correlations
indicates that values of this magnitude may have significance in this
region of the flow. In connection with vortical motions lifting from the
sublayer, the rapid decrease in this correlation indicates that, on
average, the force due to the fluctuating u and wz exists only out to
about y+ - 25. Plausibly, after this y+ location these motions would

start to undergo complex three dimensional interactions, and thus would

118

begin to lose their lifted vortex sheet-like and/or hairpin vortex-like
character. This interpretation is apparently consistent with the

interpretation of the S(wz) and K(wz) profiles in Section 3.2.

As shown in Appendix 3.1 <uwz> appears in the equation for the
transport of the diagonal terms of the Reynolds stress tensor. Thus in

this respect the motions characterized by their uw content may be

2
considered the active motions pertinent to the transport of the normal
stresses. Figure 3.30 indicates that Townsend's proposed scaling for the
active motions apparently holds for <uwz>. Furthermore, examination of
Figures 3.32 and 3.33 indicates that the (uwz)' profiles also tend to
follow this scaling. The scaling region for these profiles extends to
approximately the edge of the logarithmic region of the mean velocity
profiles. In contrast, it is interesting to note that the normalization
of <uwz> by the function y/uf, shown in Figure 3.31, produces a universal
profile only for y+ less than about 20. In general however, it seems safe

to say that the mean and rms of the motions responsible for uwz follow

inner-type scaling.

3.4 DISCUSSION AND CONCLUSIONS - RELATIVE TO CHAPTER 3

Numerous results were discussed in this chapter. Some of these will
now be summarized and discussed in the context of an interaction between
the motions of the inner and outer regions. One is referred to Sections

3.2.4 and 3.2.5 for further summary information.

It is well established (see Chapter 1) that not only is the wake
region of the mean velocity profile strongly Reynolds number dependent

for R0 less than about 3,000, but so are the instantaneous motions of the

119

outer region. A general result of the data presented in this chapter is
that many of the turbulence statistics (and especially those related to
momentum transport normal to the wall) exhibit a Reynolds number
dependence across much of the layer that is most evident for R0 less than
about 3,000. It is tempting to associate these two results through an

interaction between the motions of the inner and outer regions.

Important results indicating the universality of the spanwise
vorticity statistics under an inner variable normalization for y+ < 50
were given. Furthermore, this apparent universality in ”2 did not imply
the same for <uv> or <sz> for y+~> 15. This leads to the conclusion that
away from the wall one cannot associate the spanwise vortical content of

the turbulent motions with their ability to create and transport <uv>.

With respect to <wwz>, evidence was given indicating that with
increasing Reynolds number these motions exhibit a decrease in size with
respect to total thickness of the flow (i.e. outer variables) and an
increase in size with respect to the thickness of the viscous sublayer
(i.e. inner variables). Given that the ratio, uT/Ug, and the total
thickness of the flow change relatively little with Reynolds number,
these results indicate a decrease in the absolute scale of the motions
characterized by vwz with increasing R0. These changes in scale lend
themselves to the interpretation that at low Reynolds numbers, due to
their larger scale relative to the width of the flow, the motions
responsible for the transport of <uv> interact with each other more
significantly. This increased level of interaction may decrease the
average coherence of these motions, and thus interfere with the creation
of <uv>. under this interpretation, due to a relative decrease in scale

at higher Reynolds numbers these motions interact less. This

120

interpretation (coupled with the universality of wz under inner variable
scaling) tends to be supported by the apparent y/uf scaling found for the
rms vwz profiles. An alternate possibility is that at low Reynolds number
(again due to their larger relative scale) the motions responsible for
the transport of <uv> interact differently with the larger scale outer
region motions than they do at higher R0. This possibility would be
indicative of a change in the interaction between the inner and outer

region motions.

Nearer the wall the <uv> and <vwz> data provide evidence that under
an inner normalization the different Reynolds number profiles merge. This
leads to the conclusion that near the wall the average ”2 content of the
turbulent motions can be associated with their ability to create and
transport <uv>. Combining this with the observed universality of the wz
statistics out to y+ 3 50, suggests a physical picture in which the
initial lifting of a hairpin-type motion is dependent on the spanwise
vorticity content of the initiating mechanism, but not the ability of the
initiating mechanism to create <uv>. The apparent universality of the
<uv>+ and <.vwz>+ profiles for y+ < 15 also lends support to the
hypothesis that there are a limited number of turbulent stress producing
mechanisms in this region of the flow. These mechanisms are probably

associated with the lifted vortex sheet-like and/or hairpin vortex-like

motions, as well as the motions responsible for their formation.

In an average sense it was found that the streamwise gradient terms
of equation 3.2 were negligible relative to velocity vorticity
correlation terms. However, the rms values of these terms were found to
be at least as large as (vwz)'. This indicates that instantaneously these

motions could be quite important. In the context of an inner/outer

121

coupling through the convection of small scale vortical motions embedded
in a larger scale and relatively irrotational motion, it is tempting to

associate these streamwise gradient terms with the latter.

The data of Section 3.2.2.5 support the suggestion that the vortex
force due to the introduction of a high speed velocity perturbation at
the outer edge of the sublayer is a viable mechanism for initiating the
observed lifted shear layer and/or hairpin vortex-like motions. In a very
general sense this plausible initiating mechanism associates a cause and
effect relationship between sweeps and ejections. In terms of the
influence of convecting vortical motions above the sublayer, it was
deduced that small scale motions that have spanwise vorticity of sign
opposite that of the mean would be most effective in imposing a
distortion force normal to the wall on the sublayer vorticity
distribution via the uwz vortex force. In terms of essentially
irrotational outer region sweeps interacting with the sublayer, the
vortex force mechanism, in general, predicts that the size of the lifted
sublayer motion is determined by the surface area of the sublayer
affected by the sweep. Thus this mechanism may provide a link between the

observation of relatively large scale lifted shear layers and the direct

influence of the outer region flow.

CHAPTER 4

O P I L STRUC OF SPANWISE VORTICITY IN URB ENT BO Y
LAYERS

4.1 ISSUES PERTINENT TO AN INNER/OUTER INTERACTION

Very little data exist that are relevant to understanding the
spatial structure of the vorticity field in turbulent wall flows.
Kastrinakis (1977) performed two point streamwise vorticity correlation
experiments in a turbulent channel flow using a four wire Kovasznay-type
probel. These measurements, which were for spanwise probe separations,
revealed peak negative correlation values between -0.075 and -0.15 for
A2... separations equal to about 40 and y+ values less than about 30. These
significant negative values are readily interpreted as strong evidence
for the existence of counter-rotating streamwise vorticity containing
motions near the wall. However, for greater distances from the wall the
zero crossing to negative correlations disappears. This may indicate that
the hairpin vortex-type motions believed to be responsible for the
counter-rotating ”x begin to lose their coherence by y+ 3 30. For y+ less
than roughly 100 the two point ”x correlations become essentially zero at
separation values of Az+ 3 60. Further away from the wall these

correlations maintain non-zero values for separations out to Az+ 3 120.

 

1Even though it was later shown that significant errors in the
instantaneous measured values of w are associated with this probe,
relevant to long-time statistics Foss and Wallace (1989) conclude that
”comparison with other measurements ... indicate that these errors, which
can instantaneously be very large, may statistically be small". It is
therefore believed that these correlation curves are reasonably accurate.

122

123

Contrary to the popular belief in the dominance of streamwise
vorticity containing motions near the wall, the vorticity field for y+ S
10 is predominantly oriented in the spanwise direction. In this region,
the ratio wz'/|02| is about equal to 0.4 (see Figure 4.9), and the ratio
wx'/wz' is in the range 0.3 to 0.5, see for example, Lee et al. (1974) or
Kim et al. (1987). (Due to the no-slip condition wy' is significantly
less than either wx' or wz' in this region of the flow.) Therefore, for
y+ S 10 ([02] 2 wz')/wx' ranges from at least about 3 to at most about
12. Farther from the wall, the importance of the mean vorticity
diminishes (see Figure 4.9), and by y+ 3 40 all three rms vorticity
components are about equal, see Balint et al. (1987a). In the outer

region the mean contribution is negligible, and all three rms components

are about equal, again see Balint et al. (1987a).

The predominance of the spanwise vorticity very near the wall
argues for the validity (at least in an average sense) of the sublayer
vortex sheet approximation discussed in Chapters 1 and 3. This type of
sublayer vorticity distribution predicts that spanwise vorticity
correlations in this region remain non-zero for greater Az separations
than they do further from the wall. The diminished importance of the mean
vorticity and the nearly equal rms values of the fluctuating components
indicates that away from the wall the average vortical motions are highly

three dimensional.

Pertinent to the results in Chapter 3, the results of this chapter
shed light on the issue of whether large scale spanwise vorticity
containing motions (on average) exist. In terms of an interaction between
the motions of the inner and outer regions, this issue is important

because it gives a rough but direct indication of the scales of vortical

124

motions involved. The success or failure of plausible mechanisms for
boundary layer control would undoubtedly be dependent on their ability to
effectively interact and/or alter these scales of motion. In terms of
Townsend's active/inactive motions decomposition, the issue of organized
large scale vortical motions is relevant to the validity of identifying
the outer region as being characterized by inactive motions which are, on

average, essentially irrotational.

In Chapter 3 evidence was presented suggesting the existence of
intermittent motions in the near—wall region that have positive spanwise
vorticity (i.e. of sign opposite that of the mean vorticity).
Furthermore, in examining the vortex force mechanism for the initial
lifting of vortical sublayer motions it was deduced that small scale
positive vorticity containing motions convecting above the sublayer would
be most effective in generating a local lift force on the sublayer
vorticity distribution. Evidence pertaining to the issue of positive ”2

in the near—wall region is also presented in this chapter.

4.2 EXPERIMENTAL PROCEDURES AND CONDITIONS

Most of the details pertaining to the experimental procedures,
equipment and factors affecting data quality as they relate to the
experiments of this chapter were documented in Chapter 2. Therefore,
Section 4.2.1 will document and describe only those aspects of the two
point spanwise vorticity correlation experiments that are additional to
the single point experiments described in Chapter 2. In Section 4.2.2 the
conditions of the two point experiments are given, as well as information

pertinent to relating these experiments to the single point experiments.

125

4.2.1 Additional Equipment and Procedures

The data acquisition, signal conditioning, and probe positioning
equipment used in the correlation experiments included all of the
equipment described in Chapter 2. However, since the correlation
experiments employed a second spanwise vorticity probe additional

hardware was required.

The additional spanwise vorticity probe was operated by four TSI
model 1755 anemometers at an overheat ratio of 1.7. The electronic noise
level associated with these anemometers was found to be comparable to the
electronic noise associated with the DISA 55M01 anemometers used to
operate the other probe. Stable operation of the T51 anemometers at the
two lower Reynolds numbers of the single-point experiments (i.e. R0 -
1,010 and 2,870) required the manual adjustment of their "trimming"
capacitance. While the effect of increasing this capacitance increased
the low velocity end of the operating range, it also reduced the

frequency response of the anemometers.

