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This is to certify that the

dissertation entitled

The Velocity and Vorticity Fields of a Single
Stream Shear Layer

presented by
Scott C. Morris

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Engineering

 

 

 

 

MMrofessor

97690”?
0

Date

 

61%»? 2002,

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

PLACE IN RETURN BOX to remove this checkout from your record.
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DATE DUE

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6/01 cJCIRC/DateDuepGSm. 15

 

THE VELOCITY AND VORTICITY FIELDS OF A SINGLE

STREAM SHEAR LAYER
By

Scott C. Morris

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

2002

ABSTRACT
THE VELOCITY AND VORTICITY FEILDS OF A SINGLE
STREAM SHEAR LAYER
By
Scott C. Morris

Measurements of velocity and vorticity have been acquired in a large scale single
stream shear layer. These data provided information regarding the nature of turbulent fluid
flow at a relatively high Reynolds number. The Reynolds number dependence of phenom-
ena is important given that many technological applications involve Reynolds numbers
that are higher than those which are typically achieved in the laboratory. The use of the
vorticity vector as an alternative variable to the velocity vector provides additional insight
into the nature of these phenomena. Single and multi point measurements of both the
velocity and vorticity have allowed the examination and comparison of the different scales
of motion. The research project was effectively divided into two parts. The first involves
the very near separation region in which a turbulent boundary layer separates at zero pres-
sure gradient at a 90 degree edge. The measurements have shown the existence of a "sub-
shear" layer.. The vorticity that participates in the first instability is found to originate
from the very near wall region. At larger y values. the separated turbulent boundary layer
convects downstream adjacent to the sub-shear layer. "seemingly unaware" of the separa-
tion. The second part of the research involves measurements in the developed region of
the shear layer. Several conclusions have been drawn from these data. For example, it has
been shown that the dimensionless vorticity fluctuations do not scale in a self-similar way,
but increase with Reynolds number. The cause of this is related to the large scale motions
of the flow that lead to nonzero spatial correlations in vorticity over relatively large length

scales.

DEDICATED TO THE MEMORY OF MY FATHER, JOHN E. MORRIS

 

ACKNOWLEDGMENTS

I would like to thank my committee members. Charles Petty, Ahmed Naguib,
Manoochehr Koochesfahani, C.Y. Wang, and John Foss for their contributions and com-
ments on this research. Special thanks goes to my advisor, John Foss, for providing me
with an environment in which creativity and ideas can be explored.

I would also like to thank a number of my colleges whom I have worked with over the
last several years. This includes Dr. Alptekin Aksan. Dr. Paul Hoke, Doug Neal, Richard
Prevost, Al Lawrenz. Matt Maher. and many others who helped with the assembly of the
shear layer facility.

Lastly, I thank my wife Yvonne. and son John. Research would not be worth the trou-

ble if it weren’t for family.

TABLE OF CONTENTS

List of Figures ................................................................................................................... vii
List of Tables ..................................................................................................................... xii
Nomenclature ................................................................................................................... xiii
1 .0 Introduction .............................................................................................................. l
2.0 Experimental Apparatus and Techniques ................................................................. 8
2.1 The Single Stream Shear Layer Facility ...................................................... 8

2.2 Hot-wire Techniques and processing ......................................................... 11

2.2.1 Calibration ..................................................................................... 11

2.2.2 X-wire probe calibration and processing ....................................... 13

2.2.3 Multi-sensor Probe Configurations and Processing ...................... 15

3.0 Turbulent Boundary Layer to Single Stream Shear Layer .................................... 28
3.1 Introduction ................................................................................................ 28

3.2 Upstream Boundary Layer ........................................................................ 31

3.2.1 Boundary Layer - Shear Layer “Communication” ........................ 31

3.2.2 Boundary Layer Statistics Near x=0. ............................................. 33

3.3 Point Statistics And The “Sub-shear Layer” .............................................. 37

3.4 Shear Layer Diagnostics From Entrainment Measurements ..................... 41

3.4.1 Convection Speed of the Coherent Motions .................................. 42

3.4.2 Spectral Properties of the Entrainment Stream .............................. 44

3.4.3 The Initial Instability ..................................................................... 47

3.5 Spanwise Correlation Field of the Near Separation Region ..................... 51

3.5.1 Example realizations in the irrotational stream ............................. 52

3.5.2 Correlation Coefficient Data .......................................................... 54

3.6 Summary and Conclusions ........................................................................ 55

4.0 The Developed Single Stream Shear Layer ........................................................... 88
4.1 Introduction ................................................................................................ 88

4.1.] Motivation and literature survey .................................................... 89

4.1.2 Measurement program ................................................................... 92

4.2 The Velocity Field ..................................................................................... 94

4.2.1 Integral constraints on the mean flow ........................................... 94

4.2.2 Velocity in Self-Similar Coordinates ............................................. 97

4.2.3 Profiles of the velocity gradient variance .................................... 103

4.3 Statistics of the Vorticity Field ................................................................. 109

4.3.1 Examples of Vorticity Time Series .............................................. 109

4.3.2 Histograms of Vorticity ................................................................ 110

4.3.3 Profiles of the Vorticity RMS ...................................................... 1 13

4.4 Analysis of the Reynolds Stress ............................................................... 116

4.5 Turbulent Kinetic Energy ......................................................................... 120
4.5.1 Dissipation Measurements. .......................................................... 121
4.5.2 Measurement of Kinetic Energy Budget ...................................... 127
4.6 The Scales of Turbulent Motion .............................................................. 129
4.6.1 Correlation measurements of velocity and vorticity .................... 129
4.6.2 Spectral representation of velocity and vorticity ......................... 137
4.7 Circulation Density .................................................................................. 143
4.7.1 Experimental Configuration ........................................................ 147
4.7.2 Circulation Density Results ......................................................... 148
4.8 Conclusions .............................................................................................. 150
Appendix A .......................................................................................................... 193
REFERENCES .................................................................................................... 197

vi

LIST OF FIGURES

Figure 1.1 Idealized schematic of the single stream shear layer geometry. ....................... 7
Figure 2.1 Primary flow delivery system .......................................................................... 21
Figure 2.2 Schematic of SSSL facility ............................................................................. 22
Figure 2.3 Geometry of single hot-wire probe ................................................................ 23
Figure 2.4 Schematic of Calibration Facility .................................................................... 24
Figure 2.5 Sample calibration and fit, y=0 degrees. ......................................................... 24
Figure 2.6 Geometry of the X-wire probe ........................................................................ 25

Figure 2.7 Example of the locating algorithm of the intersection point for X-wire
processing. ....................................................................................................... 25

Figure 2.8 Predicted values of angle (7) from re-processed calibration data at

nominal values of 18, 0, and -30 degrees. ...................................................... 25
Figure 2.9 Schematic of spanwise vorticity probe ............................................................ 26

Figure 2.10 Schematic of micro-circulation used for the spanwise vorticity calculation..26

Figure 2.11 Schematic of the streamwise vorticity probe. ............................................... 27
Figure 2.12 Schematic of the multiple X-wire probe. ...................................................... 27
Figure 3.1 Schematic representation of near separation region. ....................................... 63
Figure 3.2 Boundary Layer Mean Velocity. ...................................................................... 63
Figure 3.3 Boundary Layer Velocity RMS ....................................................................... 64
Figure 3.4 Mean and RMS of vorticity at separation. ...................................................... 64
Figure 3.5 Spectrum of vorticity fluctuations at y/5=0.6. ................................................. 65
Figure 3.6 Sample flow visualization of the near separation region. ............................... 65
Figure 3.7 Figure of flow regions; not to scale. ................................................................ 66
Figure 3.8 Mean velocity profiles for 0<x/80< I 00 .......................................................... 66
Figure 3.9 Contour plot of mean velocity, ....................................................................... 67
Figure 3.10 RMS of velocity for 0<x/00<100 .................................................................. 67
Figure 3.11 Contour plot of velocity RMS ...................................................................... 68
Figure 3.12 Streamwise evolution of velocity histogram at constant y/Bo=2.0. .............. 68
Figure 3.13 Momentum thickness vs. x/Go. ...................................................................... 69
Figure 3.14 Maximum velocity gradient vs. x/Go ............................................................. 69
Figure 3.15 Peak value of velocity r.m.s as a function of streamwise location. ............... 70

vii

Figure 3.16 Convection velocity of spanwise motions. ..................................................... 70
Figure 3.17 Fourier transform of all 12 streamwise locations measured. ......................... 71

Figure 3.18 Sample of the autocorrelation function in the entrainment stream,

at x/Go=7. l 7 .................................................................................................... 71
Figure 3.19 Streamwise variation in peak frequency. ..................................................... 72
Figure 3.20 Non-dimensional frequency vs. x/Go ............................................................. 72
Figure 3.21 Time series samples from x/00=0.73. ........................................................... 73
Figure 3.22 Autocorrelations of the time series at x/00=0.73. .......................................... 73
Figure 3.23 Histogram of observed frequencies from short time autocorrelations. ......... 74

Figure 3.24 Comparison between tanh function shown and the velocity data at

x/Go=0.73. ...................................................................................................... 74
Figure 3.25 Schematic of 8 wire rake in shear layer. ........................................................ 75
Figure 3.26 Velocity fluctuations at x/00=3.53. y/Bo=-3.7. .............................................. 76
Figure 3.27 Velocity fluctuations at x/00=12.0, y/00=-5.37 ............................................. 77
Figure 3.28 Velocity fluctuations at x/Goz40.5, y/Goz- 10.9. ............................................ 78
Figure 3.29 Velocity fluctuations at x/Bo=84.2. y/00=—19.6. ............................................ 79
Figure 3.30 Spanwise correlation coefficient of the data shown in Figures 3.26
through 3.29. .................................................................................................... 80
Figure 3.31 Spanwise correlation field at x/Go=3.53. ...................................................... 81
Figure 3.32 Spanwise correlation field at x/Oo=12.0 ....................................................... 82
Figure 3.33 Spanwise correlation field at x/Go=40.6 ........................................................ 83
Figure 3.34 Spanwise correlation field at x/Go=87.2 ........................................................ 84
Figure 3.35 Correlation coefficient at Az/Go=2.08. ........................................................... 85
Figure 3.36 Correlation coefficient at Az/Go=4.16. See Figure 3.31 for description. ...... 86
Figure 3.37 Correlation coefficient at Az/Go=6.24. See Figure 3.31 for description. ...... 86
Figure 3.38 Correlation coefficient at Az/Bo=8.32. See Figure 3.31 for description. ...... 86
Figure 3.39 Correlation coefficient at Az/Oo=14.5. See Figure 3.31 for description. ...... 87
Figure 4.1 Schematic of control volume and coordinate system used in Section 4.1.
Note that the growth rate is not shown to scale. ............................................ 161
Figure 4.2 Illustration of the physical interpretation of 5* in the shear layer ................. 161

Figure 4.3 Schematic of the un-tilted coordinates (left and the tilted coordinates

(right) for use in similarity analysis .............................................................. 162

viii

Figure 4.4 Streamwise mean velocity fm) ..................................................................... 162

Figure 4.5 Lateral mean velocity: gm) in tilted and untilted coordinates. ..................... 163
Figure 4.6 Root mean square of the streamwise velocity fluctuations ........................... 163
Figure 4.7 Root mean square of the lateral velocity fluctuations ................................... 164
Figure 4.8 Root mean square of the spanwise velocity fluctuations .............................. 164

Figure 4.9 Profile of correction factor for streamwise and transverse velocity gradients

(see equations 4.25 and 4.26) ........................................................................ 165
Figure 4.10 Profile of velocity gradient fluctuations: du/dx ............................... i ........... 166
Figure 4.11 Profile of velocity gradient fluctuations: dv/dx ........................................... 166
Figure 4.12 Profile of velocity gradient fluctuations: dw/dx .......................................... 167
Figure 4.13 Profile of velocity gradient fluctuations: du/dy ........................................... 167
Figure 4.14 Profile of velocity gradient fluctuations: du/dz ........................................... 168
Figure 4.15 Profile of velocity gradient fluctuations: dv/dz ........................................... 168
Figure 4.16 Ratio of velocity gradient variances for x/0(,=384. ..................................... 169
Figure 4.17 Ratio of velocity gradient variances for x/0(,=675. ..................................... 169

Figure 4.18 Example of vorticity time series at x/0(,=707, n20. Periods of irrotational

fluid are observed. .......................................................................................... 170
Figure 4.19 Example of vorticity time series at x/0(,=7()7, n20. ..................................... 170
Figure 4.20 Example of vorticity time series at x/0(,=707, n=3.9. .................................. 170
Figure 4.21 Histogram of vorticity from conditions A,B,C, and D; see Table 4.1. ........ 171
Figure 4.22 Comparison of vorticity histogram between points of high and low
interrnittency. ................................................................................................. 17 1
Figure 4.23 Histogram or spanwise vorticity in the atmospheric boundary layer at
Reit;2500. Solid line represents the curve fit given by equation 4.87. ......... 172
Figure 4.24 The RMS of the spanwise and lateral vorticity fluctuations at x/90=384
(Re0=7. l x 104). ............................................................................................... 173
Figure 4.25 The RMS of the spanwise and lateral vorticity fluctuations at x/00=675
(Re6=1 .2x 105). ............................................................................................... 173
Figure 4.26 RMS of streamwise vorticity ........................................................................ 174
Figure 4.27 Measured and calculated Reynolds stress. .................................................. 175
Figure 4.28 Gradient of the Reynolds stress. .................................................................. 175
Figure 4.29 Measured velocity-vorticity correlations. .................................................... 176
Figure 4.30 Measured dissipation from compensated energy spectra: E1 1(k 1) ............. 176

Figure 4.31 Measured dissipation from compensated third order structure function ..... 177

Figure 4.32 Comparison of the profiles of several dissipation estimates at x/80=675. 177

Figure 4.33 The measured budget of turbulent kinetic energy. ....................................... 178
Figure 4.34 Velocity autocorrelations from three 1] locations ........................................ 179
Figure 4.35 The autocorrelations of Figure 4.34show in semilog format ...................... 179
Figure 4.36 Autocorrelations of u,v, and w at x/0(,=675, 11:0. ....................................... 180
Figure 4.37 Cross Correlation between probe (at) located at 11:453. and probe (b)
located at 11:1 .26. ......................................................................................... 180
Figure 4.38 Cross Correlation between probe (at) located at n=3.08. and probe (b)
located at nz-O. 194. ...................................................................................... 181
Figure 4.39 Cross Correlation between probe (a) located at 11:1.76, and probe (b)
located at n=-1.52. ......................................................................................... 181
Figure 4.40 Cross Correlation between probe (a) located at n=.434. and probe (b)
located at n=-2.84. ......................................................................................... 182
Figure 4.41 Cross Correlation between probe (a) located at n=-0.888, and probe (b)
located at n=-4.16. ........................................................................................ 182
Figure 4.42 Spanwise correlation of velocity at x/Bolel. ............................................ 183
Figure 4.43 Spanwise correlation of velocity at x/00=650. ............................................ 183
Figure 4.44 Autocorrelation of spanwise and lateral components of vorticity ............... 184
Figure 4.45 Autocorreltion of spanwise vorticity in semilog format ............................. 184
Figure 4.46 Close view of the zero crossing region of the (02 autocorrelation. .............. 185

Figure 4.47 Spanwise correlation of u and (slat Autocorrelations of (oz and u are

included for comparison ..................................................................................................... 185
Figure 4.48 Correlation between high speed irrotational velocity fluctuations and (02 at

11:0. ................................................................................................................ 186
Figure 4.49 Measured and modeled one dimensional energy spectra: E1 1(k1) ............. 187
Figure 4.50 Measured and modeled one dimensional energy spectra: E22(Kl) ............. 187
Figure 4.51 Measured and modeled one dimensional energy spectra: E33(K1) ............. 188

Figure 4.52 Spectral coherence function H12. Note the Mt] rage is changed from
previous figures. ............................................................................................. 188
Figure 4.53 Autospectral function of vorticity. Model spectrum is from isotropic
relation to E1 1. .............................................................................................. 189

Figure 4.54 Compensated vorticity spectra to show energy contribution to enstrophy

integral. ......................................................................................................... 189

Figure 4.55 Schematic examples of y for: (a) a thin uniform sheet of vorticity, (b) a sinuso-
idally perturbed shear layer, (c) a rolled up shear layer into Gaussian vortices.
The P(y) curves on the right show a likely pdf that would be experimentally re-
covered for each condition with Gaussian noise. ........................................... 190

Figure 4.56 Schematic of showing the finite area of integration for the

calculation ofy .............................................................................................. 191
Figure 4.57 Experimental configuration for 7 measurement .......................................... 191
Figure 4.58 Histogram of measured 7 values. Note that the measured vales have

been corrected by a factor of 1.19 to account for the limited field of

view; see Section 4.7.2. ................................................................................. 192

Figure 4.59 Peak vorticity measured by various researchers. ......................................... 192

xi

 

LIST OF TABLES

Table 4.1: Measurement conditions data shown in Figure 4.21 ..................................... 1 1 1
Table 4.2: Dissipation estimates .................................................................................... 126

Table 4.3: Summary of zero crossing points of correlation functions acquired at n=0. .154

xii

A,B,n
E

El 14322333

fm

ReO

R69

t>l<

u,v,w

NOMENCLATURE

Coeficients of hot-wire calibratirm. Chapter 2 only.

Voltage output of the anernometer, Chapter 2 only.

Components of the one dimensional velocity spectra. Chapter 4 only.

Preffered frequency observed in irrotatoinal flow.

Coherence spectra

Wave number in the x direction

Ratios of velocity gradient variances.

Magnitued of the velocity vector in the x-y plan

Normalized correlation function.

Reynolds number of the boundary layer at separation

Reynolds number based on local momentum thickness

Dimensionless time deley used in autocorrelation funciton

Convection velocity of large scale motions measured in the irrotational

stream

Freestream velocity

Effective freestream velocity for scalin g of the first instability

Components of the velocity vector

xiii

X,y,Z

8*

As,An

wx,my,mz

Velocity of the entrainment flow

Carteasian coordinates originating from the sparation point

Displacement thickness

Separation of straight wire sensors in the vorticity probe.

Size of the microcirculation domain used in the vorticity calculation.

Dissipatoin

Angle of the velocity vector with respect to an X-wire probe, Chapter 2

only.

Circulation density, Chapter 4 only.

Three dimensional wave number.

Momentum thickenss of boundary layer at separation

Local momentum thickness

Effective momentum thickess for scaling the first instability

Spreading parameter.

Components of the vorticity vector

xiv

1.0 Introduction

The study of fluid mechanics at high Reynolds number is quite unique among the physical
sciences. This uniqueness can be partially characterized by the full knowledge of the gov-
erning equations of fluid motion without a complete understanding of the phenomenon,
despite over a century of intense research. One source of the difficulty in turbulence
research is the number of different phenomenon that occur with different boundary condi-
tions. As a result, theory, modeling, or experiments which characterize a single given set
of flow conditions will usually fail to provide similar insight for a geometry which is dif-

ferent from the original.

Most studies in fluid mechanics can therefore be categorized as belonging to one of two
research “modes.” The first, is to choose a set of boundary conditions which is relevant to
an important engineering problem. For example. most research in turbomachinery would
fall into this category. The second mode, is to chose a small number of canonical flows in
which the physics of the fluid motions can be studied in great detail. One goal of these
fundamental studies is to incorporate the acquired knowledge into the applied studies, and

ultimately into engineering design.

This dissertation follows the second approach, and utilizes the single stream shear layer as
the canonical flow field of interest. This geometry is. in the author’s opinion, a very sim-
ple one which is applicable to the wide range of potential applications. An idealized sche-
matic of the planar flow geometry is shown in Figure 1.1. The irrotational freestream
enters the domain from the left, with a smooth wall boundary layer developing on the only

bounding wall. The boundary layer undergoes transition to a turbulent boundary layer, and

develops with a zero pressure gradient (constant velocity) free stream. After a specified
length, the boundary layer plate ends with a 90 degree angle as shown. This defines the
streamwise location of the separation point which marks the transition from a wall
bounded shear flow to a free shear flow. The free shear flow is characterized by a zero
pressure gradient free stream on one side, and fluid which is “entrained” into the shear
layer from the other side. From this basic description of the flow field, it is considered to
be apparent that many of the important flow phenomena of applied fluid mechanics can be

studied in the single stream shear layer.

It can be observed from Figure 1.1 that the only length scale of the problem is given by the
length of the boundary layer plate. An equivalent characterization which is often preferred
is the momentum thickness of the boundary layer at separation (80). This, along with the
specification of the free stream velocity (U0) and the kinematic viscosity of the working
fluid (v) completely characterizes the boundary value problem. The only parameter is the

Reynolds number, Re“: U 0 /v. One of the unique features of the present research

a 0
project is the large scale of the experimental facility which was constructed in the Turbu-
lent Shear Flows Laboratory at Michigan State University. This facility, which is
described in detail in Chapter 2, uses air as the working fluid and has the characteristics:

UO=7.1m/s, 60:9.62mm. This provides a Reynolds number at the separation point of

Reo=4650.

Note that the dimensionless entrainment velocity (VG/U”) is not an independent parameter.
This is because the boundary conditions specify that the free stream velocity (and pres-

sure) are streamwise invariant. That is, BUD/8x = Eli/ax: 0 .There exists a “natural”

value of ve/U0 that will allow the shear layer to grow in a zero pressure gradient environ-

ment. The magnitude of the entrainment velocity is therefore not known a priori, and must
be determined experimentally. This may seem odd at first, that the boundary conditions of
an experiment are not known before the experiments are started. To clarify why this is the

case, consider a finite entrainment domain with vc=(). This represents the backward facing

step geometry in which the separation streamline must diverge away from the high speed
fluid and re-attach to a bounding wall, leading to a recirculation region in the entrainment

side and an adverse pressure gradient condition in the free stream region. In contrast, if ve

were to be forced to be arbitrarily large, the shear layer would deflect towards the high
speed fluid, thereby accelerating the shear layer in a favorable pressure gradient condition.

The “natural” value of V6 is that which would be observed if the entrainment region were

of infinite extent for a finite height primary flow. Since a truly infinite domain is not
experimentally possible, the entrainment flow is “fed“ into the shear layer at the rate
which provides the zero pressure gradient conditions. The experimental determination of

the entrainment velocity is described in Chapter 2.

There exists a large number of research papers in the literature which have focused on the
single stream shear layer. In addition, there is a distinctively larger body of literature
which examines the two-stream shear layer. The latter flow field uses a thin splitter plate

to divide two parallel fluid streams with different velocities: U, and U2. The shear layer is

created by the trailing edge of the splitter plate. It is incorrect to think of the single-stream
shear layer as the limiting case of a two-stream shear layer with the slower velocity
reduced to zero. The reason for this is the direction of the entrainment stream, which, in

the single stream shear layer is directed perpendicular to the primary flow direction. In the

two stream shear layer the entrainment phenomenon caused by the high speed flow will
“pull” fluid from the second flow region, leading to a complicated and undesired flow
field. The difference between parallel and perpendicular entrainment streams has been
examined by Schmidt et a1. (1986). In summary, despite many useful similarities, the sin-
gle stream shear layer is a distinct flow field from the two stream flow, and not simply the

limiting case of U220.

As already noted, there are a number of research papers in the literature which have docu—
mented the physical characteristics of the single stream shear layer. These include Liep—
mann and Laufer (1947), Freymuth (1966), Wygnanski and F iedler (1970), Champagne et
al. (1976), Hussain and Zedan (1978), Hussain and Zaman (1985), Mehta and Westphal
(1986), Haw et a1. (1989), Bruns (1990), Bruns et al. (1991), Foss (1994). Given the large
amount of information already available on the subject. it is appropriate to establish the
reason for the continued interest in this flow field. It is noteworthy, in this context, that
Wygnanski and Fiedler in 1970 found it necessary to justify why they found it “worth-
while to reconsider this flow”. Following an additional three decades of research, an
updated justification is given by outlining several of the open questions which still exist in

the literature whose answers are of considerable interest.

The affects of the inflow boundary conditions on the downstream flow are of interest in

the study of both single and two stream shear layers. The dynamics of the flow field in the
very near separation region has not been documented or properly understood. As a result,
most differences that are noted between the results of different researchers using different
shear layer facilities are attributed to the differences in the inflow conditions. In addition,

the inflow conditions for most experimental facilities are either laminar, or low Reynolds

number tripped boundary layers. These facilities are therefore not representative of the
flow separation phenomenon that exist in engineering applications which are character-

ized by very high Reynolds numbers.

The experimental facility used in the present work provides a unique opportunity to “fill

the gap” in the available literature. This is because of the relatively large scale of the facil-
ity, which provides excellent spatial resolution for measurements (599(xz0) z 100mm) in
a moderately high Reynolds number boundary layer (Re0=4650). A detailed experimental

investigation of the near separation field is presented in Chapter 3. This research project is
self-contained, and distinctive from the downstream measurements. Therefore, Chapter 3
is complete in the sense that an introduction, motivation. background, literature review,

experimental results. and conclusions are provided.

The next portion of this dissertation, Chapter 4. describes measurements that were
acquired in the self similar region of the flow field. This chapter is also self contained with
introduction, literature review, etc. There are numerous outstanding questions in the
research literature which have motivated these measurements. A first example regards the
nature of shear layers in terms of how the stochastic variables of interest scale with down-
stream distance. It has long been known that linear growth and self-similarity are standard
features of free shear flows. However, the scalin g of quantities such as dissipation, vortic-
ity fluctuations and their spectra have not previously been examined. Again, the present
facility provides a unique opportunity to investigate these issues because of the relatively

large scale, and hence adequate spatial resolution of the measurements. It will be shown in

terms of the vorticity spectra that true self-similarity is not achieved. Rather, the dimen-

sionless vorticity fluctuations increase with Reynolds number.

In addition to the novel research issues which have been addressed in this dissertation,
many of the previously documented features of shear layers have been repeated in this
work. The reason for this repetition is (i) to validate the present facility, and (ii) to validate
claims made in previous research efforts. In doing so, this volume represents a more com-

prehensive study of the flow field than that which has been previously reported.

The outline of this document can be summarized as follows. The experimental facility and
measurement techniques which are common to all of the experiments are described in
Chapter 2. The two “halves” of the research project are represented by the near separation

region, and fully developed region, and are described in Chapters 3 and 4 respectively.

 

 

UO 1“)
_) We,
'—> r—>
——) 1—9
I; R"
_) :

 

11111111111111111111111111111111111111111111111111111

Ve

Figure 1.1 Idealized schematic of the single stream shear layer geometry. The thick line
represents the only solid boundary. The dotted lines bound the region of non-zero vortic-

ity. The velocity vectors are schematic, and not shown to scale.

2.0 Experimental Apparatus and Techniques

The experimental apparatus and methods used in this dissertation are described in this
chapter. First, the Single Stream Shear Layer Facility (SSSL) will be described. This facil-
ity was designed and assembled by the author, and is documented in Section 2.1. This

information can serve as a reference for future research which will utilize the facility.

Hot-wire anemometry was the primary experimental method for the measurement of
velocity. The basic sensor geometry and calibration technique are described in Section
2.2.1. X-wire probes were used to measure one or two components of the velocity vector.
The algorithm for obtaining the fluid speed and flow angle from the two sensors of the X-
wire is outlined in Section 2.2.2. Multi-sensor arrays of wires were used to obtain velocity

gradient and vorticity information. These are discussed in Section 2.2.3.

2.1 The Single Stream Shear Layer Facility

The Single Stream Shear Layer (SSSL) facility is located in the Turbulent Shear Flows
Laboratory at Michigan State University. The majority of the facility including the test
section was constructed at an elevated level (its floor is ~2.1m) above the laboratory floor.
The primary flow intake and delivery system were located at the floor level as shown in
Figure 2.1. An axial fan accelerates the laboratory air from rest. The fan exit is connected
to a 1m x 1m square section. Acoustic dampening material was placed 1.4m upstream
from the blower entrance to limit the propagation of blower noise into the room and test
section. The fan used a constant speed AC motor attached directly to the 48” diameter
blades. The exit of the axial fan housing was connected directly to a diffuser section that

expands to a 2m x 2m cross section. In this diffuser. flow dividers (not shown) were used

to limit flow separation and provide relatively equal flow distribution. The air is delivered
through two sets of 90 degree turning vanes which first turn the flow direction vertically
upwards and then horizontally at the elevation of the test section. Standard Sum furnace
filters were attached to the first set of turning vanes to eliminate dust from the free stream
flow. A second acoustic dampening wall was placed at the back wall of the turning vane

section as shown.

Following the second turning vane section, a set of flow conditioners were installed to
provide a uniform velocity with minimal unsteadiness to the test section. This condition-
ing section consisted of a 25mm long honeycomb section with 3mm tube diameters, fol-
lowed by three sections of stretched steel screen material. The screens were made from

0.060” wire mesh with 40 wires per inch (0.635mm spacing between wires).

The outflow from this conditioning section is connected directly to the inlet of the main
tunnel entrance. This is shown as “primary inflow" in Figure 2.2. The height (i.e., “out of
the page”) of the entire working section of the tunnel was 1.96m. The width of the primary
flow inlet was 1.96m wide. A planar contraction was placed downstream from the inlet
which accelerates the flow to a width of 1.1m. The elliptical shaped leading edge of the
boundary layer plate then splits the primary flow such that the boundary layer from the
curved wall of the contraction is “bled” off and returned to the laboratory as shown. This
point of the tunnel and all downstream locations are at nearly zero gage pressure with
respect to the laboratory pressure. The majority of the flow moves through the 0.98m wide
boundary layer development section. The boundary layer plate is 5.74m long, with a free
stream velocity of 7.1m/s. The boundary layer plate was laminated with 1mm thick PVC

sheeting to provide a smooth boundary condition. The boundary layer plate ends at the

separation point which was created using a machined 90 degree edge. The free stream
velocity was measured during all experiments by monitoring the differential pressure drop
across the planar contraction. This pressure difference was acquired during all experi-

ments and found to vary by less than 0.5% for the data reported in Chapters 3 and 4.

The shear layer test section begins at the separation edge. The entrainment flow is pro-
vided by four 48 inch (1.2m) diameter axial propeller type fans with an AC speed control-
ler. This fan type is designed to move large volumes of air at relatively low pressure rise
(~25Pa). For this reason, bypass holes were created such that the fans would operate at
design conditions to prevent unsteadiness from blade stall. That is, most of the air moved
by the fans is recirculated back to the laboratory. The “plenum” between the fans and the
flow conditioning is maintained at a pressure which is slightly higher than the atmospheric
value. The entrainment flow conditioning is identical to the primary flow conditioning.
The length of the entrainment screens is 9.5m which defines the working length of the
shear layer test section. The location of the final entrainment screen is y=-2.75m, using the
coordinate system shown with origin at the separation edge. The exit of the tunnel is open
to the laboratory. The nearest obstruction to the outflow is a building wall located 6.1m
from the tunnel exit. This provided sufficient room to turn the exiting flow without dis—

turbing the shear layer.

The entrainment velocity was set using an AC frequency control unit to drive the electric
motors connected to the entrainment fans. An experiment was conducted to identify the
correct motor speed in order to create a shear layer with a zero streamwise pressure gradi-
ent. Specifically, data were obtained using a single hot-wire probe in the free stream at two

streamwise locations (x=0.5m and 4.0m) in the high speed irrotational flow. The controller

10

frequency was adjusted until the hot-wire readings at these two locations indicated the

same velocity. This implies dUO/dx= 0, i.e., a zero pressure gradient condition.

A traverse system was constructed to support and move the hot-wire sensors during data
acquisition. This system used a wheel and track for movement in the streamwise direction
(x) and a precision lead screw and stepper motor drive for the transverse (y) direction. The
probes were mounted on a rotating shaft which was also motor driven. This permitted

angular positioning of the sensors for calibration and data acquisition.

