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6/01 c:/ClRC/DateOuo.p65-p.15

__———

Stochastic Estimation of the Flow Structure
Downstream of a Separating/Reattaching Flow
Region Using Wall-Pressure Array Measurements

By

Mohamed Ibrahim Daoud

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

2004

ABSTRACT

Stochastic Estimation of the Flow Structure Downstream of a Separating/
Reattaching Flow Region Using Wall-Pressure Array Measurements

By

Mohamed Ibrahim Daoud

This study examines the spatio-temporal characteristics of the surface-pressure
fluctuations and associated flow structures in the developing flow downstream of the
reattachment point of a fence-with-splitter-plate flow. The investigation focuses on
understanding the wall-pressure field characteristics, and the flow sources responsible for
its generation in the non-equilibrium boundary layer originating from the
separating/reattaching shear layer associated with the flow over the fence, using a wall-
pressure database that was simultaneously acquired with X-hotwire time series. This is
motivated by guiding efforts to predict and/or control flow-induced noise and vibration in
applications involving flows downstream of appendages and surface protrusions.

Characterization of the wall-pressure data alone showed that the wall-pressure
fluctuations were dominated by large-scale downstream-traveling disturbances that were
generated upstream in the separated shear layer. Notwithstanding this dominance, the p'
signature of these structures decayed with increasing downstream distance as the vortices
underwent a relaxation process while the contribution of eddies, associated with the
development of a "sub-boundary layer", became more significant with increasing
downstream distance. In addition, wavenumber-frequency-spectrum results showed that
pressure signatures of all wavenumbers and frequencies were associated with flow

disturbances that travel downstream with the same convection velocity.

Finally, multi-point Linear Stochastic Estimation of the flow field based on
instantaneous wall-pressure information confirmed the dominance of wall-pressure
generation by the passage of the outer-shear layer vortical structures and their mutual
interaction. Examination of the linear source term in Poisson's equation of the pressure in
conjunction with the stochastically-estimated velocity field revealed two mechanisms for
p' generation associated with the quasi-periodic vortex passage. One mechanism was
related to sources localized at the height of the vortex centers in the outer-shear layer,
which represented direct generation of p' caused by the strong vortex-induced

disturbances in the outer part of the flow. The other was located near the wall (y/6 < 0.2,

where 5 is the boundary layer thickness) and resulted from interaction of the weak near-
wall disturbances generated by the vortex passage with the strong mean wall shear. It is
important to realize that while it is likely that the former mechanism could be modeled by
the wall-pressure field associated with quasi-periodic passing of vortices embedded in
uniform inviscid flow, the latter mechanism requires proper account of viscous effects

near the wall and associated mean shear.

DEDICATED TO MY MOTHER AND THE MEMORY OF MY FATHER

iv

ACKNOWLEDGMENT

I would like to deeply acknowledge my family, advisor, committee members and
colleagues, because this thesis work could not have been done without their support
and/or contribution.

I owe a great debt of gratitude and appreciation to my little family, Nermeen,
Mariam and Omar, who have had selflessly sacrificed to stand by and assist me by any
means to overcome the toughest moments when I became stuck between a rock and a
hard place. My lovely family, whose age is the same as that of this thesis work, has made
my whole life much easier due to their understanding and support. Also, I would like to
express my love and respect to my mom who also has greatly suffered because of my stay
away from her. I would like also to thank my sisters and brother, my brothers in law, and
my nieces and nephews for their love and support.

I would like to express my special appreciation to my advisor, Dr. Ahmed Naguib,
whose guidance and directions have been greatly supportive of my work. I wish also to
emphasize that his warm friendship and brotherhood-like relationship with his students
and me myself has been the thrust of my Ph.D. rocket.

I would like also to thank my committee members Dr. Giles Brereton, Dr. Dennis
Gilliland and Dr. Manooch Koochesfahani whose advice and recommendations were
greatly helpful throughout my thesis work.

Finally, I wish to thank my colleagues whom I have worked with at the Flow
Physics and Control Lab. This includes Laura Hudy, Yongxiang Li, Antonious
Aditjandra, and Chad Stimson.

This work has been partially supported by NSF grant CTS-0116907.

TABLE OF CONTENTS

LIST OF FIGURES ............................................................................ viii
LIST OF TABLES ............................................................................. xv
NOMENCALTURE ............................................................................ xvi
1. INTRODUCTION ........................................................................... 1
1.1. Literature Review ................................................................ 3
1.1.1. Separating, Reattaching and Developing Flows .................. 3
1.1.2. Characteristics of the Turbulent Wall Pressure ................... 13
1.1.3. Stochastic Estimation ................................................. 18
1.2. Motivation ......................................................................... 22
1.3. Objectives ......................................................................... 23
2. EXPERIMENTAL SET-UP AND METHODOLOGY ................................ 25
2.1. Experimental Set-up ............................................................. 25
2.1.1. Wind Tunnel Facility ................................................. 25
2.1.2. Test Model ............................................................. 28
2.1.3. Instrumentation ........................................................ 31
2.2. Experimental Methodology ..................................................... 40
2.2.1. Static Pressure System ................................................ 40
2.2.2. Microphones ........................................................... 42
2.2.3. Tufts ..................................................................... 50
2.2.4. Hotwire Sensors ........................................................ 55
3. WALL-PRESSURE RESULTS ........................................................... 76
3.1. Mean—pressure Distribution and the Reattachment Length ................. 77
3.2. Fluctuating Pressure Distribution .............................................. 79
3.3. Autocorrelation ................................................................... 82
3.4. Power Spectra ..................................................................... 87
3.5. Cross Correlation ................................................................. 96
3.6. Wavenumber—Frequency Spectrum ............................................ 103
4. VELOCITY-PRESSURE ANALYSIS AND STOCHASTIC
ESTIMATION ............................................................................... 107
4.1. Velocity Characteristics ......................................................... 107
4.1.1. Characteristics of the Boundary-Layer Mean and
Turbulent Velocity Profiles ........................................... 107
4.1.2. Velocity Spectra ....................................................... 123
4.2. Velocity-Pressure Characteristics .............................................. 134
4.2.1. Velocity/Wall-Pressure Cross-Spectra ............................. 134
4.2.2. Velocity/Wall-Pressure Cross-Correlations ....................... 145
4.2.3. Conditionally-Averaged-Data Analysis ............................ 156
4.3. Stochastic Estimation ............................................................ 165

vi

4.3.1. Principle and Equations .............................................. 167

4.3.2. Comparison between ML, SL and SQ .............................. 174

4.3.3. Stochastic Estimation Results and Wall-Pressure Sources ...... 184

5. CONCLUSIONS AND RECOMMENDATION FOR FUTURE WORK ........... 207
5.1 . Conclusions ........................................................................ 207

5.2. Future Work ....................................................................... 211

A. Derivation of Equation (2.6) ............................................................... 213
B. Data Acquisition Settings .................................................................. 214
REFERENCES .................................................................................. 216

vii

Figure 1.1.

Figure 2.1.

I.)
I)

, 2' '.
( 1‘4;
c.
-1
(1;

(I
I.)
U)

._.' I.
r.
a
._.,
(b

Figure 2.4.

J!

Figure 2.-.

Figure 2.6..
Figure 2.
Figure 2.3.
Figure 2.9.
Figure 2.1.

Figure 2.1

Figure 2. ii

Figure 2.1

Figure 2.1J

FightelH

FiEllie 7 1.

 

Figure 1.1.

Figure 2.1.
Figure 2.2.
Figure 2.3.

Figure 2.4.

Figure 2.5.

Figure 2.6.
Figure 2.7.
Figure 2.8.
Figure 2.9.

Figure 2.10.
Figure 2.11.

Figure 2.12.
Figure 2.13.

Figure 2.14.

Figure 2.15.
Figure 2.16.

Figure 2.17.

Figure 2.18.

Figure 2.19.

Figure 2.20.
Figure 2.21.

LIST OF FIGURES

A schematic of an ideal two-dimensional flow over a

splitter-plate-with—fence ....................................................... 2
Schematic of the wind facility (dimensions in meters) .................... 26
A schematic of the test model ................................................ 28
A frontal picture of the model showing the blockage effect

of the fence ....................................................................... 30
A schematic of the instrument and middle plates depicting

the locations of the pressure taps and microphones ........................ 30
A picture of the fence-with-splitter-plate model inside

the wind tunnel .................................................................. 31
An image of a tufi attached to the I-plate .................................... 33
An image ofone ofthe microphones used in the array... . 35
A schematic of the microphone driving circuit .............................. 35
A schematic of the X-probe .................................................... 36
Images showing top and side views of the X-probe ........................ 37
Images of the traversing mechanism and the LVDT:

overall (left) and close-up (right) views ...................................... 39
LVDT calibration ............................................................... 39
Mean-pressure-coefficient distribution for the top

and bottom sides of the splitter plate .......................................... 42
Microphone installation in the Instrument and Middle

plates .............................................................................. 43
A picture of the microphone-calibration setup .............................. 45
PWT calibration results: pressure ratio (top) and

phase shift (bottom) ............................................................. 47
Frequency response of a microphone/hole assembly ....................... 48
Mean sensitivity (Km) of themicrophones (top) and

Corresponding time-delay (bottom). Microphone #1

is the most upstream microphone ............................................ 51
Cross-talk check of Microphone # 13 (Mic. #1

is the most upstream one) ...................................................... 52
A sample wide-aperture tufl image ........................................... 54
FFP of the reattaching flow versus the x-location of the tuft . . . . 55

viii

 

 

 

figm“

hguel‘

 

Figure 2.22.
Figure 2.23.
Figure 2.24.
Figure 2.25.
Figure 2.26.
Figure 2.27.
Figure 2.28.

Figure 2.29.

Figure 2.30.
Figure 2.31.
Figure 2.32.

Figure 2.33.
Figure 2.34.

Figure 2.35.

Figure 3.1.

Figure 3.2.

Figure 3.3.

Figure 3.4.

Figure 3.5.

Figure 3.6.

A schematic of the X-wire probe and angles definition .................... 57

An image of the yaw calibration setup ....................................... 59
Schematic of the yaw calibration setup ...................................... 62
Yaw calibration: voltage results ............................................... 62
Sample calibration data for the X-probe hotwires .......................... 63
Yaw calibration data after conversion of voltage to velocity ............. 63
Yaw-calibration data and corresponding curve fits for hotwire

#1 (top plot) and hotwire #2 (bottom plot) ................................... 64
A photograph of the X-wire probe and its reflection in the

splitter plate ...................................................................... 66
A close—up image of the X sensor ............................................. 67
Leading edge for X- and single-wire in-situ calibration ................... 69
Boundary-layer streamwise mean-velocity profiles from

single- and X-wire measurements at x/Xr = 3.05 ............................ 71
Boundary-layer RMS streamwise-velocity profiles from

single- and X-wire measurements at x/Xr = 3.05 ............................ 72
Pressure spectra of the microphone immediately upstream

(x/XIr = 3.0) of the X—wire for three y locations of the wire ................ 74
Pressure spectra of the microphone immediately downstream

(x/Xr = 3.11) ofthe X-wire for three y locations ofthe wire 75
Streamwise distribution of the mean pressure coefficient from the

present study compared to Hudy et al. (2003) .............................. 78
Streamwise distribution of the coefficient of the RMS pressure

fluctuation ........................................................................ 80

A comparison between the va streamwise distribution of Hudy
et al. (2003) and the present work with that of Farabee and
Casarella (1986) ................................................................. 81

Autocorrelation coefficient at five different streamwise
locations downstream of the reattachment region for
Re = 7600 and 15700 ............................................................ 84

Contour maps of the autocorrelation coefficient for all 16
Microphones and the two Reynolds numbers; Re = 7600

and 15700 ........................................................................ 85
A comparison of the auto-correlation results at zero time

delay and five different streamwise locations downstream

of the reattachment region for Re = 7600 .................................... 86

ix

 

 

Figure 3.1

Figure 3.1

 

Figure .1_

”we 4.

iERre 4

Figure 3.7.

Figure 3.8.

Figure 3.9.

Figure 3.10.

Figure 3.11.

Figure 3.12.

Figure 3.13.

Figure 3.14.

Figure 3.15.

Figure 3.16.

Figure 3.17.

Figure 3.18.

Figure 4.1.
Figure 4.2.
Figure 4.3.
Figure 4.4.
Figure 4.5.
Figure 4.6
Figure 4.7.
Figure 4.8.

Figure 4.9.

A full contour map of the p’ autocorrelation coefficient for the
separating/reattaching (Hudy et al. 2003) and the present

recovering flows .............................................................

Spectra of the wall-pressure fluctuation for Re = 7600:

logarithmic (top) and semi-logarithmic (bottom) plots

Spectra of the wall-pressure fluctuation for Re = 15700:

logarithmic (top) and semi-logarithmic (bottom) plots .

Definition of the frequency used in the splitting procedure

Of p' spectrum ...............................................................

The ratio between hi gh- and low-frequency pressure

fluctuations energy ..........................................................

Cross-correlation results at five different locations for

Re = 7600 and 15700 ........................................................

Cross-correlation results for all 16 microphones and

Re = 7600 (tOp); and 15700 (bottom) .....................................

Plot for extraction of the convection velocity for the two

Reynolds numbers of 7600 and 15700 ...................................

Contour map of the cross-correlation coefficient for all

16 microphones and Re = 7600 ...........................................

Contour map of the cross-correlation coefficient for all

16 microphones and Re = 15700 ..........................................
Wavenumber-frequency spectrum for Re = 7600 .......................

Wavenumber-frequency spectrum from the separating/

reattaching flow of Hudy et al. (2003) ...................................
Wall scaling of boundary-layer mean-velocity profile .................
Shear-layer scaling of the boundary-layer mean-velocity profile ........
115
... 117
119
... 120
121

Wall scaling of the boundary-layer ulrms profile ..........................
Shear-layer scaling of the boundary-layer urms profile ..................
Wall scaling of the boundary-layer vrms profile .........................
Shear-layer scaling of the boundary-layer Vrms profile ..................
Wall scaling of the boundary-layer Reynolds stress profile ...........

Shear-layer scaling of the boundary-layer Reynolds stress

profile .........................................................................

Streamwise-velocity spectra at y/8 = 0.05, 0.125, 0.25, 0.375,

0.5 and 0.625 and g = 2.05 .................................................

87

89

90

94

95

98

99

100

102

102
104

106
110
114

123

125

 

Figure 4.1

Figure 4.1

Figure 4.1
Figure 4.2
Figure 4.".

Figure 4.1

Figure 4.1

Figure 4.1

Figure 4.1

Figure 41

Flgllre 4.:

FigUre 4.:

 

Figure 4.10.

Figure 4.11.

Figure 4.12.

Figure 4.13.

Figure 4.14.

Figure 4.15.

Figure 4.16.

Figure 4.17.

Figure 4.18.

Figure 4.19.

Figure 4.20.

Figure 4.21.

Figure 4.22.

Figure 4.23.

Semi-log plots of the streamwise-velocity spectra at y/6 = 0.05,

0.125, 0.25, 0.375, 0.5 and 0.625 and E = 2.05 ............................. 128
Normal-velocity spectra at y/5 = 0.05, 0.125, 0.25, 0.375, 0.5

and 0.625 and E = 2.05 .......................................................... 128
Semi-log normal-velocity spectra at y/6 = 0.05, 0.125, 0.25,

0.375, 0.5 and 0.625 at E = 2.05 .............................................. 129
Boundary-layer velocity cross spectra at y/8 = 0.05, 0.125,

0.25, 0.375, 0.5 and 0.625 and E = 2.05 ...................................... 131
Semi-log plot of the boundary layer velocity cross spectra

at y/6 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625 and E = 2.05 132

Cross-spectra between the streamwise velocity (at E = 2.05)
and wall-pressure (at E = 1.33) for y/5 = 0.05, 0.125, 0.25,
0.375, 0.5 and 0.625 ............................................................ 137

Semi-log plot of the cross-spectra between the streamwise
velocity (at E = 2.05) and wall-pressure (at E: 1.33) for
y/5= 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625 ................................ 137

Cross-spectra between the streamwise velocity (at E = 2.05)
and wall-pressure (at E = 2.0) for y/S = 0.05, 0.125, 0.25,
0.375, 0.5 and 0.625 ............................................................ 138

Semi-log plot of the cross-spectra between the streamwise
velocity (at E = 2.05) and wall-pressure (at E = 2.0) for
y/5= 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625 ................................ 138

Cross-spectra between the streamwise velocity (at E = 2.05)
and wall-pressure (at E = 2.33) for y/8 = 0.05, 0.125, 0.25,
0.375, 0.5 and 0.625 ............................................................ 139

Semi-log plot of the cross-spectra between the streamwise
velocity (at E = 2.05) and wall-pressure (at E = 2.33) for
y/5= 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625 ................................ 139

Cross-spectra between the normal velocity (at E = 2.05) and
wall-pressure (at E = 1.33) for y/5 = 0.05, 0.125, 0.25, 0.375,
0.5 and 0.625 ..................................................................... 141

Semi-log plot of the cross-spectra between the normal velocity
(at E = 2.05) and wall-pressure (at E = 1.33) at for y/8 = 0.05,
0.125, 0.25, 0.375, 0.5 and 0.625 ............................................. 141

Cross-spectra between the normal velocity (at E= 2.05) and
wall-pressure (at E = 2.0) for y/6= 0.05, 0.125, 0.25, 0.375,
0.5 and 0.625 ..................................................................... 143

xi

Figure 4.:

Figure 4.2

Figure 4.2

Figure 4.:

Figure 4.:

Figure 4.2

Figure 4.3

Figure 4.3

FiElite 4.3

 

Figure 4.24.

Figure 4.25.

Figure 4.26.

Figure 4.27.

Figure 4.28.

Figure 4.29.

Figure 4.30.

Figure 4.31.
Figure 4.32.

Figure 4.33.

Figure 4.34.

Figure 4.35.

Semi-log plot of the cross-spectra between the normal velocity
(at E= 2.05) and wall-pressure (at E= 2.0) for y/5= 0.05, 0.125,
0.25, 0.375, 0.5 and 0.625 ..................................................... 143

Cross-spectra between the normal velocity (at E: 2.05) and
wall-pressure (at E= 2.33) for y/8= 0.05, 0.125, 0.25, 0.375,
0.5 and 0.625 ..................................................................... 144

Semi-log plot of the cross-spectra between the normal velocity
(at E= 2.05) and wall-pressure (at E= 2.33) at for y/6= 0.05,
0.125, 0.25, 0.375, 0.5 and 0.625 ............................................. 144

Contour plots of the cross—correlation coefficient at zero time

shift between the streamwise (top plot) and normal (bottom

plot) velocity and the wall-pressure at the location of the ten

downstream most microphones (fi'om E = 1.33 to 2.33) 147

Maps of the cross-correlation coefficient between the streamwise
velocity at E = 2.05 and wall-pressure at E= 1.33 (top plot), E= 2.0
(middle plot) and E: 2.33 (bottom plot) ....................................... 151

Maps of the cross-correlation coefficient between the normal
velocity at E = 2.05 and wall-pressure at E= 1.33 (top plot),
E= 2.0 (middle plot) and E= 2.33 (bottom plot) .............................. 152

The averaged cross-correlation between the normal-velocity
and wall-pressure in the range of y/f) = 0.375 to 0.625 for
E=1.67,1.89 and2.11 .......................................................... 155

Plot for extraction of the convection velocity from Rvp results ........... 155

Contour plots of the cross-correlation coefficient at zero-

time shift between the low-pass filtered streamwise (top

plot) and normal (bottom plot) velocity and the wall-

pressure at the ten downstream most microphones

(from E= 1.33 to 2.33) .......................................................... 158

Maps of the cross-correlation coefficient between the low-

pass filtered streamwise velocity at E = 2.05 and wall-

pressure at E = 1.33 (top plot), E= 2.0 (middle plot) and

E= 2.33 (bottom plot) ............................................................ 159

Maps of the cross-correlation coefficient between the low-

pass filtered normal velocity at E = 2.05 and wall-pressure

at E= 1.33 (top plot), E: 2.0 (middle plot) and E = 2.33

(bottom plot) ..................................................................... 160

Wall-pressure, <p>, and conditionally-averaged mean-removed
velocity-vector field associated with positive (top plot) and
negative (bottom plot) wall-pressure peaks at E = 2.05 .................... 161

xii

 

Figure 4.3
Figure 4.4
Figure 4.;
Hgm44

HE‘ure 4.4

Figllre 4.;

Figure 44

503644

(I

 

Figure 4.36.

Figure 4.37.

Figure 4.38.

Figure 4.39.

Figure 4.40.

Figure 4.41.

Figure 4.42.

Figure 4.43

Figure 4.44.

Figure 4.45.

Figure 4.46.

Comparison between the mean streamwise-velocity profile and
the conditionally-averaged streamwise-velocity profiles associated
with positive and negative wall-pressure peaks ............................. 164

Wall-pressure, <p>, and conditionally-averaged velocity-vector

Field associated with positive (top plot) and negative (bottom

plot) wall-pressure peaks, viewed in a frame of reference

moving with 0.8 ano ............................................................ 166

Correlation coefficient between the measured and stochastically-
estimated streamwise- (top plot) and normal- (bottom plot)
velocity using ML, SL1, SL2 and SQ2 ........................................ 176

Comparison between the measured and stochastically-estimated
streamwise-velocity spectra using ML, SL1, SL2 and SQ2
for y/5 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625.. .................................. 179

Comparison between the measured and stochastically-estimated
normal-velocity spectra using ML, SL1, SL2 and SQ2
for y/6= 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625 ............................. 180

Comparison of the conditionally-averaged and stochastically-

estimated streamwise-velocity using SL2 and SQ2 for y/8 =
0.05, 0.125, 0.25, 0.375, 0.5 and 0.625 ...................................... 182

Comparison of the conditionally-averaged and stochastically-

estimated normal-velocity using SL2 and SQ2 for y/5 =
0.05, 0.125, 0.25, 0.375, 0.5 and 0.625 ....................................... 183

Wall-pressure time series at E = 2.0 (bottom of each plot) and

the associated stochastically-estimated velocity vector field (top

of each plot) using ML for three consecutive time windows,

viewed in a frame of reference translating with velocity of

0.81Uco (largest vector = 0.25Uw) ............................................. 187

The stochastically—estimated velocity vector field using

ML viewed in frames of reference translating with three

different velocities of 0.76U00 (top plot), 0.81Uco (middle

plot) and 0.86U..o (bottom plot); largest vector = 0.25Uco

(note that the "x" and "0" locations are fixed for all three

plots to help identify the variability in the vortex and

saddle-point locations, respectively) .......................................... 189

Vorticity and associated stochastically—estimated velocity
vector field at three consecutive time windows, viewed in
a frame of reference translating with 0.81Uno ................................. 193

Pseudo-instantaneous linear pressure source (q L) flooded

contour maps and associated wall-pressure for three
consecutive time windows ...................................................... 198

xiii

 

Figure 4.-

Figure 4.-

Figure 4.-

Figure 4.7”

 

Figure 4.47.

Figure 4.48.

Figure 4.49.

Figure 4.50.

The conditionally-averaged normal velocity and its streamwise
gradient associated with negative wall-pressure peak for

y/5= 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625 .............................

Pseudo-instantaneous linear pressure source ((1%) flooded

contour maps and associated wall-pressure for three consecutive

time windows .................................................................

RMS profile of the linear pressure source calculated from

stochastically-estimated and measured velocity time series . ..

RMS profile of the linear weighted pressure source calculated

from stochastically-estimated and measured velocity time series

xiv

201

203

.. 205

206

 

Table B. 1

 

LIST OF TABLES

Table B. 1. Data acquisition settings ......................................................... 215

XV

1".

I .‘G‘V’lm
-.- .

 

.1"
..mw rV‘r'»
I . V‘ "V. 4.9

Abbrei'i:
AD

AC

CCD

 

FFP

FPaC L
HPF
H117
l-Plate
1.4

l d
LSE
LVDT
ML

M‘Piale

PDF
Ply
PM

031:

 

Abbreviation

A/D
AC
CCD
DC
FFP
FFT

F PaCL
HPF
HW
I-Plate
kx-f
1/d
LSE
LVDT
ML
M-Plate
MS
PDF
PIV
PWT

QSE

NOMENCLATURE

Analog to Digital

Alternating Current

Charge Coupled Device

Direct Current

Forward Flow Probability

Fast Fourier Transform

Flow Physics and Control Laboratory
High Pass Filter

Hotwire

Instrument Plate
Wavenumber-Frequency
length-to-diameter ratio of hotwire
Linear Stochastic Estimation

Linear Variable Differential Transformer
Multi-point Linear Stochastic Estimation
Middle Plate

Mean-shear pressure-source term
Probability Density Function

Particle Image Velocimetry

Plane Wave Tube

Quadratic Stochastic Estimation

xvi

1015

SAR

51.

SP1.

Q
S

N.

I'll.

 

A: in: & :5

ALE-qua 0;

.eg,

B.

RMS

SL

SPL

SQ
TM
TT

TTL

Au, lin & Av, lin

Au, quad & Av, quad

Bu&Bv

Cp, max & Cp, min

Root Mean Square

Signal Attenuation Ratio

Single-point Linear Stochastic Estimation
Sound Pressure Level

Single-point Quadratic Stochastic Estimation
Turbulence-Manipulation section of wind tunnel
Turbulence-Turbulence pressure-source term

Transistor—Transistor Logic

Half of the plane-wave-tube width

Stochastic Estimation coefficient of the linear term

LSE coefficients of the streamwise and normal velocities,
respectively

QSE coefficients of the linear term in the estimation of the
streamwise and normal velocities, respectively

Stochastic Estimation coefficient of the quadratic term

QSE coefficient of the quadratic terms in the estimation of the
streamwise and normal velocities, respectively
High-pass-filter capacitance

Speed of sound

Mean-pressure coefficient

Maximum and minimum mean-wall-pressure coefficient values,

respectively

xvii

 

 

0&C.