Following this adjustment to the anemometers, a standard square-
wave test was performed in the free stream of the R0 - 1,010 boundary
layer. The results of the test indicated frequency responses ranging
between 3.3 and 3.5 kHz. This response is somewhat less than the better
than 5 kHz found for the DISA anemometers under the same conditions. The
effect of this decreased frequency response was felt to be negligible
however, since the maximum Kolmogoroff frequencies in the R0 - 1,010 and
2,870 flows have been estimated to be about 130 Hz and 300 Hz

respectively.

An additional set of (previously custom built) DISA low-pass

126

filters were used to remove the high frequency electronic noise from the
T81 anemometers. Unlike the nearly continuously adjustable Krohn-Hite
filters, these low-pass filters allow for only a few discrete cut-off
frequency settings ranging from 1,000 Hz to 5,000 Hz. Thus in using the
1,000 Hz setting an adherence to the Nyquist criterion could not be
maintained. However, inspection and comparison of the output signals from
the two sets of anemometers/filters indicated that there were no
discernable differences in noise levels. This is probably because the
electronic noise on the signals prior to filtering was solely at high
frequencies. As was previously found for the Krohn-Hite filters, the
channel-to-channel phase shifts between the signals entering the DISA
filters were very small (less than 2 Spa). The phase shifting between the
signals from the two sets of anemometers/filters was also found to be

about the same small value.

The traverse gear for the Ay separation experiments was the same as
that used in the single probe experiments. The stationary probe in the Ay
separation experiments was mounted directly to the floor of the tunnel,
and was positioned at the proper y location via thin metal shims. By
using the cathetometer (described in Chapter 2), the stationary probe was
observed under experiment conditions. No detectable vibration of this
probe (due to being directly attached to the floor) was observed. The
initial y distances of the probes from the floor, and the initial Ay
distances between the two probes were also measured using the

cathetometer.

An additional 3-D traverse gear was built and installed for the A2
probe separation experiments. This mechanism, shown in Figure 4.1

consists of a drilled-out aluminum base that is mounted on two 3/4 inch

127

aluminum rods, and is traversed in the spanwise direction by a 32 turns
per inch worm gear. Mounted on this base is a vertical traverse mechanism
that allows for probe positioning in the y direction by a 24 turns per
inch worm gear. Finally, mounted on this y-traverse is 3/8 inch stainless
steel rod that extends approximately 16 inches in the upstream direction.
A :0.5 inch fine adjustment of this extension rod in the x direction is
accomplished via an axially mounted spring-loaded micrometer. The probe
is mounted at the end of this extension rod. This results in the
measurement location being approximately 24 inches upstream of the z-

traversable base.

Due to this extension rod, the possibility existed for unwanted
vibration. To test for this, the probe was (once again) examined through
the cathetometer at the experiment flow speeds. As with the stationary
probe in the Ay separation experiments, no discernable vibration was
detected. The good agreement with the statistics from the other probe

(see Section 4.2.2) further supports this conclusion.

A 0.254 mm (0.01 inch) division scale and a pointer were mounted
such that the relative position of the z-traversable base could be
measured. The x and y-traverses were used, with the aid of the
cathetometer, to initially position the z-traversing probe at the same x
and y locations as the stationary probe. The stationary probe in the A2
separation experiments was mounted on the y-traverse used in the single
probe experiments. The initial Az distance between the two probes was
determined by placing a 0.254 mm (0.01 inch) division scale flat on the
floor beneath the probes. By lowering the two probes near the wall and
looking from overhead the distance between the probe centers was

measured. The x-array and parallel-array configurations for the A2

128

separation experiments were such that the for one ”2 probe the x-array
was offset in the +2 direction while for the other probe the x-array was
offset in the -z direction. The two spanwise vorticity probes were then
oriented such that the two parallel-array elements were interior to the
two x-array elements. The probe center was then defined to be the
parallel-array support prong closest to its "partner" x-array contained

within a given ”2 probe.

The method of determining the initial Az separation was
significantly less accurate than in the Ay separation experiments. A
conservative estimate of the uncertainty associated with this method is
felt to be approximately $2 scale divisions or $0.5 mm; which translates
to about $1.0Az+ and $2.4Az+ at R9 - 1,010 and 2,870 respectively. The
given x-array and parallel-array configuration was used because on

average au/ay dominates av/ax in the near-wall region.

Based upon the preliminary results of Klewicki and Falco (1986),
the smallest achievable probe separations (in wall units) at R0 - 4,850
were too large to yield significantly non-zero correlation coefficient
values. Therefore it was decided to run the correlation experiments only
at the two lowest Reynolds numbers. In order to increase the integration
time of the R0 - 1,010 experiments for a given sample size, the sampling
rate of the analog-to-digital converter was reduced by a factor of 2 from
the value given in Table 2.1. Over the range of y+ positions of the
correlation experiments, the sampling rate ranged between about 2 and 5
times the local Kolmogoroff frequency. The sample size of all of the data
records was 4.8x106 points (6x105 points in each channel). The
corresponding sampling times resulted in averaging over TUg/6 3 6,700 and

5,000 at R0 - 1,010 and 2,870 respectively. All other data reduction

129

procedures for the correlation experiments were the same as given in

Chapter 2.
4.2.2 Experimental Conditions

4.2.2.1 Correlation Experiment Overview

In order to compare with the results of the single probe
experiments, as well as for convenience in data processing, the two point
spanwise vorticity correlation experiments were run (nominally) at R9 -
1,010 and R9 - 2,870. The positions of the stationary probe and Reynolds
numbers for these experiments are given in Table 4.1. All of the
correlations to be reported are for stationary probe positions in the
inner region of the boundary layer. As one can see, for y+ < 10 only the
lower Reynolds number data were experimentally available. Note also that
for the Ay separation experiments the moving probe was always at a

greater distance from the wall than the stationary probe.

4.2.2.2 Matching the Reynolds Numbers of the Single Point Experiments
In order to use the friction velocities from Table 2.1 in
normalizing distances in the correlation coefficient profiles presented
in Section 4.3, the Reynolds numbers had to be matched. This section
presents data relevant to the accuracy at which this Reynolds number

matching was accomplished.

Figure 4.2 presents mean velocity profile data acquired from the
moving probe of the Ay separation experiments. In this figure, the data
from these correlation experiments (solid symbols) were normalized using
the friction velocities from Table 2.1, and are overlaid on the single
point velocity profile data of Figure 2.2 (open symbols). As one can see

(except for a few points) in terms of mean velocity the Ay correlation

130

experiments matched Reynolds number with the single point experiments

quite well.

Assessment of the degree to which the correlation experiments
matched the Reynolds number of the single point experiments was also
performed by comparing the rms spanwise vorticity profiles. Figure 4.3
shows the moving probe data from the Ay separation experiments overlaid

on the R9 - 1,010 and 2,870 w ' profiles of Figure 2.14. (Figure 2.4

2
shows some of the Ay correlation experiment stationary point uwz'/uf2
data.) Figures 4.4 through 4.8 present comparisons between the wz' data
of the A2 separation experiments and the relevant data of Figure 2.14. In
all of these figures the correlation experiment data are normalized by
the friction velocities given in Table 2.1. The comparison in Figure 4.3
with the single point profile data provides further evidence that these
correlation experiments had about the same Reynolds numbers as the
profiles of Figure 2.14. For the most part, the same can also be said for
the A2 separation data shown Figures 4.4 through 4.8. Note however, that
the data in Figure 4.6 indicate that two attempts of the stationary probe

at y+ 3 30 experiment were made at R0 3 1,010, and both failed to

identically match the Reynolds number.

Other useful information can also be extracted from Figures 4.4
through 4.8. By comparing Figures 4.4, 4.5 and 4.6 with Figures 4.7 and
4.8 one can see that, in general, the point-to-point scatter and the
differences with the single point profiles are less in the higher
Reynolds number experiments. This indicates that it is relatively easier
to match Reynolds numbers as R0 increases. Note further that in general
the stationary probe in all of these figures tended to give slightly

higher values than the moving probe. These differences are presently felt

131

to be indicative of the probe-specific biases associated with the

measurement of wz'.

In general, the above results indicate that the Reynolds numbers of
the correlation experiments were nominally the same as in the single
probe experiments. Therefore, in the data presentation of Section 4.3 the
friction velocities of Table 2.1 are used in normalizing the distances

from the wall and the probe separations.
4.2.3 0n Interpreting Spanwise Vorticity Correlations

The data presentations in the following section feature spanwise
vorticity correlation coefficients that are derived from the fluctuating
”z signals of the two probes. Unlike the streamwise and normal
components, in a boundary layer the spanwise vorticity component has a
significant non-zero mean value, Oz 3 ~8U/ay. Therefore in order to
interpret two point ”z correlations in the context of instantaneous
motions one must take into account the mean vorticity at any given

location.

Figure 4.9 shows the ratio of the rms spanwise vorticity as
measured in the R9 - 1,010 boundary layer to the mean vorticity as given
by the formula of Van Driest (1956) and shown in Figure A2.3.l. As can be
seen, for y+ less than about 20 the mean spanwise vorticity is greater
than the rms. Thus, near the wall the presence of positive vorticity in
an absolute sense indicates the presence of positive fluctuations greater
in absolute value than the mean vorticity at that location. For y+ values
greater than 20 the rms is greater than the mean. For large enough
distances from the wall (in the outer region) the fluctuations dominate

the mean.

132

The Reynolds decomposition leaves both the mean and fluctuating
vorticity divergenceless. From an instantaneous physics point of view
however, it is easier to understand and interpret vorticity in terms of a
total (mean + fluctuating) quantity. This is because the geometry of
"real" (as opposed to Reynolds decomposed) vortical motions in a flow
depends on the absolute value of the vorticity. From a statistical point
of view, the description in terms of the vorticity fluctuations alone
seems most desirable. This is because from this viewpoint one gains an
indication of the average spatial extent of the time dependent vortical
motions (note that the mean vorticity is indefinitely correlated).
Undoubtedly, the most complete understanding is gained by examining the
data from all points of view, while the correct interpretation makes

physical sense regardless of the point of view.

4.3 TWO POINT SPANWISE VORTICITY CORRELATIONS

This section presents the two point fluctuating spanwise vorticity
correlations listed in Table 4.1. As stated in the previous section these
correlations were run at the nominal Reynolds numbers of R0 -l,010 and
2,870. Since all of these correlations are for probe positions within the
inner region, only an inner variable normalization of probe separations

is given. Also in examining these data it is very important to remember

that, as with all self-correlated variables, the nggmglizgg value of the
correlation is 1.9 for zero separation.