2.2 Hot-wire Techniques and processing

This section will describe the details of the hot-wire calibration and processing methods
used in this work. A schematic of a single sensor is shown in Figure 2.3. The active por-
tion of the sensor was a 1mm long, 5am diameter Tungsten wire. The end regions were
electro—plated with 30pm copper and soldered to stainless steel supports. The wires were
connected to DISA 55m anemometers. The frequency response of the sensors at 7m/s was

typically greater than 20kHz.

2.2.1 Calibration

The wires were calibrated using a newly constructed calibration facility shown schemati-
cally in Figure 2.4. The unit used a constant speed cross-flow blower and throttle to vary
the exit velocity from 0.5 to 12m/s. Screens, filters and honeycomb were placed upstream
of a contraction to achieve a low disturbance free stream for calibration. The contraction
exit flow dimensions were 2.5 x 24 cm which provided sufficient area to calibrate up to 10
probes simultaneously. The probe holder was connected to a stepping motor which varied

the angle of the X-wire probes with respect to the jet exit.

The calibration technique used a continuously variable “quasi-steady” method. That is, the
sensor was subjected to a velocity which was varied continuously from shutoff to maxi-
mum flow over the duration of approximately 40 seconds. Data samples of the hot-
wire(s), pressure transducer, and thermistor were acquired at a sample rate of 500Hz.
These data were used to fit the equation

52 = Am + BmQ”“” 2.1
where E represents the output voltage from the anernometer, and Q represents the exit
velocity, and y represents the pitch angle with respect to the probe axis. This velocity was
calculated from the pressure measurement using Bernoulli’s equation. Note that this
method assumes that the unsteady effects of the continuously varied velocity are not suffi-
cient to cause error in the steady Bernoulli equation. This assumption was tested by per—
forming several standard (steady state) calibrations for comparison with the quasi-steady
results. These experiments have shown no measurable differences between the two types
of calibratoins.The primary reason for using the quasi-steady method is the speed of cali-
bration. For example, single wires were calibrated in less than one minute. The x-wire
probes, for example, were calibrated at 13 angles which required approximately 12 min-
utes of total calibration time, which include angle changes between measurements. If the
sensors were calibrated with the alternative technique: establish the flow angle, record
time series data for a period of 10 seconds at each flow speed, then an X-wire probe would

take typically 45minutes to calibrate.

An example of a wire calibration and fit is shown in Figure 2.5. The quality of the fit was
determined from the standard deviation between the measured velocity and the predicted

velocity from equation 2.1. In all experiments, this was found to be less than 0.05m/s. If a

12

sensor/anemometer were found to have a higher value for the standard deviation, the
probe was not used. The sensors were calibrated before and after each experiment. If the
predicted velocity from the two calibrations changed by more than 2%, the data were dis-
carded and the experiment was repeated. In most experiments, the velocity calculated

from the “pre-calibration” and “post-calibration” agreed to better than 1%.

2.2.2 X-wire probe calibration and processing

X-wire probes were used for the simultaneous measurement of two velocity components.
The probe geometry is illustrated in Figure 2.6. Two sensors were oriented at nominally
:45 degrees to the probe axis. The spacing between the sensors was 1mm. The output

from the two sensors is denoted as E1 and E2. The goal of the calibration and processing

algorithm is to obtain a transfer function between these voltages and the desired variables:
the speed (Q) and angle (7) of the fluid velocity with respect to the probe. The optimal

transfer function, which provides the most accurate estimate of Q and 7, has been the sub-
ject of considerable research. Techniques which are used in the literature vary from the

very simple cosine law to very complex algorithms such as mapping techniques; these are
described by Bruun ( 1995). Since all techniques seem quite accurate for small angles, say,
[71 < 12 degrees, it is the largest angles where the differences between algorithms has the
most effect. Because the angles found in the shear layer can fluctuate considerably, a new

measurement technique was developed which provides the most reliable speed and angle

estimates for 171 < 36 degrees.

The present technique, is a modification of the “speed-wire/angle-wire” (SWAW) tech-

nique proposed by Foss (1981). The SWAW algorithm uses the sensor more parallel with

the velocity vector to determine the angle (7), and the sensor which is more perpendicular
to the velocity vector is used to obtain the speed (Q). An iterative algorithm is used which
converges to the correct (Q,y) pair. The present technique is similar, although the iterative

nature of the SWAW technique was replaced with a “curve intersection” method.

The new algorithm is described as follows. First. the X-wire probe is calibrated using the
spin—down technique (described in Section 2.2.1) in the range —36 < y< 36 degrees, in 6

degree intervals. This calibration results in 13 sets of (A,B,n) values for each sensor. The

experimental data are then acquired as time series values of the voltage pair (E1,E2). From
each of the recorded (EI,E2) pairs, equation 2.1 can be solved for Q1(E1,‘y,-), Q2(E2,y,-)
where i=1,2,...13. That is, Q1 and Q2 represent predicted velocities which are calculated
for each sensor using the wire voltages (E1, E2). and the calibration coefficients for

i=1,2...13 representing the 13 angles used in the calibration. Figure 2.7 shows an example
of the 13 predicted velocities that were calculated from a voltage pair. The “true” value of
(Q,‘y) is now represented by the intersection of the two curves shown. This intersection

point is found by fitting the neighboring points of each curve with a second order polyno-

mial fit, and then solving for the intersection point.

One method of testing the accuracy of the calibration/processin g algorithm is to calculate
the speed and flow angle of an X-wire probe placed a known angle with respect to a steady
flow. The calculated velocity agreed with the actual value to within 2% at angles less than
18 degrees, and within 4% at angles less than 36 degrees. Figure 2.8 shows the results of
time series data of the predicted flow angle with a probe placed at +18, 0, and -30 degrees

with respect to the jet of the calibration unit. The standard deviation of the error in pre-

dicted angle was typically less than 0.5 degrees. and often as good as 0.2 degrees. These
errors reflect the contribution of electronic noise to the processed time series. Errors due to

calibration drift were typically less than 1 degree.

2.2.3 Multi-sensor Probe Configurations and Processing

The previous sections have described the calibration and processing algorithms for both
single wire probes and X-array probes. In the following three sub-sections, probes which
utilize combinations of single and X-arrays to compute vorticity and velocity gradients are
described. The use of multi-sensor probes leads to a number of complications regarding
the uncertainty of the measurements. These can be dependent on the probe configuration,
probe-scale to flow-scale effects, and the presence of electronic noise in the time series.
Because the measurement techniques described were not developed as a part of this dis-
sertation, they will be described only briefly along with the estimated uncertainty. Addi-
tional information can be found from references to those communications in which the
developed techniques are described. These references provide more comprehensive uncer-

tainty evaluations.

2.2.3.1 Spanwise Vorticity Probe

A schematic representation of the spanwise vorticity probe, or “Mitchell” Probe, is shown
in Figure 2.9. This probe consists of an X-array with two parallel wires. These were cali-
brated using the procedures described above. The time series output from the probe is the
magnitude of velocity in the plane of the probe as determined by the two straight wires:

(ql, qz), and the angle (7) determined from the X-array. This section will review the

method by which the time series values of (q,, (12.7) were used to calculate the spanwise

vorticity. The probe geometry and processing algorithm were originally developed by
Foss and co-workers, and can be found in the references by Foss and Haw (1990), and

Wallace and Foss (1995).

The data processing algorithm can be summarized as follows. The average of the two
recorded velocities (q,+q2)/2 is used as a convection velocity to create a parallelogram
shaped micro-circulation domain using a local (in time) Taylor’s hypothesis; see Figure
2.10. The height (Ay) of the microcirculation is fixed by the physical separation distance
between the two straight wire sensors. The length of the domain (As) was set to be roughly

equal to Ay. Specifically, from the discrete time records of ql(tn), etc., the sum was com-

puted for each tn as:

 

Asn(m) = 2.2

 

l l'=""‘“(ql(t,)+r‘12(t,-))At
2m +1 2 '

i=n-m
The value of m was increased, starting from m: 1, until As" > Ay. This defines M, the
integer number of time records needed to create the one half of the micro-domain. The

value of M for each calculation time (tn) would vary depending on the flow speed. Typical

values of 2M were between 6 and 20 time steps.

The mean value of a given quantity during the interval In _ M < t < tn + M is denoted by the

brackets ( > . For example, the angle of the microcirculation parallelogram was the mean

value of yduring the time period of the microcirculation:

i=n+M
Z 70,-) 2.3

i=n-M

l
2M+l

 

<r,,> =

Once the microcirculation domain has been “created“. the average vorticity of the domain
is calculated from the velocity circulation around the domain. The magnitude of the veloc-

ity can be projected onto the “top” and “bottom” of the micro-domain as
q,-cos(y(ti)—(y>) and qz-cos(y(t,)—(Y>) . and then integrated along the length circulation
domain. The normal velocity component is calculated from v = (qI + q2)( siny)/2 at

both the upstream and downstream sides of the domain. For each point of the time series

(n), the vorticity is calculated from:

 

l §(-‘7d§)= VUH+M)—v(tn—M)_<q2>—<ql>

= (AS)(A,v) As Ay 2'4

<w,,>
The approximation cos( (y(tl.)—(y>) z 1 ) was made in order to decrease the computational

time required to process the time series. This assumption is justified based on the realiza-
tion that the differences between the instantaneous values of y( t,) and (7) were less than 5

degrees. Since the cosine can be factored out from the q. and q2 terms, the approximation

cos(lyj < 5) z 1 is reasonable.

A detailed uncertainty analysis of the vorticity probe and processing algorithm has been
given by Foss and Haw (1990). One of the primary sources of error was found to be due to
the presence of electronic noise in the time series data. The contribution of electronic
noise to the uncertainty is reported to be of order 20 ( l/sec’) in Foss and Haw (1990). This
is well supported by the measurements of vorticity in the free stream of the shear layer

where little or no vorticity is assumed to exist. Specifically, at x/0(,=384, values of

(Bz < 25(1/sec) were measured at the transverse locations (n>3.8).

2.2.3.2 Streamwise Vorticity Probe

A schematic of the streamwise vorticity probe is shown in Figure 2.1 1. This probe uses 8
senors configured as four X-arrays in a square pattern. From this, two of the opposing X-
wires will provide time series of (VI, v2), separated by a distance of Az=5.6mm. The other
two X-wire will then measure time series of (WI. wz). separated by Ay=5.6mm. The vor-
ticity calculation is then straightforward:

- Ay A2.

 

The uncertainty of this probe is dictated by the ability of the X-wire processing to recover
the true value of the lateral component of velocity. The errors will be due to electronic
noise and calibration errors. Electronic noise was shown in Section 2.2.2 to contribute to a
random error of typically 0.2 degrees. At a flow speed of, say. 7.0m/s, this would lead to a
random error of roughly 24mm/s in the normal component of velocity. If these errors are

Gaussian distributed and equal for the four X-wire probes. then the standard deviation of

the sum (w2 — wl — v2 + vl) is twice that of the single measurement. That is the standard

deviation for (it)x in an irrotational flow would be 2 . (0.024)/(0.005) = 9.8(1/sec) . The

measurements support this conclusion. For example. at x/0 =384, values of

()

(I)Z < 8 (1/sec) were measured at the transverse locations (n>3.8).

Additional details regarding the use and uncertainty of this probe configuration can be

found in Kock am Brink and Foss (1993).

2.2.3.3 Multiple X-wire Configuration

A four wire sensor was configured as two adjacent X-wire probes in a single probe body;

see Figure 2.12. The same probe configuration was used in the measurements made by

George and Hussein (1991), and by Lui (2001). This probe is similar to the streamwise
vorticity probe discussed in the previous section, in that closely spaced X-wires are used
to determine the lateral derivative of a velocity component which is normal to the stream-
wise direction. In this case, the probe was used to estimate the mean squared value of
dv/dz. This measurement was required for the dissipation estimate which uses the

assumptions of axisymmetric turbulence; see Section 4.5.

The four sensors shown were arranged such that wires 1 and 2 were processed as a single
X-wire, with the output vl. Similarly. wires 3 and 4 are processed to provide v2. The
desired velocity gradient is simply the difference between these, divided by the spatial
separation, which was 3.0mm. With this probe separation, the electronic noise from the
anemometers leads to an RMS of the velocity gradient of about 16 ( l/sec) in the free

stream of the shear layer.

The finite separation between the two X-wire probes can lead to an underestimate of the
true velocity gradient. In the present flow field, the separation between the X-wires was of
order 1511K, where “K represents the Kolmogorov length scale of the flow. Wallace and
Foss (1995) review several references which discuss the attenuation due to finite probe
scale. Figure 2 of Wallace and Foss (1995) indicates that there is reasonable agreement
between different authors on this issue, and that the present configuration likely underesti—
mates the true RMS of avflz by roughly 50%. However, the present data indicate that the
performance of the probe is likely better than this estimate. Specifically, the variance of

av/dz was found to be larger than the isotropic value, which is twice the variance of

an; see Section 4.1. Given these already larger than expected values for the variance

of £9sz , it is unlikely that significant attenuation has occurred in the measurements.

SCFE’E’VWS

 

 

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entrance to
test sectbn

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acoustt {fi’m-m J axml Fan

a

35?: z‘ -. T_"-::i
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Figure 2.1 Primary flow delivery system

21

PRIMARY lNFLOW / 11111111

FLOW
CONDITIONERS
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9.75M :— *‘
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Figure 2.2 Schematic of SSSL facility

22

/ Copper Plating
ilk/0951010 M/

// g.\ ,

 
 

 

 

 

 

 

 

 

 

Figure 2.3 Geometry of single hot-wire probe

Cross—Pkkafiower

stepper motor For
probe angMacflfive \\

[fill/f ‘ /
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motor For
/\ C7 throtfle contro

\\\\\ pflanar contractmn

probe support

 

Figure 2.4 Schematic of Calibration Facility

23

 

    

 

 

 

20 I j r 1 1 T Y I
E2 18 r 1:.2=7.334+4.142v”44 j
16 L _
14 C- —
. 1
18 A l A l . l 1 l r
.50 1.00 1.50 2.00 2.50 3.00

Vl'l

Figure 2.5 Sample calibration and fit, 7:0 degrees.

 

 

 

H0 ”11‘.

9:9:910XOXOXOX 1919:9192" { XOXO' 3:97.93 0 O
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I 20202.202020939339323:X101917 Viva O 0

Figure 2.6 Geometry of the X-wire probe

24

 

30.0

 

3* .
'5 25.0 ~ -
g .
§ 20.0 — —
8 ~ .
g 15.0 — J
.D .
8 10.0 -
a. .

5.0 — -

A‘JAA‘gLLlAAAJLLA‘LAALLALLL

00 1 1 1 l 1 1
-40 -3O -20 -10 0 10 20 30 40

 

 

 

angle 00

Figure 2.7 Example of the locating algorithm of the intersection point for X-wire process-

ing.

 

 

 

 

 

 

36
30

g- 24

an 12

cc 6

B 0 " q

.2 -6 r j

g -12 - _

A. -18 P _
-24 _— _
-3O mavzfievrwr; 1:»; A w T v A __ _ Aswww m ,1 __
-36 L 1 1 1 1 1 1 r 1 i

1.0 3.0 5.0 7.0 9.0 11,0

Velocity (m/s)

Figure 2.8 Predicted values of angle (7) from re-processed calibration data at nominal val-
ues of 18, 0, and -30 degrees. The standard deviation of these measurements was 0.18,

0.17, 0.21 degrees, respectively.

25

 

 

 

 

 

1+ rammii
{03332222101033z~ {0101010103103 0 O
182:2:23232323232339;Wm» 0 O T
1 4 mm
13.032010202020203!0Y0Y0Y0Y0‘1'awe O O —-1-
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Figure 2.9 Schematic of spanwise vorticity probe

 

 

 

 

 

 

 

tn-M t n-l t“ trwl JCrHM

Figure 2.10 Schematic of micro-circulation used for the spanwise vorticity calculation.
The velocities q, and q2 are known at the points represented by the dark circles on the top

and bottom of the domain, respectively. The normal component of velocity, v, is known at

the open circles at the left and right sides of the domain.

26

56mm

 

 

---1 lb— 1.0mm
4
O

 

l“— 4.0mm

 

 

 

 

 

'\
,1

C
C
C
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OKOZQ
29.9‘ ‘D
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A.
91.1. '
02.1.
z.. 3.3.

v
V

 

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v

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1.91..

 

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v
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6

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0'.

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o.
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Figure 2.11 Schematic of the streamwise vorticity probe.

27

Figure 2.12 Schematic of the multiple X-wire probe.

3.0 'lhrbulent Boundary Layer to Single Stream Shear Layer - The initial

transition region.

3.1 Introduction

A detailed investigation of the near separation region of the single stream shear layer was
conducted. A schematic representation of the area of interest and coordinate system is

shown in Figure 3.1. Measurements were acquired in the region 0<x/00<100. This study is

distinctive with respect to the existing literature because of the relatively high Reynolds

number of the separating boundary layer (Re9=4650), and because the large 0 value

(9.6mm) permitted detailed measurements to be acquired in the very near separation
region. This flow region is important both fundamentally and technologically. The region
can serve as the archetype flow for a backward facing step, a jet exhausting from a plane
surface, and other flows characterized by the 90 degree interaction between a turbulent

separating primary flow and the induced secondary or entrained flow.

A high Reynolds number separating boundary layer can be viewed as the transition region
between two well studied canonical flow fields: the flat plate turbulent boundary layer and
the single stream shear layer. This leads to two possible perspectives from which to view
the acquired data. The first perspective is to view the region shown in Figure 3.1 as a
boundary layer with a sudden change in the wall boundary conditions. The data then serve
to answer the question: how does a turbulent boundary layer react to this abrupt
“removal” of the wall boundary condition? A similar question was addressed using a
moving wall geometry in Hamelin and Alving (1996, hereafter referred to as HA). It will

be demonstrated in the following sections (in agreement with the results of HA) that the

28

“outer region ” of the separated boundary layer remains statistically identical to the
upstream (canonical) boundary layer for several integral length scales in the streamwise
direction. The insensitivity of the outer region of the boundary layer from the effects of
separation is remarkable, and is expected to be of interest to researchers who study bound-

ary layers as well as separated flows.

The second viewpoint considers this region to be the developing region of the shear layer.
The data then explain: how does a self-similar single stream shear layer develop from its
parent boundary layer? Implicit in the ideas of self similarity is the idea that the initial
conditions are “forgotten” (Townsend, 1974). The effects of inflow conditions on the
downstream region of a shear layer have been studied by, for example, Bradshaw (1966),
Hussain and Zedan (1978), Slessor et a1. (1998), and Browand and Latigo ( 1979). The ref-
erences report experiments in which the upstream boundary condition is modified, and
measure the “effect” on the developed downstream flow. In contrast, the present effort
investigated the evolution of the initial turbulent boundary layer as it transitions into a
self-similar shear layer. This region has been extensively studied for the case of laminar
boundary layers at separation. For example, Ho and Huerre (1984) provides a thorough
review and a substantial list of both single stream and two stream shear layer references.
Many of these studies consider the near separation region using linear stability theory. The

results of the present work will be compared and contrasted to this established theory.

Most previous shear layer studies have been conducted with boundary layer Reynolds
numbers less than 1000. The use of lower Reynolds number boundary layers is most often
justified based on the idea that the details of the initial conditions are unimportant in the

fully developed flow. However, this is still a topic of some debate. The present measure-

29

ments will show that the Reynolds number is of substantial importance in the near separa-
tion field. For example, an important feature of the transition from boundary layer to shear
layer is the change in length scales. The turbulent boundary layer is known to have both

“outer” and “inner” scales, corresponding to the integral length (0) and viscous wall units
(v/ut) respectively. The ratio of these scales varies as 0uT/v~ Re9(7/8)(Bejan 1995). In con-

trast, the fully developed shear layer is typically characterized only by its integral scale.
This change in length scales from boundary layer to shear layer will be shown to be an

important factor in the physical attributes of the near separation field. Many technologi-
cally important devices have Reynolds numbers (UOG/v) that are very large, say 106 or

greater. Since many of the phenomena described in this chapter are distinctive to higher

Reynolds numbers, the conclusions could be important in engineering design.

Section 3.2 presents statistics of the boundary layer at separation. Section 3.3 introduces
the concept of the “sub-shear layer” and it presents single point statistics acquired in the
region downstream of separation. In Section 3.4 measurements in the entrainment stream
are used to characterize shear layer features such as the convection velocity and average
passing frequency of the large scale motions of the flow. Section 3.5 presents spanwise
correlation measurements in the sub-shear layer region. It should be noted that these sec-
tions present the experimental data with a limited amount of discussion. This format was
chosen so that the many different types of data acquired could be presented prior to the

conclusions which are provided in Section 3.6.

30

3.2 Upstream Boundary Layer

The study of the near separation region was begun with detailed measurements that were
acquired near the separation point (XzO). These data and their interpretation serve to docu-
ment the inflow boundary conditions to the separated flow discussed in the next sections.
The boundary layer portion of this study was separated into two experiments. First mea-
surements to identify the “communication” between the shear layer and the boundary
layer were executed; see section 3.2. 1. Second, point statistics in the boundary layer were
measured with enough detail to ensure that a nearly canonical zero pressure gradient
boundary layer existed and that it compared well with standard measurements and correla-

tions. These data are presented in Section 3.2.2.

3.2.1 Boundary Layer - Shear Layer “Communication”

The communication between the downstream shear layer and the boundary layer was
investigated prior to the determination of the stochastic features of the boundary layer at
separation. The purpose of this experiment was to answer the question: “do the pressure
fluctuations from the shear layer significantly alter the characteristics of the boundary
layer at separation?”, Foss (1998). In other words. do the large scale spanwise coherent
motions in the shear layer cause a measurable change in the boundary layer characteristics
upstream of separation? A previous study discussed in Holmes et a1. ( 1992) has indicated
that unsteady effects in the shear layer that originates from a laminar boundary layer, do
cause fluctuations in the upstream flow field. That is, “yes", the shear layer does influence
the separating boundary layer for laminar separation. The present data will show that this

is not true for the present (high Reynolds number turbulent) boundary layer.

31

The velocity field in the entrainment stream near separation is greatly influenced by the
large scale motions which exist in the separated turbulent flow. If these motions also affect
the boundary layer flow before separation, it is intuitively reasonable that a non zero cor—
relation between velocities in the boundary layer and in the entrainment stream would pro-
vide evidence that the shear layer was effecting the boundary layer. It is noted, however,
that statistical quantities such as linear correlations to not indicate causation, and should
only be used as an indicator of such effects. Simultaneous hot-wire measurements were
taken in both the entrainment stream and the boundary layer flows to test this hypothesis.
Four hot-wire sensors were placed in the entrainment stream near the separation point
with the sensors aligned with the z-direction such that they would recover the components
of velocity in the x-y plane. Note that the w component should be quite small in the essen-
tially 2-dimensional entrainment region. The entrainment probes were then traversed

through a range of locations in the region 0<x/00<20, y<0 area. An X-wire probe was
placed in the boundary layer in the range 0<y/0(,<10, -4<x/00<0, such that the u and v

components of velocity were recovered. These data were recorded for 40 seconds at
5,000Hz. The normalized cross correlation is a function of the discrete time delay vari-

able, defined as:

 

v,(tn) - v2(tn + 1:")
R(1.'n) = ,_ . 3.1

 

where v] is the u or v component of velocity measured from the X-wire, and v2 is one of

the signals from the entrainment probes, and the overbar represents the time average. The

tilde symbol (~) represents the RMS value of that signal.

32

The results of these calculations have shown nearly zero correlation values for all of the
above measurement locations. That is, there was no measurable correlation between the
two signals for any of the spatial locations measured. A second calculation was performed
in which a low pass filter was used to allow only frequencies less than, for example,

fc=100Hz (where f. 0(0)/U“ = 0.135 ) to be correlated. This type of filtered cross-correlation

has been shown to be effective in detecting non-zero correlations in highly turbulent flow
fields which contain broad-band spectra information; see, e.g., Naguib (1992). These
cross-correlations, however, were also found to be negligible in magnitude for all of the

acquired data.

It is inferred from the above observations that the upstream “communication” is not mea-
surable in a separating turbulent boundary layer. It is therefore reasonable to assume that
the measurable characteristics of a high Reynolds number turbulent boundary layer
upstream from a fixed separation point with zero pressure gradient are identical to that
which would exist if the separation was not present. These conclusions are further sup-
ported by the results of the following section, which presents detailed velocity measure-

ments for x--0.

3.2.2 Boundary Layer Statistics Near x=0.

The single point statistics of the boundary layer in the very near region of separation were
measured using a single hot-wire sensor. The probe was calibrated and traversed through
the boundary layer at the streamwise locations x/00= -0.1 and +0.1. Each data point was
acquired at a sampling frequency of 5,000Hz for 30 seconds. The sensor was aligned in

the z direction, such that the magnitude of the velocity in the x-y plane was recovered:

33

q(t) = Jul + v2 3.2

Most authors do not make the distinction between it and q in single probe hot-wire mea-

surements. This is because the v component is small compared to u in wall bounded flows.
The emphasis on 4 in the present work is important because the surveyed region: y<0, will
be dominated by the v component of velocity. In the region (sz, y>0) the approximation

qzu is nearly exact.
The time average and standard deviation of the velocity measurements will be denoted by
a and 2'] respectively. The profiles 21(x= i 0.10”, y) are shown in Figure 3.2. Note that

these data are normalized by the standard viscous units in log-linear coordinates. The wall
shear stress was computed from the slope of a linear fit of the data points closest to the

wall at x/00= -0. 1. The standard log-linear region can be observed through the range
40<y+<800. This region roughly approximates the standard fit q+=2.5*ln(y+)+5.5. The
present data are best represented by q+=2.3*ln(y+)+6. 7 for 30<y+<500.

It is of interest to compare the two velocity profiles which have been acquired. The mea-
sured x/00=i0.1 mean values agree for all locations of y*>5(0.25mm). The insert to the

figure shows the deviation of the two traverses for y“<5 in linear coordinates since nega-
tive y values were measured at the downstream location. The deviation between the two

profiles indicates a strong streamwise gradient in velocity given that the x displacement of

the sensor for the two traverses was 1.8mm. For example, Ail/Ax z 400( 1 /sec) at y=0.

The integral statistics of interest, which were calculated from the mean profile, were the
momentum thickness: 0:9.62mm, and the displacement thickness: 5*=12.6mm. The Rey-

nolds number based on momentum thickness was Re9=4650. Additional parameters are

34

the friction coefficient and the shape factor, which are defined as C,- = 2(141/U”)2 and H=5*/

0, respectively. These were determined to be C,~=0.00295 and H =1 .31.

The 21(x= i 0.10", y) profiles in viscous units are shown in Figure 3.3. The observations

that can be made from the profiles of 2'; are similar to those already mentioned for 2']. That
is, differences between the two profiles can only be observed in the very near wall region.
Note that the maximum value for the x/0=-0.1 data is (”l/uT = 2.74. Other researchers

(for example Klewicki and Falco (1992)) have found a peak in the fluctuation intensity to
be ~2.9 for similar Reynolds numbers. This discrepancy can be explained by the relatively

large sensor length used in the present study. Specifically, the hot wire length in viscous
units was 71+: 20v/uT, where Klewicki and Falco (1992) recommend a sensor length of

less than lOv/uT in order to avoid spatial averaging of the velocity fluctuations.

The spanwise vorticity in the boundary layer is also a variable of interest. The magnitude

 

 

of the time averaged vorticity in viscous units is approximated by 111—: = ant/8y“ . The slope
of the mean velocity profile shown in Figure 3.2 was computed; see Figure 3.4. The stan-
dard deviations of the fluctuations in spanwise vorticity: 05:, were measured using the

four-wire vorticity probe described in Chapter 2; these data are also shown on Figure 3.4.
The measured values of r15Z are consistent with the data presented in Klewicki et a1. (1992)

and references therein. These data show that both positive and negative values of vorticity

were realized for y+>20. This can be compared with the near wall region wherein only one

sign of vorticity that is consistent with that of the mean shear is observed. The vorticity

35

fluctuations normalized by the local mean value: (115/ jail have been plotted for the region

10<y+<800 as the insert to Figure 3.4. Note that the values of (IL/[5| are uncertain for

y+>800 because both the numerator and denominator are approaching zero. The data show
that the relative fluctuation levels of vorticity increase approximately logarithmically (lin-
ear in semi-log coordinates) throughout the “log region” (10<y"<600) of the boundary

layer.

Qualitative aspects of the vorticity filament topology can be inferred from these measure-
ments and the solenoidal condition of the vorticity vector field. At a small distance from
the wall the filaments are necessarily perpendicular to the local shear stress vector, and
parallel to the surface. The highly fluctuating values of single signed spanwise vorticity
near the wall indicate that the near wall filament topology is relatively simple. In contrast,
the measurements farther from the wall (y+~20) indicate that the vorticity filaments must
be sufficiently complex or “twisted” such that vorticity values of both sign are realized at
a given y location. Farther from the wall, the statistics of the velocity and vorticity fields
exhibit many of the properties of isotropic turbulence; see, for example, Saddoughi and
Veeravalli ( 1994) for examples of velocity statistics. An example of this from the present
measurements is the vorticity spectrum at location y+=400 ( u/UU = 0.77) as shown in Fig-
ure 3.5. These data are presented along with a model spectrum which was derived using
the assumption of isotropic, homogeneous turbulence, (see Anonia et a1 ( 1998) and Sec-
tion 4.6). The variables have been normalized using Kornogorov variables defined in
Chapter 4. The agreement between the measured and isotropic spectra support the idea

that the vorticity filaments in the outer region of the boundary layer are not directly

36

effected by the wall. These features of the vorticity field will be considered in the follow-

ing sections which describe dynamics of the near separation region of the boundary layer.

3.3 Point Statistics And The “Sub-shear Layer”

A flow visualization image of the near separation region is shown in Figure 3.6. This digi-
tal image was acquired using a Kodak 1k x 1k camera and a 1min thick laser light sheet
oriented in the x-y plane. The boundary layer flow was seeded using an oil-fog system
located upstream of the boundary layer leading edge. The unmarked entrainment flow pro-
vided the contrast between the two fluid streams. Specifically, the marked primary fluid
appears white, and the clean entrainment fluid appears dark in the figure. Note the scale
and orientation with respect to Figure 3.1. Specifically, recall that the boundary layer
thickness 5 is of order 100. This picture is not repeatable in detail because the flow is tur-
bulent in this region. However, the general features represented and described are com-

mon to all realizations.

Several interesting observations can be made from this visualization. First, a streakline

from the separation point can be easily identified. This shows a “wavy” or “flag-flapping”
type of behavior. The first disturbance occurs at a streamwise location of xz00. The “size”
of this motion appears to be of order 0.500. This observation is consistent with the “com-

munication” results of Section 3.1.1. That is, it would seem unlikely that the pressure fluc-
tuations from this first or downstream disturbances would influence the upstream
boundary layer given their relatively small scale. Note also that the asymmetry of the
shear layer is evident from the movement of the marked fluid into the entrainment region

(y<0), whereas very little clean fluid is observed for y>0. The visually apparent “growth”

37

of the shear layer is clearly at a length scale which is significantly smaller than the bound-

ary layer thickness.

Based on these observations, and equipped with the data to be presented in this and subse-
quent sections, several distinct regions of the flow field have been identified. These are
shown schematically in Figure 3.7. These regions and their defining characteristics will
provide a conceptual framework and vocabulary which will be useful in describing the
results to follow. The “canonical turbulent boundary layer” is the region which exhibits
stochastic values of the velocity field that reflect the evolution of a boundary layer that is
still attached to a flat plate. The “sub-shear layer” represents the turbulent region where
the flow is measurably different from that which would be present if the bounding wall
were present. The region marked “significant irrotarimialflzwtrmtions” represents the
domain where measurements of the velocity fluctuations exhibit essentially no high fre—
quency (small scale) motions. It is inferred that these motions are driven only by the
unsteady pressure field created by the shear layer. Lastly, the transitions between sub-
shear layer and the developing shear layer are not well defined. Hence no such boundaries

are drawn in Figure 3.6.