K&Km

kick:

 

Dt

K&Km

k1&k2

patm

Fluctuating wall-pressure coefficient

Cross-correlation coefficient of the wall-pressure
Cross-correlation coefficient between the measured and
stochastically—estimated streamwise and normal velocities,
respectively

Half of the wind tunnel test-section height

Voltage output of hotwires

Voltage output of Linear Variable Differential Transformer
frequency

Yaw-response functions of hotwire #1 and #2, respectively, of the
X-wire sensor

Half of the total fence height

Fence height above the splitter plate

Frequency-dependent and mean microphone sensitivity,
respectively

Yaw-response constants of hotwire #1 and #2, respectively, of the
X-wire sensor

Streamwise wavenumber

Correlation time delay expressed in number of data points
Total number of data points in a time series

Fluctuating wall pressure

Root mean square of the wall-pressure fluctuations

Atmospheric pressure

xviii

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Reference mean static-pressure upstream of the model
mean static-pressure at the pressure taps

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Reynolds number based on the step height

Reynolds number based on local momentum thickness
Acoustic RMS pressure ratio

Auto-correlation coefficient of the wall-pressure

Position vector and its magnitude, respectively, of the pressure-

observation point
Temporal cross-correlation coefficient between the streamwise-

and normal-velocity, respectively, and wall-pressure
Spatial cross-correlation coefficient at zero time shift between the

streamwise- and normal—velocity, respectively, and wall-pressure
Time

Time reference of the stochastically-estimated velocity vector field
Total flow velocity for X-wire calibration

Mean streamwise velocity

Conditionally-averaged mean-removed streamwise and normal
velocity, respectively

Wall-scaled mean streamwise velocity, u/ uI

Stochastically-estimated velocity

xix

 

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respectively

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Effective velocity of hotwire

Effective velocities of hotwire #1 and #2, respectively, of the X-
wire sensor

Mean streamwise velocity at the location of the Reynolds stress
peak

Root mean square of streamwise-velocity fluctuations
Voltage output of microphones

Root mean square of normal-velocity fluctuations
Streamwise and wall-normal coordinates, respectively
Coordinates of the pressure-observation point

Mean reattachment length

Coordinates of the pressure source

Wall-scaled y location, yut/v

y location of the Reynolds stress peak

BOX/Um)

Yaw-angle offset of hotwire

Dimensionless wall-pressure spectrum, (Dpvpr/(l/Z p U002)2
Boundary-layer thickness

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Cross-spectrum of the wall-pressure with the streamwise and

normal velocity, respectively
Cross-spectra between the streamwise velocity and wall-pressure

at the ten downstream-most microphone locations

Streamwise- and normal-velocity auto— and cross—spectra,

respectively
Cross-spectra between normal velocity and wall-pressure at the ten

downstream-most microphone locations

Characteristic length of the dominant eddies

Acoustic wavelength

The characteristics size of the pressure signature associated with
the dominant flow structures in the outer-shear and sub-boundary
layers, respectively

Kinematic viscosity of air

Rotational-traverse yaw angle

Density of air

Time delay

Time-delay of microphones

Time—shift of cross-correlation peaks

Angular frequency, 21rf

Spanwise vorticity of the conditionally averaged velocity field

xxi

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(x-x,yxr

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1. INTRODUCTION

Separating/reattaching flows contain very energetic structures, which generate
large wall-pressure fluctuations. These fluctuations are a direct representation of the
excitation forces produced by the turbulent flow on the underlying surface. If such
excitation takes place at frequencies and wavenumbers of one or more of the underlying
surface's resonant modes, unwanted vibrations and noise will be generated. Investigating
and understanding the wall-pressure-field characteristics in both the spatial and temporal
domains is important to predict and/or control such undesired effects. Moreover, wall-
pressure measurements can be used as a non-intrusive technique for capturing the
turbulent flow activities above the surface that are responsible for the wall-pressure
generation. In this context, the wall-pressure signature can also be used to gain better
understanding of the turbulence processes that occur in wall-bounded flows.

The present investigation examines the surface pressure spatiotemporally by
means of a wall-microphone array in the developing flow downstream of the
reattachment zone of the flow over a fence-with-splitter-plate. Furthermore,
simultaneous measurements of the velocity field and wall pressure are conducted for the
purpose of investigating their relationship. The simultaneous data are also used as a tool
to examine the flow structures associated with the generation of various wall-pressure
signatures using Linear and Quadratic Stochastic Estimation methods based on multi- and
single—point wall-pressure information.

The flow geometry of a fence-with-splitter-plate is illustrated in Figure 1.1. The
characteristic features of such a complex flow field may be described in terms of five

overlapping flow zones. In zone I, a freestream flow approaches the fence. At the tip of

'_:n. ..
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60.99
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> . ..
1 I X
1' ..r:' i 1
Fence Splitter-plate : l
, J 11 ‘ 111 1 2 IV v

Figure 1.1. A schematic of an ideal two—dimensional flow over a splitter-plate-with-fence

the fence, the flow is forced to separate forming a free shear layer and recirculating flows,
zone H. Due to entrainment, the shear layer grows in the downstream direction till it
reattaches on the surface of the splitter-plate forming an energized reattachment zone
(zone III). At the reattachment point, a portion of the flow goes upstream into the
recirculation region, while the other goes with the downstream flow. As a result of the
impingement of the shear layer on the splitter-plate’s surface, the vortical structures that
have been energized in the free shear layer create high-pressure fluctuations. As these
structures travel downstream, they are exposed to a continually weakening mean-flow
shear and, hence, undergo a relaxation process, progressively loosing their energy in zone
IV. In parallel, the newly created strong mean shear near the wall produces energetic
small-scale turbulence similar to that found near the wall of turbulent boundary layers.
The "border" between this small-scale near—wall sustained turbulence, on one hand, and
the continually-decaying, yet still energetic, large-scale vortices that were "born" in the

separated shear layer upstream splits the flow in zone IV into two main regions: sub-

 

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zone IV

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boundary layer and outer-shear layer, respectively. This terminology will be used
throughout this document to designate these two different regions that "stack" in the y
direction to form a non-equilibrium, or developing, boundary layer region. Given
sufficient streamwise length, this boundary layer develops into an equilibrium, or fully-
developed, turbulent boundary layer in zone V. The focus of the present measurements is
zone IV.
1.1. Literature Review

Because the present study investigates a flow field that is transitioning from a
reattaching free-shear-layer state to an equilibrium-boundary—layer one, reviewing and
understanding the main physics of typical separating/reattaching as well as equilibrium-
boundary-layer flows are important. These flows have been extensively investigated in
the literature, and in order to review them comprehensively the discussion will be
prohibitively lengthy. Therefore, given the specific scope of the present work, the review
provided herein focuses mainly on investigations that address the relationship between
the velocity-field and associated wall-pressure signature. In addition, a few selected
studies dealing with fundamental flow physics pertaining to both flows are also
considered.
1.1.1. Separating, Reattaching and Developing Flows

Many researchers have investigated separating/reattaching flows, in order to
characterize the main flow features in the recirculating-flow and reattaching-free-shear-
layer zones. Cherry et al. (1984) conducted simultaneous wall-pressure and velocity
measurements in addition to smoke flow visualization in the separating/reattaching flow

of a blunt-face splitter plate. Pressure was measured using two pressure transducers,

 

while V
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while velocity was measured using a conventional hotwire. Their data showed a
dominant low frequency in the wall-pressure fluctuations near separation, which they
attributed to the successive processes of expansion and contraction of the recirculation
zone beneath the free shear layer, or the so called flapping motion. Furthermore, their
simultaneous smoke flow visualization and pressure measurements downstream of the
reattachment point revealed that negative wall—pressure-peak events were synchronized
with the passage of what appeared to be the cores of vortical structures while those
associated with positive pressure peaks occurred in the vicinity of inward-flow regions
between the vortices. In addition, their Spanwise measurements showed that the free-
shear-layer structures became three dimensional soon after separation, but this three
dimensionality did not seem to be influenced by the reattachment process as the
structures "impinged"/interacted with the wall.

The flapping motion of the free shear layer in separating/reattaching flows has
been reported by several researchers in addition to Cherry et al. (1984). Driver et al.
(1987) studied the flapping motion observed in a reattaching free shear layer of a
backward-facing step in order to determine the frequency and spatial extent of the
flapping motion. To this end, they used thermal-tuft measurements at different x
locations downstream of the step, in addition to velocity and wall-pressure measurements
employing a wall-flush pressure transducer mounted at 5.5 step heights downstream of
the step. Driver et al. explained the reasoning for the flapping motion as a disorder of the
shear layer that arose when a vortical structure escaped the reattachment zone. Such a
detachment process of a vortical structure reduced the engulfed reverse-flow by the

separation bubble causing it to collapse momentarily, which increased the angle of

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impingement and, consequently, the curvature of the shear layer. Therefore, they stated
that a larger streamwise pressure gradient was created at reattachment, forcing the bubble
to expand back again. Driver et al. also estimated the amplitude of the flapping motion to
be less than 20% of the shear layer thickness. However, they stated that the motion was
of relatively low energy and might be ignored. Lee and Sung (2002) suggested a similar
explanation of the free-shear-layer flapping motion. They stated that the recirculation
zone increased linearly in size as the shear-layer vortical structures grew in size and
moved closer to wall. The expansion of the recirculation zone continued till a vortical
structure left the free shear layer, after which the recirculation zone abruptly shrank in
size causing a sawtooth-like behavior of the instantaneous location of the reattachment
point.

Eaton and Johnston (1982), and later Kiya and Sasaki (1985), suggested the
imbalance between rates of flow entrainment by the shear layer and reinjection at the
reattachment point was the main source of the enlargement and shrinkage of the
recirculation bubble. Furthermore, Kiya and Sasaki stated that such an imbalance was
caused by the breakdown of the Spanwise vortices in the shear layer. Spazzini et al.
(2001) studied the unsteady behavior of a backward facing step flow using skin friction
measurements and flow visualization. Their results exhibited a strong correlation
between the growth and successive breakdown of the secondary re-circulation bubble (at
the comer of the step) and the flapping motion of the free shear layer. This led them to
hypothesize that the flapping motion was linked to the behavior of the secondary bubble.
On the other hand, Heenan and Morrison (1998) conducted experiments to passively

control the low-frequency, buffeting, wall-pressure fluctuation downstream of a

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backward-facing step using a reattachment surface with different permeability. They
found that the flapping motion totally vanished when a permeable reattachment surface
with length extending from the step to 0.56 of the mean reattachment length, X,, was
employed. The permeable surface apparently inhibited the recirculating flow and
associated upstream convection of disturbances produced at reattachment.

Lee and Sung (2001) made laboratory measurements of wall-pressure fluctuations
in the separating/reattaching flow over a backward-facing step. They investigated the
spatiotemporal statistical properties of the wall-pressure fluctuations using a 32 electret-
microphone array in both the streamwise and Spanwise directions. Based on the
wavenumber-frequency spectra of their data, they suggested that the shear-layer vortical
structures were modulated by the flapping motion of the free shear layer and moved
downstream with a convection velocity of 60% of the freestream velocity. In a more
recent work, Lee and Sung (2002) introduced a new spatial filtering technique, which
they referred to as Spatial Box Filtering (SBF). This approach was basically utilized to
"isolate" the wall-pressure signature of certain streamwise wavenumbers. Using the
zeroth and second modes of the SBF, they could adequately isolate the fluctuations
corresponding to the flapping motion from those corresponding to the passage of the
vortical structures generated in the free shear layer. The data showed good agreement
between the wavelength of the second mode of the SBF and the vortical structures
streamwise spacing, which was approximately half of the mean reattachment length.

The separating/reattaching flow over a fence-with-splitter—plate model was first
studied by Arie and Rouse (1956) and further investigated by many authors. For example,

Castro and Haque (1987) reported detailed measurements within the

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separating/reattaching flow of a fence-with—splitter-plate model. They stated that their
motivation for selecting this flow geometry was the attractiveness of having a thin
laminar boundary layer at the separation point. The thin boundary layer is created due to
the strong favorable pressure gradient towards the edge of the fence that is produced by
the stagnation flow on the upstream face of the fence. Thus, the effect of the boundary-
layer thickness at separation on the flow was negligible and the reattachment length
would be only affected by the blockage ratio, h/Dt (h being the fence height above the
splitter plate and Dr is the wind tunnel test-section’s half width. Smits (1982) extensively
investigated the relation between the blockage ratio and the mean reattachment length,
Xr). Castro and Haque’s measurements demonstrated that the turbulent structures of the
separated shear layer and the plane-mixing layer were quite different. They reported that
the shear-layer grth rate was neither linear nor equal to that of the plane-mixing-layer,
being initially rather higher but reducing gradually as reattachment was approached. This
was also associated with a continuous increase in turbulence energy all the way to
reattachment, followed by a relatively rapid fall thereafter.

Hussain and Zedan (1978) also investigated the effect of the initial boundary layer
state (laminar or turbulent) and Reynolds number on the flow characteristics of an
axisymmetric free shear layer. They could show that the flow features were independent
of Reynolds number, but dependent on whether the initial boundary layer is laminar or
turbulent. In particular, they observed that while the separating boundary-layer was
initially either laminar or turbulent, its momentum thickness showed independence of the

spread rates, similarity parameters, and evolution of the shear layer. In contrast, those

values

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values showed significant dependency on whether the initial boundary layer was
turbulent or laminar.

Recently, Hudy (2001) and Hudy et al. (2003) compiled a database of wall-
pressure-array measurements for studying the spatiotemporal character of the surface
pressure within the separating/reattaching flow region in a fence-with-splitter-plate flow.
They could distinguish two regions, which are defined based on the mean reattachment
length of the separated shear layer. In the first region, from the fence to 0.25X,, large-
scale disturbances dominated the signature of the wall-pressure. These disturbances,
which were associated with the shear layer flapping, were found to convect in both the
upstream and the downstream directions with a convection velocity of 0.21Uw. In the
second region, which was located beyond 0.25X,, smaller time-scale structures were
found to be responsible for the generation of the wall-pressure fluctuations. These
structures corresponded to the free-shear-layer vortices and traveled with an average
downstream convection velocity of 0.57Um. Hudy et al. could also relate the flapping of
the free shear layer to an absolute instability zone, or self-sustained oscillator, near the
middle of the recirculation region. They suggested that due to such a self-sustained
oscillator, the separation bubble continuously underwent processes of expansion and
contraction leading to the free-shear-layer flapping motion.

As mentioned above, the flow downstream of reattachment is a non-equilibrium
boundary-layer flow, in which the shear-layer vortices undergo a relaxation process as
they travel downstream. Several authors have investigated this flow seeking better
understanding of how the flow relaxes from its shear-layer—like state to a boundary-layer

one far downstream. For example, Bradshaw and Wong (1972) used existing

 

 

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experimental results on the low-speed flow downstream of steps and fences in addition to
their own measurements downstream of a backward-facing step to demonstrate the flow
nature in the separation/reattachment and relaxation (developing) zones. They defined
three degrees of perturbations; i.e., weak, strong and overwhelming perturbation to
categorize different separating/reattaching flows. The weak perturbation is that caused
by a minor disturbance such as a mild change in pressure gradient or wall roughness,
which does not significantly change the flow velocity or length scale. On the other hand,
the strong perturbation resembles the flow of a boundary layer over a very small notch or
cavity that significantly alters the turbulent structures in the flow, while the
overwhelming perturbation is one that changes, all together, the flow from one "species"
to another; e.g., a shear layer changing to a boundary layer. According to the
classification of Bradshaw and Wong, the typical backward-facing-step flow involves
two overwhelming perturbations: one when the flow changes from a boundary layer to a
free shear layer at separation, and the other, at reattachment, when the flow switches from
the latter to the former state back again. However in the case of a thin boundary layer,
such as in the fence flow, the first perturbation may be ignored assuming that the free
shear layer starts developing at the separation point.

Bradshaw and Wong (1972) conducted their experiments employing a 0.13 step-
height-thin laminar boundary layer at separation. They observed a marked deviation in
the mean-velocity profile of the boundary layer downstream of reattachment from that of
an equilibrium boundary layer. This deviation manifested itself as an overshoot, closer to
the wall, and undershoot, away from the wall, of the profile relative to the universal log-

law. Bradshaw and Wong suggested that the main reason for the deviation was the

 

 

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disproportionality between the heights of the dominant turbulent structures above the
wall and their length scales. This disprOportionality (i.e., length scale ~ y), upon which.
the log-law is based, was attributed to the practically constant length scale (of the order of
the step height) of the free shear layer vortical structures at reattachment, except in the
inner layer very near the wall. Such a deviation from the equilibrium boundary layer was
observed as far as 52 step heights (approximately 8.6X,) downstream of the step.

Chandrsuda and Bradshaw (1981) made hotwire measurements, which extended
up to two reattachment lengths downstream of the step, of the second- and third-order
products of the turbulent velocity components behind a backward-facing step. Their
measurements showed that the free shear layer in the separation/reattachment zone with a
thin initial laminar layer was not sensitive to its initial conditions. Unlike Castro and
Haque (1987), Chandrsuda and Bradshaw reported that the free shear layer was not
greatly different from a plane mixing layer with a uniform external stream. Moreover,
they observed that the shear layer underwent a rapid change near the reattachment zone,
which was related to the confinement effect of the wall on the vortical structures. This
wall effect also led to an attenuation of the normal component of the flow velocity and
the transport of turbulence energy towards the wall.

F arabee (1986) and Farabee and Casarella(l986) conducted measurements of the
wall-pressure field underneath two separated/reattached boundary-layer flows: over a
forward- and backward-facing step. They found that the process of separation and
reattachment of a turbulent boundary layer produces very large wall-pressure fluctuations.
In the backward-facing step, the wall-pressure fluctuations were higher than those of the

forward-facing step and equilibrium boundary layer by factors of five and ten,

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respectively. Although the energetic flow structures responsible for the generation of the
wall-pressure decayed as they were convected downstream by the mean flow, they were
still identifiable in Farabee and Casarella’s measurements at locations up to 72 step
heights downstream of the backward-facing step. The latter observation, which sustains
the observations of Bradshaw and Wong (1972), indicates that the upstream history of a
boundary layer could be significant on the aero/hydro-acoustic features of the boundary
layer over substantial streamwise distances.

Castro and Epik (1998) also conducted measurements in a developing boundary
layer downstream of a separating/reattaching flow downstream of the leading edge of a
blunt flat plate (the same model used in Cherry et al. 1984). Castro and Epik reported
that the log-law did not exist immediately beyond the mean reattachment point. They
also observed a slow development process in the inner sublayer; although it was faster
than the development of the outer part of the flow. Nevertheless, Castro and Epik argued
that, notwithstanding the faster rate of development of the inner region, the rate of
development of the outer part of the flow determined the overall rate of development of
the whole flow.

Recently, Song and Eaton (2002) investigated the separation, reattachment and
recovery regions of a boundary layer flow over a curved ramp. Their flow exhibited
similar structural features to those of the backward-facing step, including the roll-up of
vortices downstream of separation followed by their partial distortion at reattachment.
Farther downstream, a non-equilibrium boundary layer existed and underwent a similar
relaxation process to that discussed above. Song and Eaton (2002) also investigated

Reynolds number effects, varying the Reynolds number by changing the air density and

11

 

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freestream velocity. They found evidence of the flapping motion of the free shear layer.
They also reported that the remnants of the separated flow vortical structures dominated
the outer layer in the recovery region. Moreover, this outer portion of the flow seemed to
recover to the equilibrium state at much slower rate than that of the near-wall flow.

On the other hand, a few investigations studied the developing boundary layer
flow downstream of the separating/reattaching flow over a fence-with-splitter—plate.
Ruderich and Femholz (1986) investigated the flow over a fence—with-splitter—plate
model, and carried out mean and fluctuating velocity measurements using hotwire and
pulsed-wire anemometry. Similar to Castro and Haque (1987), they selected this flow
geometry because of the negligible effect of the upstream boundary layer on the flow
downstream of separation. Another attractive feature of this flow field was the
elongation of the recirculation region due to the steep angle of the velocity vector at
separation, which improved the measurement resolution. Ruderich and F ernholz
measurements were used to estimate the mean-velocity, Reynolds shear stress and
Reynolds normal stress distributions in the wall-normal direction. The mean-velocity
distribution of their flow downstream of reattachment exhibited a deviation from the
equilibrium boundary layer similar to that reported by Bradshaw and Wong (1972) and
Farabee (1986) in the back-step flow. Also, consistent with Farabee and Casarella (1986)
and Bradshaw and Wong (1972), Ruderich and Femholz (1986) could not observe any
evidence of the equilibrium boundary layer even at the end of the splitter plate (68 step
heights). Interestingly, Ruderich and Femholz's study is one of the very few studies that

found no evidence of the free-shear-layer flapping motion.

12

 

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1.1.2. Characteristics of the Turbulent Wall Pressure

In the past few decades, there have been many investigations of the fluctuating
surface-pressure field in turbulent flows. Although the solution of the mathematical
problem of turbulence (the closure problem) depends on the ability to understand the
physics of, and successfully model, the Reynolds stress terms, understanding the relation
between the pressure and velocity fields is important for predicting the unsteady flow
forces acting on a surface for devising solutions for flow-induced noise and vibration
problems and controlling the flow field. Basically, turbulent pressure fluctuations are
related to the velocity field of incompressible flows through Poisson’s equation, obtained
by taking the divergence of the Navier-Stokes equations. Poisson’s equation is given by

(e.g., see Willmarth 1975 and Kim 1989):

1
—V2p=-—ui‘j up, (1.1)
p

where p is the pressure, p is the fluid density and u, is the total velocity vector. By
applying the Reynolds decomposition for a two-dimensional mean flow with only one
important mean-shear component, and considering only the turbulent component of the

pressure, equation (1 .1) simplifies to:

(1.2)

where u is the mean streamwise velocity, the prime denotes the mean-removed, or
turbulent, quantities, and x and y are the streamwise and wall-normal coordinates,
respectively. Equation (1.2) shows how the wall-pressure fluctuations are a function of
the flow "sources" in the turbulent flow field. The source terms consist of the mean shear

(MS), or linear term, and turbulence-turbulence (TT), or nonlinear terms (e.g., Chang et

13

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al. 1999). Because the MS term represents the interaction between the mean shear and
the fluctuating-velocity gradients (first term on the right hand side of equation 1.2), it
changes instantaneously in response to changes in mean-flow conditions and is called the
rapid term. On the other hand, because the TT term represents the interactions between
the fluctuating-velocity gradients, it is affected only once the turbulence has had a chance
to adjust to the new mean-flow conditions, and therefore this it is called the slow term
(Kim 1989).

Ideally, it is desired to measure the pressure fluctuations inside a turbulent flow
using a non-intrusive technique to avoid introducing any error into the measurements.
However, measuring the pressure fluctuations in a turbulent flow without interfering with
the flow field is not possible to date. Therefore, pressure fluctuations measurements have
been limited to the wall in the case of wall-bounded flows. Such a measurement
technique is non-intrusive, which made wall-pressure measurements attractive to many
researchers who aim to gain better understanding of both the turbulent structures and
their relationship to the wall—pressure signature.

Typically turbulent wall-pressure fluctuations are characterized using statistical
methods, either conditional or long-time averaged. The former are used to study the
spatiotemporal features of the wall-pressure using its ensemble-averaged data relative to
the occurrence of strong positive or negative pressure peaks, while the latter include RMS
values, spectra, probability density function (PDF), and higher-order moments of the
wall-pressure. Below, an account is given of the wall-pressure characteristics in
separating/reattaching, developing-boundary-layer and equilibrium-boundary-layer flows

based on existing literature.

14

 

1. Sep‘.

 

(Farah
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slou- re

 

I. Separating/Reattaching Flows

As mentioned earlier in 1.1.1, several investigations have studied the wall-
pressure characteristics in separating/reattaching (e.g., Cherry et al. 1984, Driver er al.
1987, Lee and Sung 2001 & 2002, and Hudy et al. 2003) and developing-boundary-layer
(Farabee 1986, and Farabee and Casarella 1986) flows. The former set of investigations
have reported that the wall-pressure beneath the recirculation bubble is dominated by the
shear-layer flapping motion very close to separation. Farther downstream, the vortical
structures in the free shear layer grow in size and approach the surface till they impinge
on the wall at reattachment. As a result, the wall-pressure signature is dominated by the
shear-layer vortices, and the wall-pressure fluctuation level increases with increasing
downstream distance till it reaches a peak slightly upstream of reattachment, as discussed
by Cherry et al. (1984) and Hudy et al. (2003). Similarly, Farabee and Casarella (1986)
reported that the wall-pressure signature was dominated by the free-shear—layer vortical
structures and their level remained higher than those of the equilibrium boundary layer
flow even at 72 step heights (z12xr) downstream of the back step. They attributed that to
slow relaxation of the vortical structures.

The physical relation between the large-scale vortices in the reattaching free shear
layer and farther downstream in the non-equilibrium boundary layer and the wall-
pressure signature has been an interest for many researchers (e. g. Cherry et al. 1984, Lee
and Sung 2001, Lee and Sung 2002, Kiya and Sasaki 1985). Their investigations have
shown that negative peaks in wall-pressure fluctuations are associated with the passage of
large-scale vortex cores. On the other hand, positive peaks were found to occur beneath

the downward inrush of freestream fluid inbetween the vortical structures.

15

 

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A few studies have investigated the spectral characteristics of wall-pressure in
separating/reattaching flows, revealing that as the flow approaches reattachment its wall-
pressure signature becomes more dominated by the free-shear-layer vortical structures
(e.g., Lee and Sung 2001, Hudy et al. 2003, and Farabee and Casarella 1986).
Additionally, Lee and Sung (2001) observed a build up of a frequency range in the wall-
pressure spectra with a slope of —7/3 as the flow progressed towards and downstream of
reattachment. This agrees with the study of George et al. (1984), who reported —11/3 and
—7/3 spectral slopes associated with the MS and TT terms, respectively, in their solution
of Poisson’s equation in a free shear flow. Lee and Sung (2001) suggested that farther
downstream of reattachment, the TT interaction term became more prominent due to the
enhanced small-scale activity. Moreover, they concluded that the wall-pressure
fluctuations are largely attributable to the free-shear-layer vortical structures even
downstream of reattachment.

II. Fully-Developed Turbulent Boundary Layers

Most conceptual models of near-wall coherent structures in turbulent wall-
bounded flows are based on the theme of a horseshoe (hairpin) vortex that forms locally
at the wall, with the legs of the horseshoe trailing upstream of the arch (e.g., Thomas and
Bull 1983 and Lueptow 1997). The horseshoe vortex structure is associated with ejection
of slow speed fluid away from the wall (typically referred to as Q2 event) and inrush of
high-speed fluid towards the wall (or Q4 event). Thomas and Bull (1983) found evidence
that the wall-pressure positive peak was associated with a sudden step-like rise pattern,
even in the outer part of the boundary layer, in the streamwise velocity fluctuations,

which reflected a large-scale strong sweep event. On the other hand, Farabee and

16

 

50 UFC t‘

lilal Ft

 

 

bound

 

 

Casarella (1991) established the scaling law for the low-, mid-, and high-frequency
regions of the wall-pressure spectra. Their results showed the existence of two
"wavenumber" groups: a high wavenumber group that was associated with turbulent
sources in the logarithmic region of the boundary layer, and a low wavenumber group
that represented the large-scale turbulence contribution from the outer region of 'the
boundary layer.