4.3.1 Probe Separations Normal to the Wall

'Figure 4.10 presents the results of the Ay separation experiments

for the stationary probe positioned at y+ 3 100. As can be seen, all of

133

the lower Reynolds number data in this figure exhibit significant
positive valuesz. For a probe separation of Ay+ 3 14.5 the R9 - 1,010
correlation coefficient is about 0.41. For greater probe separations
however, the correlation decreases quite rapidly. At Ay+ 3 60 the

correlation coefficient has decreased to about 0.12.

All of the higher Reynolds number data in Figure 4.10 are
essentially zero; a result that is in striking contrast to R0 - 1,010
data of this figure. Using the correlation as a rough measure of the

average scale of organized w containing motions, one might conclude that

2
these motions become smaller relative to an inner normalization with
increasing Reynolds number. Caution should be taken in making such
conclusions however, since other factors, such as the range of spatial
orientations a given vortical motion is likely to acquire at a given
Reynolds number, could also contribute to the observed zero correlation
values. On the other hand, it is probably safe to say that none of the

data of Figure 4.10 support the (on average) existence of large scale

spanwise vorticity containing motions in the region 100 S y+ S 200.

Figure 4.11 presents the results of the Ay separation correlations
for the stationary probe near y+ - 30. The R9 - 1,010 data in this figure
show positive values for probe separations less than about 25 wall layer
units and significant negative values for greater separations. A peak
negative correlation coefficient of -0.l42 for a probe separation of Ay+
3 42 is observed. This rather modest negative value increases in

significance if one considers that at the given probe positions (y+ 3 30

and 72) the ratio of the product of the mean vorticity values to the rms

 

2Actually two other data files were taken at larger Ay+, but later these
files could not be read from the tape.

134

vorticity values is about 0.26. Thus if normalized by the local mean
vorticity values this correlation would reach a value of about -0.56;
giving a fairly strong indication that the most significant contributions
to this correlation probably involve positive total spanwise vorticity at
one of the probes. After the negative peak, the correlation coefficient
decreases to a value of about -0.04 at a probe separation of Ay+ 3 100.
For reference with the data of Figure 4.10 the correlation is equal to

0.1 at a probe separation of Ay+ 3 55.

As in Figure 4.10, the R9 - 2,870 data of Figure 4.11 show

significantly smaller magnitudes than the R0 - 1,010 data for a given

Ay+.

Furthermore, the negative peak in the higher Reynolds number data
has apparently shifted to a smaller Ay+ value. As stated above, caution
should be taken in associating these observations strictly with a change
in scale. Qualitatively, the high and low Reynolds number data in Figure
4.11 show the same trend of a zero crossing to negative values before
eventually becoming zero. In connection with Figure 4.11 it is also worth
noting that in previous measurements at R0 3 4,850 Klewicki and Falco

(1986) found essentially zero correlation in experiments with a

stationary probe at y+ 3 30 and probe separations as close as Ay+ 3 50.

Figure 4.12 presents the results of the Ay separation experiments
for the stationary probe near y+ - 15. The R0 - 1,010 correlation
coefficient data in this figure, which are negative for all measured
values, show an increasingly negative trend for decreasing probe
separations. The trend in these data suggest that at the smallest probe
separation the correlation has yet to reach its negative peak. The value
of the correlation coefficient at the probe separation of Ay+ 3 18 is -

0.325. At a separation of Ay+ 3 65 the correlation coefficient has

135

decreased in magnitude to -0.10. As with the low Reynolds number data of
Figure 4.11, the zero crossing to significantly negative values indicates
the presence of organized opposing sign fluctuating spanwise vorticity

interactions.

In general, the R0 - 2,870 correlation coefficients in Figure 4.12
show much better agreement with the R0 - 1,010 data than in either
Figures 4.10 or 4.11. However, at the smallest Ay+ separations the high
Reynolds number data are still about 50% lower. As with Figure 4.11, the
high and low Reynolds number correlation coefficient profiles are
qualitatively the same, although it is undetermined whether the negative
peaks in the two curves occur at the same Ay+ value. The value of the R0

-2,870 correlation at the closest probe separation of Ay+ 3 38 is -0.11.

Figure 4.13 presents the Ay separation correlation coefficient
profile for the stationary probe positioned at y+ - 7.5 in the R0 - 1,010
boundary layer. As can be seen, this correlation is negative over the
entire range of measured values. At the point of smallest probe
separation the correlation coefficient is about equal to -0.42.
Furthermore, as with the data in Figure 4.12 the trend in this profile
gives no indication that -0.42 is the peak negative value. As the probe
separations increase the correlation coefficients decrease quite rapidly.
At a Ay+ separation of about 55 the correlation coefficient has reduced
in absolute magnitude to 0.1. The correlation coefficient profile in
Figure 4.13 is very similar to that in Figure 4.12 in both shape and

magnitude.

Note that the negative trends in both Figures 4.12 and 4.13 are

unlike the negative trend in the R0 - 1,010 data of Figure 4.11 in that

136

the negative peak value occurs at much smaller Ay+ separations. Given
that the correlation is 1.0 at zero probe separation, this result
indicates that the region of positive correlation (for probe separations
smaller than those which are experimentally attainable) also becomes
smaller. Given a process such as that proposed by Falco (1983, 1987) in
which vortical fluid is initially lifted from the sublayer through an
interaction with positive spanwise vorticity containing motions external
to the sublayer, it is tempting to interpret the correlations of Figures
4.12 and 4.13 as being representative of the interaction between the
motions arising from the sublayer and the (+)wz motion, and to then
interpret the correlation of Figure 4.11 as being representative of the
average structure of the initiating positive spanwise vorticity

containing motion.

The strong negative correlation for small probe separations in
Figures 4.12 and 4.13 indicates the average existence of closely spaced
positive and negative spanwise vorticity fluctuations in the near-wall
region. In order to further understand the contributions to this strong
negative correlation, a two dimensional probability distribution was
constructed from the individual fluctuating ”2 records that generate the
data point of closest separation in Figure 4.13. Figure 4.14 shows this
probability distribution in relation to the coordinates of the
fluctuating signals from the upper and lower probes, as well as in

coordinates relevant to the absolute sign and magnitude of these signals.

Relative to the fluctuating spanwise vorticity coordinates, Figure
4.14 clearly shows that the major contributions to the negative
correlation coefficient come from fourth quadrant-type motions. These

motions feature positive sign fluctuating ”2 being measured at the upper

137

probe, in conjunction with negative sign fluctuating ”2 being measured at
the lower probe. In a relative sense, this was the type of motion
identified in Chapter 3 as being most effective in producing the vortex

lift force on vortical sublayer fluid.

Note that in the fluctuating coordinates all quadrants contain
probability contours. This fact might lead one to physically questionable
interpretations regarding the nature of the vortical motions involved.
For example, contours in quadrant 2 readily lead to the interpretation of
counter-rotating ”2 containing motions with the positive sign vorticity
occurring nearer the wall. When observed relative to the absolute
coordinates of Figure 4.14 however, this interpretation is clearly shown
to be incorrect since in an absolute sense positive vorticity is
essentially never seen at y+ - 7.5. This finding of a unidirectional “z
vorticity component (in an absolute sense) near the wall further supports

the proposed sublayer vortex sheet model introduced in Chapter 1.

Figure 4.14 also gives important information pertinent to the
presence of positive sign spanwise vorticity containing motions in the
near-wall region. As can be seen, in absolute coordinates the wz-wz
probability distribution is contained entirely in quadrants 3 and 4. As
discussed above, in an absolute sense the lower probe measured only
negative spanwise vorticity (i.e. of the same sign as the mean). The
upper probe data however, show a significant number of probability
contours that are positive in an absolute sense, and that approach very
near to the peak of the distribution. This figure shows that the

probability represented by wz-w combinations in which positive w

2 2

containing motions are at the upper probe is approximately 1/4 the total

probability. Furthermore, the shape of the probability distribution

138

indicates that the presence of positive vorticity at the upper probe is
most likely to be coupled with negative vorticity at the lower probe that
is of magnitude greater than the local mean. This observation is readily
explained by a physical process in which a positive wz containing motion
(possibly embedded in a larger scale high speed front) convects above the
sublayer. Under this process, the lower probe measures a positive
velocity fluctuation commensurate with the presence of the positive wz
containing motion above the sublayer. Through the no-slip condition at
the wall, this results in a negative ”2 fluctuation. The upper probe, due
to its intersection with the convecting vortical motion measures a
positive ”2 fluctuation greater in magnitude than the local mean. Recall
from Figure 4.9 that at y+ - 20, wz' 3 Oz. The interaction between the

sublayer vorticity and the convecting vortical motion would probably also

serve to intensify the respective w in these interacting motions. It is

2
worth recalling that this type of interaction is precisely the mechanism
identified as being most effective in producing a local vortex force in

the positive y direction on the sublayer w containing motions.

Z

4.3.2 Spanwise Probe Separations

Figure 4.15 presents the results of the Az probe separation
experiments at y+ 3 30. The results of the two low Reynolds number runs
are in excellent agreements. As can be seen, all of the non-zero
correlations in this figure are positive. Furthermore, unlike the Ay
probe separation results, the high and low Reynolds number data in this
figure agree quite well. The peak value of the low Reynolds number

correlation coefficient is about 0.31 at a probe separation of Az+ 3

 

3Recall that in both of these experiments the Reynolds numbers are about
the same, but are slightly less than R9 - 1,010.

139

17.3. At a probe separation of Az+ 3 30 this correlation decreases to
0.1. The R9 - 2,870 correlation coefficient of Figure 4.15 reaches a peak

value of about 0.075 at a probe separation of Az+ 3 43.

Figure 4.16 presents the A2 probe separation experimental results
for y+ - 15.3 at R0 - 1,010 and y+ - 20.4 at R0 - 2,870. The results in
this figure are very similar to those above in that the correlations are
either positive or essentially zero. Furthermore, as in Figure 4.15, the
agreement between the high and low Reynolds number correlation
coefficients is good. The R0 - 1,010 correlation coefficient attains a
value of 0.39 at a probe separation of Az+ 3 17.5. At the same Reynolds
number the normalized correlation has reduced to 0.1 at probe separation
of Az+ 3 40. The R9 - 2,870 correlation coefficient attains a value of

0.1 at Az+ 3 43.

Like the correlations in Figures 4.15 and 4.16, the R0 - 1,010 Az
probe separation results at y+ - 6.7 in Figure 4.17 are either positive
or essentially zero. Furthermore, consistent with the trend between the
correlations at y+ 3 15 and 30, the results in Figure 4.17 give still
higher correlation coefficient values for the same Az+ separation. This
trend indicates that on average the spanwise extent of correlated
instantaneous ”2 containing motions increases as the wall is approached.
The correlation coefficient of this figure is about 0.62 at a probe
separation of Az+ 3 17. The value of this correlation is 0.1 at a probe

separation of Az+ 3 47.