Point statistics were acquired using a single hot-wire probe which was traversed across the
shear layer at 12 streamwise locations. The data were acquired at 5 kHz for 60 seconds. If
the characteristic velocity is taken as U0/2, and if the shear layer width is taken to be
60(x), then the 60 seconds represents 500 to 2000 shear layer widths for the 12 locations.
The sensor was aligned in the z direction to recover the time series of q as described in

Section 3.1.2. Note that in the high speed free stream the sensor records q=u=U0, and in

the entrainment stream the sensor records q=v=v(,. The time average velocity (21) and the

38

standard deviation of the fluctuations (6]) were then computed from the time series values.
The mean velocity is shown in Figure 3.8 as an .r-_v plot, and in Figure 3.9 as a contour
plot. The values of Z] are shown in Figures 3.10 and 3.1 1 in x-y and contour formats,
respectively.

A striking feature observed in these data is the streamwise invariance of the boundary
layer statistics. As a specific example, the mean profiles are identical to the boundary
layer profile for y/00>2 in the range 0<x/00<29. The profiles of Z], are streamwise invari—
ant for y/00>4 in the range 0<x/00<29. This can be observed by the horizontal (i.e.,

streamwise invariant) contours in Figures 3.9 and 3.1 l, or by close examination of the
individual profiles in Figures 3.8 and 3.10. Several histograms of velocity are given in
Figure 3.12 for the points: (x/00,y/00)=(0.73, 2). (7.2, 2) and (19, 2). The similarity of
these histograms provides a specific example of the nominal streamwise invariance of the

velocity field.

It is noteworthy that these results are consistent with those of Hamelin and Alving (1996),
referred to as HA. In this reference, a moving belt was used to remove the inner layer of an

Re9=2800 turbulent boundary layer in order to study the stochastic evolution of the outer

scales. The present work is similar in that the streamwise evolution of the boundary layer
is measured after an abrupt change in wall boundary conditions. In the case of the moving

wall boundary condition, the mean flow field was streamwise invariant for y/00>2 in the
range 0<x/00<34 (see Figure 5 of HA). The RMS of the velocity fluctuations at a given y

location was streamwise invariant for y/00>4 in the range 0<x/0(,<30 (Figure 7 of HA).

39

These regions of streamwise invariance are nearly identical to those realized in the present

experiment.

Two variables of interest were calculated from the mean velocity profiles: the momentum
thickness of the shear layer 0(x) and the maximum slope of the mean velocity. The calcu-
lation of 0 requires the integration of velocity across the shear layer. Because the magni-
tude of the velocity was measured as opposed to strictly the u component (see equation
3.2), an arbitrary cut-off of q/ U0 =0.05 was used to end the integration on the low speed
side. This is a common technique used by Foss (1977) and Hussain and Zaman (1985).
These results are shown in Figure 3.13. The growth rate in the region x/00>60 was found
to be 0.035 as measured previously by Foss et al. (1977). Note that Hussain and Zaman
(1985) found a linear growth rate of 0.032. The second variable derived from the point sta-
tistics that will be of interest in subsequent sections is the maximum slope of the velocity

profile. These data are shown in log-log format in Figure 3.14 for streamwise locations

(1 .8<x/00<550). This extended domain shows that the function (dq/dy)max~l/x0'79

matches well throughout the sub—shear layer region as well as further downstream.

A variable of interest is the maximum value of 2] obtained at a given streamwise location.
This information was extracted from the data shown in Figure 3.10, and plotted in Figure
3.15. The most interesting feature of these data is the local maximum value realized at x/

00:2. This peak occurs at a lateral position of y/0(,=0. 15. Previously, the existence of a

local maximum has only been observed in shear layers with a laminar separating boundary

layer, whereas the tripped or high Reynolds number boundary layers lead to a monotonic

increase in 2); see Hussain and Zedan (1978). The present data support the concept that the

40

sub-shear layer region near the separation point is a viscously dominated flow region

which shares many of the characteristics of low Reynolds number separations.

3.4 Shear Layer Diagnostics From Entrainment Measurements

A ubiquitous feature of the single stream shear layer is the existence of large scale coher-
ent motions. These motions are characterized by a correlation length in the spanwise
(homogeneous) direction that is many times larger than the local shear layer thickness.
The existence, physical makeup, and dynamic significance of these large scales are still
topics of debate and research. The origin of these motions is believed to be from the
inflectional instability of the shear layer velocity profile. The flow visualization and point
statistics described in the previous section indicate that this instability does not influence
the entire separating boundary layer. Rather, it is likely that the influence of these motions

is limited to the shear layer region.

A simple method of detecting these large scale motions is to acquire velocity time series in
the irrotational flow field near the shear layer. Many studies have made use of the fact that
the time signature of velocity in both the high speed and low speed streams can be used to
identify features of the large scale motions within the shear layer; see for example, Brow-
and and Weidman (1976), Koochesfahani et al. ( 1979). Browand and Trout (1985), or
Narayanan and Hussain (1996). The efficacy of this technique is based upon the pressure
fluctuations which are mechanistically linked to the motions. This leads to easily detect-

able fluctuations in the velocity field of the irrotational flow surrounding the shear layer.

The following subsections will utilize the q( t) data that were used to create the point statis-

tics in section 3.3. Specifically, at each of the 12 streamwise positions (see Figure 3.8), a

41

single y location on the entrainment side of the shear layer was selected which exhibited
notable fluctuation levels with minimal turbulent intermittency. These locations were
determined by visually inspecting the time series data of q(t). If high frequency fluctuation
were not present, the flow was assumed to be relatively irrotational. The locus of points

which satisfied this criterion can be described by the line y/00=0.0259(x/00)-4.0. The fluc-

tuation intensity at these points was approximately (7/ U0 z 0.02.

3.4.1 Convection Speed of the Coherent Motions

A variable of interest is the “convection velocity” of the spanwise motions, Uc. Many

investigators of both single and two stream shear layers have found the convection veloc-
ity to be the mean velocity of the two streams with equal density. See for example, Dimot-

akis (1986) and references therein. The ratio would be predicted to be Uc/Uo=0.5 in the

single stream shear layer. The time averaged convection velocity was measured from time

series data of two hot-wires placed in the entrainment stream with a streamwise displace—
ment Ax. The lateral locations where q(y) z: 0.05 U“ were determined to have significant
signal amplitude in the irrotational region. The time-delayed cross correlation of the two

hot-wire signals was computed from these time series to locate the time delay to the first

observed positive peak: At. The average convection velocity is defined as UC=Ax/At.

Figure 3.16 shows the normalized convection velocity 2Uc(x)/U(,. Of interest is the low
value of 0.54 near x=0 and the increase to the expected value of 1.0 for x/00>120. This is

inferred to be a result of the sub-shear layer phenomena where only the vorticity from the

near wall region participates in an inflectional instability. Several statements can be made

42

to support this inference. The first observation is that the velocity gradients are very steep
in the region x=y=0 which lead to length scales which are considerably smaller than the
local integral scales. Since the length scale of the most amplified motions will scale with
the velocity gradient at the point of inflection, it is reasonable that only the fluid in the y=0

region will participate in the instability.

A second argument to support this idea is related to the vorticity field of the turbulent
boundary layer upstream of separation. It is known that the filaments of vorticity in the
very near wall region (say y+<1) are parallel to the wall, and are perpendicular to the wall
shear stress vector. In contrast the outer region of the boundary layer does not have this
constraint which leads to a vorticity field with a high level of disorganization. It can be
inferred from these observations that the “organized" near wall vorticity participates in the

first instability, whereas the “disorganized” outer motions do not.

It is possible to calculate “how much ” of the time averaged vorticity participates in the
first instability based on the above observations. Given the relatively small contribution of
the y component of velocity to the vorticity in the near wall region. the mean streamwise

velocity can be written as:

L
17(y) = jwdy 3.3
0
The velocity scale of the first instability is observed to be 0.54U0 from Figure 3.14 near

x=0. This corresponds to y+<42 at x=0 in terms of the mean velocity as observed from

Figure 3.2. Evaluating equation (3.3) at y+<42 shows that 54% of the vorticity which sep-

arates from the boundary layer participates in the first instability. The growth of the sub-

43

shear layer in the streamwise direction indicates that an increasing amount of vorticity
from the boundary layer participates in the coherent motions as the flow evolves in the

streamwise direction. This is also supported by the slow increase in UC to the nominal pre-

dicted value of 0.5U(,.at large x/00 locations.

3.4.2 Spectral Properties of the Entrainment Stream

The spectral properties of the near entrainment stream were used to investigate the coher-
ent motions of the sub—shear layer region. The power spectral density (the Fourier trans-
form of the autocorrelation) was computed for the 12 streamwise locations (identified in
Section 3.2),. These data are shown in “log-log” format in Figure 3.17. The amplitude of
each data set was shifted vertically in order to distinguish the data sets on one figure.
These data show a distinct local maximum, or “bump” in the spectrum for all streamwise

locations except x/0(,=0.73. For example, the hump at x/00=29.5 occurs at 10Hz. Nearly
identical results for the range 37<x/00<4800 can be found in Hussain and Zaman ( 1985)

hereafter, referred to as H-Z. It can also be observed that the frequency, where the local

maximum occurs1fm, decreases with downstream distance. This frequency can also be

identified using the first peak of the autocorrelation function. This leads to a similar,
although not necessarily the same, identification of the preferred frequency. Figure 3.18

shows an example of the autocorrelation function for x/00=7. 17. It should be noted that at

all locations, the “hump” does not imply that the signal is periodic. These humps should

be interpreted only as a preferred frequency at that streamwise location.

The frequency valueszfjn (normalized by 00 and U0) were calculated from both the spectra

and autocorrelations for the region l.8<x/00<650; see Figure 3.19. The frequency data

44

from H-Z in the region 37<x/00<4800 (taken from Figure 5 of that reference) are also

shown in Figure 3.19. Note that the boundary layer Reynolds number at separation was

Re9=428 in H-Z. The agreement is quite good in the overlap region between these data

sets, which indicates that the dimensionless frequency is not a strong function of the Rey-

nolds number of the separating boundary layer.

The frequency data can also be made non-dimensional using the free stream velocity and
the local momentum thickness. That is, a non-dimensional frequency: f" can be defined as

f*=fm0(x)/UO. These data are shown in Figure 3.20. Thef“ value measured at the location
x/00=600 is also shown on the figure. The functional dependence off“ can best be inter-

preted by considering the functional dependence of both the momentum thickness and the
frequency from the respective curve fits of the data. Although the momentum thickness
does not grow linearly in the very near separation region, the data shown in Figure 3.13

can be roughly approximated by:

x
— 0.0312(5) + 1.079. 3.4

() 0

0(x)
0

 

in the region 0<x/00<100. A fit of the frequency data is represented by:

1319. _ 0.133

_ . 3.5
U0 (x/00)0'714

 

The dimensionless frequency becomes:

/.*(X) :: [ET—W = 0-0041(X/90)0'286

0

+ 0.143(x/00)-0'714. 3.6

Equation 3.5 is shown as the solid curve on Figure 3.20. This presents two limiting regions
of interest: the very near separation region, and the limiting behavior for large x. The

larger values of x/00 are clearly dominated by the first term of equation 3.5. This equation

45

would predict anf* which is slowly increasing for large x/00 values because the momen-

tum thickness grows linearly, and the exponent of the frequency dependence is -0.7 1. This
is in contrast to the expected result that the dimensionless frequency would be constant. It
is also noted that the largest downstream distance measurement provided f"(x/

00:600)=0.026. This is slightly larger than the apparent value of 0.021 from the x/00<100
data, although this difference is close to the uncertainty of the measurement. Note also that
H-Z measured a value of 0.024, although the uncertainty in those measurements seems to

be considerably greater.

The x dependence off*k has two possible explanations: either the/"k values are dependent
on x position, or the shear layer is still “relaxing” from the initial upstream boundary con-
ditions (i.e., the turbulent boundary layer). The latter of these explanations is likely the

correct one. It can be observed from the H-Z data in Figure 3.19 that a fm~llx dependence

approximates the data for x/00>1000, wherefjn~ l/xo'7| as measured in the present data,
seems to correlate well with the H-2 data for x/0(,< l 000. It is possible that the present

shear layer would also tend towards a 1/x dependence for larger x values given the agree—

ment withe the H-Z data in the region where the data overlap.

The non-dimensional frequency in the limit of small x values is also of interest, particu-
larly with regards to the sub-shear layer described in the previous sections. In terms of

equation 3.6, the second term dominates the magnitude off* for small x/00. This is

because the frequency grows exponentially for small x, and the momentum thickness

approaches the value at separation. A preferred scaling might be considered by relating the

frequency to the local maximum velocity gradient: fjmlu) = fm/(du) . Note that

(1 y m ax

46

this definition is equivalent to using the “vorticity thickness” scaling. These data are also

shown on Figure 3.20. This definition was motivated by the fact that both variables follow

a power law ~x'" as shown in Figures 3.14 and 3.19. In fact, the exponent of the maximum
velocity gradient: n=0.79 closely matches the values of 0.71 measured for the frequency
dependence. The gradient based definition provides a dimensionless frequency with con-

siderably less streamwise dependence than f" for x/0(,<20.

3.4.3 The Initial Instability

The frequency of the initial instability has been the focus of numerous studies (see, e.g.,
Ho and Huere (1984)). The understanding of how the boundary layer vorticity is re-dis—
tributed downstream of separation is of considerable interest both fundamentally and tech-
nologically. Manipulation and periodic forcing of the boundary layer at separation can
take particular advantage of the first instability in order to control the unsteady dynamics

of the shear layer.

The near separation region of a laminar shear layer is known to be unstable to small ampli-
tude perturbations due to Kelvin-Helmholtz instability. Linear stability theory has been
used by many authors to understand and predict the frequency and growth rate of the first
instability; (see, e.g., Ho and Huere (1984)). There are several important assumptions of
the linear theory which often include parallel flow, laminar flow with infinitesimal distur-
bances, and a specific velocity profile such as a hyperbolic-tan gent function. Although
these assumptions are not physically realized in any shear flow, the theory predicts the fre-
quency of the fundamental instability for various geometries with great success. The effi-

cacy of this theory for shear layers and jets was reviewed by Ho and Huere (1984) and

47

Thomas (1991). The Strouhal number of the most amplified frequency is defined as

St 0 = me / U 0. Note that some references use the average velocity in the case of two

()
stream shear layer. This would change the definition by a factor of two; the referenced val-

ues have been divided by two where appropriate for comparison to single stream results.

In contrast, the effects of laminar-turbulent boundary layer states and Reynolds number on
the initial instability have not been previously been extensively investigated. Although the
initial development region for laminar flow has been studied extensively, most shear layer
studies which have a turbulent boundary layer do not consider the initial instability of the
shear layer. Ho and Huere (1984) comment that the most amplified Strouhal number

changes from 0.016 for a laminar flow to 0022-0024 for turbulent flow due to “presently
unexplained reasons.” The remainder of this section will explain their observation in terms

of the sub-shear layer phenomenon presented in Section 3.2.

The single sensor hot-wire data described in the previous sections was used to observe the
frequency of the initial instability. Using the spectral information shown in Figure 3.17 it
can be observed that the time-averaged spectra of velocity shows a local maximum at the

streamwise locations x/0021.87. Although these time series are not periodic, the local

maximum is interpreted to represent the preferred frequency at that location. However, the
lack of a local maximum in the spectra for x/0(,=0.73 does not imply that a preferred fre—
quency does not exist. The flow visualization shown in Figure 3.5 indicates that the first
instability does occur in this region. This has motivated a more detailed examination of the

time series data at this location to identify if a preferred frequency exists at this location.

48

Several segments of time series data are shown in Figure 3.21. These data show that
pseudo-periodic motions do exist, although the frequency, phase and amplitude vary con-
siderably during a relatively short time period. This explains why the spectra failed to
show a local maximum value at a particular frequency. This result is also physically rea-
sonable given that the vorticity, which is convected from the viscous region of the bound-

ary layer, is highly unsteady and highly perturbed by the turbulence in the “log region”.

An alternative method of determining the frequency content of a signal is to look for the
time delay of the first local maximum of the autocorrelation. The inverse of this time will
indicate the frequency. The autocorrelation of the entire signal, along with several short
time autocorrelations is shown in Figure 3.22. The time series data were divided into 0.1
and 0.23ec segments for analysis. The autocorrelation was then calculated for each of
these time segments. A distinct local maximum was found in the autocorrelation function
for 51% of the 0.1sec data and for 37% of the 0.2sec data. A histogram of the observed
frequencies is shown in Figure 3.23. The mean frequency from the 0.1sec samples was
130Hz. The mean value from the 0.2sec samples was 146Hz. This value corresponds

well with fm=124 which is calculated from equation 3.4. This agreement adds confidence

that a preferred frequency does exist for the initial instability, and that its value can be

accurately determined.

The next step in the determination of the initial Strouhal number is to choose velocity and
length scales to non-dimensionalize the indicated frequency. The standard choice of U0
and 00 would not be appropriate since it is only the inner most wall region that is partici-

pating in this first instability. It seems illogical to use. for example, the boundary layer

49

momentum thickness (00) to scale a phenomenon which does not involve the entire
boundary layer. The new scales of velocity and length will be U0” and Befito imply that it

is the effective velocity scale and only the effective momentum thickness that lead to the

first instability and roll-up.

A logical choice for the velocity scale is Ucff =0.54U0. This is justified by the apparent

convection velocity of the first instability discussed in Section 3.3.1. The effective
momentum thickness of the sub—shear layer was determined by fitting the hyperbolic tan-

gent profile to the mean flow measurements acquired at x/00=0.73. The data were fit to the

equation:

 

QO’) _ if; ( .v
U0 — 2 [I '1' tanh Z—GJ-D] 3.7

in the range 0<F1 <0.56U0. The mean velocity data and equation 3.6 are in good agreement
over this range; see Figure 3.24. This equation was chosen specifically because the linear
stability theory calculation of the most amplified wave number is derived from the tanh

profile. The effective momentum thickness was determined to be 06ff=0.5mm (Go/Gm~

=19.2) by a least squares fit between the data points and equation 3.6.

Finally, the valuesfmz 1 30112, Ucff=0.56U0=4.06m/s, and Oeff=0.5mm lead to Sto=0.0165.

The uncertainty of this measurement is estimated to be :10%. This value is in excellent
agreement with both linear theory, and all existing literature for laminar shear layers. This
indicates that the highly unsteady and turbulent sub-shear layer is likely dominated by a
viscous, linear stability mechanism. This result is both unique and surprising given the

strict assumptions which are required by the theoretical calculation of the most amplified

51)

Strouhal number. It is argued that the methods by which the frequency, length, and veloc-
ity scales were chosen are logical, and not arbitrary. Additionally, only high Reynolds
number separated boundary layers will have such a large discrepancy between the true and

the effective momentum thickness values.

3.5 Spanwise Correlation Field of the Near Separation Region

The data which have been presented thus far have been statistics acquired at a single x-y
plane. Because the flow is stochastically homogeneous, these data are representative of
the entire flow field excluding the side wall boundary layers. In addition to these data, it is
of interest to document the multi-point correlation of the velocity field in the spanwise
direction. For example, the velocity measurements taken in the entrainment side of the
shear layer indicated that motions exist within the shear layer with sufficient coherence to
be detected well into the irrotational flow. The spanwise extent of these coherent motions
is of considerable interest. For example, fluid-structure interactions will depend greatly on
the coherence of the pressure field created by the large scale motions. That is, a flow with
a high level of coherence will tend to cause large fluctuating forces on mechanical struc-
tures because the net force is the integral of the pressure field. The pressure fluctuations
from smaller scale motions will tend to cancel in magnitude leading to smaller fluctua-

tions in the applied forces.

The data were acquired using a rake of 8 single wire sensors in the flow aligned with the z

direction. The sensors were equally spaced at Az/0(,=2.05. A schematic of the rake and its

orientation is shown in Figure 3.25. These probes were traversed through the near separa-

tion region using the same experimental grid as shown in Figure 3.8. Specifically, the rake

51

was traversed through the y direction at 12 streamwise coordinates. The time series data

were recorded for 60 seconds at a data acquisition rate of 5000Hz at each spatial location.

The presentation of these results is divided into the two following sub-sections. First, sev-
eral short time segments of velocity from the low speed side will be presented in Section
3.5.1. These data will help to orient the reader to the data acquisition procedures and they
will assist in the interpretation of the results. Secondly, the zero time delay cross-correla-
tion was calculated for all the (x,y,Az) positions measured. These data are described in

Section 3.5.2.

3.5.1 Example realizations in the irrotational stream

Four locations in the domain were selected on the low speed side of the shear layer corre-

sponding to x/00=3.53, 12.0, 40.5, and 84.2. The respective y/00 locations (-3.7, -5.37, -

10.9, and -19.6) correspond to the positions that were used in the determination of the
spectral content of the entrainment stream; see Section 3.4.2 and Figure 3.17. The time
coordinate was non—dimensionalized using the local value of the peak frequency as shown

in Figure 3.17. That is, 71.5(time)"‘fm for Figures 3.26 through 3.29. This represents the data

in terms of the integer number of averaged wave periods.

Time series data from these locations are shown in Figures 3.26 through 3.29. The begin-
ning of each time record shown in these figures was chosen arbitrarily (i.e. when the A/D
system was activated) and was not intended to isolate any specific observed features.

Longer time records yield qualitatively similar representations of the flow field. The con-
tour plots show the fluctuating component of velocity from all 8 sensors normalized by the

standard deviation of the respective signals. The legend for the gray scale shows that light

52

shading implies positive fluctuations in velocity, and dark shading indicates negative fluc-

tuations. For each of the contour plots, the time traces from the wires located at z/00=2.05

and 12.4 are shown for comparison.

A feature observed in these figures is the growth in the spanwise coherence of the entrain-

ment velocity as x is increased from 3.5300 to 84.20”. Specifically, Figure 3.26 shows

clear evidence that the motions which led to these fluctuations do exhibit a coherence in
the spanwise direction, although the spatial extent is limited. Figure 3.27 could be
described similarly, although the spatial extent of the coherent motions has obviously
“grown”. It is instructive to reiterate the idea of the sub-shear layer when considering
these spanwise correlations. As a specific example, recall that at the streamwise location

x/00=12, the flow field for y/00>2 is stochastically identical to the boundary layer.

In order to directly contrast this large spanwise coherence with the boundary layer flow,
the correlation coefficient for these four locations as well as one position within the

boundary layer (at x=0, y=0.500) are shown in Figure 3.30. It is clear from these measure-

ments that the larger spanwise coherence is developed downstream of the separation point,
and within the sub-shear layer region. The length scale of the correlation measurements is
clearly many times greater in the region near the sub-shear layer compared with the

boundary layer, even at small x/00 values.

The continued growth in coherence is also seen in Figures 3.28 and 3.29, where the irrota-
tional fluid motion is beginning to appear nearly two dimensional. These measurements
closely parallel those of Bowand and Trout (1980, 1985, collectively referred to as BT)

where 12 wires were placed in the low speed side of a two stream shear layer facility. They

also found the correlation field to increase in the streamwise direction. The short time real-
izations of the velocity fluctuations shown in BT look very similar to those shown in Fig-

ure 3.29.

3.5.2 Correlation Coefficient Data

The correlation coefficient R(Az) was calculated (as shown in Figure 3.30) for each of the
(x,y) locations (see Figure 3.9 for the grid of data points used). The data set can be repre-

sented by the “volume” of points (x,y,Az). The visualization of these data was made possi-
ble by “slicing” the volume in planes perpendicular to the x and 2 directions. First, the

streamwise locations x/00=3.53, 12.0, 40.5, and 84.2 (i.e., the positions used in Figures

3.26-3.29) were selected. Figures 3.31 through 3.34 show contours of the correlation coef-
ficient at these locations. The mean velocity profile is included with each of the figures for
reference. Figures 3.35 through 3.39 show contours of R at specific Az values. The solid
boundary surrounding the contour plot represents the spatial extent where the data were
acquired. The dashed boundary represents the contour line where the fluctuation levels are
1% of the free stream velocity (see Figure 3.10). Correlation data are shown in the region

where the fluctuations exceed 1% of U0.

Several important features of the flow field can be observed from these measurements.
First, the central region of the shear layer shows quite low spanwise correlation values.
This is in contrast to the low speed irrotational flow measurements which exhibit a large
correlation length with respect to the local length scales. Although it is apparent that the

irrotational fluctuations are caused by the motions existing in the turbulent fluid, the corre-

lation measurements in the central region of the shear layer are “washed out” by the high

levels of small scale turbulence.

A second feature observed from these data is the difference between the high and low
speed sides of the shear layer. Specifically, the correlation length grows smoothly and con-
tinuously from the separation point, whereas the high speed side shows a discontinuity in
the data near x/0(,=55. These data are consistent with the concept of the sub-shear layer,
and also provide insight as to how the sub-shear layer evolves into a fully developed shear
layer. Specifically, the vortical fluid which occupied the outer region of the boundary layer
does not participate in the instability which leads to the large scale spanwise motions. As
the flow progresses in the streamwise direction. more of this vortical fluid is drawn in to
“participate” with the coherent motions. The dramatic change in the correlation field near

x/00=55 indicates that the motions have become strong enough to have a measurable

effect on the high speed irrotational fluid. The time averaged convection velocity of the

coherent motions at this location is approximately 82% of the final value of 0.5U0.

3.6 Summary and Conclusions

A number of observations can be drawn from the collection of data described in the pre-
ceding sections. These data have revealed information which does not currently exist in
the fluid mechanics literature. As stated in the introduction, the unique features of the cur-
rent facility that have allowed these discoveries are the high Reynolds number at separa—

tion (4650) and large length scale (00:9.6mm) of the flow. For comparison, Hussain and

Zaman (1985) studied a similar geometry, albeit the boundary layer Reynolds number at

separation was 428, and the momentum thickness 00:0.5mm. Many of the features

described in this chapter have been at scales of the order of 00 or smaller which would
make them unnoticeable in a facility with a small value of 00. In the following, the main

conclusions are given in a numbered format followed by a discussion and interpretation of

the relevant data sets.

1. The stochastic properties of the boundary layer at separation (x=0) appear to be unaf-

fected by the flow field downstream of separation.

This conclusion is supported by several of the experiments. First, the zero cross-correla-
tion magnitudes between velocities measured in the boundary layer and in the entrainment
stream is a strong indication that a causal relationship does not exist between the two
regions. The lack of correlation in the low-pass filtered time series further supports this
result. The reported stochastic properties of the boundary layer show that the moments of
the velocity and vorticity field are in good agreement with the existing literature on
boundary layers without separation. If the shear layer did have an effect on the boundary
layer at separation, it was not apparent in any of the point statistics or correlation measure-

ments.

The streamwise invariance of the statistics in the outer region of the boundary layer pro-
vides additional evidence to support this conclusion. In other words, not only do the shear
layer fluctuations not effect the boundary layer at x=0, there is no measurable effect on
much of the y>0 region for a considerable length downstream. This will be discussed fur-

ther as conclusion number (2) below.

The fact that a high Reynolds number boundary layer which separates at a 90 degree edge

is not affected by the separation is important for several reasons. If the separation region

56

was to be modeled using a computational method. the upstream boundary conditions
could be completely specified at x=0 by the stochastic properties of the approach bound-
ary layer. If in contrast, the separated flow field were to effect the stochastic properties
upstream of separation, then the inlet conditions to the computation would have to be

specified at some x<0, rather than at x=0.

It is likely that the boundary layer to shear layer communication is quite dependent on the
Reynolds number. Laminar and low Reynolds number turbulent flows at separation are
likely to be more sensitive to the perturbations caused by the downstream motions. A
future study of this topic may attempt to measure an objective indication of the communi-

cation for a range of Reynolds numbers.

2. The outer region of the boundary layer is stochastically streamwise invariant for sev-

eral integral lengths downstream of separation.

This streamwise invariance of the boundary layer properties is quite evident in all of the
data identified above. For example, the contour plots showing statistics in the x-y plane
(Figures 3.9, 3.1 l, 3.35-3.39) all show invariant properties of the flow from in the region
0<x/00<50. This adds information to a body of literature which exists on the topic of inner
and outer scales in the boundary layer; see, for example Klewicki (1989), and Marusic and
Perry (1990). The present observations indicate that not only are the outer scales not
dependent on the local production of near wall turbulence, but that the flow is not even

effected by the large scale fluctuations of the shear layer measured in the y=0 region.

3. The separation region can be described by the “sub-shear layer” in which only the near

wall vorticity participates in the initial shear layer instability.

57

The region which defines the sub-shear layer can most easily be identified by the wedge

shaped region defined by unns/Uo>0'07 on Figure 3.1 1. Many of the salient features that

are associated with canonical shear layers have been identified in this region. It can also be

observed from the u /U0>0.07 contour on Figure 3.1 1 that the “spread angle” of the sub-

rms

shear layer and of the fully developed shear layer are the same.

An important feature of shear layers is the presence of coherent motions which develop
from the inflectional instability mechanism (see Ho and Huere (1984 for a review). These
motions are known to move in the streamwise direction with a well defined convection
velocity. In fully developed single stream shear layers, this velocity is known to be

approximately 2anv/Uo=l .0. The convection velocity in the sub-shear layer region is
shown in Figure 3.16. These data indicate a dramatic rise in the convection velocity from
20.54 at separation to 1.0 near x/00=120. It is inferred from these data that only the near
wall vorticity is “rolling up” or participating in the inflectional instability. A quantitative
estimate of the fluid domain which participates in the initial roll-up has been identified as

that below the q/ U0 2 0.54 isotach. Equivalently, this bound is given by y+<42 for the

present boundary layer. As the sub-shear layer grows into the outer part of the boundary
layer, more of the vorticity is pulled into the coherent motions thereby increasing both the

width of the sub-shear layer and the convection velocity.

The present understanding of the vorticity field within a turbulent boundary layer is con-
sistent with the current findings. That is, the stochastic and instantaneous properties of the
motions which define the inner and outer scales of the boundary layer are significantly dif-

ferent. The near wall motions are described by, for example Adrian et al. (2000), and Pan-

58

ton ( 1999). In these references, the near wall region is understood to be comprised of
groups of coherent motions which define localized shear layers. These result in “vortex
packets” which evolve into the outer scales of motion. It is intuitively reasonable that the
near wall region with a high level of organization of vorticty would be subject to inflec-
tional instability immediately downstream of separation. In contrast, the motions farther
from the wall are usually considered to be more “disorganized” and not closely correlated
with the motions very near to the wall. The current measurements of spanwise vorticity
fluctuations at x=0 support this viewpoint. Again, it is intuitively reasonable that these

motions would not participate in the roll-up phenomenon.

4. The recognition of the sub-shear layer allows a physical explanation for why shear
layers with a turbulent boundary layer at separation approach self-similarity at smaller

x/00 values compared with shear layers with a laminar separation.

The streamwise location where the flow field becomes self-similar has been found to be
quite dependant on the state and Reynolds number of the boundary layer at separation. For
example, Bradshaw (1966) recommends a value of x/0(,>1000 for self similarity given a
laminar separation. Hussain and Zaman (1985), Bruns (1990), and the present study have

found that x/00>100 is sufficient for the mean velocity to exhibit self—preservation when
the separating boundary layer is turbulent.