Johansson et al. (1987) used conditional-average data in turbulent boundary
layers to show that the buffer region structures are responsible for the generation of large
positive wall-pressure peaks, which indicated a link between wall-pressure peaks and the
turbulence-producing mechanisms. Johansson et al.’s data indicated that the pressure-
peak amplitude was found to scale linearly with the conditional-velocity amplitude
indicating the dominance of the MS source term in generating the wall pressure.
However, Kim (1989) and Chang et al. (1999) found that in htrbulent channel flow the
slow and the rapid pressure fluctuations are of equal importance very near the wall.

Over several decades, many researchers have used scaling arguments to show that
the wall-pressure spectrum ((Dpvpv) for a certain frequency range should obey a power-law
type behavior; i.e., (Dp'pr ~ 00", where n is a function of the frequency range of the power
spectrum. Bradshaw (1967), Panton and Linbarger (1974), Farabee and Casarella (1991),
Gravante et al. (1998), and Chang et al. (1999) have shown that the wall-pressure power
spectrum of a turbulent boundary layer should exhibit a fall-off rate of a)“ in the middle
range of frequencies. However, Gravante et al. (1998) showed that the frequency extent

of this region decreased or even disappeared with decreasing Reynolds number, which

17

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agreed with Kim (1989), who reported the nonexistence of the —1 region in their low
Reynolds number channel flow.

On the other hand, Monin and Yaglom (1975) theorized a spectral fall-off rate of
(0’73 in the inertial subrange of the wall-pressure power spectrum, corresponding to the
pressure signature of locally isotropic turbulence. The results of Gravante et al. (1998)
could not reveal a substantial —7/3 spectral range. Earlier, Farabee and Casarella (1991)
also noted the absence of the -7/3 range from their spectral measurements and attributed
this to the spatial-resolution limitation of their pressure sensors. Finally, in the high-
frequency region of the turbulent wall-pressure spectrum, Blake (1986) theorized an 00‘5
behavior that was independent of Reynolds number when scaled using wall, or viscous,
variables. The decay rate of 03—5 has been also reported in several investigations in the
literature of the turbulent boundary layer; e. g., Gravante et al. ( 1998) and Chang et al.
(1999).
1.1.3. Stochastic Estimation

As will be outlined in the objectives of this work, stochastic estimation is used
here as a tool to estimate the flow velocity field from its wall-pressure signature.
Stochastic estimation was first used by Adrian (1977 & 1979) to characterize the
conditional eddies of isotropic turbulence. He examined the existence of the conditional
flow structures by computing the estimated velocity u,(x+r, t) from a known velocity u(x,
t); i.e., <u,(x+r, t) i u(x, t)>, which was referred to as “conditional eddies”. Basically,
Adrian proposed that the estimated velocity (u,) could be expanded in a Taylor series of u,

and then truncated at a certain order. When only the first term is included in the series,

18

 

 

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the estimation is referred to as Linear Stochastic Estimation (LSE) while if the first two
terms are contained in the expansion, Quadratic Stochastic Estimation (QSE) is obtained.

In the 19805 and 19905 many researchers employed stochastic estimation as a tool
to estimate the turbulent velocity field from known velocity and/or wall-shear stress at a
particular point in space. Tung and Adrian (1980) examined the conditional eddies in
isotropic turbulence by estimating the velocity field around a point where the velocity
was specified. They concluded that LSE was a satisfactory method for studying the
qualitative large-scale features of the conditional eddy field. On the other hand, Adrian
and Moin (1988) could characterize quantitatively the large—scale organized structures of
a homogenous turbulent shear flow using LSE based on the velocity and the deformation
tensor at an arbitrary point. They applied the linear estimate to the turbulent field data
generated from direct numerical simulation, DNS. They found that using the Reynolds
shear stress as the event in the estimation, the largest contribution to the Reynolds stress
were associated with second- and fourth-quadrant events.

Guezennec (1989) applied the stochastic estimation technique to a fully-
developed turbulent boundary layer. His study depicted the advantage of the technique in
obtaining conditional averages from the unconditional statistics to estimate the
spatiotemporal characteristics of the second- and fourth-quadrant events in the boundary
layer. He also extended the usage of the stochastic estimation technique, based on a wall-
shear-stress condition, to single- and two-point conditional averages. Although, the
results showed that the difference between the linear and quadratic estimation was

minimal, it appeared that the quadratic estimation included additional information.

19

 

in the
comp?
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that t?
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appro.

ICITTIS.

recons
bound.
the o:
(LSE)
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Guezennec also found that using higher estimation orders resulted in higher noise
in the estimation, which indicated that the quadratic estimation represented a “good
compromise” between the convergence of the series expansion and practical limitations,
as indicated earlier by Tung and Adrian (1980). On the other hand, Brereton (1992)
presented a procedure to assess the accuracy in the stochastic estimation model of the
time-delayed conditional-averaged velocity in a turbulent boundary layer. He showed
that the inclusion of higher-order terms in the estimation biases the stochastic model to
provide better representation of rarer events. Therefore, Brereton introduced a new
approach, which was based on tailoring the estimation model to include negative-order
terms, seeking better accuracy in estimating more-frequent events.

In addition, Choi and Guezennec (1990) used stochastic estimation as a tool to
reconstruct the conditional structure and to examine their asymmetry in a turbulent
boundary layer for various levels of Spanwise velocity perturbations near the wall. On
the other hand, Bonnet et al. (1994) introduced a linear stochastic estimation
(LSE)/Proper Orthogonal Decomposition (POD) complementary technique for
identifying structures in an axisymmetric jet and a 2-D mixing layer. Furthermore,
Glauser et al. (1999) used the LSE/POD as a low-dimensional model to estimate the
large—scale unsteadiness of the reattachment region in an axisymmetric sudden expansion.

Estimating the flow field features using their wall-pressure signatures has been
also sought by a few investigators. Naguib et al. (2001) were the first to seek to
understand the interrelation between the flow field structures and the turbulent wall-
pressure through stochastic estimation. They assessed the ability of stochastic estimation

to capture the conditionally averaged flow field associated with negative and positive

20

 

wall-pressure peaks by conducting simultaneous measurements of the wall-pressure at a
single-point and velocity in a turbulent boundary layer. The comparison between the
conditional and stochastic estimation (linear and quadratic) results showed that the linear
stochastic estimation based on the wall-pressure did not converge to the conditional
average, and that it was necessary to include the quadratic term. The latter was
inconsistent with the findings of earlier investigations employing stochastic estimation
e.g., Adrian et al. (1987) and Guezennec et al. (1987). However, the discrepancy was
attributed by Naguib et al. (2001) to the use of the wall-pressure as the estimation
condition. Specifically, Naguib et al. demonstrated that the need for the inclusion of the
quadratic term in the estimation was attributed to the effect of the TT pressure source
term.

Recently, Murray and Ukeiley (2003) have demonstrated the potential for multi-
point stochastic estimation, based on wall pressure, to accurately capture the
instantaneous flow field in a resonating cavity flow. The success of the estimation in the
cavity flow is believed to be due to the highly organized nature of this kind of flow and
the well-defined phase relationship between the pressure and the vortical structures in the
shear layer, which dominate the flow. In particular, feedback of the pressure disturbances
due to periodic vortex impingement on the downstream lip of the cavity to the upstream
lip sets the entire flow field in a highly organized resonant state. In a latter work, Murray
and Ukeiley (2004) also employed the same estimation procedure in a non-resonating
cavity flow. Although no direct comparison with the instantaneous flow field was
provided in this case, the authors demonstrated that the estimation captured the essential

features of the instantaneous flow structure.

21

 

 

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Two other example studies that employ surface measurements for flow field
estimation are those by Taylor and Glauser (2002) and Schmit and Glauser (2004).
These studies employ a combined LSE/Proper Orthogonal Decomposition (POD)
technique for estimating the flow field above the wall. In the first work, wall-pressure-
array measurements were successfully employed to capture low-frequency fluctuations of
the instantaneous velocity field associated with the flow over a backward-facing ramp.
On the other hand, Schmit and Glauser (2004) employed an array of dynamic strain
gauges mounted on the wing of a Micro Air Vehicle to estimate the instantaneous flow
field in the immediate wake behind the wing.

1.2. Motivation

This work was motivated by the interest to understand the relation between the
flow field and wall-pressure in the non-equilibrium flow downstream of a
separating/reattaching flow. In particular, with the exception of Farabee and Casarella
(1986), no other study of the wall-pressure beneath the flow downstream of a
separating/reattaching flow was found. Furthermore, Farabee and Casarella employed
only two-point measurements in their work. In the current investigation, simultaneous
measurements of the wall-pressure field p'(x,t), using a sixteen-microphone array, and
the velocity field, employing an X-hotwire sensor, will be undertaken. The coupled array
and velocity measurements will allow:

1. Determination of the space-time characteristics, including the fi'equency-
wavenumber spectrum, of the wall pressure field. The documentation and
understanding of such characteristics is important to flow-induced noise and

vibration problems since it is the combined spatial and temporal characteristics

22

 

1.3. C

 

2.

SJ

that determine if the wall-pressure field excitation of the wall will lead to
substantial vibration and noise generation.

Multi-point Linear Stochastic Estimation (LSE) of the velocity field based on wall
measurements. The estimation is motivated by examining the flow structure and
wall-pressure sources associated with various types of wall-pressure signature.
Ultimately, such examination may aid in developing structure-based models of
the flow pressure sources, allowing simplified computations of the wall-pressure
field in engineering calculations of flow-induced noise and vibration problems

downstream of appendages and other separated flows in engineering devices.

1.3. Objectives

1.

The specific objectives of this work may be listed as follows:

Construct a 16-point fluctuating-wall-pressure-sensor array and develop an
appropriate calibration procedure. Integrate the sensor array into the fence-with-
splitter-plate setup of Hudy (2001) beneath the non-equilibrium boundary layer
downstream of reattachment.

Characterize the wall-pressure field through one- and two-point statistics as well
as the wavenumber-frequency spectrum.

Conduct simultaneous measurements of the wall-pressure and flow velocity using
the microphone array and X-hotwire sensor, respectively.

Examine the velocity-field characteristics and its relation to the wall-pressure
field utilizing various conventional and conditional statistical analyses.

Estimate the "pseudo-instantaneous" flow field associated with typical

instantaneous spatial wall-pressure patterns using multi-point wall-pressure based

23

' a ' 3."

 

descri
preset
bound
result:

FCCOII‘.

 

LSE. The estimation is expected to not only provide some sense of the variability
in the characteristics of the flow structures associated with typical wall-pressure
signatures, but also to lead to better understanding of the physical nature and
relative importance of the flow pressure sources. It is noted here that such a
detailed study into the nature and significance of the wall-pressure-generating

flow structures have not been conducted to date in the flow field considered here.

The remainder of this work consists of four main chapters. Chapter 2 provides a
description of the experimental set-up and methodology, Chapter 3 contains a
presentation of the wall—pressure results, while Chapter 4 includes a discussion of the
boundary layer characteristics, pressure-velocity correlations and stochastic estimation
results. Lastly, Chapter 5 highlights the main conclusions of the present work and

recommended future work.

24

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in the
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2. EXPERIMENTAL SET-UP AND METHODOLOGY

This chapter describes the experimental set-up, techniques and methodology used
in the present work. Four measurement techniques were used to investigate the flow field
of interest: static-pressure taps, flow visualization using tufts, microphones and single/X-
hotwire probes. Two different sets of data-acquisition hardware were used to acquire the
experimental data resulting from application of these various techniques. A full
description of the set-up, methodology and data-acquisition hardware is provided in
subsequent sections.

2.1. Experimental Set-up

The experimental set-up consisted of a fence-with-splitter-plate model, an open-
circuit wind-tunnel facility, the sensing instrumentation and data-acquisition systems.
Each of these constituents is described in details in this section.

2.1.1 Wind Tunnel Facility

The present experiments were conducted in a low-speed open-circuit indraft wind
tunnel1 facility in the F low Physics and Control Laboratory (FPaCL) at Michigan State
University. The wind tunnel consisted of four main sections (see Figure 2.1): the
turbulence-manipulation section, contraction section, test section and diffuser and blower
section. The overall dimensions of the wind tunnel were approximately 7.5 m long by
2.0 m high and 1.2 m wide.

The turbulence-manipulation (TM) section directed and conditioned the air flow
to the contraction section. The main purpose of this section is to produce uniform mean

velocity profile and to break up the large-scale eddies in the flow into smaller ones so that

 

' The wind tunnel was originally designed and built at Northwestern University by Steve Snarski and
Richard Lueptow, and was moved to MSU in 1999.

25

 

CFOSS

 

 

they can dissipate prior to entering the contraction section. The overall dimensions of the
TM unit were 1.15m x 1.15 m in cross section by 1.06 m in length. It consisted of 10
separable wooden frames. Each of the first five frames supported an l8-mesh aluminum
window screen while each of the last five ones supported 30—mesh stainless steel screen.
A 25,000-cell aluminum honeycomb element (6.0 mm cell size, 50.0 mm length) was
also housed in the filth section with the 18-mesh screen.

The contraction section accelerated the flow through a contraction ratio of 10.821.
The section had a 1.15 m-wide square inlet cross-section and a 0.35 m-wide square outlet
cross-section. The total length of the contraction section was about 1.5 m and it was built

of plywood and fiberglass with a Formica inner surface.

Turbulence Manipulation Section Contraction Test Diffuser/Blower
(Screens and Honeycombs) Section Section Section
0508 -
5 x 0.125 u
5 x 0.0875 ~ < ‘ 0966 ' 7 2'77 if- «048* 1.35 >

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1. Schematic2 of the wind facility (dimensions in meters)

The test section was 2.77 m long with inlet cross-section matching that at the exit
of the contraction. The walls of the test section had a divergence angle of 0.13 degrees
with respect to the centerline. The angle was selected to compensate for the displacement

thickness of the boundary layer growing on the test-section walls in order to minimize the

 

2 Done by Chad Stimson, Michigan State University, East Lansing, MI 48824

26

streamwise pressure gradient for most of the wind tunnel operating velocity range. The
test section was made of plywood with a Formica inner surface supported by an
aluminum external frame. One of the test section walls contained three windows that
were made out of clear plexiglass. Each window was 0.9 m long, 0.3 m high and 12.5
mm thick. The windows were mounted such that their inner faces were flush with the
inner test-section wall.

The fourth, and the last, component was the diffuser/blower section. The section
was used to decelerate and turn the flow horizontally by 90 degrees towards the blower.
The section was made out of 14 gage steel exterior case with perforated 22 gage
galvanized steel interior flow surface and fiberglass acoustical wool inserted between the
two. The relatively high contraction ratio (10.8) combined with the 90-degree bend and
the perforated steel/acoustical lining were responsible for reducing the frequency and the
strength of the blower noise, which is a potential source of contamination of the wall-
pressure measurements. The high contraction ratio helped to reduce the blower rotation
speed for a given wind tunnel velocity, and hence reduce the frequency of the noise
generated due to the passage of the blower blades. On the other hand, the geometry and
sound-absorption lining of the diffuser helped to attenuate the upstream—propagating
noise resulting from the blower-blade passage.

The air flow in the tunnel is induced by an airfoil-vane centrifugal blower, type 27
SQA from Chicago Blower Corporation. The blower was driven by a dual belt drive and
Minarik 180 Volt, 3hp permanent magnet DC motor, model number 504-36-043A with
speed range of 30-1800 rpm. The speed control was provided by General Electrical

rheostat controlled AC motor-DC generator set with variable frequency controller of the

27

 

bloyy

indic

Pitot

’Q
.—
Id

1 see
symr
were
plate
1W0

regar

 

blower speed. The controller has a remote-operating keypad with a digital frequency
indicator. The frequency indication was calibrated against flow velocity measured by a
Pitot tube in the test section to facilitate setting of the freestream velocity.

2.1.2. Test Model

The fence-with-splitter-plate model of Hudy (2001) was used in the present work

(see Figure 2.2). The design of the model, which was constructed from aluminum, was
symmetric with respect to top and bottom. The total length and the width of the model
were 160h and 44h, respectively; where h = 8 mm is the fence height above the splitter
plate (or "step" height). In order to improve the two dimensionality of the mean flow,
two endplates were attached to the two sides of the splitter-plate (for more details

regarding the model design, see Hudy 2001).

     
 
  

Extensions ‘ Fence

‘ \
Flow \ - Instrument Plate
Middle Plate
Endplates \
T " \i\‘\ I I);
‘y ' - \‘a Tail Plate
Z/ \x

28 Pressure Taps

l

. E

16 Microphones a

(’1

c5
1

_ <0 f
_, 0:115
/

Figure 2.2. A schematic of the test model

28

 

 

rahc

expl.

 

tneas
than

pres;

Inods
Skelc
F1911?

ificthi

 

The total fence height (2H = 2h + splitter plate thickness) was 35 mm. This fence
height was selected to produce reasonable flow blockage while maintaining a long
reattachment length to "stretch" the flow field and facilitate spatial resolution of the
measurements in the x direction. Smits (1982) defined a blockage ratio (h/D.)
representing the ratio between the step height (h) and half of the test section height (D1).
He found that the blockage ratio resulted in flow acceleration around the fence, which
decreased the mean reattachment length as the blockage ratio increased. The blockage
ratio of the present model was 4.5% (see Figure 2.3 for visual illustration). As will be
explained later, the reattachment length (X,) of the present work was estimated using tuft
measurements, which produced an Xr value of 180 mm (22.5h). This is slightly shorter
than Xr of Hudy et al. (2003) because their blockage ratio (2%) was lower than the
present study.

For the current study, an array of 16 microphones and 56 pressure taps (28 on
each side of the model) were mounted in the splitter plate. The splitter plate consisted of
a half-inch skeleton sandwiched between 3.175 mm-thickness aluminum plates. These
plates included an instrument plate (I-plate) and a middle plate (M-plate) on each side.
The I-plate was 51h (406 mm) long while the M-plate was 76h (610 mm) long. The tail
plate (T-plate) was used to reduce plate-wake disturbances by gradually reducing the
model thickness to zero. The T-plate length was 32h (254 mm). The volume within the
skeleton was used for microphones insertion, as well as wires and tubes connections.
Figure 2.4 shows schematic drawing of the “top” view of the model. A corresponding

picture of the model installed in the wind tunnel is shown in Figure 2.5. Note that,

29

. ' V ‘ 'v' " a: ‘Vn77'7'fiirm'.

 

Flt

 

PICS I

 

 

 

 

 

Figure 2.3. A frontal picture of the model showing the blockage effect of the fence

Support Test section
Members top wall Endplates
.................. . \ ._._.._ .\
........ \ ,\\ \.\..
q 7 1600 mm
‘ 406.4 mm __
20 mm _ _‘
28 Pressure taps
Flow 4 ~ ' a»,
, I ‘ 300 mm > 16 Microphones
Fence
I 1
Instrument plate Middle Plate

Figure 2.4. A schematic of the instrument and middle plates depicting the locations of the
pressure taps and microphones

30

images in this thesis are presented in color. The 28 pressure taps and six microphones
were installed in the l-plate while the middle plate contained the rest (10) of the
microphones. The pressure taps were 9.5 mm center-to-center apart while the

microphones were 20 mm apart with the microphone array stretching from 300 mm

(3 7.5h) to 600 mm (75h) downstream of the fence.

....
n

M-plate

-1
i ,
l
1
i
.1

 

Figure 2.5. A picture of the fence-with-splitter-plate model inside the wind tunnel

2.1.3. Instrumentation

1. Static pressure

Static pressure measurements were used to align the model with the flow direction
in order to achieve flow symmetry on both sides of the splitter plate. As mentioned

earlier, there were 56 pressure taps (28 on each side) mounted flush with the l-pate

31

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' A" (F.

 

surt'i
appr
pnet

Ufifll

scar
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surface. The inner diameter of each tap was 1 mm and the outer diameter was
approximately 1.5 mm. Each of the taps was connected with a long Urethane tube to a
pneumatic connector (48D9M-l/2) of a pressure scanner. The scanner, designed and
manufactured by Scanivalve Corporation (48D9-1/2 Scanner Oiless Design with 48
ports), had a 100-psi range and was driven by a rotary solenoid (48D9M-1/2) coupled
with a homemade solenoid-actuation circuit (for more detail regarding the circuit design
the reader is referred to Hudy 2001). The solenoid was made to "step" from one pressure
port to the next by feeding a 5V control signal to the actuation circuit from an analog-
output channel of a PC-based D/A converter. The stepping and pressure-data-acquisition
algorithms were synchronized by containing them within the same data acquisition
program.

To measure the static pressure relative to some reference, the output port of the
scanivalve was connected to a 0-1.33 kPa Baratron pressure transducer (model 223BD)
with an output range of 0-1 Volt. The output port of the scanivalve was connected to the
negative side of the pressure transducer while the positive side was connected to a
pressure tap at the entrance of the test section, upstream of the model, to provide a
reference pressure (p,) for the measurements. The first port of the scanivalve was left
open to the atmosphere while the following ones were connected to the tubes from the
pressure taps. In this manner, the “home position” of the scanivalve gave the differential
pressure between the atmospheric and reference pressure (pm - pr) while the following
positions yielded the differential pressure between that acting at the corresponding
pressure tap and the reference pressure (ps - pr). Forty of the 56 pressure taps (28 taps on

the top side and 12 on bottom) were used to align the model with the flow direction.

32

 

 

1
E‘
,.
. a

ll.T‘

esuh
dime
that}
in d:
yam

thetH

caPlU:

 

 

II. Tufts Visualization

Tufts were used to visualize the flow direction close to the wall in order to
estimate the reattachment length, which is a relevant length scale for non-
dirnensionalizing the investigated flow quantities (e.g., Ruderich and Femholz 1986). A
black-yarn tuft was attached to the surface of the I-plate at different streamwise locations
in the reattachment zone. The tuft was constructed from a 10 mm-long piece of black
yarn attached to a short thread that was in turn attached to the I-plate using a tape, freeing

the tuft to follow the flow. An image of the tuft is shown in Figure 2.6.

 

Figure 2.6. An image of a tuft attached to the I-plate

A CCD camera (Sony XC-75) coupled with a flame grabber system was used to

capture the tuft image. The camera had a '/2- inch CCD image sensor with a resolution of

33

nun
pert
the

inta;
alin

lll..

 

the

Pan'

the
in th
Sensi

dElf

ofth
T6813
then

Emd I

768 x 494 pixels. The frame grabber was a National Instruments IMAQ PC1-1408 board
(for more information regarding the board specifications, the reader is referred to the
subsection V below in the current section, 2.1.3). The idea was to capture enough
number of images in order to statistically determine the dominant flow direction from the
percent of the time that the tuft is imaged in the up or downstream direction. Because of
the high reflectivity of aluminum, black yarn was used to increase the contrast of the
images and make it easier to determine the tuft location by looking for a dark area against
a bright background.
111. Microphones

A microphone array consisting of 16 Panasonic microphones was used to measure
the wall pressure fluctuations downstream of the separating/reattaching flow. The
Panasonic microphone (WM-6OAY) was an omnidirectional back electret condenser
microphone cartridge. Figure 2.7 shows an image that illustrates the geometry of one of
the microphones used in the array. The microphone was 6.0 mm in diameter and 5.0 mm
in thickness. The sensing diaphragm was exposed to the flow through a 2 mm round
sensing hole. The nominal sensitivity was specified by the manufacturer to be —42 i 3
dB for a bandwidth of 20 — 20,000 Hz.

Figure 2.8 shows a schematic of a homemade circuit that was used to drive each
of the microphones. The circuit was powered by 9 V DC power supply connected with a
resistance of 2.2 k!) in parallel with the microphone. The output of the microphone was
then high-pass filtered to remove any DC component in the output (V0). The capacitor (C)

and the resistor (R2) of the high-pass filter were selected for a cut-off frequency of 0.16

34

Hz, which was sufficiently low to block the DC without removing any significant

fluctuating-pressure energy.

 

Figure 2.7. An image of one of the microphones used in the array

I F=0.luF

R,=2.2 kn
V0
+

R2=10 M9 9 v Mic.

2.]

 

 

 

 

_I_

Figure 2.8. A schematic of the microphone driving circuit

35

 

IV.

inye

COlTl

 

 

TU

 

IV. Hotwire

Single- and X-hotwire probes were used to measure the velocity in the
investigated flow. The single hotwire was used to measure the streamwise (u) velocity
component for estimating the boundary-layer mean- and turbulent-velocity profiles. On
the other hand, the X-hotwire was used to measure the streamwise and wall-normal
velocity components simultaneously with the microphone-array data. The probe design
was similar to that of a straight X-probe except the four prongs were bent by an angle of
30°, as shown in Figure 2.9. Bending the prongs in this manner enabled positioning them
as close as 0.5 mm (or less) to the surface of the splitter plate. Furthermore, by making
the bent part lower than the tip of the probe support, the significance of any disturbances
that might be created by the probe support is reduced. Photographs of the probe can be

seen in Figure 2.10.

. .7 * ___i ,7 7 7,. l/x/T
’ Dimensions in mm ./
.. ,./"'/i
Q ’1,’ /
x???“ g “ /
e / '/ / ./
f2 / ,,// /,
° x// if \ ,/ 1’ I ,//
y-level of the tip of 31 /,/ /,/
the probe support / X / \\
\ ,/ ,/ .
/ , /. \\\
Probe Support

 

Figure 2.9. A schematic of the X-probe

36

 

norm
consi
brack

frams

hOfiZt

 

Kid in
C0510}:
“Hire :-
Rater.
“615 1';
SEFPC
motor

 

 

Top View

   

Figure 2.10. Images showing top and side views of the X-probe

The hotwire probes were operated by a Constant Temperature Anemometer, CTA,
from DANTEC (Model 54T30) with an Over Heat Ratio (OHR) of 1.6. The output
signal was low pass filtered using a built-in filter with a 10 kHz cut-off frequency.

A traversing mechanism was used to traverse the hotwire probes in the wall
normal (y) direction. Figure 2.1] shows an image of this traversing mechanism, which
consists of a toothed rod geared with a stepper motor and assembled to an aluminum
bracket. The figure illustrates how the traversing mechanism was fixed to a supporting
fi'ame that was built out of uni-strut structural elements next to the test section. A
horizontal slot was machined in the Plexiglass window to allow protrusion of the toothed
rod into the test section. The hotwire probe was clamped at the end of this rod using a
custom adapter. The position of the probe was then controlled by feeding TTL square
wave pulses to the stepper motor from a digital I/O controller interfaced with a PC. The
traversing mechanism was calibrated against a dial-gauge to determine its resolution. It
was found that the probe translated a distance of 58.8 mm for each revolution of the
stepper motor. This corresponded to a traversing resolution of approximately 74 um per
motor step (since 800 pulses, or steps were required to complete one full motor

resolution).