140

4.4 DISCUSSION AND CONCLUSIONS - RELATIVE TO CHAPTER 4

In general, the correlations with probe separation normal to the
wall indicate the increased significance of opposing sign spanwise
vorticity interactions with decreasing distance from the wall.
Furthermore, by comparing the correlations in which the stationary probe
was at y+ 3 7.5 and 15 with those at y+ 3 30, it is evident that the
position of the negative peak in the correlations nearer to the wall
occurs at a probe separation about three times as small as in the
correlation profile in which the stationary probe was at y+ 3 30. Thus
nearer the wall the average positive and negative fluctuating spanwise
vorticity interactions become more closely spaced. This is probably
indicative of the motions responsible for the interactions nearer the
wall (say y+ S 20) being of a different nature than those occurring away

from the wall (say y+ > 20).

All of the A2 separation correlations were positive over the range
of y+ values explored. Thus it is safe to say that the average spanwise
spatial structure of ”2 does not feature counter-rotating motions; from
either a fluctuating or total wz point of view. Furthermore, for a given
probe separation, generally higher magnitude correlation values were
measured in the A2 separation experiments than in the Ay separation
experiments. On average this indicates a greater spatial coherence in the
spanwise direction. As discussed below, this may be associated with a
smaller number of probable spatial orientations that organized ”2
containing motions achieve in the spanwise direction as opposed to in the

direction normal to the wall.

One may use the correlations as a rough measure of the average

141

scale of the instantaneous spanwise vorticity containing motions at a
given Reynolds number. In connection with this, the Ay separation
correlations suggest that while the average nature of the motions
contributing to the correlation changes quite dramatically with distance
from the wall, the spatial extent of these motions normal to the wall is
insensitive to the distance from the wall. This can be seen by examining
the position in each R0 - 1,010 correlation coefficient curve at which
the magnitude has (presumably) permanently decreased to less than about
0.1. For the stationary probe positions of y+ 3 7.5, 15, 30, and 100, the
Ay+ separations corresponding to this point are 55, 65, 55 and about 60.

In connection with the spanwise extent of w containing motions, the A2

2
separation results suggest that while the average nature of the motions
contributing to the correlation changes little with distance from wall,
the average spanwise extent of these motions decreases with increasing
distance from the wall. Once again, using the probe separation at which
the magnitude of the R9 - 1,010 correlation coefficient has (presumably)
permanently decreased to about 0.1, the corresponding Az+ values at y... 3
6.7, 15 and 30 are 47, 40 and 30 respectively. Thus between y+ - 6.7 and
30 the average spatial extent of the spanwise correlation decreases in
scale by about 36%. This trend may plausibly be due to either changes in
the average scale or orientation of contributing motions. In general
however, this trend supports the notion that the sublayer vorticity
distribution has a sheet-like character relative to the ”2 containing
motions away from the wall. Note that none of the correlation data

presented suggest the average existence, in the wall region, of large

scale motions that contain spanwise vorticity.

The observed Reynolds number dependence in the Ay separation

142

correlations is dramatic. The obvious trend in these results is for the
higher Reynolds number correlations to become zero at much smaller Ay+
values. Furthermore, this trend is more apparent for the results in which
the stationary probe is further away from the wall. In presenting these
results it was noted that caution should be taken in associating the more
rapid decrease in the correlation strictly with a Reynolds number

dependence in the scale of the contributing motions.

In Chapter 3 evidence was presented suggesting that the motions
most responsible for the <vwz> correlation increase in scale under an
inner variable normalization with increasing Reynolds number. The present
Ay correlations seem to contradict this result. However, in Chapter 3
evidence was also presented suggesting that the motions most responsible
for the <vwz> correlation decrease in scale under an outer variable
normalization with increasing Reynolds number. Assuming this to be the
case for the vortical motions represented in the Ay correlations, one may
conclude that these motions are smaller relative to the total width of
the flow with increasing Reynolds number. If one associates these
correlations with the presence of small to intermediate scale motions
convecting throughout the inner and outer regions, the above apparent
discrepancy may be resolved. That is, since these motions are smaller
with respect to the total width of the flow at higher Reynolds number,
then at higher Reynolds number they are probably more apt to take on a
greater number of orientations in space. Thus the observed Reynolds
number dependence in the Ay separation correlations may be due to an
increasing number of probable spatial orientations in the x-y plane that
the motions contributing to these correlations may acquire with

increasing Reynolds number. It is easily envisioned how a correlation

143

measured by a pair of stationary probes could be attenuated through such

a Reynolds number dependence.

The trends in the present correlations tend to support this
suggestion. In the case of the Ay probe separation experiments, with
increasing Reynolds number zero correlations occurred at smaller Ay+ as
the distance between the stationary probe and the wall increased. If one
thinks in terms of the possible orientations that local spanwise
vorticity interactions (eddies) can assume per total flow volume, then
the degradation of these correlations as a function of Reynolds number
appears consistent with the indicated trends, as well as the deduced
changes in scale (relative to 6) for the <sz> producing motions.
Furthermore, it was shown in Figure 4.14 that as a probe is positioned
nearer the sublayer the probability of it measuring positive vorticity
(in an absolute sense) is greatly diminished. Given that one of the two
probes is in this region, this fact probably represents a constraint on
the types of interactions that can lead to a negative correlation
coefficient at any Reynolds number. Thus one would expect correlations at
different Reynolds number to come into better agreement as the stationary
probe is brought closer to the wall. This notion is also consistent with
the results in Chapter 3 suggesting the universality of the average wz

content of the initiating mechanisms leading to the ejection of vortical

sublayer fluid.

The A2 separation results provide evidence that it is only the

average orientation of the w containing motions in the x-y plane that

2
may be Reynolds number dependent. In these correlations, generally good
agreement was found between the high and low Reynolds number data and, if

anything, slightly higher correlation values for a given Az+ probe

144

separation were found at R0 = 2,870. Thus it appears that the average

spanwise spatial structure of the w containing motions in the near-wall

2
region is essentially Reynolds number invariant when normalized by inner

variables.

The following major conclusions are drawn from the two point

fluctuating spanwise vorticity correlation experiment results.

. In an average sense, no evidence was found that supports the
existence in the inner region of organized large scale motions that

contain spanwise vorticity.

. For the stationary probe positioned at a y+ value less than about
30, negative correlation coefficients are observed in the Ay probe
separation results. The peak in this negative correlation coefficient
increases in magnitude and shifts to smaller probe separations as the

stationary probe is brought closer to the wall.

. For probes positioned at y+ - 7.5 and 22 in the R9 - 1,010
boundary layer, examination of the two dimensional wz-wz probability
distribution indicated the statistically significant occurrence of
positive vorticity (of sign opposite the mean) at y+ - 22 coupled with

negative vorticity at y+ - 7.5 of magnitude greater than the local mean.

. From the same two dimensional wz-wz probability distribution, it
was shown that at y+ - 7.5 the probability of measuring positive

vorticity is essentially zero.

. The A2 probe separation experiments resulted in positive

correlations throughout the near-wall region.

145

. The A2 probe separation experiment correlation coefficients
decreased in magnitude for a given Az+ as the distance from the wall

increased. The Ay probe separation experiment correlation coefficients

did not.

. The Ay probe separation experiment correlation coefficients
showed a significant decrease in magnitude for a given Ay+ as the

Reynolds number increased. The Az probe separation experiment correlation

coefficients did not.

CHAPTER 5

CONCLUSIONS AND PROSPECTUS

5.1 CONCLUSIONS

Very good resolution four wire spanwise vorticity probe
measurements have been made in equilibrium zero pressure gradient
turbulent boundary layers over a Reynolds number range 1,010 S.R9 S
4,850. In Chapter 2 issues relating to the accuracy of these measurements
were examined and discussed in detail. In Chapter 3 single point
measurements were presented and interpreted in a context pertinent to
both the active/inactive motions decomposition of Townsend (1961), and an
interaction between the motions of the inner and outer regions. In
Chapter 4 two point spanwise vorticity correlation results were presented
and discussed in the context of an inner/outer interaction. In this
chapter the results of the previous chapters are discussed in relation to

the hypotheses presented in Chapter 1.

In Chapter 1 of this dissertation three hypotheses were introduced.

These hypotheses are listed below.

flypgghggi§_£1; The optimal description/understanding of turbulence

may be gained through the study of the vorticity field.

Hypothg§1§_flz; At some distance away from the wall the local
instantaneous vorticity distributions are predominantly no longer

connected to the sheet-like distributions characteristic of the

146

147

sublayer, but instead take the form of closed loops.

flypgghg§1§_£3; The essential features of the interaction between
the inner and outer region motions can be described in terms of the
interaction between the reconnected and locally three dimensional
vorticity distributions characteristic of the outer region, and the
locally two dimensional (disturbed sheet-like) distributions that

arise out of the sublayer.

Before proceeding on to a discussion of these hypotheses relevant
to the results of this study, some preliminary statements need to be
made. It is recognized that relatively few hypotheses about the nature of
turbulent flows can be undeniably proven. Therefore, in the discussion
below the results will often be posed in the context of being consistent
or inconsistent (explicable or inexplicable) in relation to the given
hypothesis. That is, do these hypotheses represent a useful and accurate

model of physical reality.

5.1.1 Hypothesis #1

Clearly, a given study is not going to prove or disprove the first
hypothesis above. In general however, numerous results from the present
investigation support the notion that through the examination of
vorticity data insights may be gained that are unattainable through

velocity data alone. A specific example will serve as evidence.

Results in Chapter 3 provide important clarifications pertaining to
the validity of Townsend's (1961) hypothesis. Recall that in this

hypothesis Townsend decomposed the motions at any point into an active

148

component responsible for the production and transport of the shear
stress, and an inactive component that does not interact with the active
component or produce/transport appreciable shear stress. Given the inner
region universality of the logarithmic mean velocity profile, Townsend
then hypothesized that the statistical properties of this active
component (again in the inner region) are universal functions of r and y.
Following Hinze (1975), the association of equation 3.2 with this
decomposition allowed for questions concerning the validity of this

hypothesis to be addressed.

The present <uv>/u,2 data (as well as that of Wei (1987) in a
channel) exhibit a clear Reynolds number dependence for y+ > 15.
Townsend's hypothesis predicts that the statistical properties of the
active component should exhibit a Reynolds number dependence consistent
with this. The present v<.vwz>/ur3 and y<vwz>/ur2 results very clearly
indicate that this is not the case. Furthermore, examination of the vw

2

spectra indicated that the most energetic motions associated with vwz
shift to lower frequencies under an inner variable normalization, and
shift to higher frequencies under an outer variable normalization with
increasing Reynolds number. These results indicate that the motions
responsible for the transport of <uv> undergo either changes in scale
and/or convection velocity with Reynolds number. If one assumes that
vortical eddies are responsible for stress transport and production, then
the present results indicating the near-wall universality of the
statistical properties of the spanwise vorticity field (up to kurtosis)
suggest that even though the average vortical content of these motions is

invariant, their stress transport signatures are not. The observed

Reynolds number dependence in the wz-wz correlations for probe

149

separations normal to the wall suggest that this may be due to either
changes in the average scale or orientation of these eddies in the x-y

plane.