These differences can be explained in terms of the length scales of the turbulent vs. lami-
nar boundary layers. The suggestion was given by Dimotakis and Brown (1976) that it is

not the downstream location as defined by a number of original integral length scales (e.g.,

00) that is important for the establishment of a self-similar condition. Rather, it is the num-

59

ber of “interactions” between the large scale motions of the flow. The number of interac-
tions will depend on the ratio of the streamwise location to the spacing of the initial
disturbance. Specifically, the number of interactions mlx) is estimated (equation 14 of
Dimotakis and Brown) to be

m(x) = log2(x/ln) 3.8

where 10 is the spacing of the initial disturbance. For a laminar boundary layer at separa-
tion, the criteria x/0(,>1000 equivalent to a value of m==4. 1n the present study the stream-
wise location x/0(,=100 corresponds to a value of m between 6 and 7, given that the
spacing of the initial disturbance was [0:00. This small value of initial disturbance length
compared with 00 is a direct result of the presence of the sub-shear layer as described in
Section 3.3.

5. The dimensionless frequency of the initial instability agrees with linear stability theory

when appropriate effective velocity and length scales are identified.

The measurement and prediction of the dimensionless frequency of the initial stability has
received considerable attention in the literature: see. for example, Ho and Huerre (1984)
and Thomas (1991). This focus is a result of the universal nature of the initial instability in
a variety of shear layer and jet flow fields. Additionally, linear theory has been quite suc-
cessful in the prediction of this dimensionless frequency for many experiments with lami-
nar boundary layers at separation. However, differences in the measured frequencies have

been found in shear layers with turbulent boundary layers at separation.

The present study has shown that the above noted differences are a result of the different

length and velocity scales that exist in a turbulent boundary layer. Specifically, a laminar

60

boundary layer can be fully characterized by the free stream velocity and momentum
thickness. In contrast, a turbulent boundary layer has outer scales (free stream and
momentum thickness) and inner scales (uI and War). The data presented suggest that only
the near wall vortical fluid participates in the initial instability at this Reynolds number. It
is then quite reasonable that the outer scales would not be functionally related to the mea-
sured frequencies. A hypothesis that can be inferred from the above statements is that the
Strouhal number will properly collapse using inner scaling. From the present data, this

suggests that Utflz l 514,, and 0,,fi~10v/ut. Additional measurements are required to test
this hypothesis.

6. Preferred frequency and spanwise correlation measurements have shown that the char-
acteristic large scale motions observed in self-similar single and two-stream shear lay-

ers are present in the sub-shear layer region.

Several investigators have studied the properties and importance of the largest scales of
motion in shear layers. These motions appear to have a large spanwise coherence and a
significant fraction of the total turbulent kinetic energy of the flow. Experimentalists have
sought to measure these features in order to better understand their role and dynamic sig-
nificance; see, for example, Hussain and Zaman (1985), Browand and Weidman (1976),

and Browand and Trout ( 1980, 1985).

The present experiments have found that a number of the features measured in both two
stream and single stream shear layers in the self-similar regions have also been observed
in the sub-shear layer region. For example, the preferred frequency exhibited by both the

sub-shear layer and self-similar shear layer can be described by the relatively simple equa-

61

tion: f,,,~x’”'7’ (for x/0(,>l ) as shown in Figure 3.19. 11 can be inferred that the physical
mechanisms which lead to this preferred frequency are present throughout the flow field
downstream of separation. Similar results were found from streamwise dependence of the
maximum slope of the mean velocity profile; see Figure 3.14. Note that the y location of

the maximum slope was found in the center of the sub-shear layer.

A third feature, observed in the sub-shear layer region. was the large spanwise correlation
length; see Figure 3.31. These data clearly show that large scale motions develop at small
x/0o values. The spanwise correlation of the velocities in the entrainment stream increases
as these motions convect downstream. These observations are consistent with observa-

tions made by Browand and Trout (1980, 1985) in the self-similar region of a two stream

shearlayen

The above noted observations justify the terminology “sub-shear layer.” Specifically, the
region exhibits characteristics which are associated with self-similar shear layers, while
the prefix “sub” is meant to describe the fact that the shear layer is growing adjacent to
what appears to be the outer region of a flat plate boundary layer. One distinct difference
noted between the sub-shear layer and the self similar shear layer is the peak value of a

which increases smoothly from 0.12U0 in the boundary layer to a constant value of nomi-

nally 0.16U0 for x/0(,>150.

62

, I . I
. 41/.

\
\\
3\\
\ \
\
k\
\\i

U,;,=7.|M/S

   
 
 

T

5=|OOMM y

“I:
/
1

. g——‘x
e=9.6MM FT?
zx/uT 20.05MM '

arrrrttrrrrtrr
VE(Y=-°<9=O.055UO

H—9.75M——1/——-—

Figure 3.1 Schematic representation of near separation region.

 

 

30
4+ ' n00: +0.1
25 — g
r . #90: -0.1
20 e l
u+=2.31n(y +)+6.7
'5 ” u+=2.51n(y +)+5.5 /
\ I,

\ /
+ “
\‘\

    

+
U=y

     

 

 

 

 

 

 

 

Figure 3.2 Boundary Layer Mean Velocity. The y+=0 point (x) on the inset figure was
added based upon the no-slip condition for clarity. Note that U0/u1=25.51

 

 

 

 

 

 

/,

5.0

 

(1.1) r 1 * ‘
-l(l.(l -5.0 101) 15.0 20.0 25.0 30.0

 

 

0.0
1

la) +I

 

 

 

 

 

 

 

 

 

 

 

 

 

071
1 V“
+ 1 1 L A._._A A ' L A 1 A l
10 100 1000
y+
Figure 3.3 Boundary Layer Velocity RMS

0.7

"'0— Mean value from slt)pc of sin gle sensor traverse
0.6 ~

"Error Bars" denote +/- 1 standard deviation of vorticity
calculated from the four wire vorticity probe time series.
0.5 ’
04 . Vorticity (RMS/Mean)
0.3 » //A'"""
0.2 ~ -1» " .11 g 1 .
'0” 2109
().l *
0.0 r
-01 e A 1+- 1 A A . A .1
10 100 1000

y+

Figure 3.4 Mean and RMS of vorticity at separation.

64

 

 

2 .
(pman/un 100 3

 

Model Spectrum

_______ Measured Spectrum '

 

 

 

 

 

 

 

00 1.0 8.0 3.0 4,0 50

Figure 3.6 Sample flow visualization of the near separation region.

65

FFEESTREAM A...

 

———>

 

 

. « ' ‘ W'—
‘ A, u I c l I 1. n..." .W_»’,..
I T LAYER M" DEVBJPING SEAR | AYE?
a
«0
. -.. a .
”“1541. Van‘- . 1.11.1.V1w
(4.7.7.373 :: 7:71.71 77-513.; 2 1,4. 5.-.: .3234}? -

l

: \\ .

I; ”Ir-u NW1‘1

4 ““32:-

’i

I

.1 ..... ,_ SIGNIFICANT
l ‘ ‘k

. E 1 51' S‘IREIIIWN”‘--- lmorAmNAL

\.

 

Figure 3.7 Figure of flow regions; not to scale.

 

 

 

 

 

1.0 ”90
——<>— 101
(Two —A— 114.4
0.8 1 “—0— 63-3
—g»— 54.2
—>(—- 40.6
0‘6 i ——+_— 29.6
——+-— 19.3
0.4 + ”‘0
+ 7.19
+ 3.54
0.2» + 1.88
+ 0.13
—-.—— 0.1
0.025

 

Figure 3.8 Mean velocity profiles for 0<x/00<100

66

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Y/Oo “.en "‘70 -00—
G‘fl; '20 030—-
005 °"°‘—--—~.
-20
l l 1 l l 1 l l l
0 50 100
XIeo
Figure 3.9 Contour plot of mean velocrty, q/ U n
0.14 X/9(x=o)
~ ’ —-o- 101
(]/IJ()(112 ' -—7§—— 844
—0— 68.8
0.10 1 _g_ 54.2
-§<—- 406
0.08 r
+ 29.6
———+——— 19.3
0.06 r
+ 12.0
+ 7.19
0.04 L
+ 3.54
+ 1.1111
0.02 ~
+ 0.73
. ,. + .01
0.00 . - - 1 - - .
-30 -20 -10 0 10 20 30

y/6(x=0)

Figure 3.10 RMS of velocity for 0<x/00<100

67

 

 

 

 

 

 

 

 

 

 

 

 

y/ 00 #011
C Q
0 V 0.13
0.11 ‘-
0.01 ‘—
\ W

 

x /00

Figure 3.11 Contour plot of velocity RMS q/ U”

 

0.160

 

 

 

 

X/9(,=0.73, y/et)=2 o’ ‘ // “90:7,2' y/90=2

U) ./ ”00:19. y/Oo-_-2
1: , ,r
.9. 1
‘53 0.120 r
N
S
<3
G)
1...
91-1
0 0.080 »
G
O
'5
Q
8
u...

0.040 r

’I O
0.000 «L—wflf--O' % I;
' 0 75 0

1.
q/Uo

Figure 3.12 Streamwise evolution of velocity histogram at constant y/00=2.0.

68

 

4.50

 

 

 

0/00

4.00 r-

i ,9/
3.50 - /
3.00 1— /

‘5}.
/ 0 ~ 0.035x

2.50 —— /

1 A/,/’/

l //
2.00 ‘ A /

/
1 /

l /

1.50 1 /
A A

A
1.00-15 1 1 ' 1

0 20 40 (10 so

Figure 3.13 Momentum thickness vs. x/00.

dq
a§1lllillt

1000

100

ll)

 

 

 

x/00

Figure 3.14 Maximum velocity gradient vs. x/0U

69

 

120

 

0.17

qma/U() 0.16 E

015*

  

0.130 r

  

 

0.125

0.120 '-

0.115

 

 

0.110I ‘ ‘ L r ‘
0.0 5.0 10.0 15.0 20.0

 

 

150 200 250 300 350

x/GO

0 50 100

Figure 3.15 Peak value of velocity r.m.s as a function of streamwise location.

 

 

 

0.60

 

0.40 —

0.20 _ < » l

 

 

 

 

 

 

0.001A1111.1111111111111111 11111
0 20 40 60 80 100 120 200 300 400 500 600 700

x/00

Figure 3.16 Convection velocity of spanwise motions. Note the break in the x-axis is used

to show the x/00<120 region in more detail.

70

x/00

 

 

 

0.73
1.87
3.53
7.17
12.0
19.2
2
=9 29.5
8 40.5 W
8" 54.0 g m
— 68.6 W"
84.2
100.1 m
ill-11mm
100le
.1 . .mnm
' ”Mm
Mum-u
0.1 l 10 100 1000

frequency (Hz)

Figure 3.17 Velocity spectra of entrainement for the 12 streamwise locations measured.

 

1.00

0.80 l—

0.60 peak location for

040 L fm calculation

0.20 e

Autocorrelation

0.00 e

-0.20 e-

 

 

 

_().40 L 1 J. l A 1 1 l 1 1 1 l ' 1 1 ¥ ' 1 1 J 4 1 1 1 1 r 1 1 1 1
0 10 20 30 ~11) 50 (10 70 80

m SCC

Figure 3.18 Sample of the autocorrelation function in the entrainment stream at x/00=7.17

71

 

 

   

 

 

 

 

 

 

 

 

10‘l 1;
f m 60 E
0 -7 _ ~1/x
10 “ g . /
; I Spectra results «’
: A Autocorrelation results
10'3 :- ———-— 0.133(x/0014’7'
lt . Hussain and Zaman (1985)
-4 l 1 1111111 L 1 1111111 lllllll l l 11
l()
10" 10' 102 103
x/GO
Figure 3.19 Streamwise variation in peak frequency.
016 . Values at
x/90=600
0.14 .
0.12
0.10 «a 0.098
0.08
0.06
0.04 .
“ <,_ 0.026
0.02 >
F l 1 L
0.0114111111211112
0 0 20 40 ()0 80 100

x/OO

Figure 3.20 Non-dimensional frequency vs. x/GO

72

Fig

We.

 

 

 

 

 

 

 

 

     

 

 

   
 

_2 1 l 1 l 1 l 1 1
0.00 0.02 0.04 0.06 0.08 . 0.l0
“rm

Figure 3.21 Time series samples from x/80=O.73.

 

1.01 . . 2

0'8 " Time averaged
_

0.4 L

0.2 1—

 

0.0 b

 

 

 

-0.2 1

. 1 1 1 1
0.000 0.005 0.010 0.015 0.020

 

time delay: I (sec)

Figure 3.22 Autocorrelations of the time series at x/90=0.73. The long time averaged as

well as 0.1 second averages from the time series in Figure 3.21 are shown.

73

 

0.15

E 0.1secsampks

I 0.2 sec samples
0.10 1~

  
    

  

     

   

’fi \‘V‘l'f'
;ra

Q
A

. - .11. -.- ,‘
.. . _. .
' _ .-4 13"“ ‘Hz‘. ‘ '.-
. . ~
' " a -2

0.05 ~

fraction of realizations

   
 
 

    

  
 

I IICICCIIIICUCI
- -.--.---.
. ‘I

'I ‘ ‘- ..|&‘ Q"

IIIIIIIIIIIIIIIIIIIIIIIIII!
______--______
r.n_.l .“v.h
.~

IlllllIlIIIlIIIIIIIIl

 

 

 

    

 

 

 

   

0.00 *
60 80

 

 

I
t—ul III-IIII-I-I-I-I-II-I-C.

00 120 140 160 180 200 220

frequency of local maximum

Figure 3.23 Histogram of observed frequencies from short time autocorrelations.

 

 

     

 

 

 

0.7 . _
0.6 E "
q/Uo T
0.5 1—
0.4 U ..
; K - ‘“ [1+tanh(y/2 Gem]
0.3 .‘." 2
0.2 Uc,1,1/U0=0.56
0.1 ; Gen/90:0.OSZ
00 h. II 1 1 1 1 l 1 l 1

 

-().6 -04 -02 0.0 l 0.2 0.4 0.0 0.8 1.0

Figure 3.24 Comparison between tanh function shown and the velocity data at x/00=0.73.

74

 

 

 

 

 

 

 

 

 

 

:21
::
L_J
:2 1:1 1 IACM
:1
:2
3:1

 

 

 

 

 

Figure 3.25 Schematic of 8 wire rake in shear layer.

75

3 2190:205

ec=0.02

 

 

W

“ 2190:1221

 

 

Figure 3.26 Velocity fluctuations at x/60=3.53, y/90=-3.7. Contour plot shows magnitude
of velocity from 8 sensors normalized by the standard deviation. X-Y plot shows time
traces taken from wires located at z/Bo=2.05 and 12.4 for the same time period. Note that

the correlation coefficient indicated in the top figure by ‘cc’ represents the long time aver-

age and not that for the time segment shown.

76

     
      

cc=0.23

 

\
1}

     
 

J

      

V \
\
\7190=12.4

 

o-Arouammumco

 

Figure 3.27 Velocity fluctuations at x/00=12.0, y/60=-5.37. See Figure 3.26 for descrip-
tion.

77

 

1: up 1 -
.31.“

.11
o
8

7100

 

NQJIUImNQID

.1

 

O

 

 

Figure 3.28 Velocity fluctuations at x/00=40.5, y/00=-10.9. See Figure 3.26 for descrip-
tion.

78

7100:105 ’2/90=12,4 CC=O.89

 

--
Ga) 4.50 0,00 150 3.0)

    

2/80

 

 

 

 

 

 

 

 

 

 

 

O-‘NO’&UIO’)\IQO
I

 

Figure 3.29 Velocity fluctuations at x/00=84.2, y/60=- 19.6. See Figure 3.26 for descrip-

tion.

79

 

 

 

 

 

.\:
0.8 » ' - (1090:1142, y002-19.6l
0.6 »» - . (x190=4().5, y00=-1().9l
0.4 L ' '

. /(Xflo=l2.0, 300:5.7)
0.2 ~ '
. (xz‘Bo=3.53. )0(,=-3.7) ' .
00 L N‘F— ' -_
1 \ (x=0. y10(,=().5)
-02 1 g 1 - . 1 1
O 5 1() 15 20

Az/GO

Figure 3.30 Spanwise correlation coefficient of the data shown in Figures 3.26 through
3.29.

80

al./U0 1.00 .

 

0.50

 

 

0.00 '

_L —L —L —L .15
O -‘ N (A) «5
f f 1 r

 

 

 

 

 

 

 

 

 

 

041003-0103me

 

Figure 3.31 Spanwise correlation field at x/00=3.53.

81

1.0

 

q/Uo
0.5

l

 

 

0.0 ‘ 1 1 1 . 1 1

4.1.14.1
O-‘NCO-h

 

 

 

 

 

 

 

 

O-‘NOD-hO‘lODVCDCD

 

 

Figure 3.32 Spanwise correlation field at x/00=12.0

82

 

1.00 r

q/Uo
0.50 r
1

 

 

 

 

 

   

 

 

 

 

141
M6013

12—

11 L

10»

91—

8» \

61

5- \

41

31 ¥JTXX

2 ~ _;

1 _

O111111111111111111111111111

 

-1O -5 0 1O 15

5
y/Oo

Figure 3.33 Spanwise correlation field at x/90=40.6

83

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9~ \‘
8. B
\007
7c
6_
4.
3’ ~~-'~—-———-0.33
2» --~047
Mom
1 ' 073
Ollllll1llllllllllllilll14111111111
-20 -15 -10 -5 O 5 10

V90

Figure 3.34 Spanwise correlation field at x/00=87.2

84

 

 

 

 

 

 

y/Oo

 

 

 

 

 

Figure 3.35 Correlation coefficient at A2190=2.08. Note that the solid boundary repre-

sents the spatial extent of the measurements (shown by the grid). The dashed boundary
represents the region where urmslU0>0.Ol. A schematic of the boundary layer mean pro-

file near x=y=0 is shown for reference.

y/Bo

 

X/00

85

Figure 3.36 Correlation coefficient at Az/60=4.l6. See Figure 3.31 for description.

___-

 

 

 

 

 

 

 

20 - fr ---
1 —>--—---"""‘""" ,x'" o.95-—-«
aw --- ----....._.-§-__ """" W032
W006 \ A 0.19

 

 

 

 

 

 

X /60

Figure 3.37 Correlation coefficient at Az/60=6.24. See Figure 3.31 for description.

 

 

 

 

 

 

 

 

 

 

y/Go

 

 

'20 ~ 0.92

 

X/Oo

Figure 3.38 Correlation coefficient at Az/9(,=8.32. See Figure 3.31 for description.

86

 

 

 

 

 

 

 

y/GO

 

-20

 

 

 

x /00

Figure 3.39 Correlation coefficient at Az/9(,=l4.5. See Figure 3.31 for description.

87

4.0 The Developed Single Stream Shear Layer

4.1 Introduction

This chapter presents the results from an investigation of the developed region of the shear
layer. The word “developed” is used to indicate that the time averaged flow variables can
be represented in a self-similar form. Specifically, any dimensionless stochastic quantity,
say, a(x,y), can be written as a function of the dimensionless transverse coordinate: a((y-

yC)/l), where yC represents the “center” of the shear layer, and [(x) is a local integral length

scale of the shear layer. In the present measurements, the momentum thickness: 0(x) was
used as the length scale of choice. In shear layers, this length scale grows linearly. That is,

0~x. This was shown to be valid in the measurements presented in Chapter 3 for x/00>6O.

These data also indicated that the profile of the fluctuation intensity (see Figure 3.15) is

approximately self-similar at streamwise locations x/9(,>300. This is consistent with the
results of Hussain and Zedan (1978) and Bruns et al. (1991). The data presented in this

chapter were acquired at downstream locations 384<x/00<7O7 '.

The following two sub-sections (4.1.1, 4.1.2) are organized as follows. The open research
questions that have been considered in the present work are reviewed in Section 4.1.1.
This will include a review of the relevant literature that will serve to outline where this
research fits with respect to the current knowledge base. The second sub-section of this
introduction will describe the experimental program. Because several different types of

information can be drawn from a single data set. the details of the experiments will be

 

1. One exception is the PIV measurements used for the calculation of the circulation density. These
data were acquired at x/90=170; see Section 4.7.

88

described completely in section 4.1.2; the data presentation in sections 4.2 through 4.8

will then refer to these experiments.

4.1.1 Motivation and literature survey

The beginning point for most shear layer studies is the development of the mean velocity
profile. Information about the self-similar nature of the mean flow variables is often
described in turbulence text books. See, for example. Pope (2000), Tennekes and Lumley
(1972), Holmes et al. (1992), and Townsend (1975). However, the discussions typically
imply a two stream shear layer geometry. In the present work, the analysis for the mean
flow variables will be given specifically for the single stream boundary conditions with
the entrainment flow perpendicular to the free stream velocity vector. This has led to sev-

eral new simplifications, and the identification of errors in the literature; see Section 4.2.

Research papers which describe single stream shear layers include Leipmann and Laufer
(1947), Wygnanski and Feidler (1970), Champagne et al. (1976), hereafter referred to as

LL, WP and CPW, respectively. These papers support the self-similarity of the measured

moments of the velocity values. For example, CPW compares the profiles of a , V , and

W for several downstream locations in their facility. In addition, they cite measurements
from LL and WF and others. Although some differences between the LL data and those of
WP and CPW are noted, these data support the assertion that the velocity field in the self
similar region of the single stream can be considered universal. That is, the results are in
agreement without significant dependence on the particular upstream and entrainment

flow boundary conditions created by the different facilities.

89

There are a number of turbulent characteristics that can be inferred from the single and
multi-point measurements of velocity time series. For example, velocity gradient statis-
tics, which are described in Section 4.2, are used to compare the predictions of isotropic
and axisymmetric turbulence. Also, WF used their velocity measurements to construct an
estimate of the balance of the turbulent kinetic energy (TKE) equation. This balance was
calculated for the present measurements as well; see Section 4.5. The value added from
the present observations comes from the higher Reynolds number and improved spatial
resolution of the present facility. This has allowed an improved ability to estimate terms
such as dissipation, which are quite difficult to accurately obtain. Also, the present mea-
surements were acquired at two streamwise locations which documents the scaling of the

TKE equation.

The statistical characterization of the scales of fluid motion were also of interest, in addi-
tion to the point measurements described above. Multi-point measurements of the velocity
field were acquired in the shear layer in order to document the spatial and temporal char-
acteristics of the velocity field; see Section 4.6. The relatively large size of the present
facility provides an opportunity to study fluid motion which range in length scale from
approximately 1.2m for the largest motions, to 0.2mm for the smallest (Kolmogorov)
length scales at the farthest downstream location of the tunnel. This has permitted suffi-

cient spatial resolution to measure many of the features of the small scale turbulence at
relatively high turbulence Reynolds number ReAzIO‘}. This range of scales and the turbu-

lence Reynolds number are at the upper end of what is usually achievable in laboratory

flows.

90

Single and multi-point statistics of the vorticity field were also acquired in addition to the
velocity measurements. The description of free shear flows is often considered based on
the vorticity, rather than the velocity field. For example, vorticity measurements in turbu-
lent boundary layers have received considerable attention; see, for example, Klewicki and
Falco ( 1996), Ong and Wallace (1998), or Andreopoulos and Honkan (2001). Similarly,
Antonia and coworkers have studied the statistical properties of turbulent wake flows, as

well as grid turbulence; see Antonia et al. (1987, 1996, 1998), and Brown et al. (1987).

Although the statistics of the velocity field in the single stream shear layer are well docu-
mented as described in the previous section, the vorticity field has received less attention;
see, for example, Foss (1994), Bruns et a1. (1991). In these references, time series of the
spanwise vorticity are acquired and discussed. The moments of the population of mea-
sured values are described, as well as autocorrelation results. Many of the conclusions
drawn from these references will be described throughout the discussion of the present

vorticity measurements; See Section 4.3.

Vorticity in two stream shear layers has been studied by Balint and Wallace (1989), and
Lang (1985) and Kim (1989). These references also describe many of the important fea-
tures of the vorticity histogram and temporal characteristics. However, there are still a
number of open questions about the vorticity field which remain unanswered by these
research efforts that will be addressed in the present measurements. For example, the seal-
ing of the vorticity field with both integral time scales and dissipation time scales will be
compared. The vorticity fluctuations will also be compared to predictions based on isotro-

pic and axisymmetric turbulence.

91

4.1.2 Measurement program

The measurement program is summarized as follows. Two spanwise vorticity probes (see

Section 2.2.3. 1) were traversed through the shear layer at the streamwise locations x/00=

384, and 675. These probes recorded time series of (1),. (l) u, v, and w. Two data files

y‘
were recorded at each spatial location. The first data set was acquired at a rate of 40kHz,

for a duration of 30 seconds. The second was acquired at a rate of lkHz, for a duration of
150 seconds at the location x/90=384, and 300 seconds at x/00=675. The purpose of the
separate data files was to acquire sufficient statistical information about both the large and
small scales of motion without excessive usage of computer disk space. The 40kHz data
were used to obtain vorticity time series, and spectral information at high wave numbers.
The lkHz data were used to obtain moments of the pdf of the velocity time series and for

spectral information at low wave numbers.

In a second and third experiment, the streamwise vorticity probe (section 2.2.3.2) and the

multiple X-wire probe (section 2.2.3.3) were traversed through the same locations as the
spanwise vorticity probe as described above. This provided time series of (1)x and Bt/dz.

The data were acquired in two sets as were the spanwise and lateral vorticity measure-

ments.

Although the above measurements were sufficient to obtain statistical convergence for the
computed moments of the pdf of velocity and vorticity, two separate experiments were
executed in order to fully characterize the vorticity pdf at four different turbulent Reynolds

numbers. Th first experiment used a single vorticity probe to measure (02 at the center of

the shear layer, at the streamwise locations x/90=384 and 707 with the shear layer facility

92

in its standard configuration. These data were acquired at a rate of 40kHz for 60seconds
per file. A total of 30 files were then obtained for a total of 30 minutes of data acquisition
for the two locations. This experiment was then repeated using the same physical probe
locations, but with the free stream velocity set to 3.5m/s. This was accomplished simply
by opening an access panel to the tunnel upstream of the first set of turning vanes. This
allowed a significant fraction of the air supplied by the axial fan shown in Figure 2.1 to
escape to the laboratory. Although this could potentially create both asymmetry and
unsteadiness in the tunnel, measurements indicated that the dimensionless velocity profile
at x=0 was unaffected by the change, except for a notably thicker boundary layer. The spe-
cific conditions of the shear layer at this new condition were not evaluated from surveys in
the tunnel. However, the goal of the experiment was only to create similar conditions with

a different turbulence Reynolds number (Rex). The comparison of the (02 histograms at

these conditions is given in Section 4.3.

93

4.2 The Velocity Field

4.2.1 Integral constraints on the mean flow

The simple boundary conditions of the single stream shear layer allow several analytical
results to be obtained for the mean velocity field. These relationships will be instructive
when considering the self-similar growth of the shear layer. Because the flow is statisti-
cally invariant in the z direction, the following equations will be assumed to be “per unit
width” of the flow. Figure 4.1 shows the control volume and coordinate system that will
be used. It is assumed that the control volume has been drawn such that the line B-D is a
streamline in the high speed side, and that the entrainment flow is perpendicular to the line
A-C. The conservation of mass for this volume can be written as

) u l” 14 V1)
—d’— —11=—L. 4.1
J: U0 } A U()( ) U1)

The x component of the momentum equation can be written as

2 JD 2
fi - “_ , = 0 4.2
Uody Uody
A C
since the mean pressure gradient is zero. Equations 4.1 and 4.2 can be added to obtain:
u2 u u2 14 V
f[-—2 — 21—] (11’ — f[_7 '- U] (1)" = —U(;L 4.3
C U0 () A U“ n ()
Noting the definition of the momentum thickness (0) and taking the limit as L —) O:

(16 v0
d—x — U' 4.4

0

This derivation has been taken directly from Bruns (1990), and has also been derived in

other references. For example Holmes et al. (1996) integrate the x component of the Rey-

94

nolds averaged Navier—Stokes equation to arrive at the same result. Equation 4.4 indicates
that the zero pressure gradient shear layer “growth-rate” is identically equal to the ratio of

entrainment velocity to free stream velocity.

A similar result can be obtained for the displacement thickness of the shear layer. The dis-

placement thickness is defined as:

5*Ejfl(l——l—l—)-y — {all d(.~-l—Jfl (1) d 4.5
A M—a—fl' UnAyzy

The equality above can be shown by integrating by parts, only if the origin of the coordi-
nate system is located at point ‘A’, which ensures that 8* is positive definite. Equation 4.5
indicates that the displacement thickness also represents the position of the centroid of the
spanwise vorticity. Although the definition of 8* is identical to that used in standard
boundary layer analysis, the physical interpretation is different. Specifically, 8* is not
actually a ‘thickness’. but rather a distance from the arbitrary origin at point ‘A’ to the
location of the centroid of vorticity. As with boundary layer analysis, the displacement
thickness also represents the location where a solid boundary could be located with a ficti-
tious uniform inviscid flow which has the same mass flow rate as the actual viscous (non-
uniform) mean velocity profile. This is illustrated in Figure 4.2. Note that the magnitude

of 8* depends on the specific choice for point ‘A’. However, in the following analysis only

the derivative d8*/dx will be considered, which does not have this dependence.

In wall bounded flows, the shear stress leads to an increased value of 8* in the streamwise
direction. In the shear layer the addition of fluid from the entrainment side causes the dis-
placement thickness to move in the negative y direction. This can be quantified leading to

a result analogous to equation 4.4. This is obtained by taking the trivial integral:

95

161 1111-101 1111 = 0 4.6
A C

and adding to equation 4.1. This results in:

) V
u (I (.1
j (l — U)dV—f(l — Z7) (1} — —U(;L. 4.7

(1‘ 0 A ()

Noting the definition of the displacement thickness (8*) and, again, taking the limit

L—)O:

(16* v11
a - 77 4'8

0
Thus, the locus of points which represents the centroid of vonicity forms a line whose
slope is equal to the negative of the ratio of entrainment to freestream velocities. The neg-

ative sign in equation 4.8 indicates the preferential “movement” of the vorticity towards

the low-speed side.

From the relations given in equations 4.4 and 4.7. the “spreading parameter” can be

defined as

V, :1:
_‘ _ d9 _ _£z0_035. 4.9

O (JO-13‘ dx

That is, the growth of the shear layer, the slope of the shear layer centroid with respect to
the high speed streamline, and the entrainment of ambient fluid into the shear layer are
necessarily equal. The value 020.035 was determined experimentally in the present facil-
ity; see Figure 3.13. Other researchers have found similar values for the growth rate of the
shear layer. For example, Champagne et a1. (1970) measured d8*/dx=0.035. Ali et al.

(1985) directly measured ve/UO=O.O35, and d6/dx=0.0352. Hussain and Zaman (1985)

obtained a similar value of d0/dx=0.032.

96

4.2.2 Velocity in Self-Similar Coordinates

The idea of self similarity in free shear flows is well documented in most standard text
books on turbulent flow; see, for example, Pope (2000), Tennekes and Lumley (1972), or
Townsend (1975). The analyses developed in these references are given for general shear
layers (i.e, both single and two stream). The present analysis will be given specifically for
the single stream shear layer. The self-similarity hypothesis is satisfied if the moments of
the probability distribution function (pdf) for all field variables can be written as a func-
tion of a single dimensionless similarity parameter. This parameter represents the lateral
position in the shear layer divided by a local length scale. In the present analysis, this will

be defined as:

 

. 5' ' t“ . .1
n 0 4 0

Note that the use of 6 for the transverse length scale of the flow is arbitrary. A general

length l~x is often used to emphasize this point. It will be shown that the relationships to

be derived will take a simpler form with the present definition.

The quantity yc represents the “center” of the shear layer. The only requirement for the
choice of yc is that the locus of points representing y=yc is an isotach (i.e., constant mean

velocity). The locus of points which represent an isotach of mean velocity can be identi-
fied as any line with a fixed slope with respect to the high speed streamlines. This slope

can be characterized by the definition of the parameter:

d Y1.-

dx . 4.11

(X

For example, if the centroid of vorticity is chosen to define 11:0 (yc=5*), then 0t=-o. Note

that most authors choose yC=y0.5=y(u/Uo=0.5). That is, the point where the mean velocity

97

equals half of the free stream velocity is defined as yc. In the present study the centroid of

vorticity was found to be at the location y(17/ U” = 0.48) .