37

.;.::~:
- u
. 11": 1‘- WI— _

 

WI

p05
yer

OUI

one

nnc

daui

6024
PYOe.
SUD;

Chan

 

I0 th

VEiCig

The toothed rod was coupled with a Linear Variable Differential Transformer
(LVDT) sensor, as shown in Figure 2.11. The sensor was used to feedback the hotwire
position to verify the actual movement of the probe. The LVDT calibration was verified
versus that of the stepper motor. Figure 2.12 depicts the calibration results, where the
output voltage of the LVDT (e) is plotted versus the displacement (y). The figure
reveals the linear character of the sensor over a range of about 50 mm. This range was
large enough to accommodate the full movement of the hotwire probe in the y direction.
The slope of the calibration line represents the LVDT sensitivity, which was 0.35 V/mm.
V. Data Acquisition Hardware

Three different data-acquisition systems were used in the present work. The first
one was used throughout the experimental procedure of the wall-pressure measurements,
including aligning the model, calibrating the microphones and acquiring data fiom the 16
microphones. This was accomplished using a National Instruments AT-MIO-16E-10
data acquisition A/D card with 16-single-ended analog-input channels and maximum
sampling rate of 100,000 samples/s (corresponding to 6,250 samples/s per channel). This
sampling rate resulted in 10 us inter-channel time delay.

The second data acquisition card was also a National Instruments A/D card (NI
6024E-ADC), which was employed for driving the stepper motor, hotwire calibration
procedure, measurement of the boundary-layer velocity profiles, and acquiring the
simultaneous velocity and pressure data. The card also had 16—single-ended analog-input
channels with a maximum sampling frequency of 200,000 Hz. It was necessary to switch
to the second A/D board because the stepper motor had to be synchronized with the

velocity measurements for automated operation of the traverse/acquire procedure. This

38

I
1
I-l
III-III

 

 

F1

up

was not possible with the first board, which did not have the hardware provisions

necessary to control the motor.

'I
o
o
..

V!

F
0
O
s
'I‘t-
0.
ll.

..

‘I
I
I
.
o

l

...
s
-

 

Figure 2.11. Images of the traversing mechanism and the LVDT: overall (left) and close-
up (right) views

      
 

 

 

10 i T j 7'1 fl j ‘V T7 77 If I fl fl fir 7 f T ' ri-T 7 I 7—T "T 7 ‘l—" 7 ifi
g e =9.813—0.35y
. o r
5_ ................ . . . .......
i 0»
6
cu
_5 g ..........
. b
r J : i ,
_lOr_‘4l',L‘hi iJri. -774. ,L Val—‘1‘. ,1 274‘___1L. 2 ,l 4L_,.li A #.__l¥.J . J—; l 4;." a
0 10 20 30 40 50

y [mm]

Figure 2.12. LVDT calibration

39

1.1

 

j in!" am“ -.- 3"

tuft

11101

yidt

flour

cha.

lite

syst

anal

2.2.

1116;:

Subs

5"
Ix)

”181‘;
mO-tf

desc'

Para
“Er;

 

The third and last computer-based-acquisition board was used for grabbing the
tuft images. The board was an NI IMAQ PC1-1408 board, which was a high-accuracy,
monochrome, PCI-based IMAQ board that supported RS-170, CCIR, NTSC, and PAL
video standards. The board was connected to a BNC break-out terminal, which included
four video-input channels and four corresponding I/O triggering channels. The triggering
channels are used to start/stop sampling of images or indicate the status of acquisition.
The PC1-1408 acquired image frames in real time and transferred them directly to the
system memory after converting their data into digital formats using an 8-bit flash
analog-to-digital converter (ADC).

2.2. Experimental Methodology

A full description of the procedures used in implementation of the various
measurement techniques in the experiments conducted here is provided in the following
subsections.

2.2.1. Static pressure system

Static pressure was only measured in the separating/reattaching flow region. As
mentioned earlier, static pressure measurements were used for the purpose of aligning the
model parallel to the freestream inside the test section. This subsection gives a
description of the alignment procedure.

1. Acquisition Settings

The acquisitions settings were selected after considering the flow features and the
parameters of the static-pressure measuring system. The main settings of the acquisition
were the sampling time to converge to the average static pressure, and the time needed

after each step of the scanivalve for the pressure inside the pressure-measuring system

40

 

' “'3‘
I.“ , Him" .

“E

W e

531‘.

err

11.

niot
r Fl
pres
ofti

floyt

 

usek

Pics

disw

 

“as

(11811:

 

(connecting tubing, scanivalve and transducer) to stabilize. Tests conducted by Hudy
(2001) using the same setup had shown that 10 seconds and one second, respectively,
were more than sufficient for these parameters. Therefore, 1000 samples of pressure
were acquired at a sampling rate of 100 Hz before triggering the scanivalve to step. After
the trigger, the program waited for one full second before commencing with data
sampling. The procedure was then repeated for all static-pressure ports to complete the
experiment.

11. Model Alignment

Model alignment was done in two steps: a coarse alignment and a fine one. The
model was coarsely aligned by connecting the tubes of pressure tap # 18 (x/h = 21.4 and
x/Xr = 0.95) on opposite sides of the model to the positive and negative ports of the
pressure transducer. The "angle of attack" of the model was then adjusted till the readout
of the pressure transducer became zero, indicating rough alignment of the model with the
flow direction.

For fine alignment, 40 of the 56 pressure taps (28 on top and 12 on bottom) were
used in the alignment procedure. The taps were used to obtain a “fuller” picture of the
pressure distribution on each side of the splitter plate. By matching the static pressure
distributions on both sides of the model, the fine alignment of the model was achieved.

Figure 2.13 shows the top and bottom static-pressure distributions after the model
was aligned. It is evident that both the top and bottom mean-pressure-coefficient (Cp)

distributions match well, where:

_ P5 ‘Pr
Cp—KPUE.’ (2.1)

41

pr:
fre
bet

dei

 

 

Fig
Split

 

‘ ail

4-
«It

,6

“a1?

p, is the surface pressure at location x, pr is a reference pressure measured with a static
pressure tap located at the exit of the contraction, p is the fluid (air) density, and U.G is the
freestream velocity upstream of the model. The results in Figure 2.13 indicate that the
model is satisfactorily aligned with the direction of the flow. In fact the largest deviation
between the two curves is 3.5% of the total Cp variation (Cp, max - Cp, min), with the RMS

deviation being 2.5%.

'0.2.'" ’ T " 2 6 i ,2 7 fl 7 7 . + v 2?
l 0 “Top . O I C O
1 ‘ ' I ‘
I Bottom .'
T i ' o
-0.4" - , f , , - f
, O
'
DO- ‘06 —._.
I
O
-0.8' -.. Q,
j o
' 0 o I . . or
' a" o
_1‘. 2 , ‘. BAA- ,1_7_ .‘ __ ; _ -.., . ,
0 5 IO 15 20 25 3O 35
x/h

Figure 2.13. Mean-pressure-coefficient distribution for the top and bottom sides of the
splitter plate
2.2.2. Microphones

As mentioned in section 2.1.3, Panasonic microphones were used to measure the
wall-pressure fluctuations. The microphones installation, their calibration, time-delay

check and cross-talk check are discussed here.

42

 

1.1

and

mi;

the

 

The

the

 

1. Installation

Figure 2.14 shows the installation of one of the microphones. To mount the
microphones, counter-bored through holes were made on the back face of the instrument
and middle plates. The smaller diameter of the counter-bored holes matched the
microphone sensing diameter (2 mm), while the bigger diameter matched the diameter of
the microphone casing. The idea was to make the sensing holes completely flush with
the surface, which was achieved by inserting the microphone from the backside of the

instrument and the middle plates.

Panasonic
FIOW I Microphone
E 02 0 '
5 ' mm >‘ < ' Aluminum
" Plate
1 . l . ._ . , I .
-‘ _ " \ \ x x a E‘
r . _ e
‘ '\. O-
E ‘ . '"y
E . . c
tn
5?. £3 Conductive
M epoxy
‘0 6.0 mm;

Figure 2.14. Microphone installation in the Instrument and Middle plates

The microphones were held in place using conductive epoxy (see Figure 2.14).
The epoxy was applied around the perimeter of the microphone casing, electrically
connecting the latter in the process to the model, which acted as a ground plane. Using

the conductive epoxy guaranteed that all microphones had a common ground. All

43

 

pa

ael

 

microphone wiring was fed through the space between the top and the bottom plates of
the splitter-plate.
II. Calibration

Although the Panasonic microphones had a nominal sensitivity of —42 i 3 dB
for a bandwidth of 20 — 20,000 Hz, they had to be calibrated to determine the frequency
response of each microphone individually after their installation in the I- and M-plate.
The calibration yielded two important pieces of information. The first was the
determination of the mean sensitivity and phase response of each microphone. The
second was to check that each microphone was mounted properly without a gap between
the microphone face and the counter-bored hole, which could result in a Helmholtz-type
resonance during measurements.

A Plane Wave Tube (PWT) was used to determine the frequency response of the
Panasonic microphones. Basically, a PWT produces plane sound waves that travel
parallel to the axis of a tube or a duct. This one-dimensional planar wave propagation is
achievable if the width of the duct is small in comparison to the acoustic wavelength, 2,.
Specifically, for a square duct with rigid walls and side length of 2a, planar wave
propagation is accomplished when it, > 4a, or f < c/4a (where f is the sound frequency
and c is the speed of sound); e.g., Kinsler et al. (1982). Thus, f = c/4a establishes the
upper limit, or cut-off, frequency of the PWT. In the current investigation, the PWT
width selected is 12.7 mm, which gives an upper frequency limit of approximately 13.5
kHz for a speed of sound of 350 m/s.

A picture of the acoustic—wave-guide setup may be seen in Figure 2.15. The tube

was made of 1.8 m long aluminum square duct with a cross-section of 12.7 x 12.7 mm2

44

 

A‘

(D

 

and a wall thickness of 3.175 mm. The centers of the microphones were located along
the centerline of the PWT. Opposite to each microphone, a plug for mounting a reference
Larson & Davis (L&D) l/4-inch microphone (with known response) was made in the
upper wall of the PWT (see Figure 2.15). A sound generating system (function generator,
amplifier and speaker) was used to calibrate the Panasonic microphones. The system was
used to excite the microphones over a broad range of frequencies using white noise

containing frequencies up to the highest frequency of interest in the experiments (z 3.5

l

Spun 1\(‘l‘

l—plule

\lirltlle plate

 

Figure 2.15. A picture of the microphone-calibration setup

Before using the PWT for calibration purposes, it was necessary to check the
accuracy of the calibration procedure. The test was conducted using two L&D

microphones, where the first microphone was mounted at the upper wall of the PWT,

45

 

\K

6X1

fro

the

des

 

 

 

while the second was mounted in a plug in the side of the PWT. In this manner the
pressure ratio (R) and the phase shift (0) between the acoustic pressures acting on the top

and sidewalls of the PWT could be determined, where:

—
i

(p'nm).0p and (p'rms),,-de are the RMS values of the pressure fluctuations on the top and side

 

R (2.2)

wall of the PWT, respectively.

The results provided in Figure 2.16 are the average for two locations along the
PWT, corresponding to the beginning and end positions of the microphone array. Figure
2.16 (top plot) shows the amplitude uniformity of the sound waves over the cross section
of the PWT, as demonstrated by the Rp value of one (within 10%) for frequency range
extending up to 10 kHz. The planar character of the sound wave can also be confirmed
from the phase shift, 0 (bottom plot). The plot shows that 0 stays within a few degrees in
the same frequency range. Collectively, the amplitude and phase results show that the
desired planar sound waves have been achieved inside the PWT.

Using the above calibration procedure, the frequency response of the
microphone/hole assemblies could be determined. Figure 2.17 depicts the frequency
response of one of the assemblies. The top plot in the figure shows that the sensitivity of
the microphone (K) was flat and fell within :2 dB of the estimated mean sensitivity (Km,
which is obtained by averaging over all frequencies within the calibration range) over the
frequency range 50 < f < 5000 Hz. In addition, the plot of the phase shift between the

Panasonic and reference (L&D) microphone depicts a negligible phase delay. The results

46

 

01

01

~—

 

 

 

 

TWULMWdi C

verify the appropriateness of the frequency response of the microphone after embedding
it in the counter-bored holes of the I- and M-plates.

Figure 2.18 (top plot) shows the mean sensitivities of all microphones. The range
of sensitivities extends from 8.5 to 12.5 mV/Pa (—41.4 to —38 dB relative to a sensitivity

of 1.0 V/Pa; the nominal value reported by the manufacturer is —42 i 3 dB).

 

L5

4

 

 

H
trii—FiitTttti—rtnrr

...........................................................................................

(15

 

 

ALI—LlidLiiJlllLll

 

200

100 L

I I I I

 

0 [degree]

I I I I

-100 '—

| | T

 

 

 

 

-200
0 2000 4000 6000 8000 110

f [Hz]

Figure 2.16. PWT calibration results: pressure ratio (top) and phase shift (bottom)

47

 

:2.— .4—\.4_ 7:222‘; C

 

K/Km [dB]

0 [degree]

 

20 1

15

-10

-15

 

................

............................................................................................

l . 4
. . 1 .
1 . I l
,_ ............................. . ............................................ , ........................
>— . r . a
. l I
l . .
1 n
1 l

l l a
. I 9 I
r u I l
_5_. .............. . ...... . ......... . .....................................................................
t . .
t . y r

............................................................................................

Figure 2.17. Frequency response of a microphone/hole assembly

48

.....................................................................

##LIA._L_A~L 1214.1

III. Time-Delay Check
The time-delay of the microphone output voltage relative to the measured
pressure was deduced from the phase shift information (see Figure 2.17; bottom plot)

using the following equation:

“39’ 23
Td-dw (-)

where, Td is the time-delay in seconds, a) = 21tf (f is the frequency in Hz) and 6 is the

phase shifi angle in radians.

The deduction of the time-delay was based on estimating the average slope (if!)

in the plot of the phase shift versus frequency. The estimation indicated that Id was z 10

us, which was more than an order of magnitude (z 66 times) smaller than the average
convection time between two successive microphones (z 667 us), calculated based on the
highest U0o of 30 m/s. Figure 2.18 shows the time delays of all microphones.
IV. Cross-Talk Check

Because of the relatively large number of signals acquired simultaneously in this
study, checking the cross-talk among different acquisition channels was necessary before
sampling data. The cross-talk check was conducted by exciting only one of the
microphones by a sound source and determining the square of the RMS pressure
measured by all other microphones relative to the square of the RMS pressure measured
by the excited channel. The procedure, which utilized a Larson and Davis (L&D) hand-
held calibrator (model CAL 200), was repeated for all microphones in the array. The

calibrator was capable of generating sound waves at a frequency of 1 kHz and one of two

49

 

 

3!

TE

Sound Pressure Levels (SPL) selectable by a switch: 94 and 114 dB (relative to a
reference pressure of 20 uPa). To ensure that sound was only applied to a single
microphone, a short, flexible Urethane tube was connected to the cavity of the L&D
calibrator on one end, while the other end was pressed against the top of the excited
microphone only. Note that the test conducted in this manner enabled assessment of the
cross-talk of the whole system end to end; i.e., including the microphones while.
embedded in the I— and M-plates, the driving/conditioning circuitry, the data acquisition
terminal block, and the A/D card.

Figure 2.19 shows the cross-talk test results for the case when microphone # 13
was excited. This case is presented here because it had the most (i.e., worst) cross-talk
amongst all 16 microphones. The data used in the plot was obtained by exciting
microphone # 13 using the calibrator, as mentioned earlier, while simultaneously
acquiring data from all 16 channels. A signal attenuation ratio (SAR) was then calculated
by dividing the square of the RMS pressure measured at each channel by that of channel
13. The plot shows that the ratio is zero dB (decibels) at microphone # 13, which is
consistent with the fact that microphone # 13 is the excited microphone. The strength of
the signals measured by the other channels was at least two orders of magnitude (SAR < -
40 dB) less than the source-signal. That indicated that the cross-talk among channels was
negligible.

2.2.3. Tufts
Tufts were used to visualize the near-wall flow in order to estimate the

reattachment length (X,). As mentioned earlier, Xr is considered an important length

50

scale for characterizing separating/reattaching flows. The procedure employed for using

tufts to evaluate Xr is presented here.

 

 

 

 

 

 

 

 

 

 

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10‘“ fi ---------- O ------------ ------------------------------------------------------
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& :Illounnq nnnnnn one nnnnnnnn ‘po-o-n-r: ; T---=-q' ----- ‘ -',_ ---u: nnnnnnnn Q nnnnnnnn Ill-IIII‘: IIIIII coo-nun..-
> : . I
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8,...n-..-....-, ................................ J ..... .-.-.-W.-., .......... . ..... L ...... ..... _'1
Z c r , 2 ¢ 26
7 F 1 . . I
I; 6%... .. ...... ................................ _,
:9 P . ' _
—' I. O 5 I
45’ , ; 9 o -
a) ' . : .
g 4“,... ................. . .. ................................... ‘l
E--‘ ' : 3 I
_ . l
t : . I
2_ ..................................... - ....... I ...................................................... j
; I
0 I I I I I I l I I I I I I I l
1 2 3 4 5 6 7 8 9 10 ll 12 13 14 15 16
Mic#

Figure 2.18. Mean sensitivity (Km) of the microphones (top) and corresponding time-
delay (bottom). Microphone #1 is the most upstream microphone

51

SAR [dB]

-30;
'40:” , . . . .
-60 - ~ - 7.1 .,, - -t v: .* Li W . 2 fig H . a z, A .
2 7
Mic #

Figure 2.19. Cross-talk check of Microphone # 13 (Mic. # 1 is the most upstream one)

1. Procedure

A 10 mm-long black-yam tufi was attached to the surface of the I-plate using tape
at different streamwise locations in the reattachment zone. The streamwise location of
the point of attachment (root) of the tuft was determined using the image shown in Figure
2.6. The root point was defined as the point at which the free part of the thread
intersected with the edge of the tape. The corresponding x location of that point could be
determined from knowing the x location of the closest pressure taps captured in the image
(see Figure 2.6).

The required number of tufi images was determined by testing the effect of the
number of the acquired images on the estimated Forward Flow Probability, FFP (to be
explained later). The test results showed that 5000 images were enough for the FFP to
converge within 5% percent, which was determined by acquiring 10,000 images. During

recording, the tuft was illuminated using a fiber-optic strobe light synchronized with the

52

frame grabber system, while the shutter of the CCD camera was set to "always-open"
mode. The synchronization was accomplished by triggering the strobe light using a
signal generated from the image-grabbing LabView program through the triggering
channel of the image acquisition board. This enabled “freezing” of the tuft location in
each image.

A sample image of the tufi is shown in Figure 2.20. The image was captured after
enlarging the aperture of the CCD camera lens relative to that used to capture the image
shown in Figure 2.6. The idea was to force the background of the image (the aluminum
surface) to be completely bright, leaving the black tuft highly visible in the foreground.
In processing the images, the main idea was to determine to which side the tuft was
located, relative to a "splitting line", which passes through the root of the tufi, as
exemplified in Figure 2.20. If the tuft was to the left of the line, this would indicate a
backward (upstream) flow, and vice versa. By using a MatLab program, each image was
split into right and left "halves" based on the splitting line. Then, the average intensity of
pixels in each half was calculated. The program was then able to indicate on which side
the tuft was located by detecting the image half with lower mean intensity. The
information from all 5000 images was used to determine the mean flow direction.

II. Forward Flow Probability (FFP)

Forward Flow Probability (FFP) is defined as the probability of; i.e., the fraction
of the time that, the flow is in the forward (downstream) direction. If FFP equals 1.0, the
flow is always forward. A zero FFP, on the other hand, reflects a flow direction that is
always in the upstream direction. Accordingly, the mean point of reattachment is located

by finding the x location where FFP equals 0.5.

53

 

§\Splitting

Line

 

 

 

 

Figure 2.20. A sample wide-aperture tuft image

The F FP information was obtained for eight different x locations and the results
are shown in Figure 2.21. The FFP value varies from around 0.1 to 0.9 at x locations
extending from 161 mm to 199 mm downstream of the fence. The data for all eight
locations were curve-fitted using a third-order polynomial function (the solid line in
Figure 2.21) and the location of FFP = 0.5 on the fitting curve was found at
approximately 180 mm downstream of the fence. For comparison purposes, the
reattachment length was also estimated from the data presented by Smits (1982) for X,
versus blockage ratio for a similar flow geometry. This yielded an X, value of 168 mm
(which is 7% less than that obtained from the tuft data). Additionally, similar data from a
shorter S-mm tuft yielded an X, value of 183 mm, which is within 2% of the value found
from Figure 2.21. Considering all three values, it appears that the uncertainly in the

estimated X, value is of the order of 5-10%.

54

—" *_ ~m--.~A.?_L -- _ ......hh

 

 

] llllIllVrfTrlellfiTfilljlfiIfifT—TIITjTTYTT—T
l

l

 

 

 

 

 

 

 

i 1
fl 0 5 _ l
LL. ' - I
I i
I _
I I
O P- 1 1 L; l l l l [J J l l 1 l l i 1 1 l 4 l J 1 A L l l l l l l l l j l l l J
160 165 170 175 180 185 190 195 200
x [mm]
Figure 2.21. F FP of the reattaching flow versus the x-location of the tuft
2.2.4. Hotwire Sensors

Both a single- and X-hotwire probes were used to measure the flow velocity in the
developing boundary layer at a single x location downstream of the reattachment point.
The procedures for the magnitude and yaw calibration as well as for determination of the
probe height above the wall are presented below.

I. Procedure

A 5 um-diameter annealed tungsten wire was used to construct the hotwire
sensors. First, a short length of the wire (z 50 mm) was electroplated with copper in a
copper sulfate solution, except for a short bare length in the middle that formed the
sensing length of the wire. This sensing portion was 1 mm long, resulting in a length-to-
diameter ratio (l/d) of 200. After plating, the wire was directly soldered to the probe

prongs, resulting in a typical sensor resistance of around 5 ohms.

55

The procedure adopted to measure the flow velocity using the hotwire probes may
be summarized as follows (note: for the single wire, only procedure steps 2 through 5 are
pertinent):

1. Conducting yaw calibration of the X-wire probe to determine the angle a1+a2
between the two wires forming the X configuration (see Figure 2.22 for definition
of the angles), and identify the relation between the effective cooling velocity (U,)
for each of the wires and the velocity components u and v.
2. Positioning the probe at the desired y location above the wall and determining the
angles of the wires of the X-probe relative to the x axis
3. Calibrating the probe in-situ at the specific streamwise location where
measurements are to be conducted (pre-calibration)
4. Conducting velocity measurements
5. Calibrating the probe after measurements (post-calibration)
II. Yaw Calibration
Yaw calibration means determining the relation between the effective wire
cooling velocity magnitude, on one hand, and the magnitude of the velocity vector and its
angle relative to the wire, on the other. Such a calibration can be accomplished by
changing the yaw angle of the X-wire probe over a certain range in a known-velocity
(direction and magnitude) flow and recording the output of both wires at every angle.
Using the resulting voltage versus angle information, the sought relation can be
determined. For a single-wire probe, the wire is aligned perpendicular to the flow
direction, experiencing the most cooling influence and resulting in the maximum output

voltage for the given velocity magnitude. In this case, the angle between the velocity

56

vector and the normal to the wire (in a plane containing both the wire and velocity vector)
is zero. If there is an angle between the flow direction and the normal to the wire, the
wire primarily 'feels' the cooling effect of the component normal to the wire, with very
little cooling resulting from the component parallel to the wire axis. The velocity
corresponding to the net cooling influence is referred to as the effective cooling velocity
(Ue). Ideally, for a wire with an infinite length-to-diameter (l/d) ratio, the cooling effect
of the velocity component parallel to the wire is zero and the relation between U, and the
actual velocity is a “cosine law”. For example, under ideal conditions, wire #1 of the X-
probe shown in Figure 2.22 experiences a cooling velocity of u cos(90-0t1) associated

with the x component of the velocity (u).

 

U / q
' ‘ v

n uFC _,
Wire #1 Wire #2

   

Figure 2.22. A schematic of the X-wire probe and angles definition

57

In reality, the length-to-diameter (l/d) ratio is finite and the cooling influence of
the velocity component parallel to the wire causes a deviation from the cosine law. In
this case, the effective velocity should be related to the actual velocity by a more general
yaw response function (F2(90-a); the square exponent of F indicates positive definiteness)
instead:

U. = U F2(90-oc) (2.4)
Where, U is the flow velocity magnitude and a is the angle between the wire axis and the
velocity direction.

The form of the yaw-response function used in the present study is that proposed
by Champagne (1965):

F2(90-a) = [0032(90-0t) + k2 Sin2(90-a)]“2 (2.5)
where, k is a constant that is determined from yaw calibration. Note that when k equals
zero, the cosine law is recovered.

Figure 2.23 shows an image of the yaw-calibration setup that was used in the
present study. The setup included a rotating stage with an angle indication of one degree
resolution. The rotating stage, which was used to control the yaw angle of the probe, was
installed above the upper wall of the wind tunnel test section, and was connected to the
probe holding fixture in the wind tunnel via a 1/4" threaded rod that passed through a slot
in the test section ceiling. The fixture holding the probe mainly consisted of a horizontal
aluminum bar and a vertical cylindrical rod that were almost of the same length of 0.15 m.
Together with the X-wire probe, these elements formed a U shape, as evident in Figure
2.23. The main purpose of the U-shape design was to keep the X-wire sensing part on

axis with the rotating stage, thus allowing a change in the yaw angle while keeping the

58

center of the X-wire at the same location. This is demonstrated more clearly in Figure

2.24.

- was” mm.