In support of Townsend's hypothesis and as might be expected, for
the present flows the streamwise gradient terms associated with the
inactive motions in equation 3.2 were found to be negligible. However,
examination of the intensities of these gradients indicated that
instantaneously these terms may be quite important. Furthermore, the w -

2

”z correlation results did not support the average existence of large
scale motions in the inner region that contain spanwise vorticity. Also,
the profiles of both of the inner-type non-dimensional functions,
v(vwz)'/<uv>3/2 and y(vwz)'/uf2, showed good agreement across the inner
region at the different Reynolds numbers. This result, coupled with the
lack of agreement between the <vaz>+ profiles, was interpreted to
indicate that the organizational features of the motions responsible for
stress transport are Reynolds number dependent. In relation to the normal
stresses, both the y<tuwz>/ur3 and v(uwz)'/ur3 profiles provided evidence
for inner region universality. Also, in general, the velocity vorticity

correlation data suggest that Townsend's proposed scaling may become

increasingly valid at higher Reynolds numbers.

The above results give an example of how the examination of
vorticity data addressed questions posed through the examination of
velocity data. In this instance, issues relating to the Reynolds number
dependence of the Reynolds stress profile were approached though the
examination of velocity vorticity correlation data relevant to the
transport of the stress. As a corollary, it is also interesting to note

that Wei (1987) hypothesized that the non-universality of the inner

150

region velocity intensities and Reynolds stress profiles is due to a
Reynolds number dependence in the creation of streamwise vorticity. The
obvious means by which to address this hypothesis would also be through
the examination of vorticity data. Given the limited spatial extent of
coherent motions, descriptions in terms of vorticity appear to be optimal

since the dynamics associated with vorticity transport are local in

space.
5.1.2 Hypothesis #2

Several results provided direct evidence supporting the on average
existence of approximately two dimensional sheet-like vorticity
distributions very near the wall. As might be expected however, only
indirect evidence (subject to interpretation) was provided supporting the
existence of highly three dimensional reconnected vortical motions away

from the wall. These results will now be summarized.

The two dimensional probability contour map of Figure 4.14 provided
strong evidence indicating that instantaneously the vorticity vector at
y+ - 7.5 always has a spanwise vorticity component of the same sign as
the mean vorticity vector. Furthermore, the simple analysis in Section
4.1 indicated that for y+ S 10 the average instantaneous vorticity field
is highly oriented in the spanwise direction. These two results are seen
to provide direct evidence in favor of two dimensional sheet-like
vorticity distributions existing near the wall. Furthermore, the trend in
the wz-wz correlations for spanwise probe separations indicated a
decrease in the scale of the correlation of about 361 as the distance

from the wall increased from y+ - 6.7 to about 34. This provides evidence

supporting the hypothesis that, on average, regardless of the actual

151

scale of the proposed two dimensional sheet-like vorticity distributions,
in a relative sense they will always appear as such in comparison with

the spanwise vorticity containing motions away from the wall.

The S(wz) profiles exhibited a distinct negative peak near y+ - 40,
and then decreased in magnitude for greater distances from the wall. The
R0 — 1,010 K(wz) profile increased in magnitude with increasing distance
from the wall out to near y+ - 40, and then decreased for greater
distances from the wall. However, the higher Reynolds number K(wz)
profiles continued to increase out to y+ 3 100, and for greater distances
from the wall showed only a slight decrease. These results, for y+ < 40,
were shown to be consistent with an average physical process in which
negative spanwise vorticity containing motions arise from the sublayer.
Given the evidence of previous studies, these are probably in the form of

hairpin vortex-like and/or lifted vortex sheet-like motions.

The fact that the higher Reynolds number kurtosis profiles do not
decrease for y+ > 40 coupled with the rapidly decreasing significance of
the mean vorticity for increasing y+, provides evidence for the
intermittent existence of positive spanwise vorticity containing motions
in the near-wall region. Furthermore, direct evidence supporting the

statistically significant presence of positive sign w containing motions

2
at y... 3 22 was given in the two dimensional probability map of Figure

4.14.

The vorticity intensity results of Balint et al. (1987a) give
strong evidence supporting the existence of highly three dimensional
vortical motions away from the wall. The presence of positive spanwise

vorticity containing motions in the inner region provides indirect

152

evidence for the existence of reconnected vortical motions. This result
is deduced from the fact that regardless of virtually any orientation
that a vortex ring possesses in space, it will have both positive and
negative spanwise vorticity containing sections. In contrast, very
specific orientations of hairpin-like and/or vortex sheet-like motions
are required (namely an inversion) in order for these motions to possess
positive ”2' Furthermore, the evidence of Falco (1974), (1977), (1983),

(1987) also lends strong support for the existence of vortex ring-like

motions in the inner region.

The evidence discussed above suggests that the lifting sublayer
motions reach a maximum relative intensity at about y+ - 40. Furthermore,
the <uwz> data presented in Chapter 3 indicate that this average vortex
force is essentially zero for y+ > 25. Therefore, it is presently
believed that by y+ 3 50 the lifted sublayer-type of motions begin to
lose their coherence and/or evolve into a more (locally) three

dimensional motions.

5.1.3 Hypothesis #3

The major conclusions concerning the third hypothesis come from the

two point wz-wz correlation experiments. These are now summarized.

None of the correlations provided evidence for large scale ml
containing motions in the wall region. Given that the inner/outer
interaction involves spanwise vorticity containing motions, these motions
must therefore be (on average) of small to intermediate scale. As
discussed above, the correlations with spanwise probe separation

decreased in scale with increasing distance from the wall. This result

provides evidence that the vortical motions in the sublayer are

153

relatively two dimensional in relation to the vortical motions above the
wall. The magnitude of the correlations with probe separation normal to
the wall (at R0 - 1,010) were relatively insensitive to the position of
the stationary probe from the wall. This result is consistent with the
notion of organized vortical motions of relatively the same scale
convecting throughout the wall region. Analysis of the Ay correlation
with the stationary probe at y+ - 7.5 indicated the frequent occurrence
of positive sign vorticity fluctuations of magnitude greater than the
local mean vorticity at the upper probe (y+ 3 22) in combination with
negative vorticity fluctuations at the lower probe. Given the above
association of positive sign vorticity with the likelihood of ring-like
motions, this result provides relatively strong but indirect evidence for
the given hypothesis. Furthermore, this result is remarkably consistent
with the numerous studies of Falco concerning the interaction of
convecting vortex ring-like motions with the sublayer. The totality of
these results support Willmarth's suggestion relating the observed
sublayer structure with the presence and interaction with convecting

small scale vortical motions.

In connection with the direct influence of sweeps (presumably
originating in the outer region) on the motions near the wall, the uwz
vortex force results predict that the size of any given lifted sublayer
motion will be essentially the size of the high speed region above the

sublayer. The importance of this plausible mechanism however, has yet to

be fully determined.

154

5.2 PROSPECTUS

Extensive further study is necessary to fully verify the present
hypotheses. Fortunately however, the utility of the existing data base
has not been fully realized. Examination of the wz probability
distributions as a function of distance from the wall should be able to
locate the position in the boundary layer at which positive sign
vorticity first appears. Furthermore, comparison of these p.d.f.s at
different Reynolds number should shed further light on possible Reynolds
number dependencies. Spanwise vorticity spectra are presently being
examined. From the two point measurements, further analysis of the two
dimensional p.d.f.s should go far in clarifying the statistical
importance of particular wz-wz combinations. Specifically, it is of

interest to determine whether the positive Az correlation at y+ 3 30 has

significant positive-positive 0 contributions. Certain space-time

z
correlations may also be of interest. Similar processing of the velocity
and Reynolds stress data from the two point experiments should also yield
important results. In order to gain a better understanding of the average
instantaneous motions associated with, say, high Reynolds stress

production, conditional sampling of both the single point and two point

data has yet to be done.

The optimal verification of the given hypotheses however, would
come from directly observing and quantifying the instantaneous vortical
motions. To approximate this goal, numerous experiments combining two
view flow visualization movies and two point spanwise vorticity
measurements have been performed. In these experiments, which are similar

to those of Falco (1974), Head and Bandyopadhyay (1981), Signor (1982)

155

and Lovett (1982), the boundary layer is seeded at multiple locations
with vaporized mineral oil. This oil vapor is then illuminated by a laser
beam which has been split and spread into sheets perpendicular and
parallel to the wall such that side and plan view slices of the boundary
layer are visible. Also, in order to add increased depth of field to the
plan view, overhead flood lighting was employed. Through the use of
mirrors, high speed two view 35mm movies are then taken. Simultaneous
with the movie and with the probes in its field of view, two point
spanwise vorticity data are acquired'. Also in the field of view of the
camera is a digital clock which is triggered by the data acquisition

equipment, and thus allows the visual and hot-wire data to be correlated.

While the combined data from these experiments are just now being
examined, some preliminary results purely based upon the visual data have
been acquired. These results, which are pertinent to the nature of the
motions lifting from the sublayer, are derived from a 15 second2
realization (movie) at R9 3 1,010. In this movie, the laser sheet
parallel to the wall is centered at y... 3 12, and thus the plan view

motions are most visible at this location.

The goal of this particular visual analysis was to gain information
relating to the nature of the motions arising from the sublayer. To do
this, specific features of the visualized motions were recorded. The

following are a subset of these features.

. The maximum observed streamwise length of the lifted motion as

measured from its intersection point with the sublayer

 

1The optimal overheat ratio for the operation of hot-wires in the oil fog
environment has been previously determined, see Burkhardt (1982).
2This time corresponds to 797uT2/v.

156

. The acute angle the lifted motion makes with the wall when it has

reached a position of y+ - 15

. The position at which the head of the lifted motion is observed

to start to roll-up
The total sample size of the preliminary visual data to be discussed is
27 events. Thus, on average, a lifting sublayer motion was observed about
every 29.5 t+.