The analysis thus far has utilized the standard coordinates in which the x direction is

aligned parallel to the boundary layer plate. This will be represented by the standard or
“untilted” coordinates (x, y). It will be useful in the following sections to allow the coor-

dinate system to rotate such that the streamwise (x) direction coincides with the isotach

used in the definition of ye. This is termed the “tilted” coordinates (3c, 5)) ; see Figure 4.3.

In the (52, 5’) system, (1:0 by definition, and the boundary conditions will depend on the
choice of ye. This is contrast to the standard coordinates in which the value of or depends

on the choice of yC, and the boundary conditions are fixed.

4.2.2.1 Mean Velocity Components

The streamwise component of velocity can be written in terms of the similarity variable
as:

17(x, y) = U” -f(n). 4.12

The boundary conditions for the (x, _v) coordinate system are
f0]: -°°) = 0J0]: °°) = l- 4.13
In the (5c, 9) coordinates using 0t=-0', the boundary conditions are:

f(fi= -oo) = sin(atan(0t)) = -0.0340f(fi= 00) = cos(atan(0t)) = 0.9994.4.l4

Because the magnitude of or is small, these boundary conditions are nearly identical to

those of equation 4.13. Hence, the difference between the tilted and untilted coordinates

does not represent a significant effect in the consideration of f(n) .

98

The streamwise velocity component was measured at the streamwise locations x/00=384,

and 675 using an X-wire probe. The mean velocity calculated from these time series is
shown in Figure 4.4. The curve fit shown was taken from Bruns (1990) using a different
experimental facility. The “collapse” of the data onto a single curve indicates that the
streamwise component of velocity does have a universal form for the single stream shear

layen

The time average lateral component of velocity can be written as:
V(x,.v) = GU.,-g(n) 4.15
where the factor of 0' is used so that g(n)~0( l ). The relationship between the two similar-

ity functions f and g can be determined from the two-dimensional continuity equation:

a.a_
8x By-

The spatial derivatives of n are given by:

0. 4.16

021-1 021-5% 91
ay‘e’ax‘ ems) 4'17

Equation 4.16 can be expressed in self-similar form by combining equations 4.12, 4.15,

and 4.17:
. _ .. g
g _ f (n + o) 4.18
This can be integrated to give:
or . .,
.401) = 8(—°°)+3,f(n) + JD 51/ (:14;- 4.19

The last term can be integrated by parts to give an equivalent expression:

99

or . .
.401) = 8(—°°) + (5 +0./m) — Mama 4.20
The boundary conditions for gm) are determined as follows. For the untilted coordinates,
again, yc=8*, and Ot=-O‘ are used. The integral on the right hand side of equation 4.19 is

zero because the origin is defined as the centroid of f" . The boundary conditions for gm)

become:

8(n=-°°) = 1, 8(n= °°) = 0 4.21

which is consistent with equation 4.15. In the tilted coordinates however, 00:0, and the

value of the integral on the right hand side of equation 4.19 will depend on the choice of
ye. For yC=8*, the integral is again zero, and the boundary conditions become:

s(fi=-°°) = 1101: co) = 1. 4.22
The value of g(fi= 00) is also apparent from the vector decomposition of the freestream
velocity shown in Figure 4.3. The functions gm) and g(fi) were evaluated from equation
4.19, and are shown in Figure 4.5. Note that only the boundary conditions at n=— 00 were
enforced. The fact that the correct values of n(+oo) were obtained is an indication that

f(T|) was measured accurately, and that the analysis above is consistent.

Close examination of the functions gm) and g(fi) provide information about the shear
layer which is not observed in the streamwise velocity profile. In the untilted coordinates,

17 decreases on the low speed side of the shear layer which represents the mean stream-

wise acceleration (011/311 > O) of the entrainment fluid. Also, gm) < 0 is observed for

l()()

n > —0.38 . This indicates a net flow of fluid, which originated on the high speed side,

towards the center of the shear layer.
A consistent but different view of the flow field can be understood from the tilted coordi—
nates. An interesting feature of the g(fi) profile is the positive lateral mean velocity com-

ponent observed for all f] . Thus, to an observer aligned with the centroid of vorticity, fluid
is observed to move from the low speed side to the high speed side of the shear layer. In
this view, the shear layer entrains and “extrains” fluid at the same rate, as already observed

from the boundary conditions (equation 4.22).

Although the ‘tilting’ of shear layers with respect to the free stream flow has been recog-
nized throughout the literature, the failure to recognize the non—zero value of or in the for-
mulation of the self-similar equations has resulted in incorrect statements and conclusions
in the literature. For example, Holmes et al. (1996) derive the self-similar equations for a
single stream shear layer. Because the movement of the shear layer into the low speed side
is not explicitly accounted for, the equations imply (1:0. That is, the equations are derived

in a tilted coordinate system. A contradiction is then encountered (see page 54 of Berkooz

et al.) in which the statement is made that g(fi= 00) = 0. The argument is made that
“extrainment is never physically observed.” The resulting analysis (e.g., equation 2.23 of

Holmes et al., 1996) is therefore incorrect.
Note that geometrical reasoning (based on equation 4.9) argues that the isotach aligned

with the streamwise direction occurs at f] = l . Note however, that this istotach is not a

streamline. In the untilted coordinates, a small negative value of \7 is observed at 11:1,

indicating the movement of fluid towards the low speed side. The direct experimental

101

measurements in room coordinates show that the mean velocities which are acquired at
constant y=0 location result in 51/ U0 = 0.685 . corresponding to n=0.90. This small dif-
ference between the predicted value of n=1 and the determined value of n=0.9 is likely a
result of the growth of the displacement thickness in the boundary layer of the outer

(y=lm) wind tunnel wall.

4.2.2.2 Profiles of the Fluctuating Velocity

The probability distribution function (pdf) of the unsteady fluctuations of the three com-
ponents of velocity can be characterized by profiles of the moments of the pdf. In this sec-
tion the root mean square (RMS) of the fluctuations are considered. Higher order moments
of the velocity field in a single stream shear layer are examined in detail in several refer-
ences. See, for example, Wygnanski and Fielder (1970) or Champagne et al. (1976). The

spatio-temporal characteristics of the velocity field will be considered in Section 4.6.

The RMS of the streamwise velocity fluctuations (17) was calculated from the time series

data acquired at x/00=384 and 675. These data are shown in Figure 4.6. Similarly, the
RMS of the lateral (v) and spanwise (w) fluctuating components of velocity are shown in
Figures 4.7 and 4.8 respectively. These measurements agree well with the measurements
published in the literature. For example, the maximum value of fl/Un was reported to be
0.168 and 0.176, and 0.165 by Champagne et al. ( 1976), Wygnanski and Fiedler (1970),
and Bruns et al. (1991) respectively. The present data indicate a value of 17/ U0 = 0.17 at
both locations: x/00= 384 and 675. Similarly, the maximum value of \7/U0 was found to

be 0.12 by Wygnanski and Fiedler (1970), and 0. 135 by both Champagne et al. (1976) and

Bruns et a1. (1991). The present data indicate a value of 0.12 at x/0(,=384, and 0.13 at x/

102

00:675. The general agreement of these statistics supports the inference that the facility
provides a canonical single stream shear layer as studied in these other references, and that

a self-similar state of the flow field was achieved.
4.2.3 Profiles of the velocity gradient variance

The determination of the velocity gradient moments: (811/ 8x1.)2 is of interest. For exam-

ple, the description of the flow field in terms of the vorticity field (Section 4.2) or the dis-
sipation of kinetic energy (Section 4.4) will make explicit use of the measured values and
their relations. Secondly, the velocity gradients represent information about the smallest

scales of motion. The relationships between the various derivative moments can be used to

validate assumptions about the small scale motions.

A complete description of these moments requires the measurement of nine independent

gradients, written in matrix form as:

Bu Bu Bu
5? 5 a7.
01,. = £1" 91 ‘1’ 4.23
9") Bx Ely Bz
a_w 21» a_w
-ax By 821

 

 

In the present work, only six of these, highlighted in bold face in equation 4.23, were mea-
sured. This choice of measured quantities is partially due to experimental limitations, and
partly as a result of interest in the theories of locally isotropic and locally axisymmetric

turbulence. These hypotheses require specific relationships among these terms, which can

be validated with these measurements.

103

The measurements were obtained as follows. The streamwise derivatives were calculated

using the Taylor microscale formulation (see, e. g., Pope (2000)). For example:

 

~2

(Bu 2 u

_ = . .24
91- 2 712 4

 

where 2» is the Taylor microscale calculated from the autocorrelation function. The stream-
wise derivatives of v and w were calculated similarly. These approximations do, of coarse
make use of the Taylor’s “frozen turbulence” assumption. In regions where the fluctuation
levels are not small compared to the mean velocity, this assumption can lead to significant
errors. Lumley (1965). and later Wyn gaard and Clifford (1977) derived corrections for the

longitudinal and transverse derivative variances:

2 m 1’ a s 11 red

 

 

 

 

2 2 2
1 .' +2' +2 '
1<au/ax>1 _ I,” V W l: c 4.25
[(a '/a )zlcorrected {(2 u
u x
and
m ca 8' 11 red _.—2 T —1_
[(avi/ax)2] - U ‘ = I+(H + V2/2-l-W2) : C 4.26
[(av'/ax)2](fim I ('r (a! (:22 v

These correction factors are plotted across the shear layer at the two streamwise locations

x/00=384 and 675 in Figure 4.9. Since the individual terms on the right hand side of equa-

tions 4.25 and 4.26 have been shown to follow self-similar scaling, it is expected that these
dimensionless corrections must also. This is found to be approximately the case. However,
the small differences in the measurements on the low speed side are squared and added,

leading to more noticeable deviations between the two streamwise locations. It is of inter-
est to note that the magnitude of the correction in the low speed side is considerable, given

the high levels of turbulence intensity in that region.

104

It is expected that the velocity gradient moments, along with the statistics previously
described, should follow self-similar scaling. However. because the velocity gradients are

related to the small scale motions of the flow, care is needed in the way in which the veloc-

ity gradients are normalized. The dimensions of (Bu 1/ (it!)2 are (1/time2). Thus, a time
scale of the flow must be used to normalize these statistics. The integral time scale
150:0(x)/U0, will not properly scale the gradient statistics because to represents the largest

time scales of the flow. The proper scaling is obtained from the Kolmogorov time scale
since the spatial gradient of velocity is related to the small scale motions:

T
In = f—K L2 = ” 4.27

)

 

where 8 is the rate of dissipation of kinetic energy. The use of dissipation to scale the
velocity gradient statistics becomes obvious given the definition of dissipation in a homo-

geneous flow:
Sal-Bu,-
8 : V —— , 4.28
Bari-ax,-
Equation 4.27 relates the dissipation to the mean flow variables from the relation
3 . . . .
8 ~ U0/0. This scaling can be argued based on the d1mens1onless form of the turbulent

kinetic energy budget; see Section 4.5. Note that the ratio of the integral to dissipation

time scale is proportional to the square root of the local Reynolds number. The velocity
gradient statistics will be normalized by multiplying the measured values by 1.12; this will

be referred to as dissipation scaling.

The measured values of (ad/ax)2 for at the locations x/00=384 and 675 are shown in

Figure 4.10 normalized by dissipation scaling. The corrected and uncorrected symbols

105

refer to the use of equation 4.25. Similarly, the terms (OW/8202 and (SW/Bx)2 were cal-
culated from their respective autocorrelation functions, and are shown in Figures 4.11 and
4.12 respectively. The data indicate that the profiles of the velocity gradients are similar in
shape to the fluctuating velocity components, although the data for n<-l do not exhibit a
monotonic decrease towards the low speed side. It is assumed that these measurements
could have significant error due to Taylor’s hypothesis that are not easily corrected by
equations 4.25 and 4.26. It is also clear that the use of dissipation scaling properly scales

the data (see n>1) from the two streamwise locations into self—similar form.

The terms: (art/By)2 and (aw/am were obtained using the parallel sensors of the

Mitchell probe with 1.4mm spacing; see Figures 4.13 and 4.14. Lastly, the term

(avVaz)2 was obtained using the double X wire probe described in Section 2.2.3.3.
These data are shown in Figure 4.15. Because these measurements do not require the use

of Taylor’s hypothesis, the measurements show a more symmetric shape to the profile.

In addition to providing information about the vorticity and dissipation fields, as will be
discussed in later sections, the variances of the velocity gradients can also be used to test

the symmetry of the turbulence properties. For example, if local isotropy is assumed, then

only the term (Bu/8x)2 is independent and need be measured. This theory is reviewed in

detail by, for example, Monin and Yaglom (1975). The other eight terms are related by:

(8.1-0: 1%)2 = (3502’ 4-29

1—0 1—11-11—11—11—t-—- 2

 

 

 

 

 

106

The assumption of axisymmetric turbulence is less restrictive than the isotropic assump-

tion, as originally put forth by Batchelor (1946). The set of relations between the deriva-

tives are given by:

 

 

Ix)

(3‘92 = (851) , 4.31

 

 

 

 

 

 

 

 

19.-3”: 2 = @153"... 1...
1%,”: 2 = £893.01. 4-35

These relationships were derived and discussed by George and Hussein (1991). In order to

test the validity of these assumptions, Browne et al. ( 1987) used the ratios of the deriva-

tives:

 

 

 

 

(8.4.22 1%)2 1%- (2+5

——,K2 = 2 ,K, = ——,K4 = 4.36

(3.32 (9.02 i (9.02 1%le

If the flow field is truly isotropic, then these ratios should be equal to unity. The axisym-

 

 

 

 

 

 

metric assumption is less restrictive in that no prescribed values are required for the con-
stants. The axisymmetric turbulence condition requires Klsz, and K3=K4. The ratios of
equation 4.36 are plotted in shear layer coordinates for the two streamwise locations in
Figures 4.16 and 4.17. Note that the values with the Taylor correction factors were used.
These ratios do not clearly support either the isotropic or axisymmetric assumptions.

However, it could be argued that the values of K 3 and K4 are closer to being equal than

107

they are to being unity. The same is true for K, and K2 over much of the shear layer. The

consequence of these observations is that the additional information obtained by measur-
ing as many terms as possible will lead to improved accuracy in the predictions of

moments of the vorticity fluctuations and the measurement of the dissipation.

l()8

4.3 Statistics of the Vorticity Field

4.3.] Examples of Vorticity Time Series

Prior to the presentation and discussion of the statistical properties of the vorticity field, it
is instructive to first view segments of the time series data. This will allow several of the
qualitative aspects of the vorticity field to be observed. A sample of the spanwise vorticity

(001(0) time series is shown in Figure 4.18. These data were acquired in the center of the

shear layer: 11:0 at the streamwise location x//80 = 675. The mean vorticity at this location

was found to be —8L7/8v = —9( l/sec) . Several features can be observed from this time
series segment. First, there exist motions which appear very small in scale (high fre-

quency), with very large vorticity magnitude compared to the mean vorticity. The root

mean square of vorticity from the full time record was (I)z = l76( l/sec) . Although the
largest excursions in magnitude appear to be of relatively short time scale, there also
appear to be large regions of fluid whose vorticity is highly correlated. An example of this
can be observed near the one second mark in Figure 4. l 8. The time duration of this large
motion is approximately 0.24 seconds. Using Taylor’s hypothesis, this corresponds to a
length scale of 0.78m, or 3.59. This is a remarkably large scale vortical motion, especially
since the sign of the vorticity is positive. That is, opposite that of the mean vorticity. A fur-

ther discussion of the length scales of the vorticity field will be given in Section 4.5.

Another notable feature of the time series data shown in Figure 4.18 is the presence of
regions of fluid with near zero vorticity. It is possible that this results from large scale
motions which transport fluid from either the high speed or low speed streams across the

shear layer. In Figure 4.19, a second example of the vorticity time series is shown from the

l ()9

same data set. These data show an extended period of time in which the fluid is rotational
(lwzl > 200( l/sec)) , although there is little or no high frequencies observed. This behav-

ior was first described by Haw et al. (1989), and considered in detail by Foss et al. (1995)

in terms of the difference between “intermittency" and "activity intermittency”.

A third example of vorticity time series is shown in Figure 4.20. The location of this mea—

surement was x/90=675, n=3.9. Here, on the high speed side of the shear layer the mean

velocity is not measurably different from the free stream velocity. The vorticity is, how-
ever, still quite intermittent. One can observe from these data that small regions of highly

turbulent fluid convect past the probe.

4.3.2 Histograms of Vorticity

The population of vorticity values that occur at a given spatial location can be character-
ized by the probability density function (pdf). This provides information regarding the
scaling properties of the vorticity fluctuations. Although the shape of the pdf can be com-
pletely described by the moments of the distribution, it is first instructive to observe the
pdf at various conditions. The present experimental facility provides an opportunity to
observe the vorticity histogram over a range of Reynolds numbers and intermittency con-

ditions.

The true (continuous) pdf is approximated experimentally by the histogram of the time
series measurements. In order to “fill out” the entire histogram to best approximate the
pdf, very long time series are required to converge the statistics of the vortical motions
which occur relatively infrequently. Therefore, four time series were acquired in addition

to the traverses described above. These data were acquired in the center of the shear layer,

110

at two streamwise locations: x/00=384 and 675. and two free stream velocities: UO=3.5
and 7.1 m/s. Time series of (i)z were acquired at a rate of 4000Hz for a duration of
30minutes. A summary of the four conditions, including several of the characteristics of

the time series data, are shown in Table 4.1.

The histogram of the vorticity acquired at these points is shown in Figure 4.21. These data

Table 4.1: Measurement conditions data shown in Figure 4.21

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x/GO (Iii/Os) UL.“ U1 (33;) (I): 811:: Rel
‘ 0 0 ((02)
A 384 7.1 0.47 0.15 1 1.3 227 0.23 835
B 384 3.6 0.52 0.17 8.6 136 0.30 411
C 707 3.6 0.55 0.18 10.9 124 0.26 551
D 707 7.1 0.50 0.15 17.4 188 0.32 1210

 

 

are shown in semilog format, with each histogram shifted by an order of magnitude so that
they can be distinguished. The first notable observation is that the shape of the histograms
is similar in all four cases. This is quite remarkable given the range of Reynolds numbers

between the measurements. The shape can be characterized by the exponential tails for

large ‘sz and a lack of symmetry in the population of measurements with small |w2| .

This characteristic shape is nearly identical to the histogram of vorticity measured by
Antonia et a1. (1988) in the wake of a cylinder. In that study, the measurements were taken

in a turbulent region where the mean vorticity was non-zero, and with a turbulence Rey-

nolds number of approximately Rel: aka/v=60. The “cusp” near the (02:0 location is

111

distinctive in both the shear layer and the cylinder wake, and clearly indicates the asym—

metry of the population of measured (it)Z values.

The insensitivity of the exponential tails of the pdf to the Reynolds number is further sup-
ported by the direct numerical simulation (DNS) results of Cao et a1. (1996). These com-
putations were conducted using a cubic domain with periodic boundary conditions.
Forcing at low wave numbers was used to balance the viscous dissipation of kinetic
energy. The circulation of the flow field was calculated around small rectangular domains.
In the limit of a very small circulation domain, this calculation closely resembles the
micro-circulation algorithm used in the present experimental method (see Section 2.2.3. 1).
The population of circulation values was used to create a histogram of the realizations. It
was found that large circulation areas led to histograms which appeared nearly Gaussian.
However, as the circulation area decreased, thereby approximating a true vorticity mea-
surement, the tails of the pdf became more exponential in character, with a distinct cusp at
center of the histogram where (0:0; see Figure 3 of Cao et al. ( 1988). Naturally, the cusp is
symmetric in the computational work because of the symmetry of the boundary condi-

tions.

Time series of spanwise vorticity in a high Reynolds number atmospheric boundary layer
have also been acquired. The histogram of these data are provided here for comparison

with the shear layer results; see Figure 4.23. The flow conditions are characterized as a flat

wall boundary layer with Re9~106. The turbulence Reynolds number was determined to

be Rex=2500 at the probe location y+=3800. The ratio of the probe scale to the Kolmog-

orov length scale in this flow was 2.5. The features which are of interest in Figure 4.23

112

include the Gaussian-like region near the small values of [(1):] and the “tails” of the pdf

which have an obvious positive curvature. Further discussion of these data is provided in

the conclusions (Section 4.8).

The data and references described above vary in both geometry and Reynolds number, but

do not account for varied intermittency levels. The histograms of (it)Z at the center of the

shear layer (11:0) where the fluid is almost always rotational, and at the high speed edge
(n=3.9) where the fluid is intermittently turbulent, are shown in Figure 4.22. It is observed
that the n=3.9 histogram exhibits exponential tails similar to the 11:0 data. Note also that

the slope and magnitude of the exponential portion are not symmetric for positive and

negative realizations. The central region (Ito'I/(I) < l of the histogram at 11:39 is charac-
terized by a Gaussian like histogram (note that a Gaussian distribution is represented by an

inverted parabola in semilog coordinates).

4.3.3 Profiles of the Vorticity RMS

The unsteadiness of the fluctuating vorticity field can be partially characterized by the
RMS level of each component of vorticity. These values were obtained directly using the
time series from the vorticity probes, and indirectly using the velocity gradient informa-
tion described in Section 4.1.3. If the assumption of homogeneous and isotropic turbu-
lence is used, then the mean square of each component of vorticity is equal, and related to

the streamwise velocity gradient by:

 

(3,? = $63!)" 4.37

113

The assumption of axisymmetric turbulence can also be used to relate the vorticity to the
velocity gradients. In this case, only the spanwise and lateral components are equal, and

are given by George and Hussein (1991):

 

(If, = a: = (g—DZ +(3—‘j +(%—? 4.38

The estimates of the vorticity variances were calculated from the velocity gradient statis-

(to
MN

tics presented in Section 4.2.3 based on equations 4.37 and 4.38. These data are shown
along with the directly measured values of (i): and (I), in Figures 4.24 and 4.25. The data

were sealed with both the integral time scales (ti/U”) and dissipation time scales

( VG/ U5 ). The comparison of the two streamwise location indicate that the integral time

scale leads to values of (1)20/ U0 which increase substantially in the x direction. For exam-

ple, the maximum value of (Ike/U” increased from 4.8 at x/00=384 to a value of 7.25 at

x/00=675. The dissipation time scale leads to values of (I): lvO/U; which increased only

slightly from 0.0182 to 0.0205 for the two streamwise locations. This relative agreement
using the dissipation time scaling could be used to argue that the vorticity fluctuations

scale self similarly. However, this increase is considerably greater than the estimated

uncertainty of the measurement (roughly 3x10'4 based on sample size and probe uncer—
tainties, see Section 2.2.3.1.) This increase in vorticity fluctuation levels when using the
dissipation scales will be discussed with the conclusions of this chapter (Section 4.8),
since the explanation of this result will require the calculation of the one dimensional vor-

ticity spectral density function, which is described in Section 4.6.

114

The agreement between all the directly measured values of (by and ml with the isotropic

relation (4.37) is quite surprising, especially given the lack of isotropy shown in the vari-
ous velocity gradient statistics in Figures 4.16 and 4.17. The agreement with the axisym-
metric prediction (4.38) is also encouraging, although not surprising given the support for

the axisymmetric assumptions shown previously, as well as the observation that (byza)Z

throughout most of the shear layer.

Measurements of the streamwise component of vorticity were also acquired. Because the
measurement area of this probe is 5.6mm x 5.6 mm, it is explicit that the spatial smoothing
of these measurements will severely under-predict the actual vorticity fluctuations. For
comparison, the measured values were scaled by an arbitrary factor of 2.7, and plotted

along with the axisymmetric calculation by Hussain and George ( 1991):

—2 _ ACE gag)
(bx — 3 3 +3 82: . 4.39
These data are shown for both measurement locations in Figure 4.26. The arbitrary multi-

plier of 2.7 was only used to show that the shape of the directly measured and predicted

profiles agree well. Assuming the values from the axisymmetric calculation (eqn. 4.39)
are correct, the ratio (was was found to be 1.28 at the shear layer center, compared to

the value (by/('1)z z 1.35 found by the direct measurements of Balint and Wallace (1989) in

a two stream shear layer. Note that the nearly equal values of the spanwise and lateral
components, and the larger values of the streamwise component of vorticity, supports the

assumptions of the theory of axisymmetric turbulence over that of isotropic turbulence.

115

4.4 Analysis of the Reynolds Stress

The Reynolds averaged Navier-Stokes equations can be written as:

_aa,. ' a( P5 2 :9 ‘fi 440
lL— = —— - i'+ ll U—puiu- .

where 3;]- represents the time averaged rate of strain tensor. The modeling of the Reynolds

stress u'iu'j is the subject of considerable research: see. e.g., Pope (2000) for a review. The

physical mechanisms which lead to the Reynolds stresses are still a topic of research, since

no universal model has been found that relates these terms to the mean flow variables.

The single stream shear layer provides an interesting environment in which to study the

Reynolds stresses. One reason for selecting this flow field for study is because the only

off-diagonal component of the Reynolds stress tensor which is non-zero is u'v' . Assuming
that the mean viscous stress and pressure gradient are negligible, the momentum equation
for the streamwise direction (i=1) becomes (see, e.g., Pope (2000) page 1 l3):

aBj+VB_& _ _au'_v'_a(u'2—v'2)
Bx By _ By Bx

The final term of equation 4.41 is usually neglected in thin shear flows, which results in

 

4.41

what is usually termed the boundary layer form of the momentum equation:

[la—E.‘+\7QE — _QL——V:
Bx By_ By °

Given that the left hand side of this equation is composed of variables which have been

4.42

found to be self similar, the Reynolds stress is assumed to be of the form:

0 '

uv

hm) = —2. 4.43
0U

0

1.16

The use of o in the denominator of equation 4.43 is to make h(n)=O( 1). By combining

equations 4.12, 4.19, 4.42, and 4.43, the dimensionless stress gradient can be written as:

71
Mn) =f'{—1+J' f(§)d§}. 4.44

This equation was integrated, and plotted with the experimentally measured values in Fig-
ure 4.27.The agreement is satisfactory, and the values are similar to those measured by
Bruns et al. (1991) and Champagne et al. (1976). Because the gradient of the stress
appears in the mean momentum equation (4.42), it is instructive to look at the stress deriv—
ative as well. The profile of h'm) was determined by fitting the data shown in Figure 4.27

with a polynomial, and differentiating; see Figure 4.28.

One feature of the Reynolds stress derivative is the zero crossing at nz0.6. Consider the

terms of equation 4.42 in the untilted (x,y) coordinate system at this location. In Section
4.2.2.1 it was observed that for T]<l, Biz/By is negative, and so Bit/Bx must be positive.

Since the right and left hand sides of equation 4.42 are zero at 1120.6, then \7 must be neg-
ative as is observed in Figure 4.5. Note that this is in conflict with the comments made by
Townsend (1975). In the discussion following equation 6.10.3 of Townsend, it is stated

that the Reynolds stress gradient is zero at the location where v = 0. However, Townsend
does not correctly account for the non-zero value of 0t in his calculation of gm), which led

to this incorrect conclusion.

The gradient of the Reynolds stress can be further studied using the tensor identity:

I l _ l a t I
B—x-uiuj — —8ijklljwk+§$lljllj. 4.45
I

J

 

Using i=1 as before, and ignoring the streamwise gradients, this reduces to

117

 

Bu' v' — -—
— 2'; V (1).. — 1V (1)" o 4046
By * -

The velocity-vorticity correlation coefficients can be defined as:

v a), 11"(03,'
C = " , C = - 4.47
w

 

 

 

v (1) ~ w (n
z V (0, y

81
'<

By combining equations 4.47, 4.43, and using the scaling relationship 4.27, equation 4.46

becomes:

_..m)=[c,,3 ..,)(U1)(_3f_33_3) 10191110175117 4...

It is useful to make the approximations V = w, and (I): z (I), for a further understanding of

equation 4.48,. These appear to be reasonable approximations, especially for n>0. The dif-

ference between the velocity-vorticity correlations can then be written as:

U U oh'm)
C , — C — ..—3-[ "J " [—] 4.49
1110333 vwz [(1):];— / :8] R89

It is now instructive to evaluate the right hand side of this expression for a given point in
the shear layer, say, 11:0. The first two bracketed terms are found to be 8.7 and 50.0

respectively. The values of h ’(n)=-0.06 and 0:0.035 allow this expression to be written as

-1/2
C33,m33—Cm3 = —o.9i(R.»,) 4.50

For example, at the streamwise location x/00=675. Re9=121,000. the left hand side of eqn.

(4.50) C W m — C v w = —0.0026 . The difference between these correlation coefficients is

,r :
therefore very small, which helps to clarify the “balance” of the terms in equation 4.46.
Specifically, the vortical motions which lead to the existence of the Reynolds stress gradi-

ent is not necessarily due to the lateral fluctuations of spanwise vorticity alone. Rather, it

118

is a very close balance between very large fluctuations of spanwise (z) vorticity compo-
nent moving in the lateral (y) direction, and similar spanwise movement (2) of the lateral

component of vorticity (y).

The values of C M 0 and C v m can be determined experimentally since the time series
\' 2

data used to calculate vorticity includes a measurement of the cross stream velocity com-
ponent. These data are shown in Figure 4.29. The measured values, for the left hand side
of 4.50, show a difference between the correlation coefficient that is of order 400 times
larger than the right hand side of equation 4.50. It is apparent that the measured correla-
tions cannot be measured with the required accuracy to observe the balance of equation

4.46.

119

4.5 'Ihrbulent Kinetic Energy

The turbulent kinetic energy (TKE) equation is examined in detail in this section. This is
of interest both for theoretical and modeling considerations. It is noted that Wygnanski
and Fiedler (1970) previously constructed a budget for the TKE for the single stream shear
layer at a single streamwise location. The purpose of the present repetition of this work is
based on the higher Reynolds number, increased spatial resolution, and improved experi-
mental capabilities of the present experiment. This has permitted the measurement of
nearly all the terms in the TKE actuation, with particular emphasis on the measurement of
the kinetic energy dissipation. This work was similar in scope to that of Liu (2000) who
measured nearly all of the terms in the TKE budget for a wake flow with various pressure

gradients.

The time averaged turbulent kinetic energy is defined as:

k = gun-u"- 4.51

The time averaged transport of k can be stated as:

Dk _ B 3 p' Bu'i Bu'j _3_3an Bu'i Bu'j Bu'j
E - —§;[ui(6+k)+vuj[§;j+-a__r3.]]_uiuja—x3_v5%.??ch 3:33 4.52
(I) (II) (III) (IV) (V) (VI)

 

 

The derivation of this equation is given in, for example, Hinze (1975). Each of the terms
of the equation can be identified as follows. The left side of the equation (1) represents the
time averaged material derivative of turbulent energy, usually termed the “convection” of
energy given that Bk/BtzO. Terms (11) and (IV) represent the net work rate resulting from
pressure fluctuations and viscous diffusion, respectively. Term (III) is the convective

transport of k by turbulent motion, Terms II, III and IV are collectively referred to as the

1 20

the “diffusion” of turbulent energy. The next term (V) is the production of turbulent
energy since the same term with opposite sign exists in the energy balance equation for the
time averaged variables. Lastly, term (VI) represents the viscous dissipation of TKE. The
measurement of the convection, diffusion, and production terms are relatively straight for-
ward, and will be discussed in Section 4.5.2. The estimate of the dissipation term, how-
ever, presents a considerable challenge. It is therefore described independently in the

following section.

4.5.1 Dissipation Measurements.

The dissipation is treated separately due to the difficulty of the measurement and its
importance in both the energy balance of equation (4.52) and the scaling of turbulence

quantities treated in following sections.