 

 

Figure 2.23. An image of the yaw calibration setup

Figure 2.24 provides a schematic of the setup for yaw calibration. The angle 9
shown in the figure is that between the flow direction and a line drawn from the center of
the X-wire to the center of the vertical rod. 6 is used here to indicate the rotation of the
wire relative to the freestream. Note that because of the bending angle of the sensor
prongs, discussed earlier, when 9 = 0, the axis of the cylindrical body of the probe makes
a very shallow angle relative to the flow direction. Also note that 9 does not represent
the yaw angle (which is that between the normal to the wire and the flow velocity vector),
but 9 increments equal yaw angle increments for both wires, with the largest wire output

voltage at a given freestream velocity corresponding to a yaw angle of zero. In

59

implementing the yaw calibration procedure, the setup was initially adjusted such that the
readout of the rotating stage was zero when the horizontal rod was visually aligned with
the x direction (flow direction). The stage was then rotated through a large negative
angle (—110 degrees) to start the calibration. At this angle, the corresponding hotwire
output voltage was recorded. The procedure was then repeated for a total of 55 angles
obtained by incrementing 0 in the positive direction in increments of 5 degrees up to
+160 degrees.

Figure 2.25 shows the primary output of the yaw calibration. The figure depicts a
plot of the output voltage versus 0 for both wires of the X-probe. The results were used
to determine the zero-yaw angle for each wire; i.e., where the sensor output reaches its
peak, which was approximately 0 = —15 and 75 degrees for wires #1 and #2, respectively.
The difference between these two values equals to 180-(11—(12 (see Figure 2.22),
showing that the two wires are in fact very close to being perpendicular to each other as
intended in the construction of the probe (note that the angle between the two wires is
a1+oc2).

To determine the value of the constant k in equation 2.5, each wire had to be
calibrated individually at zero yaw angle (i.e., 9 = —15° for wire #1 and 75° for wire #2,
as discussed above). The hotwire calibration was conducted by recording the mean wire
voltage (E) at different known values of the flow velocity (U). The data, a sample of
which is shown in Figure 2.26, were fitted using a fourth-order polynomial fit of the form:
u = a0+a1E+... (further details of the calibration process is given below). Using these

polynomials, it was then possible to convert the voltages shown in Figure 2.25 to velocity

60

and replot the results as U,,/U.O (or effective to actual velocity ratio, where U.0 is the
freestream velocity in this case) versus 9 in Figure 2.27.

Figure 2.28 shows the data points from Figure 2.27 replotted as Ue/U.o versus
(90—0L) for wires #1 (top plot) and #2 (bottom plot). This is accomplished by subtracting
the 0 value corresponding to the peak wire output from all 0 values. Plotted in this
format, the data in Figure 2.28 can then be fitted with an equation of the form given by
equation 2.5. The equation has a single adjustable parameter: k. A second parameter, ore,
representing an angle offset error in determination of the 0 value corresponding to the
peak wire output (zero yaw) was also added in order to determine the zero-yaw angle
with resolution better than the 5-degree increment used in the yaw calibration. The best
values of these two parameters were found by varying each of them independently over a
predetermined range, and calculating the corresponding average absolute error between
the data points and the fitting curve. The pair of values corresponding to minimum error
for each of the wires was then selected for the best fit. For reference, the resulting fits are
displayed in Figure 2.28 with the data. The corresponding parameters are: k1 = 0.04, kg
= 0.0, and on+ocz = 93.

Once k1 and kg were determined, the following equations were used to calculate
the streamwise and wall-normal velocity components from the effective cooling

velocities of the two wires, respectively (for derivation of the equations see Appendix A):

u 2 U... F301.) + U. ma.)
F.2(90— a.) F5612) + F301.) F§(90— a.)

 

(2.6)
v _ U... F12(90-aI)-U.IF§(90-az)
F12(90- “1)F22(a2)+ 1212(0'1) F2200 “ a2)

 

61

E [V]

Vertical rod

 

Probe ‘ \
\ ‘ \7 ‘ .
.< 7 a ,3 93 m >— 'I

Center of rotation
, +

Figure 2.24. Schematic of the yaw calibration setup

1.9 ' ' ‘ ' V I V
I 180 — a, - a2 : —o— Hot-Wire #1
H—> '
1 35 . 7 —I— Hot-Wire #2

  

50 A 100 150 200

9 [degree]

Figure 2.25. Yaw calibration: voltage results

62

U [m/s]

Ue/Uao

30;

25:"

20;

1.2'

0.8‘
0.6 '
0.4 " "

0.2 ‘

 

 

+HW#1

 

 

Figure 2.26.

-150 -lOO

 

1.7 1.8 1.9 2
E [V]

Sample calibration data for the X-probe hotwires

180-(11-(12 . _HOt-WlfC#l
—l— Hot-Wire #2

JL

 

 

-50 0 50 100 150 200
0 [degree]

Figure 2.27. Yaw calibration data after conversion of voltage to velocity

63

80

 

80

90—0t

Figure 2.28. Yaw-calibration data and corresponding curve fits for hotwire #1 (top
plot) and hotwire #2 (bottom plot)

64

where, U61 and U62 are the effective cooling velocities measured by wires #1 and #2,
respectively, and F12 and F22 are given by:

F12(a1)=1cos2(a1)+ k3- sin2(0c1)]m (2.7)

F22(0t2) = [0082012) + k2 sin2(0t2)]”2 (2.8)
The method used to determine on and or; will be discussed below.

[11. Probe Installation

The main purpose of the probe installation procedure was to position the probe at
the desired location (between microphones #13 and 14, or 550 mm (3.05 X,), downstream
of the fence), and to adjust/determine the angle (in the x-y plane) between the wires of
the X-probe and the x direction. The alignment of the probe prongs parallel to the splitter
plate at that location was achieved with the aid of a small piece (approximately 25 mm by
25 mm) of a 23 um-thick polymer foil with a thin layer of aluminum deposited on one
side. The polymer side of the foil was adhered to the splitter plate surface directly below
the probe to provide a mirror image of the prongs. Figure 2.29 shows a sample of the
bent probe prongs and their reflected image. By adjusting the probe till the real prongs
and their reflected image were parallel, the probe alignment was achieved.

After the alignment of the prongs, their image was also used to position the probe
at the desired starting y location. The main idea was to measure the distance between one
of the bottom prongs (the one closest to the wall) and its reflected image. Hence, the
corresponding y location of the prong was the difference between half the measured
distance and the thickness of the foil (23 um). To measure the distance between the
prong and its image, each image had to be calibrated (estimating how many pixels in the

image per mm in real dimensions). The image was calibrated by causing the traversing

65

mechanism of the hotwire to move a known distance (typically 1 mm) and measuring the
corresponding prong displacement in pixels, yielding an image scale factor in pixels per
mm. This scale factor was then combined with the measured distance between the prong
and its reflected image in pixels to position the bottom prongs 0.5 mm above the splitter
plate surface. Given the geometry of the X-wire probe, the corresponding y location of
the center of the sensing parts of the wires was 2.0 mm (0.5 + 1.5 mm). For the single
wire probe, the same procedure was used to position the single wire at a starting y
location of 0.5 mm. Because the wire of the single hotwire was parallel to the splitter

plate, it was used as a reference to determine the y location of the probe.

 

Figure 2.29. A photograph of the X-wire probe and its reflection in the splitter plate

66

Figure 2.30 shows a zoomed-in image of the X-wire sensor. The image was used
to measure or] and or; (required for evaluation of equations 2.6 and 2.7). Once the probe
was aligned as explained in the previous paragraph, the prongs of the probe became
parallel to the x-direction. Thus, it was possible to determine on] and 012 by measuring the
angle between the wires and the prongs as depicted in Figure 2.30. The figure
demonstrates how the angles were calculated by determining the pixel coordinates of the
three points (1, 2 and 3 for or], and 1, 4 and 3 for (12) for each of the angles. The
measured values of or] and 012 were 47° and 46°, respectively. Note how the sum of these
two geometrically determined angels equals to 93 degrees, which is in very good
agreement with the value determined earlier of the angle between the two wires from yaw

calibration.

Prongs

1 mm 6) j *

 

Figure 2.30. A close-up image of the X sensor

67

IV. Hotwire Calibration

In every test, the hotwire probes had to be calibrated before (pre-calibration) and
after (post calibration) velocity measurements. It was desired to accomplish this in-situ
by positioning the probe in the freestream above the re-developing boundary layer, where
the flow is steady and laminar and the velocity is known. However, the thickness of the
boundary layer (which was about 80 mm, as will be explained later in section 2.2.5) was
larger than the farthest possible position the traversing mechanism can reach away from
the wall. Thus, it was not possible for the probe to reach the freestream with the existing
experimental setup. This problem was overcome by replacing the fence with a sharp (V-
shaped) leading edge during calibration to reduce the boundary layer thickness by
eliminating the bluff-body effect of the fence.

An actual image, along with a schematic diagram, of the “v-attachment” used in
the calibration may be seen in Figure 2.31. The image provides a side view while the
schematic yields a frontal view of the attachment. The modification of the leading edge
resulted in diminishing the thickness of the boundary layer to approximately 25.0 mm. In
order to measure the freestream velocity (which is used as a reference in the hotwire
calibration) at the specific x location where the hotwire measurements were conducted, a
pitot-static probe located in the freestream at this x location was calibrated against a
static-pressure tap in the sidewall of the test section upstream of the model prior to
installation of the hotwire. During the actual calibration procedure, the static pressure tap

was then used to infer the freestream velocity “seen” by the wire.

68

 

 

V-attachment I-plate

Flow I \ i a

   

 

 

Figure 2.31. Leading edge for X- and single-wire in-situ calibration

The procedure for the hotwire calibration was as follows:
1. The wire mean output voltage was acquired at eight different wind tunnel
freestream velocities. The corresponding velocity magnitude was measured using the

output pressure of the reference pressure tap.

69

2. To deduce the effective velocities of both wires (U61 and U62) at each calibration

point, U.o together with the known values of (11 and (12, were plugged in:

Uel = U... F12(90-oc1) (2.8)
U,2 = U... F22(9O-a2) (2.9)
3. The effective velocity of each wire was plotted versus the output voltage and the

resulting data were curve-fitted using a 4th order polynomial to provide an analytic
calibration equation for converting the wire output voltage to effective cooling velocity.
The calibration procedure of the single hotwire only included steps 1 and 3 of the above
procedure.

V. Boundary Layer Profiles

A single hotwire was used to obtain the boundary-layer velocity profiles (mean
and RMS) at the selected streamwise location of x = 550 mm (x/XIr = 3.05 and x/h =
68.75) downstream of the fence. The velocity was measured at 182 wall-normal (y)
locations from y = 0.5 to 91 mm in increments of 0.5 m. 10,000 samples were acquired
at each y location with a sampling frequency of 1000 sample/s to construct the boundary
layer profiles as shown in Figures 2.32 and 2.33.

Figure 2.32 depicts the mean-velocity profile of the boundary layer. The ordinate
is the mean velocity in the x direction (11) normalized by the local freestream velocity (U0o
= 16.2 m/s), while the abscissa is the physical y locations of the hotwire normalized by
the boundary-layer thickness based on 0.99U.o (5 = 80 mm). The shape of the profile
generally agrees with the typical shape of a developing boundary layer downstream of a
reattaching shear layer (a detailed discussion of the profile and comparison to the

literature may be found in section 4.1.1). For verification of the X-wire measurement

7O

procedure, which is substantially more elaborate than the single wire, data from the X-
wire are also plotted using open circles in Figure 2.32. The X-wire was traversed from a
y location of 2 mm to 51.5 mm (y/5 = 0.025 to 0.64) in steps of 0.5 mm (Ay/B = 0.00625).
The figure depicts a generally good agreement between the two profiles with a maximum

error of 3% of Ugo.

 

0.4- .....
o.3~ ..... -
0.2: ......
01 - smgfimé
, w , 1 ..X-Yzirez .
00 ‘02 A04 T 0.6 T 0.8 i ’ 1.2

Figure 2.32. Boundary-layer mean-velocity profiles from single- and X-wire
measurements at x/Xr = 3.05

Figure 2.33 shows the RMS values of the flow velocity in the x direction. The
ordinate represents the RMS velocity normalized by U30 and the abscissa is the same as

Figure 2.32. The RMS profile shows a plateau of high RMS value of approximately

71

° Single wire
I f: X-Wire

8 0.08

 

rms
o
o

:3 0.06 ‘ - . . .415"- ,
0.04z’ * 7 7 7 ,, 4.5%;

0.02

(i i 0.2 0.4 H 8 0:6 0.8" i 1 1.2

Figure 2.33. Boundary-layer RMS streamwise-velocity profiles from single- and x-wire
measurements at x/Xr = 3.05

0.12Uoo. Beyond a certain height, the RMS value decays as the probe moves away from
the wall. Very close to the wall, within a narrow region of y/8 < 0.025 or so, there is a
rise in the RMS value above that given by the plateau. The RMS values calculated from
the X-wire data are also plotted in Figure 2.33. The figure depicts that the two profiles
agree fairly well. The estimated maximum deviation between the two profiles is less than
1% of Ugo. This deviation, which depicts the X-wire data to be consistently lower than
that of the single-wire data, is not surprising since the X-wire averages the measurements
over a spatial sensing volume of approximately 1 m3, while the single wire averages the

measurements over a 1 mm length in the Spanwise direction (the wire dimensions in x

72

and y directions may be neglected relative to the flow scales). The larger sensor volume
of the X-wire, particularly in the y direction results in attenuation of the small-scale
energy. This is most evident near the wall, where the X-wire data decreases in value as y
approaches the wall, while the single-wire RMS data rise.
VI. X—Probe Disturbance Check

Because of the simultaneous wall-pressure and velocity measurements, it was
necessary to check if any significant disturbances caused by the presence of the X-probe
in the flow contaminated the wall-pressure data. The check was done by examining the
microphones output when the X-wire probe resided at different y locations. Figures 2.34
depicts the power spectra of the microphone voltage output for three different heights of
the X-probe (y = 4, 10 and 50 mm). It should be noted that these three locations indicate
the height of the center of the probe (the sensing part), while the corresponding locations
of the closest prong to the splitter plate were 2.5, 8.5 and 48.5 mm, respectively. Figure
2.34 shows the spectra of the pressure measured by the microphone closest to the hotwire
location (x/Xr = 3.0) on the upstream side. In interpreting the data, the farthest y location
was considered as the no disturbance case. The figure depicts good agreement among
the three spectra, which indicates that the microphone measurements are insensitive to
the hotwire location, suggesting that the presence of the probe in the flow does not create

any undesired contamination of the microphone data.

73

 

 

10

-1
.— 10 3'
N ;~
«3 T.
E. ii
I“ *-
9“ -2
'9‘ 10 I:
-3
10 .

'4: . . ... 3 3 3... , .

10 43 - ~* 3 ~ 4* .3 3 . 3 , - +- -3

10 10 10 10

f [HZ]

Figure 2.34. Pressure spectra of the microphone immediately upstream (x/Xr = 3.0) of the
X-wire for three y locations of the wire

Moreover, the same test was done for a microphone immediately downstream of
the hotwire. The test results are shown in Figure 2.35. The data confirm that the hotwire
probe effects on the pressure data can be neglected for the range of frequency of interest.
Note that the same test was conducted for all of the microphones, but only the data for
microphones #13 and #14 were presented because they were the closest microphones to
the hotwire probe, and hence they should suffer the most from any interfering effects of

the X-probe.

74

 

0
10
-10
F, 10 z:
N
a .
E:
3:.
CL -2
6‘ 10 "f
-3
10
'4: . ; . , , . .. . . _ .
10 3.1 , ,z , - , . .m.2 as i .i. m -A3 W -, _. . W. ,. 4
10 10 10 10

f [Hz]

Figure 2.35. Pressure spectra of the microphone immediately downstream (x/Xr = 3.11)
of the X-wire for three y locations of the wire

75

3. WALL-PRESSURE RESULTS

This chapter focuses on the results of the wall-pressure measurements only. This
focus is important for two reasons: (1) one of the primary goals of this work is to
understand the nature of the wall-pressure field beneath the recovering boundary layer
and compare this to the fully-developed boundary layer, on one hand, and the
separating/reattaching flow, on the other; (2) because in the following chapter the
velocity field will be estimated using Linear Stochastic Estimation (LSE) based on wall-
pressure data, understanding of the characteristics of the latter is an important precursor
to proper interpretation of the estimated velocity field. Of course, an understanding of
the wall-pressure/velocity field relationship is also important for interpretation of the LSE
results. This relationship will be explored in detail in Chapter 4 prior to implementation
of the LSE procedure.

As explained in Chapter 2, the mean wall-pressure was measured using 28
pressure taps located on the center of the splitter plate, while the fluctuating wall-pressure
signature underneath the flow was captured using a l6-microphone array downstream of
the reattachment point. It should be noted that due to the lower limit of the microphones'
operating frequency range (20Hz), all microphone data are high-pass filtered at 20Hz.
This removes any ambiguity concerning the magnitude and phase of p' in the analyzed
data. As will be demonstrated herein, the wall-pressure spectral peak is in fact captured
within the analyzed band of frequencies. Moreover, there is negligible amount of
fluctuating—pressure energy lost because of the filtering process. In the following

subsection, data-processing details and results will be presented concerning the mean-

76

and fluctuating-pressure streamwise variation, autocorrelation, power spectra, cross
correlation, and wavenumber-frequency spectrum.
3.]. Mean-pressure distribution and the reattachment length

Figure 3.1 shows a comparison between the mean-wall-pressure coefficient (Cp,
see equation 2.1) distribution from the present work and that from Hudy et al (2003).
The ordinate represents the pressure coefficient while the abscissa represents the
streamwise location of the pressure taps (x) in mm with respect to the fence. The test
model used in Hudy et al. (2003) was the same as the current one except that their
blockage ratio was 1.94% versus 4.5% in the present work. This blockage ratio is
defined based on the fence height (h). The actual blockage in the present flow (based on
the total fence height, 2H) was 9.9% (compared to 4.26% in Hudy et al.). Because of the
higher blockage ratio and consequent flow acceleration, the distribution of C], from the
present data shows a more pronounced negative valley downstream of the fence (note
that in both studies the reference pressure in Cp is taken at the beginning of the test
section, or upstream of the model). Furthermore, the pressure recovery is less than that of
Hudy et al. (2003). This pressure recovery is faster in the present study, resulting in a
shorter reattachment length (180mm, 22.5h, as found using tufts, compared to 205mm,
25.5h, in Hudy et al. 2003).

The observation that the mean reattachment length of the present work is shorter
than that of Hudy et al. (2003) agrees well with the results of Smits (1982), who
investigated the effect of the blockage ratio on the mean reattachment length. Smits’
results showed that the reattachment length decreased for larger blockage ratio.

Furthermore, when the blockage ratio of the present model is used in conjunction with

77

Smits’ data to estimate the mean reattachment length, an XIr value of approximately 21h

is found, which is shorter than, but falls within 7% of, the measured reattachment length.

3T 3 PfieSent T T T3
-0.1 '4 3- 'I-Iudy et al. (2093) 3

-0,2.--

03-, ,,,,., ,3,3i7fi177,.ln

-o.3 ~
.0.4 ~
on -0.5 ~

-O.6 r

I :.‘1T.TT..
‘9 J .....
' l
I ‘7
9
if 2‘0TT ST 25 T ”T 3.0 TT T 355

Figure 3.1. Streamwise distribution of the mean pressure coefficient from the present

study compared to Hudy et al. (2003)

Ruderich and Femholz (1982) also used a fence-with-splitter—plate model with

two fence heights above the splitter plate (h) of 11 and 22mm. The cross-section height

of the wind tunnel used with their experiments was 0.5m, while the splitter plate

thickness was 6.3mm, which resulted in total fence heights (2H) of 28.3 and 50.3mm.

Consequently, the blockage ratios based on h = 11 and 22 mm were approximately 4.4%

and 8.8%, while the actual blockage ratios based on 2H were approximately 5.7% and

10%, and the corresponding mean reattachment lengths were 22.6h and 17.2h,

respectively. Therefore, Ruderich and F emholz’s results depict an agreement with that of

Smits that the mean reattachment length decreases for higher blockage ratio.

78

Furthermore, the mean reattachment length of the present work agrees well with that of
Ruderich and Femholz (1986) for the case of h = 22mm, which approximately have the
same blockage ratio based on h as the current work. Additionally, Castro and Haque
(1987) reported a mean reattachment length of 19.2h with a 6.2% blockage ratio based on
b.

Note that due to the flow acceleration caused by the blockage effect of the model,
the local (at the location of hotwire measurements) freestream velocity will be used as U.o
for the remainder of this thesis. This local value is approximately 14% higher than the
approaching freestream velocity.
3.2. Fluctuating pressure distribution

Figure 3.2 shows the streamwise distribution of the fluctuating-pressure

I
p [1115

coefficient (C3. = 1 U2 , where p'rms is the root mean square (RMS) of the fluctuating

2 co
pressure). The figure depicts the profiles of Cpl versus g, where Q = (x-X,)/X,, for
Reynolds numbers 7600 and 15700 based on step height. The distributions depict a
decay in the pressure fluctuation with increasing distance from the reattachment point.
This suggests that the flow structures dominating the wall-pressure generation are
continually loosing energy as they travel downstream of the reattachment point. This is
consistent with the relaxation process of the shear layer vortices in zone VI (see Figure
1.1) referred to in the Chapter 1. More specifically, the high mean shear stress energizing
those vortices during their earlier development in the free shear layer (before
reattachment) slowly gives way to a much weaker mean shear as the attached shear layer
gets thicker with increasing x. Thus, the source of turbulent energy received by these

vortices becomes weaker and weaker and so does their energy as well as their wall-

79

pressure signature. Furthermore, the plot in Figure 3.2 shows that at the higher Reynolds
number p' is more energetic and its rate of decay is slightly higher than that at the lower
Reynolds number. This decay presumably continues monotonically till p' reaches a level

corresponding to that of an equilibrium turbulent boundary layer.

 

01 T ' T ' T T T' ' TT T T T TTITNW ,T‘ 3;T : , '7
I Re= 7600
009-- 4 » y .3 Re=15700
0.08
0.07
0.06
0‘“- 0.05-
0.04~
0.03
0.02- - ~ ~— —- - —
,0. 292211001129...“ ........... .
boundary layer
012%. H ,A i - .- L, ..__ .__f__l____. _ ,-__ ..___ 1,2 , 7 -j
0.5 1 1.5 2 2.5

5.

Figure 3.2. Streamwise distribution of the coefficient of the RMS pressure fluctuation

Figure 3.3 depicts the full streamwise distribution of va in both the separating/
reattaching and redeveloping flow zones at Reynolds number of 7600. The solid squares
represent data obtained in the separating/reattaching flow by Hudy et a1. (2003), while
the solid circles show those from the present work. A polynomial fit is also added to the
plot for the purpose of showing the general trend in the streamwise profile of Cp'. The
profile shows the typical behavior of p' in a separating/reattaching flow before and after

reattachment, which is represented by a rise in the wall-pressure fluctuation till it reaches

80

a peak slightly upstream of reattachment, followed by a decay during the relaxation

process downstream of reattachment.

02 T T ' T T T,if’__"',_, T 1’ T T: T i ,
I Present (Re = 7600)
018- j , I Hudy et al. 2003 (Re = 8000) »-
. 3 — Curve-fit
0.16 - - — - :1: [arabesand QaSér¢11M1986> :
0.14- 7
0.121 *
0‘“ 01

0.08

 

-1 -05 0 0.5 T 1 T 1.5 2 2.15
Figure 3.3. A comparison between the va streamwise distribution of Hudy et al. (2003)
and the present work with that of Farabee and Casarella (1986)

Figure 3.3 also depicts a comparison between the Cp' streamwise distribution of
the fence-with-splitter-plate model used in the present work with that of a backward-
facing step flow from Farabee and Casarella (1986). Both distributions exhibit the same
qualitative behavior. However, the present flow exhibits much more energetic wall—
pressure fluctuations than that of the typical backward-facing step. This is believed to be
caused by the much stronger shear across the very thin, laminar boundary layer at
separation in the fence flow, in comparison to that experienced across the much thicker

turbulent boundary layer of F arabee and Casarella (1986). Considering that the stronger

81

the shear (ii-E) the higher the turbulent energy production term MR7 $5, the strong
Y Y

shear in the investigated flow results in more energetic vortices resulting from the roll-up
of the shear layer and their associated wall-pressure signature.
3.3. Autocorrelation

In order to analyze the time scales of the flow structures producing the most

energetic pressure fluctuation, the autocorrelation of p' time records was evaluated. The

autocorrelation, Rp'p', is defined as:

 

p' 0.0 p' (m - r)
(p'...(x))2 ‘3'”

where the overbar denotes time averaging and r is a time delay. The discrete form of

 

Rp.p.(t , x) =

equation (3.1) that is implemented in the processing of the discrete-time data is given by:

 

N—m-l
Zp'1x.n)p'(x.n — m)
R,.,.(m,x)= "=0 N (3.2)
N — lml '2
(X)
“2:313

where N is the total number of data points in the time series and m is the time delay in
data samples. The corresponding time delay in seconds is estimated from r = m/fs, where
fS is the sampling frequency in samples/sec. To calculate the autocorrelation, each
pressure data series, is split into records of 4096 samples each. Since N = 223 = 8,388,608
and 220 = 1,048,576 samples for Re = 7600 and 15700, respectively, 2048 and 256 data
records are obtained for the low and high Reynolds numbers, respectively. The
autocorrelation is finally calculated by averaging the inverse Fast Fourier Transforms

(FFT) of the product of the F FT and its conjugate for all records.

82

Figure 3.4 shows the autocorrelation coefficient (Rp'p') plotted versus the non-
dimensional time delay (tUoo/Xr) for the two Reynolds numbers examined here at five
sparse locations covering the streamwise range of the measurements: E, = 0.67, 1.11, 1.56,
2.0 and 2.33. Generally speaking, there is very little change in Rp'p' for all x positions.
This is more evident in the color contour maps in Figure 3.5. These maps
represent R W values for all x positions using color contours. The constant-shade
contours are almost parallel to the 1 axis, revealing a practically “frozen” auto-correlation
function.

In Figure 3.4, a “preferred” time scale corresponding to the negative peak in Rp'p'

is suggested. This time scale represents a quasi-periodic disturbance with a period of

tUm
X

r

 

z 1.3 (peak-to-peak time delay). The dominance of this disturbance decays

somewhat with increasing 8,. This is evident from the small decay in the largest negative
Rp'p. value with downstream distance. On the other hand, Figure 3.6 depicts a plot
enlarging the region around the autocorrelation peak for the lower Reynolds number and
all five x locations. The figure shows an increasing curvature of the auto-correlation at
zero time delay with increasing 5,, which implies a decrease in the Taylor microscale.
This suggests that the smaller scale turbulence is increasingly contributing to the wall-
pressure fluctuation with increasing downstream distance from X,. Finally, it is noted
here that Rp'p' results seem to be affected very little by the Reynolds number (at least for

the small Re range covered here).