Based upon their streamwise length and the angle they made with the
wall, it became evident that two distinctly different motions arise from
the sublayer. The most prevalent of these had an average maximum
streamwise length of about 96x+ (maximum — 122x+, minimum - 50x+), and
made an average acute angle with the wall of about 37 degrees (maximum -
48.9°, minimum - 28.5'). Of the 27 observations, 19 were of this type of
motion. Thus on average this type of motion was observed every 42t+. It
is worth noting that in 12 out of the 19 observations, apparent ring-like
motions (in an orientation indicating that positive spanwise vorticity
was brought near the wall) were seen to be above the lifted sublayer
motions during some part of their development. The remaining 8 observed
lifted motions had an average maximum streamwise extent of about 217x+
(maximum - 254x+, minimum - 188x+). These motions, which had an average
frequency of occurrence of about once every 100t+, made an average acute
angle with the wall of about 14.5 degrees (maximum - 19.7', minimum -
ll.5°). During the evolution of this type of motion, a pocket-like motion
in the plan view or evidence of an initiation by or interaction with a
ring-like motion was never observed. A distinct difference in the
position at which the two types of motions began to roll-up was not
observed. The overall average position at which both types of motions

were seen to begin to roll-up was y+ 3 32.7 (maximum: y+ - 49.0, minimum:

157

y+ - 16.9). The sample size for this statistic was reduced to 22 since 5

of the higher angle events lost their coherence prior to roll-up.

Near the center of the side view portion of Figure 5.1a one can see
an example of one the high angle lifted motions. This motion, which
resembles a hairpin vortex, makes an acute angle of about 40 degrees with
the wall. At this point in its evolution its head portion has rolled up
and is centered between y+ - 35 and 40. Above and slightly upstream of
this motion one gets the hint of a vortex ring-like eddy. Further in the
evolution of this hairpin-like motion (see Figure 5.1b), the ring-like
motion can be more clearly seen. In the plan view and underneath the
ring-like motion, one can also clearly see that marker has been moved

away to form a pocket-type motion as described by Falco (1980).

Figure 5.2 shows an example of one of the less prevalent and larger
scale low angle motions. The furthest downstream of these motions, of
which there are apparently two, has just begun to roll-up at a y+
position of about 30. By comparing this figure with Figure 5.1, the large
difference in angles that the two types of lifted motions make with the
wall is easily seen. Also note that, as with all of these types of
motions observed, there is neither the presence of a ring-like motion in
the side view or evidence of a pocket-type motion in the plan view. These
motions have a striking resemblance to the lifted shear layer-like
motions presented by Jimenez et al. (1988). For reference, the probes in

Figures 5.1 and 5.2 are located a y+ - 7.6 and 22.4.

These preliminary results are encouraging with respect to the
validity of the hypotheses given in Chapter 1. The interaction

represented by the visualization in Figure 5.1 lends credence to the

158

notion of small scale convected vortices strongly interacting with the
sublayer. Furthermore, the indicated average position at which the lifted
motions begin to roll-up is consistent with the present interpretations
of the higher order spanwise vorticity statistics. The presence of the
large scale low angle lifting motions in Figure 5.2 probably indicates a
different scale initiating mechanism than in Figure 5.1. In the context
of the uwz vortex force, this may be due the direct influence of high
speed fronts originating in the outer region. Also, in comparison with
the computational interrogation of Jimenez et a1. (1988) it is
interesting to note that they found the larger scale low angle motions to
be most prevalent. The evidence here does not support that finding.
Clearly, further processing of these combined visual and hot-wire data
should clarify many issues relating to the instantaneous motions in the

boundary layer.

TABLES

159

Table 1.1 Summary information pertaining to the spatial resolution of wall-
flow investigations cited in Section 1.4.

Probe scale
in viscous units

Reynolds Probe wire wire
Studies(s) number type length spacing Ro/i+
Corrsin and R9 - 7,900 “x 70 100 113
Kistler (1954)
Kovasznay et al. transitional au/ay 24 8 ---
(1962)
Kovasznay et al. R0 - 3,100 au/ay 25 40-60 124
(1970)
Eckelmann et al. R9 3 400 wy, wz 1.75 1.75 229
(1977)
Kastrinakis R0 3 650 ”x 5 5 130
(1977)
Willmarth and R9 - 11,700 x-array 2.5 2.5 4,680
Bogar (1977)
Falco (1980) R9 — 1,068 “z 3.6 3.6 297
Kastrinakis and R0 3 1,200 ”x 9.1 11.5 132
Eckelmann (1983)
Johnson and R0 3 400 x-array 1.7 1.7 236
Eckelmann (1983)
Willmarth and R9 - 9,840 au/ay 0.35 1.6 28,015
Sharma (1984)
Balint et al. R0 - 2,100 9-wire 2.3 8.9 913
(1987a) probe
Present R0 - 1,010 ”2 1.95 1.95 518
R0 - 2,870 4.76 4.76 603
R0 - 4,850 7.82 7.82 620

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163

Table 2.4 Summary information concerning other wall-flow investigation data

Note that R9 indicates boundary
layer flow. and Rd 2 indicates channel flow. See Table 2.3 for information

pertaining to the present probe dimensions.

presented in Chapter 2 (except Figure 2.8a,b).

Figure(s)

Studies(s) [2.*]
Ueda and 5,6,7
Hinze (1975) 5.6.7
Andreopoulos ' 5,6
et al. (1984) 5.6
Purtell et al. 5.6
(1981) 5,6

5.6

5.6

5.6
Wallace et a1. 7
(1977)
Johansson and 5

Alfredsson (1983) 6

Ligrani and 5

Bradshaw (1987)

Wei (1987) 5.6
5.6.18.19

4,5

Balint et al. 14.15.22

(1987a)

Kim et al. 14

(1987)

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Ro-3 .480
R,-5,1oo
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R052 9 620
R -2 970
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N. 8.

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type

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single
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single
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x-array

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6.7

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165

Table 3.1 Summary information concerning other wall-flow investigation.data
presented and referenced in Chapter 3. Note that R, indicates boundary layer

flow. and Rd/Z indicates channel flow. See Table 2.3 for information
pertaining to the present probe dimensions.

Probe
Figure(s) Reynolds Probe scale in
Studies(s) [3.*] Symbol number type(s) viscous
units
Alfredsson 6b.7b A» R032.800 SWP 8.0
et al. (1988) l.6b.7b ‘V Rd/2-3,850 SFP 32.0
Andreopoulos 2a,3.8 C) Ro-3,624 SWP 20.9
'et a1. (1984) 6b.7b C) Ro-3.624 TWS 320.9
Balint et a1. l3b,14.15 3’ Ro-2,100 9WP 9.5
(1987a)
Barlow and' 6b.7b e 39-1.140 LDA. Ay+3l.5
Johnston (1985) Az+33.0
Erm et al.(l985) --- 6175R0SS.010 SWP 24-51
Gupta and 8.11 m Ro-l.900 XWP 3.9
Kaplan (1972) ll 0 Ro-6.SOO XWP 14.2
Kim et al. 6b.ll.l3b )( Rd/2-3.300 DNS *
(1987)
Kreplin and 6b X Rd/2-3.850 SFP 32.0
Eckelmann (1979)
Ligrani and 1.6b.7b V Ro-2.620 SWP 3.3
Bradshaw (1987)
Murlis et a1. --- 791SR9S4,750 XWP 25~30
(1982)
Perry and ’ --- 40,0005Rd/25130.000 XWP 36-140
Abell (1975)
Purtell et al. 1 X Ro-l,340 SWP 8.2
(1981)
Spalart (1988) 13b +- 3OOSRosl.410 DNS #
Wei (1987) 2b.5,ll A Rd/2-2.970 LDA 0.66
2b.3,5.6b,7b.9.11.12 E] Rd/2-14.9l4 2.76
5 0 Rd/2-22,776 3.94
2b 0 Rd/2-39.580 6.43

SWP: single wire probe. SFP: single film probe
XWP: x-wire probe, XFP: x-film probe

TWS: temperature wake sensor

LDA: laser Doppler anemometer

DNS: direct Navier-Stokes simulation
9WP: nine wire probe '

* Ay+m1n a 0.05. Ay+hax a 4.4. Az+ a 7.0, Ax+ a 12.0

# As+ 3 6.7. AxI 3 20.0, for Ay+ information see Chapter 2.1

166

Table 4.1 Summary of two point wz-wz correlation experiments.

A: Experiments with probe separations in the direction normal to the wall.

Stationary Probe Positions
Fig. 4.10 Fig. 4.11 Fig. 4.12 Fig 4.13
+
R, — 1,010 y+-100.5 y -29.s y+-13.4 y+=7.5

R, - 2,370 y+-102.7 y+-32.4 y+-l6.2

B: Experiments with probe separations in the spanwise direction.

Stationary Probe Positions
Fig. 4.15 Fig. 4.16 Fig. 4.17
R, - 1,010 y+333.5 (2) y+-15.3 y+-6.7

R, - 2.870 y+-33.6 y+-20.4

FIGURES

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a) Inner normalization
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W

250

APPENDIX 2.1 ON COMPUTING STREAMWISE DERIVATIVES

In their study Subramanian et al. (1985) derive the following
expression for a streamwise derivative with cu percent error in the

measured values of u.
<(8u/6x)2> = <Au/Ax>2 + <2ueu>/Ax2 + <O(eu)2>

This expression shows that provided the Eu are correlated with u the
error in (Bu/8x)2 as computed by a finite difference approach becomes
quite significant as the equivalent Ax under Taylor's hypothesis becomes
small. Subramanian et a1. further show, using grid flow measurements,
that the minimum Ax for which the above error term is negligible is E 3.8
mm (for the given cu of their experiment). In this appendix a
quantitative example is given showing that the least squares method of
computing streamwise derivatives of the present study is superior to a
finite difference approach in that it allows for a much smaller Ax
without sacrificing accuracy, and is more robust in the presence of
noise.

In this example estimates of <(6u/6x)2> as computed by both methods
are compared for various Ax separations and noise levels for a known
function u with correlated errors Gu' For the function we choose u =
cosan + cu with cu - AUu, where U represents random selections from
uniformly distributed numbers between 0 and l, and A is small. Using this

function, the "true" mean square derivative is:

T
<(6u/8x)2> =(412/T) sin22wxdx - 212. (A2.1.l)
o

The comparisons in Figures A2.1.1 and A2.1.2 are made between the

251

central difference approximation

(flu/6x)3 = (-u5 + 8u4 - 8u2 + u1)/(12Ax) + 0(Ax2) (A2.1.2)

and the derivative of a local 5 point least squares fit to the model

u = x + Yx + 2x2

(Bu/6x)3 = Y + 22x3, (A2.1.3)
where the subscripts in the above relations refer to the point within the
5 point window at which the derivative is evaluated (i.e. point 3 is the
center point). The comparisons presented are representative of the
phenomena observed for numerous reasonable Ax separations and noise
levels. Figure A2.1.l shows the effect of decreasing Ax for a given noise
level and Figure A2.1.2 shows the effect of increasing the noise level
for a given Ax. In both of these figures the straight line indicates the
"true" value of 212 g 19.74. Figure A2.1.1, for small Ax, shows results
similar to that of the grid flow data of Subramanian et al. If these
results are representative of what happens in turbulence data reduction,
then by using the least squares method one can expect to be able to
decrease the minimum allowable Ax approximately 2.5 times the value
permitted by the finite difference method. In any case, under the
conditions presented in Figures A2.1.l and A2.1.2, the least squares

method clearly out-performs the finite difference method.