As shown in POpe (2000), the dissipation can be expanded as:

 

 

 

2 __ BuiBu, BuI-Buj B27979 453
8 _ vsns’j _ V ijij+ijBr — £+vainj °

where 8 is the “pseudo-dissipation”. The last term of equation 4.53 is proportional to the

spatial curvature of the Reynolds stresses, and is usually considered to be negligible in

free shear flows. Therefore, as with most literature. the distinction between 8 and E is not

(BuiBui
v

J is assumed to be valid.

Bxini

 

made, and the approximation 8 =

In Cartesian coordinates the dissipation can be expanded as nine derivatives of the fluctu-

ating velocity components as:
s =13—“j 1311 11:1 13.) 1—1 1311 11211— 11:11-54

121

 

 

 

 

 

 

The direct measurement of all nine terms with adequate spatial resolution is difficult, and
rarely attempted. Dissipation is typically estimated by using a set of assumptions about the
relations between the different derivatives. Alternatively, dissipation can be estimated by
assuming a specific form of the one dimensional spectral density of the velocity fluctua-

tions.

The measurements of the velocity gradient statistics described in Section 4.2 as well as the
one dimensional velocity spectra were used to estimate the dissipation using a variety of
assumptions. The methods and assumptions for these calculations are given in the follow-
ing paragraphs. Note that each type of estimate is given a capital letter that will be used as
a subscript for e in order to clarify which estimate is being discussed. The various esti-

mates were calculated for the spatial location: x/0(,=675, n=0 for comparison; these results
are summarized in Table 4.2.
The most common method used to obtain 8 is to assume that the velocity fluctuations and

their derivatives are isotropic. With this assumption, the relationships among the terms in

equation 4.54 leads to the approximation:

 

e = mg) 4.55

The streamwise derivative variance can be calculated from the Taylor microscale using
equation 4.24, as discussed in Section 4.2.3. The lateral component of velocity v'(t) can
be used to define the transverse Taylor microscale 213,. 1n isotropic turbulence this is

related to the streamwise microscale by:

)1, = x,/f2 4.56

Two independent estimates of the dissipation can be calculated based on equation 4.55:

122

a, = 30v’3‘3—3, 4.57
and
.7”-
51: = 15v33—33. 4.58

The use of Taylor’s hypothesis is explicit in the calculation of 21“ and 21., in these expres-

sions, and can be corrected according to equations 4.25 and 4.26. These are shown in

TC TC . . . .
Table 4.2 as 8A and EB where the superscript TC indicates Taylor Correction.

In homogeneous, isotropic turbulence, the dissipation can be directly related to the span-
wise vorticity by

a, = 312(1):. 4.59
This was calculated using the variance of the spanwise vorticity as described in Section

4.3.

The next group of estimates attempt to measure several of the terms of equation 4.54 with
assumptions that are less stringent than perfect isotropy. These are referred to as the
“semi-isotropic” relationships. The first assumption is that of “axisymmetric turbulence”
as discussed in Section 4.1.3. There are a number of equivalent expressions that can be
written for the dissipation based on these assumptions. One possible expression that is eas-

ily estimated experimentally was derived in George and Hussein (1991):

., = 1.331;) ..(g_:') aggmggf]. 4...

The techniques to measure all of the terms in this expression were described in Section

 

 

 

 

4.1.3.

123

A “semi-isotropic” approximation of 8, as given by equation (14) of Browne et al. (1987),

is given by:

 

 

 

8,3. = MKS—:j + 26—13132 +(3—:)2] 4.61

A different semi-isotropic formulation was given by Wygnanski and Fiedler (1969):

 

 

= 3.11;) .1 +4.,1..1;;')212 . 3313)] 4...

where K, is given by equation 4.36.

Wygnanski and Fiedler (1970) used a semi-isotropic calculation that imposed the condi-

tion:

111—111-111-1—1-1—1

where only the last of these terms is measured. The estimate then takes the form:

so = v[(%7 +(g—l3‘j +(3—lj +(3—Dz +5@::j2]. 4.64

The next two approximations to the dissipation are indirect methods. These assume that

 

 

 

 

 

 

 

the spectral content of the velocity signal is assumed to have a universal form, from which
the dissipation can be estimated. The first of these methods, assumes that the one dimen-
sional spectrum of the velocity fluctuations satisfies the relationship:

E”(kl) = C22/3k‘1” . 4.65

over a certain range of 11,. In order for this relation to be a useful method of estimating the

dissipation, the “universal constant”, C must indeed be universal. Sreenivasan (1995)
addressed this issue using data from a wide variety of flow geometries and Reynolds num-

bers. For each data set, the dissipation was established using equation 4.57. The conclu-

124

sion from this reference is that no perceivable dependence of C on Reynolds number or
flow geometry can be observed. The mean value was found to be 0.53, with a standard
deviation of 0.055. In the discussion of these results. Sreenivasan also noted that equation
4.57 always under predicts the true dissipation, indicating that the true value of C is likely
closer to 0.50. Although this topic is still under consideration, the available evidence sup-
ports the idea that C truly is a constant at high enough Reynolds numbers, and that the
scatter in the observed values of C are likely a result of errors in the estimate of 8, and

errors in the measurement ofE” due to Taylor’s hypothesis.

Based on equation 4.65, the dissipation can be written in terms of the compensated spec-

tra:

5 3 /3

2,, = (2-kl‘/3E“(kl)) . 4.66

This assumes C=0.5, and is only valid in the inertial range of the spectra. The conpensated

spectra is plotted versus the normalized wave number k 111K? see Figure 4.30 for the spatial
location (x/0(,=675, 11:0). The data indicated roughly 2 decades where the compensated

spectra is relatively close to a constant value. The mean value taken from the range

6x10_4<kan< 6><10_2 was found to be 1.9 (1113/53).

The assumption of isotropy also leads to a unique relation for the third order longintudinal

structure function (see, e.g., Landau and Lifshitz. 1987):

 

. . 3 4
D|H(rl) = [u (x+rl)-u (1')] = —§€,r| 4.67
The interesting feature of this equation is that there is no arbitrary constant. The measure-

ment of dissipation is then only a matter of achieving a high enough Reynolds number

such that the compensated structure function D, I I is constant over a wide enough range of

r/n. The compensated structure function for the present velocity field shown in Figure

4.31. This exhibits a relatively constant value near 81:1.30 over the range 10<ran<200.

Table 4.2: Dissipation estimates

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

equation # value(m2/s3) 0.00 1.00 2.00 3.00 4.00
a, 4.57 2.06 3 l " '1'
83 4.58 2.09 3 1 1
3,3371 4.57, 4.24 1.57 3 1 3
8337C 4.58, 4.25 1.76 1 3 1
ac 4.59 1.79 1 3 J
2D 4.60 1.98 *3 I
85 4.61 2.73 l - a
2,. 4.62 3.06 1 l I r
80 4.64 1.79 1 l 1
8H 4.66 1.8 l 1 J
81 4.67 1.3 :Zi—l

 

 

It is observed from the results summarized in Table 4.2 that the different methods of calcu-
lation lead to results which can differ quite widely. The mean value of these measurements

is 1.99, with a standard deviation of 0.503. The estimate 8F was nearly two standard devi-

ations above the mean of the population, indicating that this is likely a poor method of

estimating 8. Similarly, the estimate 81 was far below the mean. Saddoughi and Veeravalli

(1994) also found that using the third order structure function led to an underestimate of

the dissipation. If the minimum (81) and maximum (8F) are excluded, then the mean value

is 1.95, with a standard deviation of 0.33. Although this does not provide a definitive cal-

126

culation of the true dissipation, it does provide insight into the variability introduced by

the various approximations that are often used in the calculation of 8.

The profile 8(11) was calculated for four of the estimates described above (82“ , (SD, SE, and
80) at the location x/0,,=675; see Figure 4.32. This example shows that the relative magni-
tudes of the estimates is consistent across the shear layer. For example, the ratio of 85 to BB

was found to be in the range l.6<£E/€B<2.06.

4.5.2 Measurement of Kinetic Energy Budget

The terms of the kinetic energy budget of equation 4.52 will be described in this section. It
is assumed that the flow is stationary (B( )/B1 = 0) . and homogeneous in the spanwise
direction (w = 0) ,(B( )/Bz = 0) . Also, the viscous diffusion of energy is typically very

small compared to the other terms, and will be ignored. Equation 4.52 can be re-written as:

 

 

 

 

_ Bk .1].- _ a .11 a .11
+ :i-I—(u'3+u'v'2+u'w'2)111(14'2V'+1"3+1"W'2E|
_ 8x2 812
~ — — Bi? _ 817 Bi?
,2 .2 . .
—3(u —v )3 +uv(a—333 +EH—8

 

The four terms in square brackets represent the convection, pressure transport, diffusion,
and production of turbulent kinetic energy, respectively. These terms have been computed,

and are shown in Figure 4.33 for the streamwise locations x/0,,=384 and 675. The mean

1

velocity components were taken from Figures 4.4 and 4.5. The velocity-velocity correla-

tions were calculated directly from the time series velocity data described in Section 4.1.

127

Lateral (y) derivatives were evaluated by fitting a smooth curve through the data. Stream-

wise derivatives were evaluated using:

M - 93—1—33 4.69

Bx B1] Bx

assuming self similar flow development and equation 4.17 for Bn/Bx. The dissipation

was calculated using the axisymmetric relationship. equation 4.60.

The pressure transport term was calculated from equation 4.68. The term therefore also
contains any residual errors from the measured terms. It is observed that the profiles of the
pressure transport term exhibit quite small magnitudes. and an irregular shape which does
not have a consistent pattern from the two streamwise locations. It is reasonable to assume
that the pressure transport is quite small, and the magnitude of the observed values are
mostly a result of the residual measurement errors. The small magnitude of the values also

gives confidence to the accuracy of the directly measured terms.

The data shown in Figure 4.33 have been normalized by 0/ U3. The agreement between
the two downstream locations is satisfactory for all the measured terms. The data also
compare quite well with the results of Wygnanski and Fiedler (1970), referred to as WF.
Note that this reference uses a coordinate system in which positive 11 values are on the low

speed side of the shear layer. The profiles of the measured values are very similar in shape
between the present measurements and WF Re = 21000. As an example of the compari-
son between WF and the present data, the ratio of the peak value of the production to the
peak value of dissipation was found to be 1.82 in the present data compared to 1.88 in WF.
However, both the diffusion and convection terms were somewhat smaller in the present

measurements, although the general shape of the curves was the same as WF.

128

4.6 The Scales of Turbulent Motion

The length and time scales of the fluid motions within the shear layer are discussed in this
section. This view of the experimental data will be instructive with regards to the dynam-
ics of the flow field. That is, the previous sections reported on the stochastic properties of
the velocity and vorticity fields related to the pdf of the measurements without regards to
the temporal and spatial scales which led to the observations. A more complete under-
standing of the fluid motions can be obtained through the study of the length and time
scales of the flow. To this end, various data sets have been acquired that will be presented
in the following sub-sections. First, Section 4.6.1 describes the correlation field of the
shear layer. This will include both single point autocorrelations as well as multi-point
cross-correlations of the velocity and vorticity fields. Section 4.6.2 will discuss the Fourier
transform of the correlation functions, i.e., the spectra. Although the information con-
tained in the correlations and the spectra are equivalent, different insights can be inferred
from these two representations of the data. Specifically, the correlation measurements lend
themselves towards the understanding of the large scale motions of the flow. For example,
the zero crossing points and integral of the correlations are readily obtained from the cor-
relations. The spectral representation of the velocity and vorticity show the contributions
of the various wave numbers (i.e., length scales). In addition, hypotheses about the univer-
sal scaling of the small scale motions of the flow can most easily be observed and tested

using the spectral representation.

4.6.1 Correlation measurements of velocity and vorticity

The correlation coefficient is defined as:

129

 

amboc + Air")
5:13 ’

Rab(A'-i) = 4.70

 

where a and b represent any two signals of interest. The spatial separation vector between
the points of interest Afr can be created using multiple sensors, or approximated using a

single sensor with a time delay and suitable convection velocity.

The discussion of the correlation field will begin with single point measurements of the
streamwise component of velocity. The separation variable used in this correlation was
created using a time delay, which can be considered to be either a dimensionless time vari-

able (t*) or a dimensionless spatial coordinate (Ax/9):

*- U0 At ’ U(. Ax
z _ “(i—é) _ —9— _ F 4.71

The use of the convection velocity UC=UOI2 in the definition (4.71) (rather than the local

mean velocity, for example) is used to represent the convection speed of the large scale
motions of the shear layer. As discussed in Chapter 3, the convection velocity of the large

scale motions was found to be U0/2 throughout the self-similar region of the shear layer.

The autocorrelation Ruu(t*) is shown in Figure 4.34 for three positions representing the

high speed, center, and low speed regions of the shear layer. Data from both streamwise

locations (x/60=384 and 675) are shown for comparison. These data indicate that the

correlation function also exhibits self-similarity. Figure 4.35 presents the same data on
semi-log coordinates to show the detailed behavior at small t*. Note that all curves
approach zero correlation values at a value of t*:=20. This indicates that the large scale
motion that are responsible for the relatively long correlation times of the velocity scale

with the local momentum thickness. This result is consistent with previous research in

130

both single and two stream shear layers. For example, Browand and Trout (1985) calcu-
lated the autocorrelation of the velocity in the high speed side of a moderate Reynolds
number two stream shear layer. Their results are nearly identical to the 11:3.1 data shown
in Figure 4.34. Specifically, the “undershoot” in their measurements was found to be
roughly -0.5, after which the correlation approached zero. Koochesfahani et al. (1979)
presents similar results, with an undershoot of -0.58 occurring at t*=8. The present data

indicate a minimum correlation of -0.43 at t*z6.
Similar autocorrelation results were reported by Dimotakis and Brown (1976) for a two

stream shear layer at a similar Reynolds number (AU G/v z 105 )to the present measure-
ments. In this reference, considerable attention is paid to the interpretation of the relatively
long correlation times observed in the high speed stream. In particular, they defined the

time delay to the first peak in the autocorrelation to be 1:0 and then define a length scale 71“,

as:

xvflouc 4.72

where UC=U0I2 represents the convection velocity of the large scale motions of the flow.

This is interpreted to be the average spacing between the large scale motions of the flow.
This length scale can be made dimensionless using the local value of the vorticity thick-

ness (50,) which was defined as:

_E(_._ . 4.73

(l) 0 max

Dimotakis and Brown (1976) found the normalized to values to be in the range 3. l<}\.v/
5055.0, with a mean value of roughly 3.8. In the present measurements, the values were

found to be in the range 3.2<Kv/5w<4.5. Koochesfahani et al. (1979), obtained a mean

131

value of 3.9 for XV/Sw Note that the numerical calculations of Moore and Saffman (1975)

predict a value of 3.5 based on the idea that a vortex can be “torn apart” by the strain field

of nearby vortices.

The autocorrelations of the three components of velocity at the center of the shear layer
are shown in Figure 4.36. From these data it is observed that the v and w components
exhibit smaller correlations at smaller separations. This is made quantitative by comparing

the Taylor microscale for each of the components. The values of KUU/XW=1.43, and

luu/ wa= 1.49 were recovered from the data presented in Figure 4.36. These values com-
pare quite well with the value of x/2 z 1.41 that would be obtained in isotropic turbulence.

The spatial correlation of velocity across the shear layer was also of interest. For this pur-
pose, two single hot-wire probes were positioned locations ya and yb, with spacing (ya-yb)/
9:3.27. The velocity measured by the probe closest to the high speed side of the shear

layer is denoted u,,, and the velocity from the probe closer to the low speed side is ub. The

time series data were acquired at a rate of 3,000Hz for 300 seconds, at the streamwise

location x/90=529. The probes were then traversed across the shear layer with the fixed

separation in the y direction. The time delayed cross correlation results are shown for five

of these positions in Figures 4.37 through 4.41.

These data clearly reveal characteristics of the large scale motions. The long correlation

times are evident as with the autocorrelation. The peak magnitude of the correlations in all
of these figures was found to be of order 0.1 or higher. This magnitude is significant given
the large separation between the probes. The asymmetric shape of the correlation function

is quite evident. This has been considered previously in the experiments of Pui and Gart—

132

shore (1979), where time delayed cross correlation data in a moderate Reynolds number
two stream shear layer were obtained. The results shown in Figure 9 of that reference are
similar to those shown in Figures 4.37 and 4.38. It was argued that the positive values
obtained at negative time delay, followed by the negative correlation at positive time delay
were caused by large scale motions which are “tilted” at an angle with respect to the
streamwise direction. If the coherent motions were perfectly round, the resulting cross
correlation would be symmetric with a negative minimum occurring at zero time delay.
The feature of the correlation function was observed in the time series data of a very low
Reynolds number two stream shear layer; see Koochesfahani et al. (1979). If the large
scale motions were tilted towards the high speed side, then the correlation measurements
would shift towards positive time delay, as is indicated in the data. It is of interest to note
that the correlations in Figures 4.37 and 4.38 are shifted such that the zero time delay
results show a nearly zero correlation value. In terms of trigonometric functions, this
would represent a “quarter-wave” shift between the two signals. The signals appear to be
shifted by a “half-wave” in Figures 4.39 through 4.41. Further evidence that the tilted ori-
entation of the large scale motions is a universal feature of shear layers was also given by
the conditionally averaged measurements of Browand and Weidman (1976) in a two

stream shear layer, and Hussain and Zaman (1985) in a single stream shear layer.

The proceeding results have commented on the temporal or quasi-streamwise correlations,
and then on the lateral correlations of the velocity field. The next set of measurements will
focus on the spanwise correlation field. Surprisingly, very few attempts have been made to
document the spanwise organization of shear layers given that many references discuss the

two-dimensionality of the large scale motions as a ubiquitous feature of the these flows

133

without making any out of plane measurements. A notable exception is that of Brownand
and Trout (1980, 1985). As discussed already in Chapter 3, these two papers present the
results of a 12 wire rake of sensors in the irrotational flow adjacent to a low Reynolds
number two stream shear layer. They found that the spanwise correlation, measured in this
way, led to spanwise length scales of order 20 times the local shear layer thickness. The
inference from these measurements, was that the large scale motions approximate two
dimensional disturbances. They do not, however, report on the spanwise correlation of the

velocity in the active portion of the shear layer.

The streamwise development of the spanwise correlation field near the separation point

was discussed in Chapter 3; see Figures 3.30 through 3.39. These results will be extended
to the fully developed region in this section. The experiments were conducted using a rake
of 8 straight wire sensors spaced in the spanwise direction. The rake was traversed across

the shear layer at x/90=101 with a maximum spanwise separation of 3.36, and at x/60=675

with a maximum separation of 49. The downstream measurements were only executed for
n>-l.8 due to limitations with the traversing mechanism. These data were acquired for

300 seconds at a rate of 3000Hz.

The spanwise correlation of velocity is shown in Figures 4.42 and 4.43. The dimensionless
data show excellent agreement between the two streamwise locations indicating that this
statistic scales in self-similar coordinates. Several observations can be made from these
figures. First, the two dimensionality of the flow in both the high and low speed regions is
evident from the high correlation values, in agreement with the Browand and Trout (1980,
1985) results. For example, for n>3.5 the correlation is 0.9 or larger for O<Az/6<4. In con-

trast, in the central region of the shear layer the correlation falls quickly to zero at a span-

134

wise displacement of roughly 1.59. Note that the contours of constant correlation value are
nearly horizontal in these figures in the range —1.5<n<l.5. In other words, the scales of
motion shown in this way are homogeneous in the y—direction in the central region of the
shear layer. Given the long time scales provided by the autocorrelations of velocity at 11:0,
the magnitude of the lateral velocity correlations, and the two dimensional nature exhib-
ited near the edges of the shear layer, one might expect that the spanwise correlation
would indicate a high level of coherence over large spanwise lengths. Figures 4.42 and
4.43 are very instructive in this regard, in showing the very limited (~1.59) length of the

spanwise correlation in the central region of the shear layer.

The focus of the discussion and data presentation thus far has been on the velocity field.
The remainder of this section will discuss correlations of the vorticity field. Section 4.2
already described many aspects of the vorticity measurements, including examples of time
series and profiles of the root-mean-square of the fluctuations. The following will explore
the temporal and spatial aspects of vorticity. This is particularly important given the com-
mon interpretation of the large scale motions as “vortical structures” (see, for example,
Thomas (1991 ), Dimotakis and Brown (1976), or Browand and Trout (1985)). These
interpretations are typically made without regards to any measurements of the vorticity
itself. Note that Hussain and Zaman (1985) and Browand and Weidman (1976) use the
curl of a conditionally averaged velocity field to represent the “coherent vorticity”. This

is, in principal, quite different from the direct measurement of vorticity.

The autocorrelation of the vorticity components (Dy and col are shown in Figure 4.44 at the
location n=0, for both streamwise locations, x/6(,=384 and 675. The same data are also

shown in Figure 4.45 in semilog format to show the detail of the correlation at small t*

135

values. In both of these figures the velocity autocorrelation from x/90=675 has been

included for reference. The data show agreement for the two streamwise locations with
both components of vorticity approaching zero correlation at t*z2.5. Note that the vortic-
ity autocorrelation falls much faster than the velocity correlation, indicating the impor-

tance of the small scales of the vorticity.

A closer view of the zero crossing region is shown in Figure 4.46. Note that there is a
small (0.04), but significant, region of negative correlation values. This “undershoot” then
returns to a zero value at t*==10. When interpreted as a spatial distance (Ax/9), t*z10 corre-

sponds to a vorticity correlation of nearly twice the shear layer thickness. Although the

correlation is extremely small (lme(t*)| < 0.05) in the range 2.5<t*<10, this still repre-

sents a significant result in that the vorticity field is represented by motions of a wide
range of scales. This is clearly important in the consideration of the large scale “struc-
tures” of the flow, which are considered to be related to the “phase correlated vorticity”,

see Hussain (1983). Further discussion will be considered in Section 4.8.

In addition to the autocorrelation of the vorticity, the spanwise correlation of (1)z was also

measured using two vorticity probes separated in the spanwise direction at 11:0. These
data are shown in Figure 4.47 along with the autocorrelation results for comparison. It is
interesting that the spanwise correlation of vorticity is nearly identical in magnitude to the
autocorrelation. This is in contrast to the velocity measurements where the spanwise cor-
relation at 11:0 approached zero at a much smaller length scale (~ 1 .59) than did the auto-

correlation (~69).

136

A final measurement related to the correlation field of the shear layer was created using a
single hot-wire sensor in the high speed region of the flow (n=3.5), and a vorticity sensor
in the center of the shear layer (11:0). The purpose of this measurement was to observe the
correlation between the large amplitude irrotational fluctuations which occur in the high
speed region, and the time series of spanwise vorticity. This correlation is shown in Figure
4.48. These data confirm that there is a measurable correlation between the irrotational
fluctuations observed outside of the shear layer and the highly unsteady vorticity field.
The asymmetric correlation data are also consistent with the concept of large scale
motions which are “tilted” towards the high speed side. Further conclusions that can be

drawn from these data are discussed in Section 4.8.

4.6.2 Spectral representation of velocity and vorticity

The spectral representation of the fluid motions are discussed in this section. The scaling
of the small scales of motion are most easily examined from this viewpoint. The most

important and complete theoretical understanding of the small scales of motion are given
by Kolmogorov (1941 ), and subsequent refinements. The K41 theory, as it is often called,
assumes that there exists a range of turbulent length scales that are not effected by the

large scale motions of the flow, and therefore can be considered locally (in wave number
space) isotropic. When the assumptions of homogeneous and isotropic turbulence is made,
many predictions can be made about the behavior and scaling of the small scale motions;

see, for example, Batchelor (1965) and Pope (2000).

There is a large number of research papers in the literature that are devoted to the experi-

mental determination of the accuracy of the predictions that follow from the K41 theory.

137

Although a review of this literature is not the intent of this work, it suffices to note that the
present experimental conditions provide a unique opportunity to compare the shear layer
to several previously published results. For example, most of the flow fields which have
been used for testing the K41 theory are decaying grid turbulence, the outer regions of a
boundary layer. or a wake flow; see Sreenivasan and Antonia ( 1997) for a review. The
shear layer is distinguished from these flows, and in some sense provides a closer approx-
imation to the flows in many engineering applications given the well organized large scale

motions which were discussed in the previous section.

The present discussion will begin with the formulation of a model spectrum. The model
proposed by Pope (2000) will be used and compared with the various measured spectra in
this section. The model is defined to be:

Em) = [Caz/3K—5/31MKL) _/'.;(v<n,.) 4.74
where K is the wave number in three dimensional wave number space. The “universal
equilibrium” is represented by the function in square brackets, where C=1.5 represents the

. 1/4 3'4 . . . .
universal constant. The value oan = 8 v / = 0.20mm was found usmg the dISSIpa-

tion value (83) obtained from equation 4.58 at the location 11:0, x/90=675. The functions
fL and fl] are given by:

11
.7
/L( KL) = [L] 4.75

,1KL2+CL

and

fn(KnK) = exp{Bcn—B[(KnK)4+c:]l/4}. 4.76

138

These are effectively curve fits to the large and smallest scale motions respectively. The

value of cL=6.78 was used as suggested by Pope. The value of L represents the integral, or
largest scale of motion, and it is specified in terms of the ratio to the Kolmogorov length

scale: L/rk . This ratio was selected to provide the best agreement between equation 4.75
and the large scale portion of the spectra, using the measured EH spectra as described
below. The resulting L value of L = 6000 - rk is roughly equal to the shear layer width

of 1.2m at x/90=675.

The values [3:52, Cn:0 were suggested by Saddoughi and Veeravalli (1994) in their mea-
surements of the spectra in a high Reynolds number boundary layer. Pope (2000) sug-
gested 13:2.1. Cn=0.4. The high wave number end of the present spectrum was fitted using

values of [325.2, cn=0.33.

The modeled one dimensional energy spectra can be calculated from E(K) from the isotro-

pic relation:

Euuhq) = 10° 13152114121211" 4.77

MK

The experimentally obtained spectrum E (Kl ) was obtained by taking the twice the Fou-

[Ill
rier transform of the auto-correlation Ruu (see, for example, Bendat and Piersol (1986)).
The wave number magnitude was calculated from the temporal frequency from

K, = 21tf/ii. These data are shown for the location 71:0, x/90=675 along with the model

spectra on Figure 4.49. The agreement is quite good for a large range of (Kan). Given

that Pope (2000) used this model spectrum to represent data from a wide range of Rey-

nolds number and flow conditions, it is assumed that the one dimensional streamwise

139

velocity spectra does indeed have a universal form. Note that the differences observed at

very high wave number are assumed to be caused by anemometer noise.

The one dimensional model spectrum of the v and w velocity components were also calcu-

lated. This is given in terms E(K) by the isotropic relation:

_ _ 1 °° E(K) (K1) 2
Evv(Kl) ‘— Ewwhcl) "' iIKlT[1 + ? JdK 4.78
The measured and model spectra for Evv and EWW are shown in Figures 4.50 and 4.51

respectively. The agreement is quite good for a wide range of wave numbers. This implies
that the assumptions which led to equation 4.78, namely, homogeneous and isotropic tur-
bulence, are reasonable in the description of the velocity spectra. Note that, as with all
experimental verification of isotropic turbulence, equations 4.77 and 4.78 are necessary,

but not sufficient to show that the flow actually exhibits local isotropy.

In addition to the auto-spectral density, the cross spectral density (CSD) is also of interest.
The normalized (CSD) is defined as the coherence function:
2

1 “1’1

..__ 4.79
Eu u . EV 1’

Huv(K1nK) =
where Pm, is the twice the (complex) Fourier transform of the cross correlation Ruv. In

truly isotropic flow, the coherence function H ( Kl 11K) = 0 for all wave numbers. Since

uv
W at 0 in the shear layer, Hm, must also be nonzero at some wave numbers. Figure 4.52
shows the coherence function at 11:0, x/90=675. These data indicate that nonzero coher-
ence is present for 1(an < 0.01 , and nearly zero values were obtained for 1(an > 0.01 .

This result is reasonable, and suggests that correlations between u and v are significant

only at low wave number, and that local isotropy is supported for higher wave number.

1 40

Note that the coherence is more sensitive to anisotropy than is the auto-spectral density,

since the auto-spectra BW and Eww are both reasonably well approximated by the isotropic

relation 4.78 over the entire range of the spectrum.

In addition to the auto and cross spectrum of velocity components, the vorticity spectra
can also be computed from the experimental measurements. Antonia et al. (1987) derived
the isotropic relationship between the one dimensional vorticity spectra and the energy

spectra:

K1 °°E(K) l °°E(K) 2 2
¢(,,z(l(l) = 3].“ —K—dK + EL. —K-[K + K1 ]d1<. 4.80
The model spectrum (equation 4.74) was used to calculate what the vorticity spectra

would be in an isotropic flow. This, along with the experimentally determined spectra of

(it)z are shown in Figure 4.53. These data are remarkable in that they are not similar to the

isotropic prediction in any way. There is considerably less energy in the high wave num-
ber range, and considerably more energy at low wave numbers than that predicted by
equation 4.80. This result is especially interesting given the excellent agreement between
the measured and isotropic vorticity spectra shown in the boundary layer of the present
facility shown in Chapter 3. Antonia et a1. (1998) also found very good agreement
between equation 4.80 and the directly measured vorticity spectra for grid turbulence, and
Antonia et al. (1987 and 1996) found the similar results in wake flow. Ong and Wallace

(1995) found the same in their boundary layer measurements.

A notable difference between the boundary layer and wake flow studies and the present
measurements in the free shear layer which created the spectra shown in Figure 4.53 is the

existence of the organized large scale motions of the flow. The relative contribution of the

141

different scales of motion to the vorticity variance can be observed by plotting the com-

pensated spectra: 19%, in semilog format:

 

. 2 00 00
((1):) =1 $1.10?!) (1K1 = j K, "1.011(1) d(ln(K1)) 4.81
0 ‘ 0 ~
The integrand of the second right hand side (Kl- ¢(‘)(Kl )) is plotted in Figure 4.54. It can

be observed from this figure that a large portion of the integral contribution is from the

low wave number range of the spectra.

142

4.7 Circulation Density

A distinctive feature of the shear layer flow field is the existence of the large scale
motions. Many previous research efforts have attempted to determine the nature of these
motions, with the hope that an improved understanding of orderly structure may lead to
improved turbulence modeling and control. The previous sections have described the
results of single and multi—point measurements in the shear layer, which support the exist-
ence of these large scale motions. In particular, the scales of the velocity correlations (Fig-
ure 4.34), the temporal and spanwise coherence of the vorticity (Figure 4.47), and the
velocity-vorticity correlation shown in Figure 4.48 all indicate that motions of relatively

large length scales are an important part of the shear layer dynamics.

The large scale motions of a shear layer are often referred to as vortical or vortex motions,
or coherent structures in the literature; see, for example, the review article by Thomas
(1991). However, information about these motions is generally limited to information
derived from two sources: flow visualization, and measurements in the irrotational fluid
near the active shear layer. For example, the studies by Dimotakis and Brown (1976),
Winant and Browand (1974), and Hileman and Samimy (2001) use results of flow visual-
ization images to obtain information about the structure of the shear layer. The measured
irrotational velocity fluctuations were used by Browand and Trout (1980, 1985). Both of
these techniques were also used in the present observations which described the large
scale motions of the near separation region in Chapter 3. However, these measurements do
not easily quantify the importance of the large scale motions, and, could lead to significant
misinterpretations about the flow field. As an example of the possible misinterpretations,

Hama (1962) has shown that even an infinitesimal disturbance to an otherwise uniform

143

 

thin sheet of vorticity will produce streaklines that “roll-up” into what appear to be large
scale vortices. Hama concludes with a comment about flow visualization data in which
images of streakline patterns are recorded. He states that “practically no truth can be
obtained as to the nature of time-dependent phenomena and that images, which one might

receive from such observations, can be entirely misleading”.