83

 

'-2T T‘-T1 0 1 T T 2 '-2TTTT-1 T 0 1T * T2

 

 

‘ ...----:.'R.e= 1579(1‘

 

TU/X
00 r

Figure 3.4. Autocorrelation coefficient at five different streamwise locations downstream
of the reattachment region for Re = 7600 and 15700

84

Re=7600

 

Re=15700

IU/X

 

Figure 3.5. Contour maps of the autocorrelation coefficient for all 16 microphones and
the two Reynolds numbers; Re = 7600 and 15700

85

Figure 3.7 depicts the color contour map of the autocorrelation coefficient for the
flow upstream and downstream of reattachment. The results show combination of those
from the separating/reattaching flow region obtained by Hudy et al. (2003), g < 0.3, at Re
= 8000, and the present results. The map shows that the auto correlation width, and
hence the dominant time scale of the wall-pressure-generating structures, remain
practically unchanged with increasing x downstream of i z —0.5. This suggests that the
flow structures dominating p' generation within the development zone trace their origin
upstream to the middle of the separation bubble: an observation that is consistent with the
idea proposed earlier that the vortices originating in the separating shear layer do in fact
dominate the wall-pressure signature in the non-equilibrium boundary layer within the
streamwise extent investigated here. The persistence of the shear-layer vortices
downstream of reattachment has been reported by Bradshaw and Wong (1972), Farabee

and Casarella (1986) and Ruderich and Femholz (1986).

9 g: 0.67 T

3 3 g=1.11 1
0.95 - V 5:156 *1
‘ ‘ 552.0
g=2.33
0.9:
h.
b.
m
0.35
0.8
0.75 - ‘7 , - 3 -
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

tU/X
m I’

Figure 3.6. A comparison of the auto-correlation results at zero time delay and five
different streamwise locations downstream of the reattachment region for Re = 7600

86

z UGO/X

 

-0.5 0 0.5 l 1.5 2

5.

Figure 3.7. A full contour map of the p' autocorrelation coefficient for the separating
reattaching (Hudy et al. 2003) and the present recovering flows

Upstream of g = —0.5, the dominant p' disturbances posses a substantially larger
time scale, as indicated by the significantly larger width of the auto-correlation. In
addition to Hudy et al. (2003), the larger-time-scale or lower-frequency has been reported
in different investigations and attributed to shear-layer flapping; e.g., Castro and Haque
(1987), F arabee and Casarella (1986) and Lee and Sung (2002); see Chapter 1.

3.4. Power Spectra

The wall-pressure power spectrum, (Dp'pl, is used for characterizing the frequency

content of the pressure signature at the wall. CD”: is calculated by multiplying the FFT

of the wall-pressure signal by its complex conjugate and then dividing the product by the

square of the number of samples in the spectrum. To reduce the random uncertainty error,

(DWI is calculated as the average of the power spectra of individual data records obtained

87

from partitioning the pressure time series. Each record contained 4096 samples, resulting
in a spectral resolution of 12.2 Hz. The total number of records was 2048 and 256,

resulting in random uncertainty error of 2.2% and 6.3%, for the low and high Reynolds

numbers, respectively. The specific equation that is used to calculate (Dprpc

¢prp'(0= NTLZ PINE-X”) 1 (3.3)

2
i=1 N

J

where P and P. are the FFT of the wall-pressure data record and its complex conjugate,
respectively, f is the associated frequency, fs is the sampling frequency, N is the total
number of samples in each record, NJ- is the total number of records, and j is the record
index.

Non-dimensional power spectra for the low and high Reynolds numbers, at the
same five locations of the autocorrelation results in Figure 3.4, are plotted in two

different forms in Figures 3.8 and 3.9, respectively. The top graph in both figures depicts
the non-dimensional power spectrum of the wall-pressure, B (where [3 = CDp'pl/(l/Z p

Uw2)2), versus the non-dimensional frequency (f Xr/Uw) using logarithmic scale for both
the ordinate and abscissa. For both Reynolds numbers, the top spectrum shows

concentration of the pressure fluctuations at low frequency as depicted from the broad

 

spectral peak close to fUX' = 0.7. This frequency value is consistent with that

documented in the literature to correspond to the passage of the vortical structures
generated in the separated shear layer upstream of reattachment. For example, the top

graph in Figure 3.8 shows a comparison between the spectrum of the present data and

88

 

 

a.~§‘5 £5.50! . .a -. ..
I

>1 ha» 12.1-

 

10 ;
5 o
1 _ “N -7/3 slope
-4 T ~ 11.3%” \
10 ‘ ‘3‘ \\ i
T ‘I‘AI“ \\ 4
~ TTlT“;‘§ \\ 3
5 “‘ ‘41.»33‘3‘ \\
10 1 x - {nix -~\ ~ - ~ 3
‘~ 1“ 5‘ ‘ a
3 TT‘x “if“ \\ T5 Slope i
-6 \s~~‘ 3 . 'fl 33533“? “ \\\ 3
ca. 10 “x “ “7, .‘y‘ * "1
.» ‘\ i_1.‘-\ 3
3 I 3 33 3 3 A 3 s 1"134339 I.“ 3
10-7: 1 §=0 (Hudy et al. 2003) 13 3 ‘13:} 35‘ l
E . a: 0.67 1 $93 3
E" = 1.11 1 .\ 3
-8 T i . §_ 1 i \\ l
10 3:1 £4.56 3 ~ ~ ~ ~ ~ * ~ ~ ~~ 13 ‘~~.«.\* - 3
1 a: 2.0 3 ; ‘ ‘s,
_ 3 a=z_-33 e , 1 * ii“ “ i
10 =53 ~ ~~ . 3 ~ ~ ~ - . L3* eeeeee .. ~ ~43
10 10 10 10
.4
x 10
2.5.~~~ ~ 3 3 3.. ereeeee+++j
* 1
1
2. - .-1
1 1
‘ 1
‘ 1
1.51~ ‘3
1
5 ‘ 1
1 1
l3 ‘
1 1
05+ ~ «
-1 7 7 W. , k, 3 _ 1 , ,, . :0 i. .1. . 1 1...- ; l l_ _, - . ’ -‘ i i,,,.. L. i 2
10 10 10 10
f Xr/U00

Figure 3.8. Spectra of the wall-pressure fluctuation for Re = 7600: logarithmic (top) and
semi-logarithmic (bottom) plots

89

-3
10 3
-4 I
10 1" 3 ,
l 3 3
T \
. \\
10's: ‘ \ _, , 1
' \\-5 slope 1
6 \\
©- 10 \\ ‘ 1
‘ \
' '- \
’ ~11 « ' ,\
-7 T ‘IT‘T‘T"‘iTTi.\\
T 1 7 7 , T .‘~'.-:\ , ,
E ‘ §= 1.11 ; * 1 .3 ..
'8 ' 1.» 1
10 1_ E1: 156 7 1 7 7 7 7. 3 _, 7 3, 133333333 _ _ 3
é: 2'0 I 1 \\
39~ §= 2.33 3 \:.‘\.J
10 _3 - -3 1 , ~10 7 ,. 77 +1 ..3 77 -2 L- a, a2
10 10 10 10
74
x 10
2.51 - 7 ~ 3 7 m 3 g g g 3
2 .
1
’ 1
1.53 3
C3 1 3
1 ;
0.5
1
.l . ,n, . 1 1 l l . i 0 . - L n . . ‘ . . 1 TT 1 3“, “‘ -' —- . 2 1 1 . 2
10 10 10 10

f Xr/Uw

Figure 3.9. Spectra of the wall-pressure fluctuation for Re = 15700: logarithmic (top) and
semi-logarithmic (bottom) plots

90

that of Hudy et al. (2003) at reattachment. The comparison shows that the pressure-
fluctuation—peak frequency found at reattachment is similar to that found farther
downstream beneath the redeveloping boundary layer, which is consistent with the auto-
correlation results discussed earlier. Similar values of the dimensionless frequency of the
wall-pressure spectral peak were also reported by a number of studies focused on the
separation/reattachment zone e.g., Cherry et al. (1984), Farabee and Casarella (1986) and
Lee and Sung (2001).

As the vortices are advected downstream by the mean flow, they appear to decay
as evident from the attenuation in the spectrum peak with increasing 5,. In contrast, the
spectral level at the high-frequency end is maintained by the smaller-scale turbulence.
Blake (1986) showed that the high-frequency end of the wall-pressure spectrum beneath
turbulent boundary layers is characterized by a slope of —5 on a log-log plot. The slope
of -5 (within approximately 5% uncertainty) is also found here at the high-frequency end
of the spectra shown in the top plot of Figures 3.8 and 3.9, respectively. Based on
Blake's analysis, the flow structures responsible for the -5 spectral region are those
residing in the buffer layer of a fully developed turbulent boundary layer. Thus, it is
possible that the existence of a -5 region in the spectra measured here is a manifestation
of the redevelopment of the inner region of the boundary layer downstream of
reattachment; i.e., that associated with the sub-boundary layer.

On the other hand, George et al. (1984) found that the turbulent-turbulent (TT)
source terms of pressure result in the establishment of a power-low-type spectral
characteristics of p' with exponent of -7/3; i.e., corresponding to a frequency range with

slope of -7/3 on a log-log plot of the spectrum. The top graph in both Figures 3.8 and 3.9

91

depict that the slope of the wall-pressure spectra is slightly shallower than -7/3 for 1 <

fX

 

' < 10 beneath the redeveloping boundary layer. However, the wall-pressure

w

spectrum of Hudy et al. (2003) at reattachment does posses a slope of -7/3 within the
same range of frequencies. Furthermore, Lee and Sung (2001) observed a slope of -7/3
in the wall-pressure spectra near reattachment, and a shallower slope downstream of
reattachment (E, z 0.35). Thus, it appears that within the separation/reattachment zone,
the wall-pressure signature reflects spectral characteristics similar to those found in a
free—shear layer. As the shear layer reattaches, and the vortical structures continue to
decay with increasing x, these characteristics gradually change, ultimately disappearing
and giving way to boundary-layer—like characteristics as x approaches infinity.

The bottom graph in both Figures 3.8 and 3.9 shows a plot of the spectral
information plotted with the ordinate on represented on linear scale (a = f B X,/ U00). The
rationale for plotting the spectra in this manner may be seen when integrating the

spectrum to obtain the fluctuating wall-pressure energy:
7 co m
p = 1d>,.,.df=1f<1>,,.dnog(m (3.4)
0 0

Equation (3.4) shows that when using a logarithmic frequency axis, the geometrical area
under the spectrum curve corresponds to the pressure fluctuations energy only if (1);,er is

multiplied by f and plotted on a linear scale. The corresponding non-dimensional

quantity is then or instead of B. The abscissa still represents the non-dimensional

 

f X . . . . .
frequency U ' . The figures confirm that the maxrmum contribution to p' occurs Within

w

a non-dimensional frequency band centered around O.7-O.9, which corresponds to the

92

passage frequency of the separated shear layer vortices obtained in previous literature
within the separation/reattachment zone as discussed above. The center of this range,
er/Uoo = 0.8, corresponds to a non-dimensional time scale of 1.25, which agrees quite
well with the dominant time scale of 1.3 identified from the auto-correlation results
earlier. Moreover, visual extrapolation of the data towards lower frequencies than those
shown in the plot suggests that only very small fraction of the overall p' energy is not
captured because of the low cut—off frequency of the microphones (20 Hz).

The semi-log plots also verify the growth of the boundary-layer fractional
contribution to p' energy. This may be examined by considering the ratio of the area
under the spectrum curve at high frequencies to that at low frequencies (p'h/p'1)2. This
ratio is estimated by splitting the spectrum of the wall-pressure fluctuation into two
portions. One is for low frequencies (due to the large vortical structures originating in the
separated shear layer) and the second is for boundary-layer contribution at high
frequencies (or small turbulent eddies). This idea of the p' spectrum beneath the non-
equilibrium flow being a composite of two spectra separated in frequency, one associated
with the outer-shear layer and the other with the sub-boundary layer, may be examined
further in Figure 3.10. The figure depicts a comparison of typical p' spectra in
separating/reattaching (Hudy et al. 2003) and equilibrium boundary layer (Gravante et al.
1995) flows. To superpose the spectrum from Gravante et a1. (1995) and the present
spectrum at g = 2.33 on the same plot, the start of the -5 region of the two spectra were

made to overlap3. On the other hand, the spectrum of Hudy et al. (2003) was re-scaled

 

3 As a side note: the equilibrium-boundary-layer literature shows the -5 region to start at f v/u,2 z 0.13,
where v is the kinematic viscosity and n1 is the friction velocity. Using this value suggests that n1 for the
current flow is 0.5 m/s

93

such that its peak coincides with the present spectrum to facilitate comparison of the

spectral shapes.

 

11 — geiflkeattEchfi = 0,1Hud_y et al.) 3

11 —- Equilibrium B.L. (Gravante et al.)

-9 3 — Present Data (2'; = 2.33) 1

10 1.. M- *4 :1 17? Jr 1 7 V ,, ;‘ f
71

0
10 10

1
1
1
1.
1

— 1

.—l
O
‘ I I .1

—O
O
H 1-
O

fX/U
1' 00

Figure 3.10. Definition of the frequency used in the splitting procedure of p' spectrum

Based on the comparison in Figure 3.10, the splitting of the spectral energy into
out-shear—layer and sub-boundary-layer components was based on the frequency at which
the spectrum changes its slope, which is indicated by fn in Figure 3.10. The choice of fn
is somewhat heuristic since the demarcation between the two frequency ranges is likely
to be not sharply defined, and one would expect a gradual switch from one spectrum to
the other with the middle range of frequencies receiving contribution from both the

smallest scales of the outer-shear layer and the largest of the sub-boundary layer.

94

 

 

. leI In Ila? or”. J... .. P1VWWU
P5055 .. .
. .

However, fn does coincide with the point at which the spectrum slope switches from the
value typical of the middle frequency range in turbulent boundary layers (z —1 , see Blake,
1986) to a more negative value that has never been observed in boundary layers at low
frequencies.

The streamwise variation of the ratio (p’h/p'l) 2 is provided in Figure 3.11 for the
two Reynolds numbers examined here. The results verify that the boundary layer
contribution to the wall-pressure fluctuation becomes increasingly important with
increasing x. Moreover, Figure 3.11 shows that the initial contribution of the small-scale
turbulence to p' energy increases as Reynolds number increases, which is consistent with
known turbulence physics. Note that the two solid curves in the figure are plotted to aid
in visualizing the trend of the data.

0.4 :1

6 Re =7600 ’1 '
0.351 ,',,' Re :‘5700 ;

 

0.3
0.251

0.21

(rah/p.)2

0.15-
0.1 "

0.05 ~ 1 ~ ~ 71
11:5 ‘ 1 " 1:5 ' i s 7.5

Figure 3.11. The ratio between high- and low-frequency pressure fluctuations energy

95

3.5. Cross Correlation

The cross-correlation coefficient (va3,!) is defined as:

 

C : p'<x.,t)p'(x.,t-r)

P'P' r I
poxmthJms

 

(3.5)

6‘i"

where subscript “0” refers to a reference microphone location (a: = 0.67), subscript
refers to a variable microphone location, the overbar denotes time averaging and I is the
time delay. The discrete form of equation (3.5) that is used to calculate Cprpv from the

computer-sampled time series is given by:

N—m—l
N Zp’<x.,n)p'(x.,n—m)
vapv(m , X,;Xo) = ":11 m (3,6)

PJ__ NI
Im1 31211209.) 211209)

where N is the total number of samples and m is the time delay between the two pressure

 

time series in data samples. Similar to the calculation of vapv, Cp'p' is calculated as the
inverse FFT of the cross-spectrum of the two. wall-pressure signals. The calculations
include the same length and number of records as those used with the autocorrelation
calculation in section 3.3. Unlike the autocorrelation where the peak correlation is found
at zero time delay, the cross-correlation peak typically exists at some time delay
corresponding to the duration needed for a given wall-pressure signature to travel from
the reference to the variable microphone. Thus, the cross correlation provides valuable
information concerning the convection characteristics of the dominant wall-pressure-
generating motion.

The cross-correlation, Cp'p', between the time series obtained from five sparsely-

positioned microphones (é = 0.67, 1.11, 1.56, 2.0 and 2.33) and that captured by the most

96

upstream microphone (i = 0.67) for Reynolds numbers of 7600 and 15700, are plotted in

Figure 3.12. In the figure, the cross-correlation coefficient Cp'p' is the ordinate and

w

 

is the abscissa. Note that the results at 2; = 0.67 represent the autocorrelation of the

microphone at that position. The five plots in Figure 3.12 reflect the streamwise
convective nature of the dominant wall-pressure-generating motion. This is seen fiom
the shift in the time delay corresponding to maximum positive (or negative) correlation
with increasing i. More specifically, as the x location of the variable sensor increases,
the correlation peak is located at an increasing negative time delay (i.e., increasing time
advance). This corresponds to a disturbance whose signature reaches sensors that are
farther downstream later in time. It is also noteworthy that the dominant time scale (i.e.,
the period between the two negative Cp'p' peaks) of the convective quasi-periodic
disturbance depicted in Figure 3.12 is the same as that observed earlier in Figure 3.4 in
the autocorrelation plot.

Figure 3.13 provides a fuller picture of the convection motion of the dominant
vortical structures for the two different Reynolds numbers. The figure depicts plots of
CW for all 16 microphones where the ordinate and abscissa are exactly as those of Figure
3.12. The figure shows that the CW peak shifts by about 0.15 on the non-dimensional
time-scale axis between each two consecutive microphones. This time shift is estimated
to be almost the same for both Reynolds numbers. To estimate the corresponding
convection velocity (UC), Figure 3.14 depicts a plot of the dimensionless time shift of the

TPU
X

r

a)

peak of vap' (

 

) versus the dimensionless distance between the corresponding

97

0.8'
i=1-_11

0.6

   

70.4177
73 72 71 0 1 2 3
1 - 1
0.8 1 as
0.6 E; 1.56 0.61 §— 2.0

 

70.4- . - 70.4 ‘
-3 -

1 ,
08*
036 {,— 2.333

04. ‘, . — Re=7600

U 032 ----- Re = 15700

0..

 

-0.2 ,
-0.4 '7
-3

‘t Uw/Xr

Figure 3.12. Cross-correlation results at five different locations for Re = 7600 and 15700

98

Re = 7600

Re = 15700

 

Figure 3.l3. Cross-correlation results for all 16 microphones and Re = 7600 (top); and
15700 (bottom)

99

2.; 7 7

1 ”1&57‘7500 * * * * ‘1 ' ' ,.
1.82 I R6 :15700 '7 _ . . . _ ,_ 3 3-1 7 .-. .Ifi
_ Line fit: Uc/Uoo = 0.81 . .
1.6 ., ..... Line fit: UC/Uco = 0.82 .

 

0 ii 0.5 i 1.5 2 .25

tU/X
poor

Figure 3.14. Plot for extraction of the convection velocity for the two Reynolds numbers
of 7600 and 15700

microphone and the reference (the most upstream) microphone, Ag. In

 

1:
this manner, the local Uc/Uoo value is given by the slope of the A: versus ”X °° curve,

I'

and an overall average convection velocity may be obtained through a linear fit to the
data over the whole range. The results depict a local slope that seems invariant with Ag,
which shows that Uc/Uoo is constant along the whole x range and equal to 0.81 and 0.82
for Re = 7600 and 15700, respectively. Those velocities are higher than the velocities
reported in the literature within the upstream separation/reattachment region. For

example, Cherry et al. ( 1984) reported convection speeds of approximately 0.5Uoo and

100

0.63Uoo from the pressure-pressure and -velocity cross-correlation and, recently, Hudy et
al. (2003) and Lee and Sung (2002) reported a mean convection velocity of 0.6U.,o in their
separating/reattaching flows. This indicates that the convection speed of the vortices
originating in the separating shear layer increases as they travel from the separation to the
redevelopment zone. These conclusions are qualitatively consistent with those of
Farabee and Casarella (1986). However, quantitatively, Farabee and Casarella showed
that the convection velocity at similar locations downstream of the reattachment point
was 0.6—0.7 Um. The reason for the higher value found in the current study is elaborated
on in section 4.2.2.

The cross-correlation information of all 16 microphones are also plotted in color
contour maps in Figures 3.15 and 3.16. The map yields the cross- correlation of all 16
microphones with respect to the most upstream microphone (f; = 0.67) for Re = 7600 and
15700, respectively. The maps show an inclined positive-correlation lobe with a peak
value at zero time-shift at the location of the reference microphone. The inclination of
the main lobe in this spatiotemporal plot provides another representation of the
convection of the vortical structures downstream of the reattachment region. In fact, the
slope of the broken line in Figure 3.15, which is aligned with the center of the positive-
correlation lobe, represents the ratio Uc/Uoo. This slope value is found to be
approximately 0.81, in agreement with the earlier results for Re = 7600. The results for
the higher Reynolds number of 15700, displayed in Figure 3.16, yield approximately the

same ratio of Uc/Uco = 0.81.

101

 

Figure 3.15. Contour map of the cross-correlation coefficient for all 16 microphones and
Re = 7600

so

tU/X

-0.25

 

Figure 3.16. Contour map of the cross-correlation coeflicient for all 16 microphones and
Re = 15700

102

3.6. Wavenumber-Frequency Spectrum

The wavenumber-frequency (kx-f) spectrum of the wall-pressure data was
calculated for Re = 7600. The spectrum was obtained by calculating the two-dimensional
Fourier transform of the spatiotemporal wall-pressure data, p'(x,t), of all 16 microphones.
1048576 samples of wall-pressure were acquired from each microphone at a sampling
frequency of 6250 samples/sec. Therefore, the wall-pressure data set could be
represented by a two dimensional array of 1048576x16 in size. This 2D array was then
broken up into 2048 sub-arrays (records) of 512x16 in size, which produced a random
uncertainty error of 2.2% in the spectral estimation. The wavenumber-fiequency
spectrum of each record was calculated by multiplying the two-dimensional FFT of each
record by its complex conjugate, then averaging the spectra of all records. The 2D F FT
operation is accomplished by doing a one-dimensional FFT transformation of the
columns of the 2D record, to transform time into frequency, followed by another 1D FFT
operation of the rows, to transform x to kx. The resulting resolution of the dimensionless
frequency and wavenumber were 0.14 and 0.6, respectively. To obtain smoother contour
plots of the outcome, the spectrum was interpolated along the wavenumber axis by zero-
padding each record in the x direction to be 512x64 in size. Thus, the increment in the
dimensionless wavenumber became 0.15.

The physical interpretation of the wavenumber-frequency spectrum may be
clarified by considering propagating harmonic waves. Since the propagation velocity (Uc)
of any harmonic wave is the product of its frequency and wavelength (1/kx), then the
slope, or f/kx, of a line connecting the coordinates of the spectrum peak corresponding to

the wave to the origin of the kx-f spectrum equals Uc. Accordingly, if the disturbance

103

being characterized contains waves with different frequencies and wavenumbers, but
propagate with the same speed, the disturbance energy will be distributed along a straight
line emanating from the origin in the kx-f plane, Wills (1964).

Figure 3.17 depicts a color contour map of the kx-f spectrum. The abscissa

represents the dimensionless wavenumber (kxxr) while the ordinate represents the

dimensionless frequency (13$). Also, the slope of a straight line drawn on the map

from the origin gives the ratio between the convection velocity and the freestream

 

C
.

velocity,

:1)

 

ky x,

Figure 3.17. Wavenumber-frequency spectrum for Re = 7600

104

Figure 3.17 depicts an inclined ridge of peaks where most of the fluctuating
pressure energy is concentrated. The highest point of the ridge corresponds to a
dimensionless frequency of about 0.6-0.7 and a dimensionless wavenumber of 0.8-0.9.
Furthermore, the broken line plotted in the figure represents the peak locus of the
spectrum ridge. This line passes through the origin indicating that all dominant wall
pressure disturbances propagate downstream with the same convection velocity
regardless of scale; i.e., the wall-pressure modes are not dispersive. This observation
along with the narrowness of the ridge suggests that a good representation of the
frequency spectra could be obtained from the wavenumber spectra by using U, to
transform x to t. Thus, it appears that Taylor’s hypothesis of fi'ozen turbulence may be
used acceptably to obtain spatial statistics from temporal ones in the late stages of the
developing boundary layer (E, > 2), where the RMS and spectra change very little and the
statistics reflect a fair degree of homogeneity in the streamwise direction. The specific
convection velocity value was found from the slope of the broken line to be 0.81. This
agrees quite well with the value estimated fi'om the cross—correlation analysis.

It is interesting to compare the results in Figure 3.17 to similar type of results
obtained upstream beneath the separation bubble. Figure 3.18 depicts the wavenumber-
frequency spectrum from Hudy et al. (2003). The spectrum shows an inclined ridge
similar to that in Figure 3.17. However, the peak locus of the ridge in this case does not
pass through the origin of the plot. This produces a wavenumber dependent convection
velocity, which indicates that the flow structures upstream of reattachment are dispersive;
i.e., flow structures of different scales travel with different speeds. Finally, it is also

noted that at very low frequencies, Figure 3.18 depicts both upstream and downstream

105

propagating disturbances. This signature, which is absent in the spectrum obtained
beneath the redeveloping boundary layer, was attributed by Hudy et al. (2003) to the

shear-layer flapping, or expansion/contraction of the separation bubble.

 

-3 -2 -1 0 1 2 3
k, x,

Figure 3.18. Wavenumber-fiequency spectrum from the separating/reattaching flow of
Hudy et al. (2003)

106

4. VELOCITY-PRESSURE ANALYSIS AND STOCHASTIC
ESTIMATION

This chapter presents an analysis and discussion of the results obtained from the
simultaneous velocity and pressure data and their usage to stochastically estimate the
velocity field from the wall-pressure signature. The chapter has three main sections. The
first section provides a presentation of mean, turbulent and spectral characteristics of the
velocity measurements at 1; = 2.05, while the second one illustrates a scenario, that has
been established in the present study of the relation between the wall-pressure and flow
structures. Analysis of the stochastic estimation work including its principle and results,
may be found in the last section.

4.1. Velocity Characteristics

Here, various characteristics that are derived from the velocity measurements
within the non-equilibrium boundary layer are analyzed and compared with the
corresponding published results for separating/reattaching, free-shear-layer and
equilibrium-boundary-layer flows.