252

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254

APPENDIX 2.2 DATA SMOOTHING PROCEDURE

As discussed in Chapter 2.4.5 a point-to-point check of all of the
velocity records was made prior to computing the velocity derivatives and
wz. This procedure was used to reduce the singular-like phenomena that
spurious isolated data points could have on derivative quantities. This

testing procedure and the corrections used once a spurious datum is

found, are described in this appendix.

The voltage to velocity conversion process was broken up into a 100
point buffering procedure in which the hot-wire voltages are initially
converted via the calibration equations into uncorrected velocities. An
rms is then computed based upon this 100 point buffer of velocities. For
a given file conversion the user selects the number of these rms levels
to be used as the smoothing criterion. Selecting a greater number of
these rms levels will cause less smoothing. During highly turbulent times
the criterion will accordingly increase, whereas during quiescent times
the criterion decreases. Thus care had to be taken in selecting an
overall criterion (i.e. appropriate for an entire data set) that worked

well in the intermittent regions very near and very far from the wall.

A failure is defined to occur when the absolute difference between
any two adjacent points is greater than the given criterion. Three basic
failure types/corrections were defined. A type 1 failure (isolated point)
occurs when point I+l fails in relation to both points I and 1+2, and
points I and I+2 do not fail. The corrective measure taken for a type 1
failure was to replace point I+l with the average of points I and 1+2. A

type 2 failure (double point) occurs when points I and 1+1 and points I

255

and I+2 fail, but points 1+1 and I+2 do not fail. The corrective measure
in this case was to adjust point l+l such that it is l/2 a criteria from
point I (+ or - such that they maintain their original orientation), and
to move point I+2 such that it maintains its relationship with point I+l.
A type 3 failure (multi-point) occurs when points I and I+1, I and I+2,

and 1+1 and I+2 all fail. The highly arbitrary correction in the case was

to set all of the points equal to the value of point I.

The philosophy behind this smoothing procedure was to eliminate
possible non-flow generated data without removing any information
provided by the turbulence. Using this as a goal, it was assumed that
(since electronic phenomena typically occurs at much higher frequencies
than found naturally in the present turbulent flows) the generic failure
type was the isolated point (type 1). Confidence was gained in this
assumption in that for any given file one could increase the criterion
until virtually all of the detected failures were type 1. Output from the
smoothing subroutine included reference to the failed points, their
original and corrected values and the failure type. By iterating through
numerous files covering the various regions of the boundary layer it was
found that a criterion of three rms levels worked well in recognizing the
maximum number of type 1 failures relative to types 2 and 3 (for the
three single probe experiments described in chapter 2). This optimal
criterion undoubtedly depends upon the given flow, as well as the
particular data acquisition equipment being used. An example of the

smoothing subroutine output is given in Table A2.1.l.

256

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257

APPENDIX 2.3 ON MEASURING THE MEAN VELOCITY GRADIENT

An alternate means of assessing the accuracy of the present
spanwise vorticity probe gradient measurements is to compare the
derivative of the mean velocity profile with the average of the
instantaneous aU/ay measurement deduced from the parallel-array contained
in the wz probe. According to the study of Bottcher and Eckelmann (1985),
probe interference effects may be compensated for in velocity
measurements during the calibration procedure, but in order to obtain
accurate gradient measurements the parallel array must be calibrated in a
gradient field. This conclusion inherently assumes that the construction
of the parallel—array is such that the flow field seen by one wire is

altered by presence of the other wire.

Before proceeding to an examination of the present aU/ay
measurements it is useful to note some of the differences between this
study and that of Bottcher and Eckelmann. The two most important of these
differences have to do with the probe construction. In their study, a TSI
l244-20W parallel array was employed. This hot-film array has 1mm long
films attached to 12.7mm long prongs which are attached to a blunt-end
3.2mm diameter probe body. This results in the ratio of the prong length
to frontal diameter (pertinent to flow blockage) to be equal to about 4.
In contrast, the parallel-array of the present study has 3mm long wires
(in which only the center millimeter is active -- thus serving to isolate
the active sensor from the prongs) supported on 20mm long prongs which
are attached to a tapered head with a frontal diameter (again pertinent

to flow blockage) of about 2.5mm. This results in 3 prong length to

258

frontal diameter of about 8. It is presently believed that these
differences greatly reduce the effect of blockage effects on the present

6U/6y measurements.

Comparisons were made between aU/ay estimates as deduced from the
mean profile, and those deduced from the average aU/ay of the parallel
array, 6U/6ylp. However, the scatter in the 6U/8y data inferred from
finite differencing the mean velocity profile was great enough that it
made for an inconclusive assessment of the performance of the parallel-
array. Thus it was decided to instead compare 6U/aylp with the semi-
empirical curve-fit of Van Driest (1956), 3U/8ylv. Note that the use of
other curve-fits such as Spalding (1961) yield essentially identical
results. This comparison for the three Reynolds numbers of the present
study is made in Figure A2.3.l for y+ S 50. As can be seen, in general,
the present results agree quite well with each other, but all of the data

under-predict the semi-empirical based result.

This trend appears consistent with the results of Bottcher and
Eckelmann (in that probe interference effects generally result in an
under-estimation of the gradient -- except for very low probe Reynolds
numbers, see below), and thus suggests that probe interference effects
may have influenced the present measurements. To further examine this
possibility, these differences were analyzed in the same manner suggested
by Figure 7 of Bottcher and Eckelmann. Figure A2.3.2 plots the relative

systematic error, E,

6U/8ylp - 6U/6yIv

aU/Bylv

E:

 

times the wire separation (Ay = 0.98mm) versus the two different probe

259

Reynolds numbers; using the mean velocity, the probe support-rod
diameter, and the tapered-head diameter, Rp and Rh respectively. When the
differences with the Van Driest model are examined in this way, the
trends in the data are seen to be significantly different from those
predicted by the Bottcher and Eckelmann study. In their study they found
that for probe Reynolds numbers greater than about 23 the quantity Ay-E
started out negative and (on average) steadily increased with increasing
probe Reynolds number to a limiting value of about -0.2. As can be seen
for the present two lowest Reynolds number flows (which fall within the
range of probe Reynolds numbers investigated in the Bottcher and
Eckelmann study) this is not the case. In these two flows the lowest
probe Reynolds number data clearly have the smallest (rather than the
largest) error.

In interpreting results presented in the form of Figure A2.3.2 one
must also consider the fact that at y+ - 50 the mean gradient is only
about 52 of the value at the wall. Therefore, any small additive bias
(in terms of ufz/v) has an amplified affect as the distance from the wall
increases. To illustrate this point, consider "correcting" the two lower
Reynolds number profiles such that each value in the distribution is
altered by an additive constant equal to the amount that makes the point
representing the smallest Ay-E values of Figure A2.3.2 have essentially
zero error. For the R9 - 1,010 and 2,870 distributions this means adding
2.75 s.1 and 7.5 s'1 to each point respectively (or about 5.2% and 2.22
of uT2/v respectively). The "corrected" distributions are shown in Figure
A2.3.3. These distributions show that a constant additive error model
does very well in explaining the deviations from the semi-empirical
result (especially for the R0 = 2,870 flow). Figure A2.3.4 further

supports this conclusion in that it shows that the Ay-E values for the R9

260

= 1,010 distribution are about equally spread around zero, and the R0

=2,87O Ay-E curve is nearly a horizontal line.

Thus, while there are differences between the present aU/aylp
distributions and that given by Van Driest, these differences seem to be
best described by an additive systematic error model, not a proportional
error model. This point is important since probe interference errors
should be proportional to value of the mean gradient -- as shown by
Bottcher and Eckelmann. Somewhat unfortunately, this finding also
excludes other proportional error sources such as the measurement of the
wire separation and/or u’. Interestingly enough however, the ratio of the
absolute "corrections" discussed above are nearly proportional to the
friction velocities of the two flows (i.e 2.75/7.5 g “7'101o/uflza7o 5
0.39). This may be a clue regarding the source of the small differences
between the two 8U+/6y+ distributions. At this time it appears that a
plausible explanation might be some small additive bias in the matching
calibration procedure described in Chapter 2. However, it is not certain
that the present results are statistically significant. Furthermore,
given the relatively small absolute "errors" involved, the association of
these "errors" with a single source presents a formidable task. Finally,
it should be noted that very little explicit data on aU/ay is in the
literature, and thus the low Reynolds number deviations from the semi-
empirical equation used for comparison (which is largely derived from
high Reynolds number velocity profile data) may be a true Reynolds number

dependence.

261

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265

APPENDIX 3.1 VELOCITY-VORTICITY CORRELATIONS RELATED TO THE GRADIENTS OF
THE REYNOLDS STRESSES IN PARALLEL TURBULENT WALL FLOWS

This appendix describes, in general, how velocity vorticity
correlations are related to the gradients of the Reynolds stresses in
wall bounded turbulent flows that are (nearly) homogeneous in planes
parallel to the wall. These relations are derived both from a tensor
identity, and directly from the equations for the mean flow. Furthermore,
using the velocity vorticity correlations measured in the present
experiments and approximate analytical relations, unmeasured velocity

vorticity correlation profiles are deduced.

The contents of this appendix have been accepted for publication as

a "Brief Communication" in the Physics of Fluids A.

266

VELOCITY-VORTICITY CORRELATIONS RELATED TO THE GRADIENTS

OF THE REYNOLDS STRESSES IN PARALLEL TURBULENT WALL FLOWS

J. C. Klewicki
Department of Mechanical Engineering

Michigan State University, East Lansing, Michigan, USA.

ABSTRACT

By using the momentum equations in "vorticity" fonn, relations between velocity-
vorticity correlations and gradients of the Reynolds stresses are established for a two
dimensional turbulent channel flow. Employing these relations, the approximate fomtu-
las of Phillips [Phys Fluids 30 1987] and recently obtained experimental data allows
for unmeasured velocity-vorticity correlation profiles to be deduced. The results herein
indicate that the contributions to the gradients of the diagonal stresses are dominated by
the correlation involving the spanwise vorticity component, while the contributions to
the off-diagonal stresses are shared almost equally between the correlations involving

the spanwise and normal vorticity components.