A technique for extracting information about the coherent motions of the shear layer not
mentioned above is conditional averaging; see, e.g., Browand and Weidman (1976), and
Hussain and Zaman (1985). Its application by these authors has made use of a trigger sig-
nal from the irrotational flow outside the active shear layer along with measurements of u
and v in the shear layer. The velocity field is then averaged using points which satisfy a
certain condition for the trigger signal. This technique is instructive, and supports the
existence of large scale motions. Since only a fraction of the total time series of velocity
(which corresponds to a specific event in the trigger signal) is used in the averaging pro-
cess, conditional averages do not provide a complete description of the flow field at all

times.

This preamble is to establish the author’s motivation to obtain a measure of the “coherent
structures” of the shear layer which is both quantitative, useful, and representative of the
characteristics of the flow. Specifically, it is recognized that they to not simply seek to
characterize a “conditional event.” The circulation density (7) of the shear layer is pro-
posed as an alternative response to this motivation. The circulation density is defined as
the integral of the spanwise vorticity along a line perpendicular to the streamwise direc-

tion made dimensionless with U0:

144

0° a(v/UO)
J ———dy 4.82

If
:— (l)d’:—
Y U0 2) 1+ 8x

—m —m

That is, 'y represents the “circulation per unit length.” In general, 7 can be a function of x,
z, and t, since only the y dependence of vorticity has been integrated. However, note that a
constant free stream velocity requires that the time average value of the integral on the

right side of equation 4.82 at a given (x,z) location must be essentially zero, given that

a(V/U,,)/ax « 1 .That is, 7 = —1.

In order to gain a better physical interpretation of the circulation density, schematic exam-
ples of y in several stages of a temporal (y-t) shear layer are shown in Figure 4.55. In the

first example, a uniform sheet of vorticity is shown in gray, with a corresponding constant
value of y=- l. The pdf of an experimentally determined population of 7 values might look
like a very narrow Gaussian distribution, with a variance corresponding to the uncertainty

of the measurement.

The second example shows the case of a laminar shear layer undergoing a two-dimen-

sional linear instability. The vorticity field during the linear and nonlinear stages of growth
have been studied in detail both numerically (see, e.g., Pozrikidis and Higdon (1985)) and
analytically (see, e.g., Michalke (1965)). During the linear stages of growth, 'yhas a sinu-
soidal shape with increasing amplitude. A measured pdf of y for this case would be “dou-

ble humped.”

The third example represents the condition in which the coherent motions roll up into dis-

crete vortex motions with a Gaussian core of vorticity. In this case, regions of nearly irro-

145

tational motion will exist between these motions as shown. A measured pdf of 7 would

have a local maximum near 7:0, and a second peak located at y<~ 1.

These idealized examples illustrate that information about the circulation density in the
turbulent shear layer will be instructive with regards to the large scale dynamics. Ideally,
time series of y at multiple (x,z) locations would permit a full characterization of the circu-
lation field. However, experimental limitations do not permit such difficult measurements.
In the present experiments, the pdf of yat a single (x,z) location was acquired. Although
far from complete, this does provide a significant amount of previously unavailable infor-
mation. For example, if the large scale motions of the flow are comprised of regions of
rotating fluid which exist between regions of relatively small circulation, then the pdf of 7
would look something like that which is shown in Figure 4.55c. Note that this result
assumes that each coherent motion contains the same net circulation. If coherent motions
were to exist with a wide range of circulation values, then the histogram would be consid-
erably wider, which could lead to a histogram of 7 values that is no longer double peaked.
As a final example, if no large scale motions exist and the turbulent fluctuations distribute
the vorticity more evenly, then the pdf would show few realizations far from the mean

value of -l.

The only reference to the circulation density of a turbulent shear layer in the literature is
found in the direct numerical simulation (DNS) of a (y-t) shear layer by Rogers and Moser
(1994). A similar definition as equation 4.82 is given, although they ignore the mean
value, such that only the fluctuations: Y are considered. However, no information about Y
is provided except to say “variations [in Y] are still dominated by small scales.” They then

apply an arbitrary two dimensional spatial filter in the streamwise and spanwise directions

146

 

 

“to extra
that larg
HOWCW

exampl

4.7.113

llie Cir
nents 0
4.56. T

size as;

The ve
(PIV)
EM .1
0f the

Streai
The s
was '
liOn ,
ure 4

Henc

been

“to extract the large-scale behavior.” This “roller parameter” (9?) is successful in the sense

that large scale quasi-two dimensional regions with 9i<0 are visible in the computed data.

However, significant spatial smoothing is required in order to observe this structure. For

example, it would be difficult, and of limited use, to derive a transport equation for (58).

4.7.1 Experimental Configuration

The circulation density was experimentally determined by acquiring the u and v compo-
nents of velocity along a narrow rectangular region. This is shown schematically in Figure
4.56. The integration defined in equation 4.82 is modified to account for the finite domain

size as:

y = ii = {CIA—in?- at) 4.83
The velocity field was measured using a two camera digital particle image velocemetry
(PIV) system. The equipment was a Dantec 2000 processor, with lk x 1k Kodak Megaplus
EA] .0 cameras, and a New Wave Research Minilase 3, NDzYag laser. The configuration

of the equipment is shown schematically in Figure 4.57. The primary and entrainment

streams were both seeded using two Roskco 1600 smoke generators.

The streamwise location of the measurement was x/90=170. The momentum thickness

was 71mm at this location. The 313mm extent of the integration area in the lateral direc-

tion was centered such that the range -2.21<n<2.21 was measured. As observed from Fig-

ure 4.4, this corresponds to a range of mean velocity values equal to 0.11<i7/U0<0.94.

Hence, 83% of the mean vorticity is captured in the field of view. Although it would have

been preferable to capture a larger percentage of the total shear layer for the calculation,

147

 

 

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oneik
farther
menus
Urdu
inaeh
whkh

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this would have required moving the apparatus closer to the separation point. However, in
order for the present results to be representative of a fully developed shear layer, locations

farther downstream are preferred. The location x/90: l 70 was chosen as a reasonable com-

promise between these two constraints.

The data were acquired in six groups of measurements. Each group contained approxi-
mately 100 data sets. The total number of PIV vector maps acquired was 615. The rate at
which the images were acquired was 0.5Hz. At this data rate, each evaluation of 'y can be

considered to be statistically independent.

4.7.2 Circulation Density Results

The magnitude of the circulation density was calculated from each of the acquired 615
data sets. The mean value was found to be '7 = —0.84. This corresponds well to the

expected value of —0.83 based on the extent of the interrogation area and the mean veloc-

ity profile. Since the true mean value of 7 should be - l, the measured values were scaled

by the factor 1.19. The histogram of the scaled measured values is shown in Figure 4.58.

The RMS of the distribution was computed to be 9 = 0.57 .The third and fourth moments
of the distribution, were 0.19 and 3.55 respectively. A Gaussian distribution with the same
mean and RMS is also shown in the figure. Note that because the scaling factor (1.19) was
used, this only represents an approximation to the true 7 distribution. Specifically, the vari-
ability of the net circulation which occurred at the edges of the shear layer was not mea-

sured.

The observed histogram of y is instructive when compared to the model shear layers

shown in Figure 4.55. That is, the nearly Gaussian probability distribution indicates that

148

the large scale motions do not fully collect the vorticity into large scale motions as in the
low Reynolds number shear layer as studied by (Koochesfahani et al., 2000). This conclu-
sion is drawn from the relatively rare occurrence of y values near zero. However, the large
standard deviation of the histogram suggests that the vorticity is very nonuniformly dis-
tributed in space. If these large excursions in 7 were spatially correlated, this could lead to
the observed streakline patterns in flow visualization studies. Further comment will be

added in the conclusions section.

149

4.8 Conclusions

This section presents conclusions that are based on the data and analysis that were pre-
sented in the previous sections of this chapter. These conclusions are presented in a num-
bered format, followed by a discussion of relevant data and examples which support each
conclusion. These have not been listed in a particular order, although the discussion will

reference previously given arguments when possible.

1. Rotating the coordinate system such that the x axis is aligned with the centroid of vor-
ticity provides an alternative formulation of the mean flow variables. In these coordi-
nates, a net mass flux of fluid across the surface which represents the centroid of

vorticity can be observed.

The derivation of the equations in both tilted and untilted coordinates has been given in
Section 4.2. It is noted that any coordinate system can be formally correct. However, if the
standard “untilted” coordinates are to be used, the movement of the shear layer centroid
away from the x axis must be explicitly accounted for in the formulation of the self-similar
equations. The coordinate rotation provides a different viewpoint of the velocity field.
This has only a negligible effect on the mean streamwise component of velocity, or the

moments of the fluctuating velocity vector. However, this rotation has a significant effect

on the boundary conditions and the shape of the mean lateral component of velocity: (\7).

It was observed in Figure 4.5 that v > 0 for all n in the rotated coordinates as introduced

in Section 4.2. That is, a positive mass flux exists across any surface parallel to the 3r axis,

which supports the second statement of the conclusion given above. Note that the surface

representing the centroid of vorticity is given by n =0, where g(fi) = 0.375 . It is inter-

150

esting to contrast this result with the observation that mass cannot cross a time steady
sheet of vorticity. That is, the unsteady character of the vorticity field must be taken into

account in order to explain this conclusion about the mean flow variables.

It would also be of interest to extend this result to two stream shear layers. For example,

the flow visualization results of Brown and Roshko ( 1974) clearly indicate the movement
of the shear layer towards the low speed side. This movement, as well as the angle of the
bounding wall, which defines the low speed entrainment rate in a two stream shear layer,

could then be correlated to the shear layer growth rate.

2. The flow field is three dimensional in nature, as illustrated by the spatio/temporal
velocity and vorticity correlation fields. However. a volume integral of the vorticity

can be written which is two dimensional.

The length scales of the shear layer can be divided into three general categories for the
purpose of the following discussion. In the turbulence literature these are usually termed
the energy containing, inertial, and dissipation length scales. Conclusions regarding the
nature of the inertial and the dissipative scales will be given later in this section. The large
scales of motion are of primary interest because they contain a large fraction of the turbu-
lent kinetic energy. They are responsible for the majority of the material and momentum
transport. In contrast to the small scales, it is considerably more difficult to develop strong

conclusions about the nature of the large scale of the fluid motions.

The present conclusion and discussion examines the large scale motions through the spa-
tial correlation of velocity and vorticity. An interesting summary of these correlations is

obtained by observing the spatial separation required for the first zero crossing of the cor-

151

relations acquired at 11:0. These results are tabulated in Table 4.3. These data show some
relative agreement that indicates that the large scale motions are correlated over a length

scale of order 2.59 both in the streamwise and spanwise directions.

The exception to this result is the autocorrelation of the streamwise velocity (Ruu(t*))

which exhibits longer correlation times. It is thought that this is due to the use of the time
delay variable rather than a true spatial correlation. That is, the large scale motions have a
temporal correlation which is quite long (sometimes referred to as “quasi-periodic”)

which leads to the long correlation time of the streamwise component of velocity. Further

support for this observation can be drawn from the comparison of (Ruu(t*)) at the loca-

tions n=0 with those acquired in the high and low speed irrotational flow; see Figure 4.34.

Specifically, the correlations decrease towards ze(o\)approximately the same rate (e.g.,

|Ruu(t*)| < 0.05 for t*>22).

A distinct point of contrast can be made when these results are compared to the spanwise
correlation of velocity obtained in the irrotational flow in both the high and low speed
sides of the shear layer; see Figures 4.42 and 4.43. These data indicate high (>0.9) correla-
tion values over length scales of order of the shear layer thickness. This is in close agree-
ment with the results of Browand and Trout (1980, 1985) in which correlation lengths of
order 209 were observed in a lower Reynolds number two stream shear layer. The inter-

pretation of the large scale motions as “two-dimensional” structures can be explored from
these results. The term two-dimensional implies 9( )/az = 0 as a necessary condition.

However, the measurements of velocity gradients shown in Section 4.2 as well as the vor-

152

ticity measurements in Section 4.3 have shown that the smallest scales of motion exhibit
many of the properties of homogeneous and isotropic turbulence.

It is therefore instructive to examine what is two-dimensional in the shear layer. The
approximation 912(0an > 3_2)/9z z 0 has been shown from the correlation measurements

shown in Figures 4.42 and 4.43. This velocity can be written in terms of the vorticity in the

shear layer as:

 

. _ 1 (“B x 7) 7
a(n > 3.2) — HUN-K r3 (dx)(d_v)(d‘.) 4.84
where the limits of integration are given by the bounding walls of the facility. The deriva-

tive of this expression can be take with respect to the z direction to give:

3% = agitHK‘Br—j—r)(ill-)(dyxdz) = 0. 4.85
That is, the volume integral of the vorticity field expressed in equation 4.84 is two dimen-
sional, even though the vorticity itself is highly three dimensional. That is, it is the integral

properties of the vorticity field which lead to the observations of two dimensional

motions.

153

Table 4.3: Summary of zero crossing points of correlation functions

 

 

 

 

 

 

 

 

 

 

acquired at 11$.

Correlation Location of first R=0 condition
Ruu(t*) 4.89
thz") 2'97
R,,,,,.(l*) 2.84
Ram} t*) 2.35
Rm‘wfl.) 2.40
RwrmfAz/O) 2-0
RW(Az/9) 1.56

 

 

154

 

3. The histogram of circulation density values indicates that the description of the large
scale motions of the flow may be more accurately stated as fluctuations in the circula-

tion density, rather than “structures” or “vortices”.

Considerable research has been devoted to the characterization and understanding of the
large scale motions of shear layers. Observations from both flow visualization and veloc-
ity measurements in the irrotational flow adjacent to the shear layer have been used to
infer the properties of the large scale motions. For example Brown and Roshko (1971)
used a shadowgraph technique to investigate the large scale structure of a two stream shear
layer. Since then, significant research efforts have focused on the characterization, predic-
tion, and control of these motions (see Thomas (1991) for a review). These results often
lead to an idealized conceptual picture of the velocity/vorticity fields which exist in high

Reynolds number shear layers.

The circulation density measurements provide informatino about how the vorticity is dis-
tributed within large scale motions of the flow. For example, the pdf of yis nearly Gauss-
ian, and not double peaked. The relatively large standard deviation of 7 values measured
indicates that considerable variability exists in the motions which convect past a given

streamwise location. Also, the occurrence of y==0 is observed to be a fairly rare occurrence.

These results are not inconsistent with the observations of the large correlation lengths
observed in the velocity and vorticity data. These latter observations suggest that the fluc-
tuations in the circulation density also have a significant correlation in both space and
time. The suggested spatial correlation of the circulation density is consistent with the

flow visualization results of Brown and Roshko ( 1971), and Dimatokis and Brown (1976).

155

That is, spatially correlated fluctuations in 7 could lead to the observed “roll-up”. Future
research would be needed to confirm this statement. Conclusion 3 above states that this
perspective of the large scale motions may be more useful than the generalization of the

flow field as containing coherent structures or vortices.

4. The energy spectra have been found to be well represented by a model spectrum, and
the assumptions of local isotropy. In contrast, the vorticity spectra do not appear to be

even qualitatively similar to the predicted spectra using the same assumptions.

The conclusion regarding the velocity spectra is apparent from Figures 4.49 through 4.52.

The model spectrum provides a close representation of the El 1(k!) data. Assuming both
homogeneous and isotropic turbulence, the model spectrum also represents both E22 and
E33 quite well. This is especially notable at the lowest wave numbers where the fluid
motions are clearly non-isotropic. The H12 data (see Figure 4.52) however, are more dis-

criminating in that km>.01 is required for local isotropy to be approximately valid.

Given these results, it is quite striking that the vorticity spectra cannot be predicted by
local isotrOpy predictions as shown in Figure 4.53. Note that the one dimensional spec-
trum of vorticity in an isotropic flow does not have a log-linear region as does the energy
spectrum. Rather, the one dimensional spectra “flattens out” as shown, leading to
decreased contribution from the small wave numbers. In isotropic turbulence the enstro-
phy spectra also represents the dissipation spectra. It is consistent with K41 theory that
small wave numbers would have minimal contribution to the enstrophy. However, the

anisotropy of the large scale motions of the flow have led to vorticity correlations which

156

are non-zero over a large spatial extent indicating that the vorticity spectra contains signif-

icant contributions from the low wave number range.

The fact that the vorticity spectra does have a log—linear region with slope of k'1 is also of
interest. This is because the compensated spectra (ki¢m) shown in Figure 4.54 is relatively
constant over a wide rage of wave numbers. This implies that the relatively large scales
also cause a significant contribution to the vorticity fluctuations. This is in contrast to the
homogeneous-isotropic result where the vorticity and dissipation spectra are related only
by viscosity, and the dissipation spectra obtains the majority of the contributions from the

high wave number range.

5. The stochastic values of the velocity, velocity fluctuations, and velocity gradient fluc-
tuations scale in a self similar manner, whereas the vorticity fluctuations increase with

increasing Reynolds number.

It is recognized that exact self-similarity cannot be achieved in a shear layer. This is

because the ratio of the smallest length scales to the largest length scales is a function of

Reynolds number. By scaling the dissipation with U3/9, the ratio of the integral to Kol-

. . 3/4 . . . . .
mogorov scales 1S found to be 9/1] K ~ Re . Stochastic quantities which scale only With

the integral scales of motion (that is, have no dependence on nK) will exhibit self-similar
scaling. Similarly, variables which depend only on the Kolmogorov length scales, with no
explicit dependence on the integral scales of motions will also scale in a self-similar way.
It will be shown in the following discussion that the vorticity fluctuations depend both on

9 and on 11K, and that this leads to a direct dependence on the Reynolds number.

157

The spectral representation of the vorticity field exhibits a log-linear region with a slope
(pm ~ [(1 . This can be observed by the roughly constant value observed in the compen-

sated spectra shown in Figure 4.54. The dimensionless variance of the vorticity fluctua-

tions can be expressed as the integral of the spectral density:

00 kan:l
((092: j(¢,,)d(k.n,,) = j (k.<i>,,)dtlog(km,.)r 4.86
0 truism/9

 

In the right hand expression, the limits of integration are modified to account for the
observation that there is little or no contribution to the integral at scales larger than the
integral scales of the f low, nor is there a substantial contribution from scales smaller than
the Kolmogorov length scale. The Reynolds number dependence of the vorticity variance

can now be concluded from the observation that the ratio of largest to smallest length

scales varies as Rem; see, for example, Pope (2000). Therefore the integrand “1%) is

roughly constant, and the lower limit of integration decreases with increasing Reynolds
number as observed in Figure 4.54. Therefore, the value of the integral increases with

increasing Reynolds number.

The maximum values of the vorticity RMS measured at 11:0, x/90=384 and 675 support

this conclusion; see Figures 4.24 and 4.25. Additional support has been found by compil-
ing the vorticity measurements of several authors in other shear layer facilities. These
include Bruns (1990), Lang (1985), Loucks (1998), and Balint and Wallace (1988). These
data are summarized in Figure 4.59. Each of the data sets reported (where more than one
streamwise location was measured) indicates an increase in the dimensionless vorticity

RMS with increasing Reynolds number. The compilation of all these data also support the

158

conclusion. The scatter observed between the different data sets is larger than that which
would be expected from the uncertainty calculations of the individual measurements. This
most likely represents the effects of the different boundary conditions of these studies,

such as velocity ratio and the state of the boundary layer at separation.

6. The pdf of vorticity is highly insensitive to Reynolds number, but quite sensitive to

the level of intermittency.

This conclusion was drawn from the comparison of the histogram of observed vorticity

values obtained for the following flow conditions: a cylinder wake at Rel=60 (Antonia et
al. (1988)), DNS calculations of isotropic turbulence at Rek=216 (Cao et al. (1996)), shear
layer measurements in the range 41 1<Rexu<l210 (Figure 4.21), and near wall boundary

layer measurements at ReAu=2500 (Figure 4.23). In all these measurements, the intermit-

tency was above 95%. That is, turbulent fluid occupied the measurement volume more

than 95% of the time.

The similarities between these histograms is striking given the 40:1 range in Reynolds
number and differences in the boundary conditions. The specific features include an asym-
metric “hump” in the range of small vorticity fluctuations, and long “tails” which extend
several standard deviations in both directions. These tails can be characterized by the dis-

tribution:

. lw'l "
P(o) ) ~ exp -B ‘T 4.87
(t)
where the exponent n characterizes the curvature of the distribution when shown in semi-
log coordinates, and [3 is a fitted constant. For example, n=2 for a Gaussian distribution,

and n=1 for an exponential distribution. A least squares curve fit of equation 4.87 to the

159

shear layer measurements gives n=0.95i0.04. Antonia et al. (' 1988) do not report a value

for n, although visual observations of their data indicate nzl.

A distinctive feature that is observed in the data shown in Figure 4.23 is the concave-up
nature of the tail region of the histogram. The fit of equation 4.87 to these measurements
gives n=0.95i-0.04. Three possible reasons are suggested to account for this observation:
Reynolds number effects, near wall effect, or probe resolution. The latter reflects the con-
dition that the ratio of the probe dimensions to the Kolmogorov length scale of the flow
was 2:1 in the boundary layer measurements, compared to 6:1 in the shear layer data. The
relatively large size of the probe in the shear layer measurements will lead to spatial aver-
aging that could cause the exponent to be underestimated. This explanation is further sup-
ported by the calculations of Cao et al. (1996). Specifically, they plot the exponent (n) as a
function of the length scale of the domain over which the circulation is calculated. The
results indicate nz0.7 for very small (highly resolved) circulation domains, and a mono-

tonic increase to n=2 for domains of the order of the inertial scales of turbulent motion.

7. The relative magnitudes observed between the velocity gradient variances, the vortic—
ity, and the dissipation indicate that the assumptions of axisymmetric turbulence may

provide a more accurate description of the flow than does isotropic turbulence.

This conclusion supports the findings of Hussein and George (1991 ). Specifically, Figures

4.16 and 4.17 indicate that the KlzKZ, and K3=K4.

160

 

 

 

 

 

 

~=>
B B
y ——>
—>
x —>
7 4
,,
.
L
ltlltlltllitlllllliilll
V9
e L a

 

l\ /1

Figure 4.1 Schematic of control volume and coordinate system used in Section 4.1. Note

that the growth rate is not shown to scale.

 

 

)—
lllllllilllllllllllllll

 

Figure 4.2 Illustration of the physical interpretation of 5* in the boundary and shear layer

161

 

 

K<
‘<>

/
S

 

 

$4 ////////////\Ln:o

 

 

My]; \

Figure 4.3 Schematic of the un-tilted coordinates (left and the tilted coordinates (right) for

use in similarity analysis

 

 

 

 

 

1.2 p
E 10 L Brunstl990) TaFU—‘A‘
U 0 x/90=370 ,K’f"
o 0.8 . 1:1 x/90=650 44%
f”
0.6 "/4.
/
. ,v’
0.4 . «1’
1 ~’"
.21”
0.2 'I’i A
.. «9’
0.0 . ‘ . 7
_4 -3 -2 -1 o 1 2 3 4
11

Figure 4.4 Streamwise mean velocity f(n)

162

Cllct
<2)

-2 11 1 2 3
Figure 4.5 Lateral mean velocity: gm) in tilted and untilted coordinates.
T I l I l l I l T T I T l l l l l l l l
a 6‘ 3 r. 0 x/90=370
r-— I ——1
.I o I. I x/90=650
1- . ' —
.__. "II ____
L.
O
a I 4
y
f— _
.— . O . .a
O
T T
1 l 1 l l 1 l l l 1 #1 I 1 l 1 1 1 I
-22 - l (l l 22 El ‘1

-0.2()

-040
-4

 

1.22()

1 .()()

().13()

().(3()

().41()

().IZ()

().()()

 

 

 

 

 

().IZ()

(1.1 (1

(1.1 2!

(1.1115

().()<1

().()()

 

 

 

 

Figure 4.6 Root mean square of the streamwise velocity fluctuations

1(331

 

 

 

 

~ (1.14 1 1 1 1 1 1 1 I 1 1 1 I 1 1 1 I 1 1 1 I 1 1 I I 1 1 1
v e . . -
_ (1.12 >—— . x/O,,=37() . : I I 0 fl
4 I
U0 f I x/B..=65() o" ' ' ‘
().l() '—‘ .74 - . ~
*— 0 I I . ‘1
( . 1 ~— _,
1( 8 . . .
._ ‘ . _
(1.06 r——- '. I _
A ‘.
*— —1
.
.
(1.114 — ' ' _.
.
h— . —<
.
(1.02 —- .—
(1.110 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 L 1 1
-3 -2 -1 (1 1 2 3 4

Figure 4.7 Root mean square of the lateral velocity fluctuations

 

~ (1.14 1 1 1 I f 1 1 I 1 r 1 I 1 1 1 I 1 1 1 I 1 1 1 I 1 1 1
w h . , ‘
U. (1.12 1-— . V9,,z370 ‘ I ' I. '-
0 F I V9..=650 "' ' ° —
(1.1(1 ~—- .. I —‘
r—— . —‘

O I I
(1.118 -— . .—
_ . I I -
(1.06 >—— I o —
I
r— . I. -
0.04 h— . I ~
9 ' I
(1.02 >— . __
I

 

 

 

000 l 1 1 1 L 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l 1 1 1 1 1

Figure 4.8 Root mean square of the spanwise velocity fluctuations

164

 

1 . . . , . . . . .
x/90: 384 (175
200 _ + _g_ correction for (du/dx)2 (Cu) d

+ {1— correction for (dv/dx)2 (CV)

T Y T T T Y r T I I ‘

    

 

 

 

 

 

1.50—

1.00

0.501111.11111111111..111L41114
-3 -2 -l () l 2 3 4

11

Figure 4.9 Profile of correction factor for streamwise and transverse velocity gradients
(see equations 4.25 and 4.26)

165

1 .4E—4

 

1.2E-4

l

1 .0E-4

1

8013-5

1

6013—5

1

4013-5 r

2.0E-5 *-

 

0.0

T I T T T I T T T I T T T I T T

x/o0 = 384 675

 

T I Tr r T I T T T

. O uncorrected

 

Figure 4.10 Profile of velocity gradient fluctuations: du/dx

1

 

 

 

 

  

 

1.25-4 .,...,.../_,,,,_..,..., ..,
2 j ‘
(dv/dx)“ /
3 1.0134 — - _ , .
(U0 /V9) 6 '
8.0E—5 ~ ’ ' .
1
6.0E-5 —
4.(1E-5 1— X/GOZ 384 675 -.\\\
—g— —0— uncorrected at,
2.0E~5 — I C] corrected
1 \-
().O ‘ ‘ 44 1 r l 1 l 1 1
-3 -2 -1 (1 1 2
11

Figure 4.11 Profile of velocity gradient fluctuations: dv/dx

166

 

 

1.5E-4 . r 1

T TTr T T T

(aw/(1x)2

(U03/v9)
1 013-4

1

   

5.0E-5 - x/eo: 384 675

_._ —O— uncorrected
I [3 corrected

 

 

 

0.0 J A A l L A A l A A A l A A A L A A A

Figure 4.12 Profile of velocity gradient fluctuations: dw/dx

4.0E‘4 T T T I T T T I T T T T T V I T T T I T T T I T T T

2 .,
(du/dy 1' 3.5134 — s

3 . 1
(U0 /V9) 1034 x/9U: 384 675
' —o— —o—

 

l
1

2.5E-4

l
1

2.0E-4

T
1

1.5E-4

l
1

1 013-4

1
1

5 015—5 - —

 

 

0.0 A A A 1 A A A l A L A l A A A J A A L

 

Figure 4.13 Profile of velocity gradient fluctuations: du/dy

167

 

2,5E'4 I I 1 I r T r I 1 1 1 I r r 1 I r r t I T T I I 1 r r

2
(du/dz)‘

3 2013-4 — _
(UO /v9)

1,513-4 _ _

1015—4 - _

1 x/9(,= 384 675
5.05.5 — —o— —o— -

 

 

 

().O A A A A A A A A A A A A A A A A A A A A A A A i A A A

Figure 4.14 Profile of velocity gradient fluctuations: du/dz

3.0E'4 1 1 1 I 1 1 1 I 1 r 1 I 1 1 fl f1 1 r T ,

{—ldV/dz 2 * moo: '384
(U03/v9) 2513-4

 

+
1

2.0E-4 A _

1.5E-4 1- -

1 013-4

1
1

l
1

5.0E-5

 

 

0.0 A A A A A A A A A A A A A A A A A A A L444 A A A A

 

Figure 4.15 Profile of velocity gradient fluctuations: dv/dz

168

 

 

 

 

 

 

   

3.00-1vrrvvitwvivnl.fir,...,..
L '1
2.50-
2.00-
1.50- _D_kl
1 ..Q.k2
1.00
—O—k3
0.50- k4
0.00 ‘
-3

Figure 4.16 Ratio of velocity gradient variances for x/9(,=384.

 

2.50 T T T I T I T I r T T I T T T I T T T I’ t l I I f T T

2.00 ~—

1.50 -

 

 

0.50 r

 

 

0.00

 

 

Figure 4.17 Ratio of velocity gradient variances for x/9,,=675.

169

 

 

1500 ‘ l I l I l

1(100

(02(t) (l/sec) 500 ll 1 , . ' l
0 '1" . - . ‘1 . i l. 1M1 1‘11 ‘ ‘m.n' __‘ 1' _ L
It ' 1 ”WIWIIV I 1 1 II 11 w 1

 

1 I
-l(100
-15(10 1 1 1 1 1
(1.0 0.5 l .(1 1.;

time(sec)

Figure 4.18 Example of vorticity time series at x/9(,=707, nzO. Periods of irrotational fluid

are observed.

 

 

 

 

 

 

 

 

200 _ f . . T - T . . . . . . - ,
100 :— h I 1
0 : 1 .1 1A A 11 A A 1\ M:
(1) z(t) (“8%) V 1“
-100 .2
-200 1 l 1 _
1 1‘11 1 :
{too 1 U 1
400 5
A_ J A AA A l A A . l A I .
3.0 3.1 3.2 3.3
time (see)

Figure 4.19 Example of vorticity time series at x/9(,=707, n=0. Note the extended period

of rotational fluid with little high frequency content near the beginning of the time record

 

11100 . , . , - , . , -

(1) z (l/sec) 5111)

-5()()

 

 

 

 

-l(1()(1 ‘ ‘ ‘ ‘ ‘
(1.00 (1.50 1.00 1.50 2.00 2.50

time (see)
Figure 4.20 Example of vorticity time series at x/90=707, 11:39. The fluid is mostly irro-

tatoinal with intermittent bursts of highly fluctuating vorticity.

1 7(1

 

 

 

 

 

Figure 4.21 Histogram of vorticity from conditions A,B,C, and D; see Table 4.1.

 

f V I V V Y I T T f r ‘1' Y Y I 1 f V I Y

-1» u
b 4
IO» ~
.4

4

y
y

y

 

 

 

 

Figure 4.22 Comparison of vorticity histogram between points of high and low intermit-

tency.

I71

 

 

 

 

,.-
0’.
'a’
f
‘4’
0",
’ .O’.O.
A l l l

-l0.0 -5.0 0.0

(Di/(Dz

Figure 4.23 Histogram or spanwise vorticity in the atmospheric boundary layer at

Rex=2500. Solid line represents the curve fit given by equation 4.87.

I72

 

0.020

e
8
CD

 

(003/ve)(”2’
0.015

I

0.010 -

1

0.005

 

 

 

0.000
-3

 

I (Dz ____ Axisymmetric
. coy _ Isotropic

Figure 4.24 The RMS of the spanwise and lateral vorticity fluctuations at x/90=384

(Re9=7. 1x104). The isotropic and axisymmetric results were calculated from the measure-

ments of the velocity gradient variances.