4.1.1. Characteristics of the Boundary-Layer Mean and Turbulent Velocity
Profiles

Figure 4.1 depicts the mean streamwise velocity profile obtained from single-
hotwire measurements. The profile has been normalized using inner (or viscous)
boundary-layer scaling and is compared with the data of Ruderich and Femholz (1986),

Bradshaw and Wong (1972), and Farabee (1986). The abscissa represents the

. . . + ur . . . .
dimensmnless height (y = L— ); where y 18 the hotwrre distance above the wall, uI 18
v

the friction velocity and v is the kinematic viscosity of air, while the ordinate represents

107

the dimensionless mean streamwise velocity (u+ = Wu.) The solid line shows the "log-
law" of an equilibrium boundary layer. Ruderich and Femholz (1986) conducted their
measurements on a fence-with-splitter-plate model with a fence height of 22 mm and
freestream velocity of 9.6 m/s, resulting in an Re of 1.4x104. The present data are
compared with theirs at g = 1.86, which is the most downstream location of their
measurements. The thickness and friction velocity of their boundary layer at that location
were approximately 130 mm and 0.4 m/s, respectively. Note that the friction velocity of
the present measurements (0.64 m/s) was estimated for the best visual agreement between
the present mean streamwise velocity profile and that of Ruderich and Femholz. This is
done in order to provide a comparison between the profile shapes. It should also be
mentioned that the same uI value is used for normalization of the turbulent velocity and
Reynolds stress profiles below. On the other hand, Bradshaw and Wong, and Farabee’s
investigations were of a backward-facing step. The step height, freestream velocity and
corresponding Re of Bradshaw and Wong (1972) were 25 mm, 24.5 m/s and 40835 while
those of Farabee (1986) were 12.5 mm, 15.3 m/s and 12750, respectively. The friction
velocities of the former and latter studies were approximately 1.0 and 0.6 m/s,
respectively.

Before proceeding further, it is important to highlight a fundamental distinction
between the backward-facing step and the present model, fence-with-splitter-plate. In the
case of a backward—facing step, a boundary layer develops over a generally long surface
upstream of separation to a laminar or turbulent state. Once the flow separates, a free
shear layer forms downstream of the step. For a laminar boundary layer condition, the

initial velocity profile is typically that of a Blasius boundary layer at the step and due to

108

the Kelvin-Helmholtz instability of the flow, vortical structures are created and energized
by the mean-velocity gradient of the free shear layer, resulting in the whole laminar
boundary layer quickly becoming a free shear layer. However, in the case of a separating
turbulent boundary layer, the recent study of Morris and Foss (2003) showed that the free
shear layer contributing to the Kelvin-Helmholtz instability develops only from a thin
layer within the inner layer of the boundary layer with no contribution from the outer
layer and its large—scale structures. The latter were also found to persist unchanged for a
large distance downstream of the separation point, before the much-smaller shear-layer
structures grow sufficiently in scale to contaminate the entire width of the separating flow.

Based on the above, it is anticipated that the flow in the separation/reattachment
region of a turbulent boundary layer consists of three main layers on top of one another: a
recirculation bubble, a free shear layer and the outer layer of the original boundary layer
(in order, from bottom to top). Such a three-layer structure may be depicted in the
measurements of Bradshaw and Wong (1972), Farabee (1986) and Song and Eaton
(2002). Downstream of the reattachment zone, the layers become: the growing sub-
boundary layer, the remanence of the fi'ee shear layer that contains the vortical structures
that were generated in the separating shear layer upstream of reattachment, and the outer
layer that contains the large-scale structures of the original boundary layer. It is obvious
that, unlike laminar separation, in this case the upstream boundary layer has a direct
fingerprint on the characteristics of the turbulence structure downstream of the step.

In the case of the fence-with-splitter-plate, a very thin laminar boundary layer is
formed at the point of separation. As a result, downstream of reattachment the recovering

boundary layer is expected to consist of two, rather than three, layers: the sub-boundary

109

layer and the remanence of the separating shear layer; i.e., similar to the case of laminar

separation over a back step.

301': g 1'" 1‘1? 2:112: .1 i, a: f I I: *1:- 1LT;.T‘"‘1’* *Fa 1*"- f’ ' ‘fi .fi.,,,_.,,,,.,.. 1
' Present (E, = 2.05)
‘71 Ruderich & Femholz (C = 1.86)

       

-7 Bradshaw& Wong (g = 2.6) 1 A513 3 3 1 1'1:
25 Farabee (g = 1.7) -- — ») ‘ -- ‘
1 A Equilibrium boundary layer (Klebanoff) ‘ v1 1 :3-
— log-law ' v ,

20 "

10 ~
5 T L
: .4“ , ..._... .44; 3. 1
0 b_ _. .__g . . .4 .. l ._~ . . __._._.. .. . .43— ..A- ._.._... _ .._3 _ I .+. ...r-.- . - 4
10 10 10 10 10
+
Y

Figure 4.1. Wall scaling of boundary-layer mean—velocity profile

Comparing with the equilibrium boundary layer, the current velocity profile
deviates from the log-law and the mean-velocity profile of an equilibrium turbulent
boundary layer (Klebanoff 1954) as depicted in Figure 4.1. In particular, the data is seen
to overshoot the log-law for y+ < 300 and undershoot it for y+ > 300. This is in
agreement with the data of Ruderich and Femholz (1986), and Bradshaw and Wong
(1972), as evident from Figure 4.1. These two studies included velocity measurements at
different streamwise locations, and hence they were able to track the redevelopment

process of the mean-velocity profile due to the growth of the sub-boundary layer

110

originating from the new viscous sublayer downstream of reattachment. They showed
that the development process continued with a slow build-up of the logarithmic law with
increasing x. Initially, closer to reattachment, the velocity profiles undershot the log-law.
As x increased, the observed undershoot moved farther away from the wall, with an
overshoot region developing closer to the wall, as seen in Figure 4.1. The authors
provided two reasons for the observed deviation from the log-law: the rapid change of the
turbulence structure near reattachment and the non-proportionality of turbulent length
scales to height above the wall.

Although the profile of F arabee (1986) in Figure 4.1, who used a backward-facing
step, does not quantitatively agree with the rest of the profiles, it qualitatively behaves in
the same way for y+ < 300. In this range, Farabee's mean velocity profile exhibits a
viscous-sublayer-like profile near the wall, followed by a log-law undershoot farther
away from the wall. For y+ > 300 F arabee’s profile significantly deviates from the others
both quantitative and qualitatively. The most important difference between Farabee's
profile and that of the other studies is that it exhibits the overshoot above the log-law
characteristic of the wake region of a fully-developed turbulent boundary layer. The
amount of overshoot above the log-law is classically known as the wake strength, Coles
(1956). It is clear that the present data along with those from Ruderich and Femholz and
Bradshaw and Wong do not possess such a wake region. A reasonable physical
explanation of the disagreement could be related to the fact that in Farabee's study, a fully
developed turbulent boundary layer was established at the point of separation, instead of
the laminar boundary layer employed in all of the other studies. As reasoned earlier in

this section, the flow downstream of separation/reattachment of a fully developed

111

turbulent boundary layer consists of three layers, and therefore the difference may be
attributed to the existence of an outer boundary layer originating from the equilibrium
turbulent boundary layer upstream of the step in Farabee's investigation.

Castro and Epik (1998) presented measurements in the flow downstream of the
mean reattachment location of the separated flow at the edge of a blunt flat plate. Their
results showed that the developing boundary layer downstream of reattachment exhibited
the same behavior as that of the present flow. More specifically, Castro and Epik's mean-
velocity profile depicted that the log-law behavior was absent immediately downstream
of reattachment. However, they observed that the mean-velocity profile started to build
up a "logarithmic region" with increase in x through an overshoot and undershoot similar
to that of the present mean-velocity profile. Although their results showed that the log-
law was established by 2‘, z 10, they stated that the turbulence structures did not develop
fully even at a streamwise location of g z 19.

Song and Eaton (2002) studied the flow of a fully developed turbulent boundary
layer over a backward-facing convex ramp. In spite of the geometrical difference from
the canonical separating/reattaching flow models (e.g. backward facing step, forward
facing step, etc.), their flow exhibited similar physics. At streamwise location of 2'; z 2.0,
similar to that where the present data are acquired, their mean-velocity profile showed a
similar redeveloping behavior represented by an overshoot and undershoot of the log-law.
Also, because the boundary layer upstream of the ramp was turbulent, the profiles
exhibited a wake region in the outer portion of the flow. Furthermore, Song and Eaton's
results showed that the flow downstream of reattachment reasonably reached the

condition of an equilibrium boundary layer farther downstream at g z 9.0. This suggests

112

that the current mean-velocity profile represent the state of the boundary layer some 25%
or so of the distance required to reach an equilibrium state.

The results in Figure 4.1 are normalized in a manner that is consistent with the
boundary-layer component of the "two-layer" structure of the present flow. Another way
to normalize the results that reflects the free-shear-layer like flow in the outer portion of
the flow would be using classical shear layer normalization. For this purpose, the y
location of the shear-layer center needs to be determined. Typically, this location
coincides with the position at which the mean velocity is the average of two velocities on
the low- and high-speed sides of the shear layer, which corresponds to half of the
freestream velocity for a single-stream shear layer. This also coincides with the location
where the mean streamwise-velocity gradient, the velocity fluctuations, the Reynolds
stress, and therefore the turbulent energy production term are maximal, e.g., see Ho and
Huang (1982). In the current flow, the mean streamwise-velocity gradient is maximal at

the wall, and hence it can't be used to locate the center of the "free" shear layer. Instead,

the position of the maximum shear Reynolds stress (W) is taken as the outer-shear-

layer center. This location also agrees roughly with that of the maximum vrms, as will be

seen later. It is also noted here that the peak W within the sub-boundary layer may be
the strongest across the whole layer. However, because of the size of the X-hotwire used
to conduct u and v measurements, it is not possible to capture this region of the flow.
Thus, the identified peak is that corresponding to the center of the outer-shear layer.
Figure 4.2 depicts a plot of the mean streamwise-velocity in shear-layer

coordinates. The abscissa is (y-yref)/x; where yrs; is the y location of the Reynolds stress

peak (y = 38 mm, or yref/S = 0.48). The ordinate is the streamwise mean velocity

113

normalized by the mean velocity at yref (uref), where [Imp/Ugo = 0.89. The figure shows the
comparison between the present mean profile with that of Ruderich and Femholz (1986).
The two profiles show good agreement in the outer portion of the flow. The agreement
reinforces the earlier observation that the free-shear—layer vortical structures still

dominate the outer part of the boundary layer.

° Fresent (g = 2.05) i 1
ii? MerichéFimholzjéf 13592 .1 1

1.4.-,—— -— — — - — — ,— —, -—----——-— —..

0.81 ~~

ref

 

u/u

0.6'- - .3,

0.2 ‘*

(5.1 i ii -0.05 0 0.105 C 0.1 i 0.15
(y-ymfyx

Figure 4.2. Shear-layer scaling of the boundary—layer mean-velocity profile

Figure 4.3 shows the root mean square, RMS, profile of the streamwise velocity
fluctuations (urms) in boundary layer coordinates. The abscissa is y+ and the ordinate is
urms normalized by the friction velocity. The figure shows a reasonable agreement

between the present data and that of Ruderich and Femholz for y+ < 300. Although the

114

present data could not resolve the peak of urms near the wall, they exhibit a rise in urms
with decreasing y+ for y+ < 70, consistent with the existence of such a peak. Ruderich
and Fernholz’s results show the peak near the wall to be located in the buffer layer of the
sub-boundary layer (y+ z 10). The profile also shows a flat region of velocity fluctuation

in the range of 70 < y+ < 1500 apparently associated with the outer-shear-layer structures.

 

3.5 s 1 1 - ~ ~ - ~
"it i: (1’-
3~ ‘ *1';;'O.fi>°fi‘.‘.9'"
2.5
2 f .1;
:3P ‘ H 4*» g A
E .1
:3
1.5 ' '
1 .1 1 \- -
7:5.
0 5 i - flPregnt (3: 2.65 k V? W 1“ _ "A _____
' i 1 ‘ Ruderich & Femholz (i = 1.86) ‘ 1* ‘~
Farabee (F, = 1.7) ‘ .2.
Equilibriwnboundary layer (Fame) - .
'.-0 .-.. I .7 L . é. 1.1.l7- ,2 1 a . .4 L . .. i2. 1 . .. l - . . l 3 . '12 .1 A . .-.-. 4
10 10 10 10 10
+
Y

Figure 4.3. Wall scaling of the boundary-layer urms profile

As discussed above, the present data and that of Ruderich and Femholz show
good quantitative agreement near the wall and disagreement in the outer part of the
boundary layer. On the other hand, when including Farabee's data, there is a qualitative

agreement between the three data sets in the sense that all results reflect the existence of a

115

 

 

near-wall sub-boundary-layer peak and an away-from-the-wall free-shear-layer peak.
The latter is most well defined in F arabee's results because the peak is less broad. This is
apparently caused by the narrower extent of the shear layer in the case of Farabee (1986).
Moreover, the distance between the near-wall and outer urms peaks is smaller for
Farabee's data: a manifestation of the smaller Reynolds number of that boundary layer in
comparison to the other two data sets, where the Reynolds ntunber based on the boundary
layer momentum thickness, R69, is 8000, 11284 and 4027 for the present study, Ruderich
and Femholz (1986) and F arabee (1986), respectively.

Finally, although the current data set does not resolve the buffer region of the sub-
boundary layer, the data from Ruderich and Femholz (1986) as well as those from
Farabee (1986) do. It is expected that both data sets should collapse given the viscous
scaling of the data. However, there is discrepancy between the two data sets with that
from Farabee agreeing well with the equilibrium boundary layer. This raises a question
concerning the inner-portion of the sub-boundary layer and how quickly it reaches
equilibrium conditions downstream of reattachment for the two different flows of
Ruderich & Femholz (1986) and Farabee (1986). Further investigation of this issue is
beyond the scope of the current work. Finally, it is useful to note here that both
investigations of Ruderich and Femholz, and Farabee reported the relaxation effect of the
flow to an equilibrium boundary layer, showing a decay of the outer peak of urms with
increasing x till the profiles monotonically reach the shape of the canonical boundary
layer profile, exhibiting only one peak near the wall.

The data in Figure 4.3 are rescaled and plotted in shear-layer coordinates in

Figure 4.4. The abscissa is the same as that in Figure 4.2 and the ordinate represents urms

116

normalized by Ugo. The figure shows that all three profiles almost collapse in the outer
portion of the flow, above the location of the peak of the Reynolds stress; i.e., the shear-

layer center. On the other hand, the present data exhibit good agreement with the data of

Ruderich and Femholz (1986) for all heights but very near the wall ((y___y:e_r_) < _()_()5 ).
x

Moreover, both profiles reveal a region of flat turbulence intensity (um/Um z 0.12) that

extends well below the shear-layer center.

0-141'77 W ' 7 7W 7 T 7 W F ’ 7’ -7 '1'. .-- T:j ;: g; _T' LT 1"" ,1
' Present (g = 2.05) i 1‘
. , 1 , 1 :2 Ruderich & Fernholz(§ = 1.86)
’ J _- <7 Farabee(§=1.7)

  

Ci» . . “fig :
01 i ‘ 0.0
. i ‘ Tl) \f’ .V,’ iv? a l 1
8 a .v ‘ ‘
:3 1 V ‘ , 1 i V ."o , I
E i '3 1
:3 0.06"* :jfv * . ' - - :77 --.- :-§w._l-0 --
. ; 3.
. 'SAO .
L. V .-- .- -1 .2--. .___.-.,---.-_-'. -u.--...__. .1 _____ --
0.04 1 I g...

0.02~ —~

-808 -0.06 -004? -002' 0 *0.02 0.04 0.06 0.08 0.1
(Y‘yrefyx

Figure 4.4. Shear-layer scaling of the boundary-layer urms profile

The root mean square profile of the normal velocity fluctuations (vrms) is plotted

in boundary-layer coordinates in Figure 4.5. Note that hereafter all results are based on

117

 

flit-III! lubE- E. .5!

Or

 

'1 1
2

velocity data measured using the X-hotwire probe as described in section 2.2.4. Unlike
um, vrms profile exhibits a clear peak in the outer part of the flow. The vrms fluctuations
are also attenuated with y change from the shear-layer centerline towards the wall more
strongly than away from the wall. This is another feature that differs from that of urms
where a flat region of urms fluctuations was observed on the lower side of the shear-layer
center. For a truly free shear layer, velocity fluctuations are expected to be damped as
one moves away from the center of the shear layer. The stronger damping of vmns on the
lower side of the shear layer in this case is consistent with the anticipated wall-damping
effect of v'.

The present vrm, profile is also compared with those of Ruderich and Femholz
(1986), and Farabee (1986) in Figure 4.5. It is evident that the present data agree well
with Ruderich and Fernholz’s vm, profile for y+ < 800. On the other hand, the profiles
don’t collapse farther away from the wall, but they show qualitatively the same trend and
approximately the same peak location. The discrepancy is consistent with the
expectation that the outer portion of the boundary layer cannot be scaled with wall
coordinates. On the other hand, F arabee’s data show significant disagreement with both
the present data and that of Ruderich and Femholz. Farabee’s profile shows the same
trend of Figure 4.3, in which his boundary layer is thinner than the present one in wall-
coordinates by approximately and order of magnitude. As discussed previously, this is

consistent with Farabee's lower Reynolds number which is proportional to the ratio of the

outer and inner scales of the boundary layer, or 8+.

118

- Present (§ 2 05)
Ruderich & Femholz (5," - 1 86)
Farabee (i = 1 7) D

2.5"" " 1.1.1.._., 7 -0- --5- ,, 1
1 O I O
3' 1 0
O. ‘
2 .3. - - ,- -
. A
C, V
p
D 1
,7 .
\E 1.5' ' .7: fl ,
> C, \/ V
v
0.51“ . H
1
O 1,] .1, ..- 7. . .1. ; l *2 .7 ..7 pi . +1 . .14. . .-3 i . ...i . .. .. 1; . .-. 4
10 10 10 10
+
y

Figure 4.5. Wall scaling of the boundary-layer vrms profile

Figure 4.6 depicts the same data in Figure 4.5 afier they are rescaled and plotted
in shear-layer coordinates; i.e., in the same manner as in Figure 4.4. The figure shows
that the peak of vrms falls on the negative side of the abscissa which is below the peak of
the Reynolds stress. Moreover, all profiles exhibit a peak approximately at the same
location in shear-layer coordinates. Generally, Figure 4.6 shows that the present data
agrees better with Ruderich and Fernholz’s than that of Farabee, although all data sets
show the same trend. The level of vrms for both the current and Ruderich and Femhotz
(1986) studies is approximately twice that of Farabee (1986), showing that the vortical
structures are more energetic in the fence-with-splitter-plate flow. This is believed to be

resulting from the more intense mean shear stress from which the vortices originate at

119

separation in the case of the fence flow, which is the same reasoning used to explain the
larger level of pressure fluctuations found in the fence flow relative to that of the back-

step flow (see Figure 3.3 and associated discussion).

0.1 ;‘ " " " " T.-- i fl; . _’;’__ g *7...
,1. O Q ' Present (5 = 2.05
0.09 A, We”; - “-1 O Ruderich & Femholz (E, = 1.86)
3.. , '0.” . V Farabee (g = 1.7)
e! ; “on” :—- “~- ~ ~
0.08 ~ , ‘67 4 . . w ..... . z. - .. . ......
‘l'.’
0.061 1- ~ »'—
8 1 741‘
E 'V " "9"- , :1
m 0.05 M ' “ "(1' _ ' ‘ — - - '- ‘- - V15; — — - _‘ ._ _ _ ,
E » 1 t .
0.04%” ~ ~» 7» \7 - 1-
0.03 ~
0.02 ~
0.01 1-»-
O 1' i 1 7 ~ - --- 7, -fi. ..‘_.____v 7 .171 -1 7 7_ _'
-O.1 -0.05 O 0.05 0.1 0.15
(y-yrefyx

Figure 4.6 Shear-layer scaling of the boundary-layer vrms profile

Another notable observation from Figure 4.6 is the fact that the data from
Ruderich and Femholtz (1986) do not collapse completely with the current data in the
outer part of the flow, even with the employment of shear-layer scaling. The fairly small
deviation (~10%) may be caused by differences in the streamwise freestream pressure

gradient due to flow blockage by the model and wind tunnel boundary layers. Finally, by

120

comparing Figures 4.4 and 4.6, it can be shown that the peak of vrms coincides with the

outer edge ( 9:141) z _0_02) of the y range where urms profile is flat.
x

The boundary-layer Reynolds stress profile PW ) is plotted in wall and shear-
layer coordinates in Figures 4.7 and 4.8, respectively. Figure 4.7 shows the qualitative
consistency between the present data and Ruderich and Fernholz’s, which exhibit the
same trend and peak location (y+ z 2000). Furthermore, general comparison of the two
profiles shows that the turbulence in Ruderich & Fernholz’s flow is more energetic than

in the present one, possibly due to pressure-gradient differences as discussed above. The

smaller y+ location and lower -1_1_'V_' values for Farabee's results are consistent with the

same observations concerning vms and associated discussion made earlier.

3,. f.».i - 74;-..gggvfigggrj .7 —.... ..--~...~. .--.jggfi .111_.,, ......q ...
' Present (5 = 2.05)
:1 Ruderich & Femholz (E, = 1.86) 1

 

Far abee (E, = 1-7) 1 :..: '9
2.5 .1, v 1 — . +. - - . . 1 1
2'“ ......... i.-- ___1 :3
N P 3 ‘ I . J
3 O o
> 1.5"“ 7 -__9 ..... Q--- .- 14
‘ 1 | .i .
:3 1 o .
c .
1 V\ “.7 . 5
1.. _____ v .. SQA/11. . 1-
1; \f . x)
. .V
05* * , ' 1, .. _.
C .
o
v.
1-.. 1,. .. ' . 7; ...1 ; .' .‘ 1. : L ”7-1%.. .1 I 1 :.1;/:,-.., t ‘.. 1.; g _‘ . ..l
0 I 2 3 4
10 10 10 10
+
y

Figure 4.7. Wall scaling of the boundary-layer Reynolds stress profile

121

Figure 4.8 depicts the — W profile in shear-layer coordinates. As stated earlier,

yref is the locus of the - W peak, which means that all three profiles are forced to have
the same peak location. Data comparison in the figure shows that there is a trend
similarity among the three profiles with quantitative differences as clarified before.

In summary, the boundary layer profiles show that the non-equilibrium boundary
layer in the redevelopment zone studied here mainly consists of a growing sub-boundary
layer near the wall and outer-shear layer that exists above it. The characteristics of the

investigated boundary layer exhibit a discrepancy with those of the fully-developed

turbulent boundary layer, as demonstrated by the shape of the um, vms and -—W
profiles. The maxima of those profiles for an equilibrium boundary layer fall in the

buffer layer (y+ < 30) as reported in the literature (e.g. Wei and Willmarth 1989). In

contrast, the boundary layer at Q = 2.05 predominantly exhibits free-shear-like

characteristics, associated with the existence of urms, vrms and —l1—;\7 peaks in the outer
layer. Those maxima, which clearly differ from those described for the equilibrium
boundary layer, are related to the energetic large-scale vortical structures that were
generated and energized upstream in the separating free-shear layer. As was concluded
from the pressure data in Chapter 3, those vortical structures seem to dominate the flow
especially in the outer layer of the boundary layer. Although a few authors investigated
the flow downstream of separating/reattaching flows (e.g. Ruderich & Femholz 1986,
Farabee 1986 , Farabee & Casarella 1986, Castro and Epik 1998 and Song and Eaton
2002), they observed that the vortical structures could be seen at downstream locations as
far as six reattachment lengths (6X,). These authors also stated that the equilibrium

boundary layer state could be reached at the location where the sub-boundary layer had

122

propagated all the way through the non-equilibrium boundary layer. Moreover, Farabee
(1986) indicated that the distance needed to reach the equilibrium boundary layer stage
was approximately 20X,. However, the results of Castro and Epik (1998) and Song and
Eaton (2002) showed that their non-equilibrium boundary layer relaxed to the

equilibrium behavior at 10Xr downstream of separation.

x 10'3
4 T“ *7 7 ' 7 7 ' V '7 _ T — : — — 7 ,'__ ,__.._._‘_—._ -.L:____.—j— ;--.'__ ; i. I-
. 'V' c: . 0 Present (5 = 2.05
l l C ‘ O Ruderich& Femholz(§ = 1.86)
3.5 ~ — ‘ ii , , * -533?“ (3 1:7) -
3 - - - _ a
00.9..
o i o .
l. . o.
2.5...» __ __ —— -. -.. ....... . ...... 1.—w—-«7- -.-- -.l
s O ‘ ' I
NEDS . .
|~> 2 T' .
‘:
I C V U ‘7 Q
o . .
1.5~~~ ~ . — .7
_ . ‘ . fl
0 w by
l I ’ “
o \ I
O 7‘ x
0.5- ~ .—'
49.08 i 6.06 2.20.04 T— -0.02 7 0* 0.67 A '034 i 0.06;! V (T08 T 0.1
(y-yrefflx

Figure 4.8. Shear-layer scaling of the boundary-layer Reynolds stress profile

4.1.2. Velocity Spectra
Here, the turbulent velocity and shear-Reynolds-stress spectra in the non-

equilibrium boundary layer at g = 2.05 are presented and analyzed in order to obtain more

123

information of the turbulent flow features during the boundary-layer relaxation process.
In order to show the effect of height on the spectral characteristics of the flow, the
dimensionless auto-spectra and cross-spectrum of the streamwise and normal velocities
((Duu, <va & (Duv) are plotted at six different heights (y/B = 0.05, 0.125, 0.25, 0.375, 0.5 &
0.625) that approximately cover the y range of velocity measurements. Furthermore,
each spectrum is plotted in both logarithmic and semi-logarithmic scales. As explained
earlier based on equation 3.2, the reason for using the semi-log scale is that the
geometrical area under the spectrum represents the spectral distribution of the turbulence
energy in this case.

The spectra are calculated by dividing the time series of velocity into 2048
records of 1024 samples each. The Fourier transform of each record is then calculated
and multiplied by its conjugate, except in the case of the cross-spectrum, where the
Fourier transform of the u component is multiplied by the conjugate of the Fourier
transform of the v component. Subsequently, the products of all records are averaged to
give the average auto- or cross-spectrum of the velocity components of interest. The

(fX

resulting frequency resolution is 12.2 Hz ____r= 0.136) with a random-error uncertainty

U

'1

of about 2%.