267

It can be shown1 that the gradients of the Reynolds stresses in an incompressi-

ble turbulent flow are related to velocity-vorticity correlations via the following rela-

tion.

a—xj'(UjUi) : —EijkuJ-wk + 3.8—£01191). (1)

In that many investigatorsz‘5 feel that the most promising approach to understanding
turbulent flows is through the study of the vorticity field, Eq. (1) is important because
it relates the mean transport of the Reynolds stresses to the vorticity field through the
indicated velocity-vorticity correlations. By setting i = l and applying Eq. (1) to a fully
developed, plane turbulent channel flow it may be readily shown that the gradient of

the 1-2 component may expressed as

 

—(uv) = w —vcu,, (2)

where u, v and w are the fluctuating velocity components in the streamwise, x, the
direction normal to the wall, y, and the spanwise direCtion, z, and 0),, is the fluctuating
vorticity component in the k direction. Thus Eq. (2) specifically shows how the gra-
dient of the component of the Reynolds stress tensor that appears in the equation for
the mean velocity is related to the fluctuating vorticity field. In this Brief Communica-
tion it is shown that Eq. (2), as well as the relations between the y-gradients of the
diagonal terms of the stress tensor represented by setting i = 2 in Eq. (1), are not only
derivable via Eq (1) but may also be derived directly from the equations for the mean
flow.

This Brief Communication begins by presenting the relevant time-averaged

momentum equations in "velocity" form for a plane turbulent channel flow.

268

Examination of these equations (along with Eq. (2)) indicates that the mean velocity,
U, is dependent on the above velocity-vorticitv correlations. Then by re-deriving the
appropriate momentum equations in "vorticity" form, the same identities involving
velocity-vorticity correlations found by setting i = 2 and 3 in Eq. (1) are deduced. Per-
taining to the i = 1 and i = 2 relations from Eq. (1), W6; and ETD; data are provided
from the recent high resolution measurements of Klewicki and Falco‘s. Then, by using
8(t't_v)/By data, and the approximate asymptotic results of Phillips7 for the gradients of
the pertinent Reynolds stresses, approximate profiles for the unmeasured terms, W):
and W, are deduced. A brief discussion concerning the physical interpretation of
these terms, and their possible Reynolds number dependence follows. A much more
complete discussion of these results, and their relationship to Other structural features
of turbulent wall-flows is given in Klewicki and Falcos.

The pertinent time-averaged x and y momentum equations (the 2 equation is ident-
ically zero) in "velocity" form for a steady, fully developed, incompressible, plane tur-

bulent channel flow can be readily found in many books on fluid mechanics (cf.

0

Arpaci and Larsen9) and are given by

 

a P d dU H
axlp] dy vE-uv] (3)
and
i .I: — ....d— —2

where lower case letters denote fluctuating quantities and upper case letters denOte

mean quantities. Examination of Eq.s (2) and (3) indicates that Eq. (3) may be rewrit-

ten as

269
13 =vfl_(11-111;—v-11,). (5)
3x 9 dy2
Thus Eq. (5) explicitly shows that the equation for the mean velocity depends on the
velocity vorticity—correlations appearing in Eq. (2).

The momentum equation in "vorticity" form is given by,

 

%] + VVZB, (6)

where a tilde denotes a total quantity. By letting it = U + u, 'p = P + p, and
(o = Q + to, expanding Eq. (6) into its component equations, time averaging, and not-
ing that for the given flow that:

o a( )/3x = a( )/82 E 0, except for P (due to homogeneity),

0V=W=Qx=Qy501

0 Q2 5 -dU/dy, and that

. 1/2d/dy(U2) = -Un,,

one arrives at the following equations.

 

_ a P d2U

11 .1.]

ii? "i —2 _ --.?LB

23y“ +v +w)+m5; wmx- aylp] (8)
var—rim;=0 (9)

Examination of Eq. (7) indicates that the "vorticity" form of the momentum equa-

tion contains the relation between the mean velocity and the velocity-vorticity

270

correlations given in Eq. (5). Comparison of Eq.s (8) and (4) reveals that there are
additional terms in the "vorticity" form of the y momentum equation. By subtracting
Eq. (4) from Eq. (8) and setting the difference equal to zero (which it must be since

both forms of the y momentum equation are valid), one gets the i = 2 result from Eq.

(1).

%%(137+§72—72)+1-1-111,—ww, = 0. (10)

 

Finally, Eq. (9) presents a relation between velocity-vorticity correlations that is not
made apparent if one only examines the "velocity" form of the momentum equation.
The “Tog/m3, and wring/m3 profiles of figures 1 and 2 are derived from the four-
wire spanwise vorticity probe measurements of Klewicki and Falco" which V1 ere made
in a very thick zero pressure gradient turbulent boundary layer. In the profiles of figure
2 one will notice the data point at y+ = 1. This data point was derived by placing a u-

measuring probe at y+ = 2 (in a R9 = 1,000 flow, see Klewicki”) and using the rela-

1

tion

VLTOT)‘ 1 _
32(y=0) : 2— %] ’
U1; “1 y—>O

which can be derived via a Taylor series about y = 0. (Note that a similar relation

 

 

incorporating the limiting value of 1713 may be used to deduce the limiting value of
vm/uf, see below). As can be seen, the vVZfii/u,3 profiles show a much greater rela-
tive variation with Reynolds number in the log-region than do the viicTz/u.‘3 profiles.
Also it is readily apparent that the Vim/m3 profiles are about an order of magnitude
larger at their peak value. Near the wall (say for y“ < 20) both figures suggest that
"wall variable" scaling causes a collapse of the data. Also, in both profiles the peak

values are seen to occur near the edge of the sublayer. This should be expected

271

however, since the mean Reynolds stress gradients are largest in this region of the
flow.

In order to use the data of figure 1 and the gradient of the Reynolds stress to
deduce the vWngluT3 profile as indicated by Eq. (2) one must first assess the validity of
the parallel mean flow assumption implied by the form of Eq. (2). This has been done
by Klewicki and Falco“, in which it was demonstrated that the u and v‘ streamwise

gradient terms of the "boundary layer" form of Eq. (2),

_3_— _——_-—— Li's 2_7
ay(uv) — we)y vwz+ 28x(v + u),

are approximately two orders of magnitude smaller than the velocity-vorticity correla-
tion terms for the given flow field. This approximate homogeneity in the streamwise
direction also allows the data of figure 2 to be used in Eq. (10) to determine the
V\V—(Dy-/ll13 profile.

Using Eq. (2), the data of figure 1, and numerically differentiating the correspond-
ing tW/u,2 profiles at each R9 allowed re} the approximate vv—v'tfigluT3 profiles given in
figure 3 to be deduced. As one can see, away from the wall the deduced vm/mi’
profiles are very similar to the \ATab'z/u,3 profiles given in figure 1. The profiles of
figures 1 and 3 indicate that the nearly zero mean transport of uv in the log-region is
due to a balance of contributions rather than all terms being zero. Therefore, altering
these correlations suggests the possibility for modifying momentum transport in the
boundary layer. Nearer the wall the scatter in the thfiyuT3 profiles is probably due to
inaccuracies caused by numerically differentiating the t'W/u,2 profiles. This is felt to be
the case due to the fact that both the [TV/U12 profiles (not shown) and the vim-gm,3
profiles given in figure 1 tend to exhibit a collapse for y+ less than about 20, and thus
so should the profiles of figure 3.

The fact that the profiles of figure 2 show very little variation with Reynolds

272

number allows one to use the approximate high Reynolds number formulas of Phillips7
for the y-gradients of the diagonal Reynolds stress terms of Eq. (10), and the R9 =
1,010 data of figure 2 in order to obtain an approximate high Reynolds number
vWCiX/u,3 profile. The deduced vW‘Gfluf’ profile of figure 4 is considerably different
from the “ESQ/u,3 profiles of figure 2. In general the data shows that \nTvZfijju,3 exhi-
bits less variation across the boundary layer than Vang/11,3. An approximate error esti-
mate on the deduced high Reynolds number “WE/u? profile may be obtained by not-

icing that in this profile the value at y+ = 1 is “=- -0.04. Given that

WTG); 1

(y = 0) = —
11,3 2‘11

 

 

the deduced negative value is clearly shown to be incorrect. Once again, based on the .
data of Klewicki10 the actual value should be about +0.02. Thus an error estimate of =
i- 0.06 is obtained.

The results herein show that the contributions to the y gradient of W are almost
equally shared between Va), and wary, while the contributions to the normal stresses
are dominated by the not)z term. Also, the fact that the measured VETZ/uf profiles
show very little R9 variation indicates that any R9 variation in the 35 and v7 gradients
of Eq. (10) are predominantly due to an R9 variation in the correlation involving (1),.
This is in apparent agreement with the recent hypothesis of Wei”, which associates
the the nonuniversality of v’/u1 with a relative increase in the creation of a)x as the

Reynolds number increases. This hypothesis, as well as other Reynolds number depen-

dent features of boundary layer structure are further examined in Klewicki and Falcog.

ACKNOWLEDGMENT

This work was supported by the Air Force Office of Scientific Research Contract
No. 87-0047, contract monitor Dr. J. McMichael. The author also wants to thank pro-

fessor R. E. Falco for his useful comments concerning this manuscript.

273

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REFERENCES

11. o. Hinze, Turbulence (McGraw-Hill, New York, 1975), 2nd ed.

21. Laufer, Ann. Rev. Fluid Mech. 7, 307 1975.

3W. W. Willmarth, in Adv. in Appl. Mech., Vol. 15, edited by C. S. Yih (Academic
Press, New York, 1975) p. 159.

4]. L. Balint, P. Vukoslavcevic, and J. M. Wallace, in Advances in Turbulence, eds.
G. Comte-Bellot & J. Mathieu, (Springer-Verlag, Berlin, 1987) p. 456.

5J. F. Foss and J. M. Wallace, to appear in
Frontiers in Experimental Fluid Mechanics, edited by M. Gad-el—Hak, (Springer-
Verlag, Berlin, 1989).

61. C. Klewicki and R. E. Falco, Rep. TSL —88-4, Michigan State University, 1988.
7w. R. c. Phillips, Phys. Fluids 30,2354 1987.

81. c. Klewicki and R. E. Falco, to be submitted to J. Fluid Mech..

9V. S. Arpaci and P. S. Larsen, Convection Heat Transfer, (Prentice-Hall, Englewood
Cliffs, N.J., 1984).

101. c. Klewicki, (unpublished) 1988.
“J. c. Klewicki and R. E. Falco, Bull. Am. Phys. Soc. 33, 2286 1988.
12T. Wei, Ph. D. dissertation , University of Michigan, 1987.

278

APPENDIX 3.2 OUTER VARIABLE NORMALIZATIONS

This appendix gives outer variable normalizations of the present
average and rms velocity, Reynolds stress and velocity vorticity
correlation profiles. The presentation of these profiles allows for the
interested reader to compare and contrast the outer variable
normalizations of this appendix with the inner variable normalizations
presented in Chapter 3. Specifically, the presentation of these data
normalizations gives the reader the opportunity to make a more complete
judgement concerning the validity of the conclusions and suggestions

pertaining to flow physics made in Chapter 3.

279

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