 

 

 

 

 

 

y 8
1 ~
~ 0020 ~ ‘ 7
w 36 52.9
3 (1/2) 1
U /v0 4
( 0 ) 0015 ~ :5 U0
~j 4
0.010 - :
1 3
0.005 - -: 2
f l
0.000Hauuu-n‘-~:u-_#-n.-fi0
-3 -2 -| 0 l 2 3 4
I (Dz [I'4Xl'4mm] ___- Axisymmetric
. (Dy [1 .4x 1 .4mm] ISOtI‘OpiC

Figure 4.25 The RMS of the spanwise and lateral vorticity fluctuations at x/90=675

(Re9=1.2x105). The isotropic and axisymmetric results were calculated from measure-

ments of the velocity gradient variances.

I73

 

 

 

 

 

0.030
N xx x/(-) =675
/’ ‘s O
(O 0.025 » ,1, ° °\\ /
6 ,-
U 3N0 (1/2) , I]. ’. . ‘\\ O \\\
( o ) 0.020 - , u ° \
4’," / 3‘ o\
'I’
0.015 - ,xfo/ \\ ;\
0.010 - ./ \°\
A}:
0005 - ‘9
. O \\
' i
0000 l 1 l l 4 l . l .
-3 -2 -1 0 1 2 3 4
T1
2.7%)x [5.6 x 5.6 mm]
Axisymmetric

Figure 4.26 RMS of streamwise vorticity. Broken lines represent the predicted values from

the measured velocity gradient terms assuming axysymmetric turbulence. The symbols

are directly measured values from the 8—wire probe, that have been multiplied by the arbi-

trary factor of 2.7.

I74

0.25 7 r T I 7 I Y I I I Y T f I I I I I r I mi! 1 I 1 1 v

 

   
     

 

 

 

b . x/00=384
-h( 77) 0.20 -

I x/00=675
0.15 —
0.10 —
0.05 —

0.00 ' .
-3 -2 -1 0 1 2 3

Figure 4.27 Measured and calculated Reynolds stress.

0.100

 

0.050 ~

 

-h '(77) 0.000 -

I

—0.050

I

-0. 100

 

 

 

0150.11L144141.1.11LL1111111111
3

Figure 4.28 Gradient of the Reynolds stress.

I75

0.300 7 T T I F r I i F F I I r I I I 1 1 l I I I I

 

 

 

 

 

 

 

 

 

 

. .
0200 " +va" ‘ x/90=384
0.100 - + 5

0.000 — -
0.300.......,...,.-,,.fifl

0.200 r- - x"’o=675
0.100 - _

0.000 - ..

41100-2 . . . -11 . 1 . (1) . . 1 i . . . 12 s L . 3 4

Figure 4.29 Measured velocity-vorticity correlations.

 

  
 

 

 

 

 

2'0 (Q /\\VI‘:11”IIII.LIEAI I III ‘ .

_ U V‘JUV IIIHI 1,1:
* * (5/31)(3/2) 1.5 - d

(2 E110(1) k1
1.0 — _
0.5 F" / ' -
/
0.0 l l llLllll 1 L Lllllll 1 1 lllllll l l l -_
10‘4 10'3 10'2 10“

Figure 4.30 Measured dissipation from compensated energy spectra: E1 1(kl)

I76

I.5() II I Y Y YTTYTT T T YTYYIII

 

-(5/4)*D 111/r1

l.()()

().fi()

 

 

().()()

 

 

r1/T1K

Figure 4.31 Measured dissipation from compensated third order structure function: D, 11

0.0025 1 U I I Y Y Y I I T V I 1 I I I

 

. ' T'C.
89 0.0020

I

—

0.0015

I

0.0010

T

0.0005

T

 

 

 

 

0.0000
-2

Figure 4.32 Comparison of the profiles of several dissipation estimates at x/00=675. The

. TC . . . . .
estimate 83 assumes isotroplc turbulence, whereas the others assume vanous sem1—1so-

tropic properties.

I 7777

0.0015

 

0.0010 P
0.0005
0.0000
-0.0005 P
-0.0010 5
-0.0015 -
-0.0020 F

 

 

 

-0.0025

 

1000 = 384 675
. O Convection

_. _ —-{‘] Dissipation
‘ A. Diffusion

 

* 0 Production
I Pressure

transport

Figure 4.33 The measured budget of turbulent kinetic energy. The ordinate represents the

magnitude of the bracketed terms in equation 4.68 scaled by 0/ U:.

178

1.0

 

 

 

 

 

 

 

x/eO
___. 384
=-l.5
Ruu(t*) 0.5 - \ ‘1 675
\
\‘l \
\ \\ ’
\\‘ _
‘T'b‘
00 . g , 7 ,aww
\ x’ T]:
$ _ 11:3 1
_O.5 | 1 1 I 1 1 1 I 1 1 1 l 1 1 1 L 1 1 1
0.0 5.0 10.0 15.0 20.0 25.0

 

1.0

Ruu(t*)
0.5 .

0.0

 

 

 

 

-05 .
10" 10

 

Figure 4.35 The autocorrelations of Figure 4.34show in semilog format

179

 

1.00

0.80

0.60

0.40

0.20

 

0.00

 

 

 

-0.20
0.0

 

Figure 4.36 Autocorrelations of u,v, and w at x/0(,:675, 11:0.

 

Ruaub

 

 

 

 

Figure 4.37 Cross Correlation between probe (a) located at n=4.53, and probe (b) located
at n=1.26.

I80

0.15

 

Ruaub 0.10
0.05

0.00
-0.05 .

 

-0.101

 

 

_0.15 1 I 1 I 1 I 1 l 1 1 1 I 1 I 1
~40 -30 -20 -10 0 10 20 30 40

 

Figure 4.38 Cross Correlation between probe (a) located at 11:308. and probe (b) located
at n=-0.194.

0.10

 

Ruaub 0.05

0.00

 

-0.05

 

 

 

_010 1 I 1 I J_ I 1 I1 1 I 1 I 1 I 1
-40 -30 -20 -IO 0 10 20 30 40

t*

Figure 4.39 Cross Correlation between probe (a) located at n=l .76, and probe (b) located
at n=-1.52.

18I

 

0.10

Ru aub 0.05

0.00

 

-().05 e '

 

 

 

_010 1 I 1 l 1 I 1 I 1 L 1 I 1 I 1
-4O -30 -20 -10 O 10 20 30 40

Figure 4.40 Cross Correlation between probe (a) located at n=.434, and probe (b) located
at n=-2.84.

 

0.3

RWNb

 

 

 

 

'—40 -30 —20 -10 0 10 20 30 40

Figure 4.41 Cross Correlation between probe (a) located at n=—O.888, and probe (b)
located at n=-4. 16.

HQ

 

 

 

 

 

 

 

 

 

 

 

Az/B
\.
01L11l111111111111111111111111olw111111111l11114
-5 -4 —3 -2 -1 O 1 2 3 4

71

Figure 4.42 Spanwise correlation of velocity at x/80=101.

4—

Az/O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 — ‘r—-~~, “x_ _fif/L—I
\__\ ,\ r/
M\ -
mo 60
-\
_0.80 0.70k
lLIIALJlLAllll- Jl llJL lT—ll
O
-5 -4 -3 -2 -1 o 1

 

Figure 4.43 Spanwise correlation of velocity at x/0(,=650.

I83

 

1.00

 

 

 

 

 

szwz 0.80 —‘
I. ,
0.60 ~ i Velocity(u)
0.40
0.20 v
0.00
-O.20 1 I 1 I 1 I 1 I L I l
O l 2 3 4 5 6
t*

Figure 4.44 Autocorrelation of spanwise and lateral components of vorticity

l.()()

 

\ ' Velocity (u)

().80

().()() t—

-_-. 384
__ 675

().40

l

().2() ’-

 

().()()

 

 

 

 

_().20 1 1 114111I 1 1 111111I 1 1 111111l 1 1 1 1
-3 -2 -
l() 10 H) 10

Figure 4.45 Autocorreltion of spanwise vorticity in semilog format

I84

().()50

 

().()30

R (1.1110 »~

(”sz

 

-().()I()

I

-().()3()

I

 

 

 

-().()5()
0.0

Figure 4.46 Close view of the

2.0 4.0 6.0 8.0

t*

zero crossing region of the (DZ autocorrelation.

 

 

 

 

 

I-00 I I CT] I I 7 I I I I I I I
Ruu(t*)
Ruu(Az/6)
0.50 ”—' ‘1’}!
R11 .(t*) ’
(1)/, \S \
..
\9
0.00 C -w a."
Rmszmz/e)
_050 1111l 1111I 1111I
10'2 10" 10“ 10

t*, AZ/G

I0.0

Figure 4.47 Spanwise correlation of u and (oral 1’1/UU = 0.5 Autocorrelations of (11)Z and

u are included for comparison.

185

().l()()

 

Raglan

0.050 I
(1.11110 ~

-().()5() »

 

4

 

 

-().|()(I
-4II

Figure 4.48 Correlation between high speed irrotational velocity fluctuations and (1)Z at

n=0.

-3()

-2()

-I()

I865

()

41(

.n.‘

I 0"

E11(k1) 105

1/4 5/4 '04
8 V 103

 

 

 

lll“ lllll‘ llll‘ llllll" lllll‘ l l

llll‘ l

l lll'll‘ Wlll" lllllq

 

I
Lit

  
 

Model Spectrum

_______ Measured Spectrum

J 111111l 1 1 1111111 1 1 1L11111 1 1 1 111111

10“ 10' 10'2 10" 100

1(an

Figure 4.49 Measured and modeled one dimensional energy spectra: El 1(k1)

E22(k1) '05

 

1/4 5/4
8 V

 

 

ll'll‘ Tlllll‘ lllllII‘ llllll‘ lll

l llll'l‘ l l llll‘ l l

lll" l

Tlll" lTllll'q 1 ll

 

LA

10-

    

Model Spectrum

....... Measured Spectrum

1(an

Figure 4.50 Measured and modeled one dimensional energy spectra: E22(K1)

I87

 

'06 f l l lllll' T l l llllll l l lllllll l l Tllllll l l l llllll l

E3309) '05
10
1/4 5/4 3
8 V

lll

I II' I.

l llllll‘ ’

 

I0’

lllllll‘ lllllll'

10
10'
10

10“

Model Spectrum

_______ Measured Spectrum
-2
I 0

10'3

 

j lllll‘ rnnrq llllll" l lllllI‘ lllm‘ l lllll‘

-4 1 1 1 11111I 1 l 1 llllll l 1 1111111 1 1 L111111 1 1 1111111 1

10‘4 10‘3 10'2 10'1 10°

1(an

IO

5.
til

Figure 4.51 Measured and modeled one dimensional energy spectra: E33(K1)

 

 

 

 

 

0.3()0 f Y ffi v v v v v r v v l fi ‘ v v Y Y V Y I
0.250 _ .4
. ' o .
0.200 - -
H12 . . .
0.150 _ 4
C . -
0.100 — o . .1
O ..
0.050 -— . . ..
.0

I. O

().()()() b I ‘ ‘ ‘l L ‘ #1 1 L441. . . 1. .EW
‘0'} 10'2 104
km

Figure 4.52 Spectral coherence function H12. Note the km rage is changed from previous

figures.

I88

 

 

  
  

 

 

 

2 _
'0 ,5» N x/00=707
1 .. .3.
10'
x/90=384
10°
Model Spectrum /
10"
I
I0-2 _6 111115 1 1.4 1 1“ “nu-2 1
IO 10‘ 10 IO‘ 10 10

lelK

Figure 4.53 Autospectral function of vorticity. Model spectrum is from isotropic relation

tOE“.

2 2
k¢wzm /un )

().()5

0.04

0.03

0.02

0.0]

0.00

 

1

 

 

x/60=707 :1 '
\ ~ . k. I

 

 

 

108(km K)

0.0

Figure 4.54 Compensated vorticity spectra to show energy contribution to enstrophy inte-

gral.

I89

Pt 7)
0.00

Y -0.50
-1 .00 >-
-I .50 I
-2.()() I - ' - r - ‘ 1 . - ‘ -

'1"

 

 

 

 

' P( '10
0.00 - '————

4‘
Y 0.50 . ‘
-100 W 3
> 1
-150 ~
-200 - - - . - - - . - . -

 

 

 

Nanci”! I I. _ PCY)
0.00
-0.50
Y -100
4.50
-2.00 r

 

 

 

Figure 4.55 Schematic examples of yfor: (a) a thin uniform sheet of vorticity, (b) a sinuso-
idally perturbed shear layer, (c) a rolled up shear layer into Gaussian vortices. The PW)
curves on the right show a likely pdf that would be experimentally recovered for each con-
dition with Gaussian noise.

I90

——9 —me m—a —_—>

 

 

 

 

‘0.0....IOOOOOOICCCOOOOOCOOO0.008

C3

1),.CIOOOOCCCOOOOOOIOOOOOOO.0.-COOS?

IIITIIIIITITITITIITTIIT

Figure 4.56 Schematic of showing the finite area of integration for the calculation of y

Lght Sheet

/

Comero l
Feud OF \Hew

 

  
   

 

§ .

Mean
Veuxflty
H PPOFHe

Area 0?
Integroflon

Comero 8
FEfld OF \New

 

L.

 

Loser

 

 

 

Figure 4.57 Experimental configuration for 7 measurement

I9I

 

Number of realizations

 

 

 

 

Figure 4.58 Histogram of measured 7 values. Note that the measured vales have been cor-
rected by a factor of 1.19 to account for the limited field of view; see Section 4.7.2.

 

  
     
 

 

 

 

0.03
(02* F Bruns (1991) O
0.02 ~—
IJOUCkS ( I008) resent meaSUI‘émenIS

‘1’ Balint and WaIIace

0.01 k ,«v

g, “Ms"
[/4/ 1,4111 1' I91“ “\I
0.00 1 1 1 1 , 1 1
2.0 2.5 3.0 3.5 4.1) 4.5 5.0 5,5 6.0

Log(Re)

Figure 4.59 Peak vorticity measured by various researchers.

 

I92

 

 

113:1:
Thhrc
semed

inge

Ther
how
enUe
thes
fine
dd.
mot
dus
gut
thn
It is
mm

0101

Appendix A

This appendix introduces a final conclusion which is not given in the body of this thesis.
This result is related to the mechanism of entrainment, and is not central to the results pre-
sented in section 4.8. It is therefore given here as a final point of interest. The conclusion

is given as the following.

The phenomenon of entrainment, defined as the movement of fluid towards the
shear layer from the low speed side, would exist even if the fluid viscosity were
zero. It is supposed that the observed magnitude of the entrainment would not

change if the fluid viscosity were zero.

The motivation for arriving at this conclusion originated from the lack of understanding of
how fluid is entrained into the shear layer. One concept is verbalized by the term “viscous
entrainment.” This implies the high speed fluid acts to “drag along” the fluid adjacent to
the shear layer through the viscous diffusion of momentum. A second, and distinctly dif-
ferent concept of entrainment is termed “engulfment” by Roshko (I979). This is an invis-
cid entrainment mechanism in which the unsteady pressure field created by the large scale
motions act to move the entrained fluid into the shear layer. As stated in the previous con-
clusion, fluid is also extrained from the shear layer to the high speed side. The analysis
given in the following paragraphs will seek to clarify physical mechanisms that lead to

this asymmetrical movement of fluid.

It is noted that Dimotakis (I986) utilized a model of point vortices to explain an inviscid
mechanism of entrainment. Although the predictions of the entrainment rate given by this

model are in agreement with many experimental results, the argument is heuristic in

I93

nature, and is too simple to describe many other features of high Reynolds number shear
layers. The present conclusion and the following discussion are not given to support any
particular model, but simply to provide a solid argument that the mechanism which causes

entrainment is an inviscid one.

The proof of the conclusion stated above begins with the results of stability theory. Specif-
ically, it has been shown by several authors (see. for example Michalke (1965)), that the
inviscid, linear stability theory implies that the shear layer is inherently unstable and that
this instability will lead to disturbances which grow spatially. Huerre and Monkewitz

(1985) have provided a general result for shear layer instability based on the velocity ratio
AU/ZU where AU is the velocity difference between the two streams and U is their aver-
age velocity. The results of Huerre and Monkewitz (I985) have shown that for values of

AU/ZU>I .3 15, the linear instability mechanism will lead to a temporally growing shear

layer; values of AU/2(7<1.315 will lead to a spatially growing shear layer. For the

present case (AU/2 0:1), this result implies that disturbances will undergo spatial

growth. The nonlinear growth that results from this will lead to a nonzero value of dex.
Equation 4.4 therefore requires that the entrainment is nonzero, just as equation 4.8
requires that the shear layer is tilted at a non—zero angle with respect to the constant veloc-

ity free stream.

It is concluded from these statements that if a shear layer could exist with a fluid which
has zero viscosity. the shear layer would be convectivly unstable. The resulting spatial
growth would lead to entrainment. However, this does not imply that the observed rate of

entrainment (020.035) would be unchanged by the hypothetical case of zero viscosity, as

I94

I!" I,
1 _.1

 

indicated by the supposition above. The following arguments will support the notion that

entrainment is an emir'lv inviscid phenomenon.

The first argument is given by the observation that the growth rate is linear. That is, dex
is not a function of the Reynolds number, and therefore neither is the entrainment velocity.
It is not correct to literally take this limit to Rezoo. because the Navier-Stokes equations
become singular. However, the physical implications of the singular nature of the momen-
tum equation are usually only important near solid walls where the boundary conditions
must be applied. In summary, since the value of the Reynolds number does not effect the
velocity of entrainment, it can be inferred that the phenomenon responsible for entrain-

ment is not effected by the viscosity of the fluid.

In addition to supporting the inferences above, additional insight can be obtained by
observing the mechanism of viscous entrainment in two model flow fields. First, the idea
of viscous entrainment can be studied from the “scraping plate” problem. This is described
as steady flow of a viscous fluid in the upper half plane (y>0), with the boundary condi-

tions (u,v)=(U0,0) at y=0, xe (—oo,oo). The entrainment rate is found to be

ve = v(x, oo) ~ A. This solution shows that a viscous boundary conditions with constant

velocity leads to an entrainment field which is dependent on the streamwise coordinate.

The second idealized flow considered is the “stretching plate” problem. The boundary
conditions are similar to the scraping plate problem. except the wall boundary condition is

accelerating spatially at a rate a. That is, u(x,o)=ax, v(x,0)=0. The solution of this problem

for large distances from the wall yields the entrainment rate: vc = v(x, oo) = x/av, where

v is the kinematic viscosity of the fluid. In this solution, the entrainment velocity is a con-

I95

 

stttttl. as I

the work
observed

That is. t

 

In sumn
through
morem
ena. wl
scale 1
the re

oi the

stant, as in the actual shear layer. It is then instructive to note that if air were to be used as

the working fluid, the acceleration parameter that would be required to generate the value

observed in the present physical shearlayer: VC 2 0.035 U0 = 0.25m/s, is a=4I60m/s/m.

That is, the wall velocity would be: u(x=l,0)=4l60m/s. u(x=2,0)=8320m/s, etc.

In summary, these model flows indicate that the induction of fluid towards the shear layer
through the viscous diffusion of momentum is not likely to be significant. Rather, the
movement of fluid towards the shear layer from the low speed side is an inviscid phenom—
ena, which is believed to be related to the unsteady pressure field associated with the large
scale motions of the flow. In this way, the present conclusion serves to motivate several of
the results that were described in the bode of this thesis regarding the large scale motions

of the flow.

I96

 

REFERENCES

Adrian, R. J., C. D. Meinhart, and C. Tomkins. 2000. Vortex organization in the outer
region of the turbulent boundary layer. Journal anluid Mechanics 422, 1—53.

Ali, S.K., Klewicki, C.L., Disimile, P.J., Lawson, I.. Foss, J.F., I985, “Entrainment region
phenomena for a large plane shear layer,” proceedings, Fifth symposium on Turbulent
Shear Flows.

Andreopoulos, Y., Honkan, A., 2001, “An experimental study of the dissipative and vorti-
cal motion in tubulent boundary layers,” vol. 439, pp. 131-163.

Antonia, R.A., Browne, L.W., Shah, D.A., (1987) “Characteristics of vorticity fluctuations
in a turbulent wake,” J. Fluid Mech, vol. 189, pp349-365.

Antonia, R.A., Shafi, H.S., Zhu, Y., (1996) “A note on the vorticity spectrum,” Phys. Flu-
ids 8 (8). P2196.

Antonia, R.A., Zhou, T., Zhu, Y., (I998) “Three-component vorticity measurements in a
turbulent grid flow,” J. Fluid Mech, vol. 374, pp29-57.

Balint, J.L., Wallace, J .M., 1988 “Statistical properties of the vorticity field of a two-
stream turbulent mixing layer,” Advances in Turbulence 2, Springer-Verlag, NY.

Batchelor, G K., ( I946) “The theory of axisymmetric turbulence,” Proc. R. Soc. Land. A
186, 480-502.

Bejan,A., 1995, Convective Heat Transfer, John Wiley and Sons, New York.

Bradshaw, P., 1966, “The effect of initial conditions on the dvelopment of a free shear
layer,” J. Fluid Mech., vol. 26, pp25-236.

Browand, F.K., Lati go, B.O., 1979, “Growth of hte two-dimensional mixing layer from a
turbulent and nonturbulent boundary layer,” Phys. Fluids, vol. 22, No. 6, pp 1101-1019.

Browand, F.K., Trout T.R., 1980, “A note on spanwise structure in the two-dimensional
mixing layer,” J. Fluid Mech., vol ”7, pp77l-781.

Browand, F.K., Trout T.R., 1985, “The turbulent mixing layer: geometry of large vorti-
ces,” J. Fluid Mech., vol 158, pp489-509.

Browand, F.K., Weidman, PD, 1976, “Largest scales in the developing mixing layer,” J.
Fluids Mech. Vol 76, ppl27-l44.

Browne, L.W.B., Antonia, R.A., and Shah, D.A., ( I987) “Turbulent energy dissipation in
a wake,” J. Fluid Mech., vol. 179, pp307-326.

I97

Brown, GL., Roshko, A., “On density effects and large structure in turbulent mixing lay-
ers,” J. Fluid Mech., vol 64, pp775-816.

Bruns, J., 1990, “Evaluation of hot-wire velocity and transverse vorticity measurements in
a single stream shear layer and its parent boundary layer,” Diplomarbeit, Technische
Hochshule Aachen.

Bruns, J .M., Haw, R.C., Foss, J .F., 1991, “The velocity and transverse vorticity field in a
single stream shear layer,” in the Eighth Symposium on Turbulent Shear Flows. Technical
University of Munich, September 9-l l.

Bruun, H.H., I995, Hot-wire Anemometry, Oxford University Press.

Cao, N., Chen, S., Sreenivasan, K.R., I996, “Properties of velocity circulation in three-
dimensional turbulence,” Phys. Rev. Let. Vol. 76, No. 4, pp 6l6-6l9.

Champagne, F.H., Pao, Y.H., Wygnanski, I.J., I976, “On the two-dimensional mixing
region,” J. Fluid Mech. Vol. 74, pp.209-250.

Dimotakis, PE, 1986, “Two-dimensional shear-layer entrainment,” AIAA Journal, Vol.
24, No. II, Ppl79l-l796.

Dimotakis, P., Brown, G, 1976, “The mixing layer at high Reynolds number: large-struc-
ture dynamics and entrainment,” J. Fluid Mech., vol 78, pp535-560.

Foss, J .F., 198 I , “Advanced techniques for transverse vorticity measurements,” Proceed-
ings, 7th Biennial Symposium on Turbulence, Univ. of Missouri-Rolla.

Foss, J .F., 1994, “Vorticity considerations and planar shear layers,” Experimental Thermal
and Fluid Sciences, Vol. 8, pp. 260-270.

Foss, J.F., 1998, Private communication.

Foss, J.F., Haw, RC, 1990, “Transverse vorticity measurements using a companct array
of four sensors,” In The Heuristics of Thermal Anemometry, ed. DE STock, SA Sherif, AJ
Smits, ASME-FED 97: 71-76, PP98.

Foss, J .F., Bohl, D.G, Kleis, 8.1., 1995, “Activity Intermittency in a two-stream shear
layer,” Proceedings of the Tenth Symposium on Turbulent Shear Flows, The Pennsylvania

State University, University Park, PA.

Freymuth, P., I966, “On transition in a separated laminar bloundary layer,” J. Fluid Mech.
Vol. 25, pp.683-704.

Hama, F.R., I962, “Streaklines in a perturbed shear flow,” Phy. of Fluids, Vol. 5, No. 6, pp
644-650.

I98

Haw, R.C., Foss, J.K., Foss., J.F., I989, “Vorticity based intermittency measurements in a
single stream shear layer,” in Advances in Turbulence 2, eds. Femholz, H., and Fiedler,
H., Springer-Verlag, Berlin.

HO, O, Huerre, P., 1984, “Perturbed free shear layers,” Ann. Rev. Fluid Mech., vel l6,
pp365-424.

Holmes, Lumley, Berkooz, I992, Coherent Structures. Symmetry, Dynamical Systems
and Turbulence. Cambridge University Press.

George, W.K., Hussein, H.J., (1991) “Locally axisymmetric turbulence,” J. Fluid Mech.
V0]. 233, pp- 1'23.

Hamelin, J. Alving, A.E., 1996, “A low-shear turbulent boundary layer,” Phys. Fluids,
vol. 8,No. 3 pp 789-804.

Hileman, J., Samimy, M., 2001, “Turbulence Structures and the Acoustic Far-Field of a
Mach 1.3 Jet,” AIAA Journal, Vel. 39, No. 9, ppl7l6-l727.

Huerre, P., Monkewitz, P.A., 1985, “Absolute and convective instabilities in free shear
layers,” J. Fluid Mech., Vol. 159, pp. 151-168.

Hussain, A.K.M.F., Zedan, M.F., 1978, “Effects of the initial condition on the axisymmet-
ric free shear layer: effects of the initial momentum thickness,” Phys. Fluids, Vol. 21, No.
7.

Hussain, A.K.M.F., Zaman, K.B.M.Q., 1985, “An experimental study of organized
motions in the turbulent plane mixing layer,” J. Fluid Mech. Vol. 159, pp.85-104.

Kim, J.H., 1989, PhD thesis, University of Berlin.
Klewicki 1989, PhD thesis, Michigan State University.

Klewicki, J .C., Falco, R.E., Foss, J .F., 1992, “Some characteristics of hte vortical motions

in the outer region fo turbulent boundary layers,” J. ofFluids Engineering, Vol. 114, pp.
530-536.

Kock am Brink, B., Foss, J .F., 1993, “Enhanced Mixing via Geometric Manipulation of a
Splitter Plate,” AIAA paper no. 93-3244, AIAA Shear Flow Conference, July 6-9,
Orlando, FL.

Kolmogorov, A.N., I941, “The local structure of turbulence in incompressible viscous
fluid for very large Reynolds number,” translated in Turbulence and Stochasitc Processes:
Kolmogorov’s ideas 50 years on, ed. Hunt, J.C.R., Phillops, O.M., and Williams, D., The
Royal Society, London, 1991.

I99

Koochesfahani, M.M., Catherasoo, C.J., Dimotakis, P.E., Gharib, M, and Lang, DB,
(1979) “Two-point LDC measurements in a plane mixing layer,” AIAA J., Vol. 17, No. 12,
ppl347-l351.

Koochesfahani, M.M., Cohn, R., MacKinnon, C., 2000, “Simultaneous whole-field mea-
surements of velocity and concentration fields using a combination of MTV and LIF,”
Meas. Sci. Technol. Vol. II, pp 1289-1300.

Lang, DB, 1985, PhD Dissertation, Cal. Inst. of Tech.

Liepmann, H.W., Laufer J., I947, “Investigations of free turbulent mixing,” NACA
Report No. 1257.

Loucks, RB, 1998, “An Experimental Examination of the Velocity and Vorticity Fields
in a Plane Mixing Layer,” Ph.D Thesis,University of Maryland.

Lui, X., 2000, Ph.D. Thesis, Department of Aerospace and Mechanical Engineering, Uni-
versity of Notre Dame.

Lumley, J.L., (I965) “Interpretation of time spectra measured in high-intensity shear
flows,” Phys. Fluids. 8, 1056-1062.

Mehta, R.D., Westphal, R.V., 1986, “Near-field turbulence properties of single and two-
stream plane mixing layers,” Experiments in Fluids, 4, pp. 257-266.

Michalke, A., 1965, “On spatially growing disturbances in an inviscid shear layer,” J.
Fluid Mech, vol. 23, pp521-544.

Monin, A.S., Yaglom, A.M., 1975, Statistical Fluids Mechanics: Mechanics of Turbu-
lence. Vol 2, The MIT Press, Cambridge.

Naguib, A., 1992, PhD thesis, Illinois Institute of Technology, Chicago, IL.

Narayanan, S., Hussain, F., 1996, “Measurements of spatiotemporal dynamics in a forced
plane mixing layer,” J. Fluid Mech., vol 320, pp7 l -I IS.

Ong, L., Wallace, J .M., 1998, “Joint probability density analysis of the structure and
dynamics of the vorticity field of a turbulent boundary layer,” vol 367, pp. 291-328.

Ong, L., Wallace, J .M., 1995, “Local isotropy of hte vorticity field in a boundary layer at
high Reynolds number,” in Advances in Turbulence V, edited by R. Benzi (Kluwer, Dor-
drecht), p. 392.

Panton, R.L., 2001, “Overview of the self-sustaining mechanicsm of wall turbulence,”
Progress in Aerospace Sciences, Vel. 37, pp. 341-383.

200

Pope, S., 2000, Turbulent Flows, Cambridge University Pres.

Pozrikidis, C., Higdon, J.J.L., 1985, “Nonlinear Kelvin-Helmholtz instability of a finite
vortex layer,” J. Fluid Mech., Vol. 157, pp. 225-263.

Pui, N.K., Gartshore, 1.5., (1979) “Measurements of the growth rate and sturcture in plane
turbulent mixing layers,” J. Fluid Mech., vol. 9l, ppl 1 l-l30.

Rogers, M.M., Moser, RD, 1994, “Direct simulation of a self-similar turbulent mixing
layer,” Phys. Fluids, Vol. 6, No. 2.

Roshko, A., 1979, “Structure of turbulent shear flows: a new look,” AIAA Journal, Vol.
l4, No. 10, PPI349-l357.

Saddoughi, S.G., Veeravalli, S.V., I994, “Local isotropy in turbulent boundary layers at
high Reynolds number,” J. Fluid Mech, vol. I89, pp333-372.

Schmidt, T.P., Ali, S.K., and Foss, J.F., 1986, “Relative efficiencies for parallel and per-
pendicular entrainment flow paths,” AIAA Journal, Vol. 24, No. 9.

Slessor, M.D., Bond, C.L., Dimotakis, PE, 1998. “Turbulent shear-layer mixing at high
Reynolds numbers: effects of inflow conditions,” J. Fluid Mech, vol. 376, pp115-138.

Sreenivasan, K.R., Antonia, R.A., I997, “The phenomenology of small-scale turbulence,”
Ann. Rev. Fluid Mech., Vol 29, pp. 435-472.

Tennekis and Lumley, I972, A First Course in Turbulence, MIT Press.

Thomas, F.O., I99 I , “Structure of mixing layers and jets.” App]. Mech. Rev, vol 44, No 3..
pp] 19-153.

Townsend, I975, The Structure of Turbulent Shear Flow. Cambridge University Press.

Wallace, J .M., Foss, J .F., 1995, “The Measurement of Vorticity in Turbulent Flows,”
Annu. Rev. Fluid Mech. 27, pp469-514.

Winant, C.C., Browand, F.K., 1974, “Vortex pairing: the mechanism of turbulent mixing
layer growth at moderate Reynolds number,” J. Fluid Mech., Vol. 63, pp. 237-255.

Wygnanski, I., Fiedler, HE, 1970, “The two-dimensional mixing region,” J. Fluid Mech.
Vol. 41, pp.327-36l.

Wyngaard, J. C., and Clifford SF, (1977) “Taylor’s hypothesis nad high-frequency turbu-
lence spectra,” J. Atmos. Sci. 34, 922-929.

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