Figure 4.9 depicts the streamwise-velocity spectra at the six different heights,
plotted using logarithmic scale. The abscissa is the dimensionless frequency (f Xrono),
while the ordinate is the velocity spectrum (Cbuu) normalized by U002. The spectra show
that with the exception of y/5 = 0.625, corresponding to the lowest spectrum, the energy

content of all other spectra is quite similar. This is consistent with the uniform

124

distribution of urms shown in Figure 4.3 for 70 < yJr < 1500 (0.02 < y/5 < 0.5). The
spectra also depict that the closer the location to the wall the more the contribution from
high-frequency fluctuation to the velocity spectra. This observation, which is more
evident in the semi-log plot in Figure 4.10, is presumably a manifestation of the
increasing importance of the small-scale turbulence associated with the sub-boundary
layer as one approaches the wall. It is noted though that, for the y values considered, the
overall contribution of these small scales to um,s is negligible compared to the lower-

frequency larger-scale structures.

   

10'2. * ~ 1 f f a 4 - -- g Q -_;~_
' y/5=0.05
y/5=0.125
. . y/8=0.25
10'3. “*~\ 1 - g . y/5=0.375 ..
‘13s, ‘ A y/a = 0.5
: 1’5 :99} f
-4
10
N 8 ;
E33 :
e -S
10 :4 “-
lo-
-71' ‘ ‘ ' ' ‘ ’ ‘ ‘ 1 9-4 L i l ,' -77 1 - 7A 1 L_ - '. .: ;.-.-_.‘
10 ”:1 ‘ ‘ ““1; ”“ 1 2
10 10 1o 10
f X/U
r (I)

Figure 4.9. Streamwise-velocity spectra at y/S = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625
and E, = 2.05

The practically-uniform u,rms energy found at five of the presented heights (y/5 =

0.05, 0.125, 0.25, 0.375 & 0.5) can be related to the mean-velocity and Reynolds-shear-

125

stress profiles presented earlier and their influences on the ntrbulence-energy-production

T7 11 du . .
term (-u V T). Near the wall, the : terrn 18 large Wthh compensates for the low
Y

 

— . . u .
Reynolds stress term (—u'v'). As the height Increases, : decreases while —u'v'
y

increases, keeping the whole term approximately the same. This scenario seems to be

valid up to the location of the Reynolds stress peak location, above which the energy

. du . . .
production term decreases because the d— terrn remains practically constant while
Y

-W decreases with height. As a result, urms decreases with further increase in y
resulting in the drop in the spectrum seen at y/6 = 0.625.
Another notable feature of the spectra in Figure 4.9 is the existence of a more

than a decade wide f ’5/3

range, implying the sustenance of an inertial sub-range. The
width of this range increases with increase in height in the boundary layer. Farabee
(1986) and Castro and Epik (1998) reported the same slope at similar streamwise location
and heights above the wall. Furthermore, Ruderich & Femholz (1986) observed the same
trend in the dimensionless streamwise-velocity spectra, which showed similarity at
different heights in the boundary layer.

To study the distribution of um, energy among different frequency ranges, (1),,“ is

plotted in semi-log coordinates as shown in Figure 4.10. The ordinate is the product

f X! (Dun
U, u;

 

 

).

between the dimensionless frequency and streamwise-velocity spectrum (

fx,

 

while the abscissa is the dimensionless frequency ( ). The spectra are all grossly

G)

126

similar qualitatively and quantitatively except at the highest location (y/S = 0.625), which

is consistent with what is observed earlier in Figure 4.9. From the general inspection of

 

the spectra, it can be observed that there is a frequency range, centered around fUX’ z

0.55, where most of the u' fluctuations are concentrated. This dominant frequency range,
which encompasses the signature of the low-frequency shear-layer vortical structures
(based on the analysis of the pressure data in Chapter 3) persists at all heights, even near
the wall. However, closer to the wall, there is also more contribution fiom small-scale

structures within the high frequency range, which is reflected in the grth of a small

 

hump in the spectra for fUX' > 5 and the lowest height.

Figure 4.11 depicts the normal-velocity spectra, which are plotted at the same
heights as in Figure 4.9. The spectra show that the normal-velocity energy increases
significantly as the height increases in the region near the wall (y/S = 0.05 & 0.125),
which highlights the attenuation effect of the wall on the normal velocity fluctuations.
This is consistent with what is deduced fi'om the vm, profile shown in Figures 4.5 and 4.6.

In addition, Figure 4.11 depicts that the spectra for y/8 > 0.25 start to show a peak at a

 

frequency of fUX' z 0.75, which is approximately the same frequency as that found in the

Q

wall-pressure spectra earlier, but higher than the frequency range of the u' flat spectral
peak in Figure 4.9. These observations further supplement the earlier conclusions that
the energy-generating flow—features are still dominated by the vortical structures

generated upstream in the separating free-shear layer.

127

 

n1 9.9.3.5.. . . : ....‘JI. .. ~ .. -..:Iwh)!...|:.. ®JP|..|L....1.!.U
o

. - . 9....

A:

 

 

 

 

61” . . . .. . . . . *1, -.W, 7..7..-,7.--.,
l y/8=0.05 l_
l y/5=0.125 ll.
1 ~ y/8=0.25 Li
1 y/5=0.375 !
i * y/6=0.5 ll
, , . ; A y/6=0.625 l
4- - — ,5 .- _ . .. . - -1- a. —.~- -

UUOO

(fX/U )o m2
r 00
- a»

l _A __l

\ v
t". , ’ W
‘ M ‘ '\_’-A A
~_‘ “3"
\A ‘
‘ ‘»A“\ . X‘ .
C“ .
\K‘:
\‘V‘u...M
l
0t .- i i I ,1 1 A l i .i‘. .7 . 7 . a; . 171 1 LL .k 1...-’.A _ki L L._L-l .l

10 100 10 102
fX/U
r 00

Figure 4.10. Semi-log plots of the streamwise-velocity spectra at y/B = 0.05, 0.125, 0.25,
0.375, 0.5 and 0.625 and E, = 2.05

-‘7

lO-g " ' " ' ' r" ' = '— ' ——‘ * __,. ’ "‘—r —f *: 1;: ' '31
h l " y/5 =0.05 l"
‘ y/6=0.125
3 . p l y/8=0.25 .
10 ; -, ~ ~ .- . ; y/8=0.375 --
. l y/S = 0.5

i - “'1‘ ’ * " l ‘ y/8 =O.625 i

M if \\

H—o—a

/
_A_.. .L. L L .._~..l. .__.

. L. ._._‘.._1._..-

VV

10:15.- _ ; L "1 _ , l 0 ..--g--.. ..i . l : x J r ; .A.- .- 1 I . 2 :....- 1. ~
10 10 10 10
fX /U
f C!)

Figure 4.11. Normal-velocity spectra at y/S = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625 and
2; = 2.05

128

Figure 4.12 provides the dimensionless semi-log normal-velocity spectra.
Grossly, the spectra show significant dissimilarity with the semi-log streamwise velocity

spectra shown in Figure 4.10 for fo' < 20. For higher frequencies (er > 20), the

U

6) co

 

 

normal-velocity spectra exhibit more or less the same behavior as u', reflecting an
increase in the contribution of the high-frequency fluctuations to the normal-velocity

energy near the wall.

6" ' "7 ’ - ‘ '7’" ' l i 7 i' T ’ "fl ' f ’7 ..' A7; ’7F_I_—“;'_ 'T’
I ; 1 , y/5 = 0.05 1
, l y/8 = 0.125 1
5 I fl | y/8 = 0.25 r
. ‘ y/8 = 0.375 1 .
1 . 1 ‘ y/6 = 0.5 l
‘ ~ 1 -- ; y/6 = 0.625 1
841- - - - -- L —- - , . ' - #1
N
Q> 1 -.-
> / . ._ l l
u— \ . .
V 2 1' 1 1 i
1 l" ' K "‘99; 3%. - ’ ‘ “
/ ‘ “;.\ ‘
l I HR“ ‘
O _ _ . 1 _1_.L_ L 1 1 . 1 _.__ __1v 1 .. 1._-i1.._.L-. . ; 1 i .l-.__ .. gnu—“- 1.4—4 1.1-1 1
10'l 10O 10l 10‘
f X /U
T 00

Figure 4.12. Semi-log normal-velocity spectra at y/5 = 0.05, 0.125, 0.25, 0.375, 0.5 and
0.625 at 2; = 2.05

The collapse of the spectra for y/5 < 0.375 and fo’ > 20 in Figure 4.12 implies

a)

 

that the energy contribution of the flow structures within that fiequency range is

 

independent of the height above the wall. On the other hand for fUX’ < 20, the spectra

a)

129

reveal a strong evidence of the dependency of spectral distribution of v' on height.
Particularly at y/5 = 0.05, the spectrum shows that the contribution of the low-frequency
shear-layer vortical structures to v' fluctuations at that height is minimal in contrast with

the contribution of the hi gh-frequency small-scale structures. The spectrum at y/5 = 0.05

 

emphasizes this observation by showing a peak at approximately fUX' =10, which is at

w

least one order of magnitude higher than the reported passage fiequency of the shear-
layer vortices.

On the other hand, Figure 4.12 also shows that as y increases the contribution of
the large-scale shear-layer vortical structures increases, which is implied from both the
grth and the Shift to lower-frequency of the spectrum peak. More specifically, the

frequency of the spectrum peak drops by almost a factor of five as y/5 goes from 0.05 to

0.125. Furthermore, the area under the spectra (which represents the energy) for ' <

 

20 increases significantly as the height increases. At higher locations (y/8 > 0.25) the

spectra show that the energy of the normal-velocity fluctuations is mainly provided by

the shear-layer vortical structures. Furthermore, for the same height range, the spectra

 

show more or less the same level of energy with a peak at fUX' z 1.0. However, the

w

energy contained in the spectrum at y/5 = 0.375 is maximal since vrms reaches its peak at

that height.

Next, the shear Reynolds stress (W) spectra, or the cross-spectra of the
streamwise and normal fluctuating velocities ((Duv) are evaluated. Figure 4.13 shows the

(DW spectra in the same dimensionless coordinates of Figures 4.9 and 4.11. The spectra

130

shown in Figure 4.13, which depict the spectral characteristics of W at different heights,
reveal that at all heights except y/S = 0.05 (the closest to the wall), the cross-spectra are
similar and the peak contribution to the Reynolds stress is at a non-dimensional
frequency of 0.55. At y/6 = 0.05, there is a reduction in the level of the Reynolds-stress
contribution at that frequency relative to the other heights. Additionally, a substantial
increase in Reynolds stresses associated with the high-frequency end is seen at this
location closest to the wall. This most likely reflects the build up of small-scale energetic

turbulence within the growing sub-boundary layer near the wall.

 

10 7 , ~ - . , ~ - . . - 1
1} 7 ya : 0.05 '
. " 1 y/8 = 0.125
10-4 :1 ._ ; 7 y/8 = 0.25 ,.
l ‘ ‘ y/s : 0.375
. ‘~.\ ‘ ya = 0.5
_51 ‘\, _ y/s = 0.625
10 . ‘f ,2 . . .
. Wu
N 8 6 ' y
I
2> 10 X
:3
e
.7.
10
1
-8
10
E
,1
10° .
10' 100 10' 10'

fX/U
r

CO

Figure 4.13. Boundary-layer velocity cross spectra at y/5 = 0.05, 0.125, 0.25, 0.375, 0.5
and 0.625 and 2; = 2.05

The semi-log plot of (13m, in Figure 4.14 also depicts the same trends as those of

Figure 4.12, but it shows more clearly that the contribution of the flow structures with

131

non-dimensional fiequencies higher than 6 to the shear Reynolds stress is practically
negligible for y/5 > 0.05. That is, the shear Reynolds stress for y/B > 0.05 is solely
associated with the large-scale motion in the fi'ee shear layer. The peak magnitude of the

corresponding cross-spectrum increases with height till it reaches a maximum at y/8 = 0.5,

before dropping down again at y/5 = 0.625. This behavior follows the trend of the —W

profiles displayed in Figures 4.7 and 4.8. Additionally, Figure 4.14 shows that the

frequency band containing most of W is centered around the same frequency at which

 

the wall-pressure fluctuation exhibit a peak i.e., fo' z 0.7.

w

10
2.5x , ~ , ...--. , . .--; ...-.g.--...l
‘ ' . ' y/8=0.05 :3
. y/8 =O.125 l.
1 "\1 ' . Y/S = 0.25 l i
2 1 ~ 7 y/8=0.375 i.
. , , was :05 l
‘ ' ‘ { 5 y/8=0.62_5_1
N 8 (I \ 3
E1 15 \
3 I
'6' .' \‘
A8 \
a; . .
X 1 '
‘CL
0.5 X;
-l 0 I 2
10 10 10 10

{X N...
Figure 4.14. Semi-log plot of the boundary layer velocity cross spectra at y/f) = 0.05,
0.125, 0.25, 0.375, 0.5 and 0.625 and E, = 2.05

The collective observations of the velocity and W spectra may be interpreted

from the perspective of Townsend’s hypothesis of active and inactive motions, Townsend

132

(1961), which was extensively studied by Bradshaw (1967) using zero- and adverse-
pressure-gradient boundary layers. Basically Townsend defined the inactive motion as
that of the large-structures that do not contribute to the Reynolds shear stress near the
wall while contributing to u'. This was verified by the measurements of Bradshaw (1967)
who reported that the large-scale structures produced Reynolds shear stress in the outer
layer but not in the inner layer. A good explanation of the reason for the inability of the
large-scale motion to contribute to the near wall W generation is offered by the
structures-based model of Perry et al. (1986). These authors envisioned the turbulence
structures of equilibrium turbulent boundary layers to consist of a hierarchy of scales of
hair-pin eddies. Treating these vortices as potential vortices, they argued that the heads
of the hair-pin eddies, which are similar to an x-y section of a Spanwise vortex, contribute
to v' only at heights above the wall that are similar to the vortex center location. Away
from the center, the vortices only contributed to the u'. Hence, because locations near the

wall would be far away from the large-scale vortex centers that reside in the outer-layer,

these vortices contribute very little to v’, and consequently to —u—'v—' , while contributing
substantially to u' near the wall. That is, the large-scale vortices are inactive near the
wall.

This analysis is consistent with the current results where Figures 4.10, 4.12 and
4.14 imply that the large-scale structures are active in the outer layer as they contribute to
both the streamwise and normal velocity fluctuations and consequently to the Reynolds

shear stress. Those structures, however, become inactive near the wall because they only

produce the streamwise velocity fluctuations while their v' and W contributions are

attenuated substantially. Similarly, the analysis of the small-scale structures

133

demonstrates that they are active near the wall because they contribute to the fluctuations
of both velocity components and the Reynolds shear stress.

Finally, it is important to note that the physical nature of the inactive motion here
is different from that referred to by Townsend (1961) and proposed by Perry et al. (1986).
In particular, Townsend’s inactive, or large-scale, structures are those that occupy the
wake region of an equilibrium boundary layer. As discussed in section 4.1.1 the non-
equilibrium boundary layer investigated here does not have a wake region. Instead, the
inactive motions here are those originating in the free shear layer at separation. It is also
emphasized that whereas Townsend’s inactive motion contribute, but does not necessary
dominate, the wall-pressure fluctuations, the inactive motion here does dominate the
wall-pressure generation process. The physical mechanism(s) by which it does so will be
elaborated upon in section 4.3.3.
4.2. Velocity-Pressure Characteristics

To investigate the relationship between the velocity field and its wall-pressure
signature, the simultaneous velocity/pressure data are explored in this section. The
exploration includes analyses of the cross-spectra and cross-correlations of the velocity
and wall-Pressure data. Furthermore, the average velocity field that is conditioned on the
occurrence of strong positive and negative wall-pressure events is obtained and analyzed
for additional insight.
4.2.1. Velocity/Wall-Pressure Cross-Spectra

The velocity/wall-pressure cross-spectrum is the Fourier transform of their cross-
correlation. Therefore, in this section the cross-spectra are investigated to better

understand the spectral properties of the velocity/wall-pressure cross-correlations; i.e., to

134

 

identify the frequencies at which strong velocity-pressure correlation is found. The
cross-spectra are calculated between the flow velocities at i = 2.05 and the wall-pressure
at three streamwise locations i = 1.33, 2.0 and 2.33. The microphones at those locations
are the 1”, 7th and 10th microphones, where the 1St microphone is the most upstream one,
and therefore, p1, p7 and p10 will be used to denote the corresponding pressure at these
respective locations. The procedure for the velocity/wall-pressure cross-spectra

calculation is the same as that used for the velocity cross-spectra, given earlier in 4.1.2.

The cross-spectra between u’ and p' at g = 1.33, 2.0 and 2.33 ((13%, (bum &(Dup,, ,

respectively) are shown in Figures 4.15 through 4.20. Figures 4.15, 4.17 and 4.19 show
the logarithmic plots of the dimensionless cross-spectra, while Figures 4.16, 4.18 and
4.20 depict the semi-logarithmic ones. Figures 4.15 and 4.16 show that the only
correlation between u' at 2; = 2.05 and p' at F; = 1.33 is produced by the passage of the
large-scale vortical structures, which correspond to the low-frequency range, while no
correlation is evident due to the small-scale structures near the wall (corresponding to the
high-frequency end). This is not too surprising given that p' measurements are more than
one boundary-layer thickness (z 1.68) upstream of that of u’.

In contrast, the logarithmic and semi-logarithmic cross-spectra of the microphone
closest to the hotwire (F, = 2.0), Figures 4.17 and 4.18, show a rise in the contribution of
the small-scale structures to the cross-spectra near the wall. This may be deduced from

inspecting the spectrum at y/6 = 0.05, which exhibits a rise in the spectra at high

 

frequencies ( fUX' > 1.5). At higher y locations, the data still resemble those in Figures

Q

4.15 and 4.16 at the same heights, but with a decrease in magnitude of the cross-spectrum.

135

The observed reduction in the spectrum magnitude seems to oppose intuition that the
correlation should increase as the distance between the microphone and the hotwire
decreases. However, the reduction mainly results from the decay of the vortical
structures energy with increasing x due to their relaxation process, which is documented
in Chapter 3 as a decay in the p'rms in the streamwise direction. That is, if the cross-
spectra were normalized by using urms and pm, there would be an increase in the cross-
spectra levels at g = 2.0 relative to § = 1.33 instead of attenuation.

The cross-spectra corresponding to p'lo or g = 2.33, which is downstream of the
hotwire location, are presented in Figures 4.19 and 4.20. The results in both figures are

generally similar to those obtained in Figures 4.17 and 4.18 for g = 2.0, including

 

evidence of some small-scale ( fUX' > 1.5) contribution to u'-p’ correlation near the wall.

w

Unlike the data at i = 2.0, though, this contribution is not only evident at y/6 = 0.05, but
also at y/B = 0.125. It is interesting to note that the range of frequencies corresponding to

the small-scale signature (1.5 < fUX' < 10) is actually at the low-frequency end of the

w

 

range that has been identified to correspond to the sub-layer pressure fluctuations. Thus,
it is believed that the flow structures corresponding to this range of frequencies although
small in scale relative to the shear-layer vortical structures, they represent the largest of
the sub-boundary layer turbulence. Since the sub-boundary layer thickens with increase
in x, these structures are expected to grow in size with downstream distance. This is

likely why the small-scale contribution to the cross-spectrum at g = 2.33 is felt as high as

136

 

- . I.;-"

 

w
. r

 

 

 

10 7 . , r 7 - ,
y/6 = 0.05 i
, y/8 = 0.125 ‘
10" ~ 2.: £\ 7 , ‘ ‘ y/s = 0.25 t!
, xv "‘ ‘3‘ y/5 = 0.375 1
‘ y/s = 0.5 1
10.5 \ . y/5 = 0.625 .‘
A 7 1R 1

m 8 '
D 10 ° rim”
0. lll ‘1 _ i
Q ' , it“ 1
Q '7 l ll) I
n'.‘ 10 1
3

9 .‘
.3 1
10 , l
.9 l
10 ' g
1 i
10'‘0 » l , i A

10 1 10° 01 102

fX/U
l’ G)

Figure 4.15. Cross-spectra between the streamwise velocity (at if, = 2.05) and wall—
pressure (at l; = 1.33) for y/5 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

x710-

 

9. ~ - - «1
y/s = 0.05 ‘
8 y/8 = 0.125 _
345 = 0.25
‘ y/6 = 0.375 ‘
7 * > “t, y/s z 0.5
”‘9 t we = 0.625
O. ‘1
Q 3
s 5 , i
\d: ‘
:I
2 4
8 ’ l
2, ./
>< 3 ft '
8-4 3’ ‘
2 "i/ 7 \
,'//
l /i
// >
r / 15,154 y_,-,;_
. _ ‘, ’51:: . , . x #
10 ‘ 10° 10' 102
f X [U
f on

Figure 4.16. Semi-log plot of the cross-spectra between the streamwise velocity (at l; =
2.05) and wall-pressure (at g = 1.33) for y/6 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

137

10
.5
I0

As

MD 10'6

Q.

Q’

Q .7
d? 10 =
5

9

.x
10

-9
10 »

-|(I
10

l , ' l
y/S = 0.05

 

y/s = 0.125 V
f ~._ ‘ y/S = 0.25 1
I" ' ‘_':‘:-.\ ya = 0.375 e
‘ § y/S = 0.5
\ y/a {0.625 11
L51;
‘Jlr a
‘. ll/‘l,y‘,‘,I‘ lv l
:3 \, 1%, z:
' l
l
l
7 1 7 i ..l
100 10l 102
fX /U
|' (1)

Figure 4.17. Cross-spectra between the streamwise velocity (at a = 2.05) and wall-
pressure (at E, = 2.0) for y/6 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

3
(f Xr/Um) (Dup7/(1/2 p Um)

Figure 4.18. Semi-log plot of the cross-spectra between the streamwise velocity (at E, —

y/8 = 0.05
y/8 = 0.125 t.
y/8 = 0.25

y/B = 0.375 1
y/8 = 0.5

y/8 = 0.625 ‘ ,

*"r‘** rrr "1

 

fX/U
f (D

2.05) and wall-pressure (at E, = 2.0) for y/8 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

138

o /(1/2p U30)

“pm

Figure 4.19

y/S = 0.05

 

y/s=0.125 1
l0 f\ y/8=0.25
fl, y/5=0.375
'/ y/5205
Io" y/8=0.625 .,
.5 1
ID A, a,
\. ' _ .
I It" .,
.7 "f \ll‘
‘0 "V‘y:
l
4
l0
-9
10
.10 4
10 -l i u 7 A i l A '2
IO 10 lo 10

fX/U
r d)

. Cross-spectra between the streamwise velocity (at g = 2.05) and wall-

pressure (at E, = 2.33) for y/6 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

)

3
co

/(1/2pU

“pr 0

(er/Ux) m

 

9 x 10 . .
y/vS = 0.05
8 y/S = 0.125 _ _
y/8 = 0.25
y/8 = 0.375
7' y/5 = 0.5
7* iy/S = 0.625 ,
6
5
4.
3 ‘-
2 y.
.5 s
f. e
l “1.
///‘f' I
/

0 -1 o v "J 1 1 ' 2
l0 IO 10 IO
fX /U

r 00

Figure 4.20. Semi-log plot of the cross-spectra between the streamwise velocity (at g =
2.05) and wall-pressure (at E, = 2.33) for y/8 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

139

y/5 = 0.125 in comparison to y/6 = 0.05 at i = 2.0. On the other hand, the disappearance
of small-scale effects in the results at i = 1.33 (Figure 4.16) is an indication of the short
“life-time” of these structures such that after registering a small-scale-pressure signature

at a = 1.33, they decay and dissipate prior to reaching the hotwire at g = 2.05.

The cross—spectra (DVD , (Duh & (I)vp are presented in Figures 4.21 through 4.26.

Figures 4.21 and 4.22 depict the cross—spectra calculated at E, = 1.33 in logarithmic and
semi-logarithmic scales, respectively. The cross-spectra are dominated by the large-scale
vortices with no evidence of small scale contribution even at y/S = 0.05. This implies the
same observation as the u'-p’ cross-spectra at the same streamwise location, that the “life-
time” of the small-scale structures near the wall is not long enough for the structures to
travel the distance between the locations of the microphone at 5 = 1.33 and the hotwire at
E; = 2.05 (A2; = 0.72, or Ax z 1.66). The spectra in Figure 4.22 also show the attenuation
effect of the wall on the normal velocity fluctuation as a decrease in the spectral peak at

locations near the wall. In addition to the peak attenuation, the cross-spectra exhibit a

 

peak at fo' z 0.7. This value is consistent with the frequency of the peak pressure

w

fluctuations found in Chapter 3. In contrast, the peak of u'-p' cross-spectra is found at

fx,

 

z 0.55 (e.g., see Figure 4.17). This shift to lower frequency of (Du-pv peak is a

manifestation of the higher u' energy at normalized frequencies lower than 0.7 (see (bum.
in Figure 4.9). Additionally, in contrast to the u'-p', the v'-p’ cross-spectra depict the

strongest correlation is at the highest location of y/5 = 0.625.

140

3
vpl/(1/2 p Um)

(D

-6
l 0

was = 0.05

 

y/S = 0.125 7
. y/S = 0.25 q
\ ..
8:, y/5 = 0.375 ‘
‘\\ y/8 = 0.5 l
A“ . g y/5 i0.625 .
~\:\f‘1x_ l
Lumpy,
i
l
o i i l l K A i 1 A 7 I ,_ “A 7 2
10 10 IO
fx /U
r no

Figure 4.21. Cross-spectra between the normal velocity (at E = 2.05) and wall-pressure (at
5: 1.33) for y/8 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

CO

1

VP

(er/Um) c /(1/2 p U3)

x10-

y/S = 0.05
1 7 ‘ y/S = 0.125 l ’
l y/S = 0.25
l‘ y/5 = 0.375 1
7 ' ' i 7 7 ‘ y/8 = 0.5
l y/B = 0.625

er/Uw

Figure 4.22. Semi-log plot of the cross-spectra between the normal velocity (at g = 2.05)
and wall-pressure (at i = 1.33) at for y/5 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

141

The cross-spectra at i = 2.0, shown in Figures 4.23 and 4.24, depict a pronounced
contribution of the small-scale structures to the cross-correlation at heights of y/6 = 0.05

and 0.125, which is consistent with the fact that the corresponding microphone location is

the closet to the X-wire. As a result, the frequency range of (1),, extends to a frequency

p!

as high as f_UX_, z 20 and 10 at y/8 = 0.05 and 0.125, respectively. Finally, the spectra in

03

Figures 4.23 and 4.24 seem to follow the same trend of those at E, = 1.33 in terms of the
spectral peak attenuation near the wall and the increase in the cross-spectrum peak with
increase in y. Moreover, there is a slight shift of the spectrum-peak's frequency from 0.7
to 0.8 at y/5 = 0.05. The peak is also seen to become progressively broader with
decreasing y. This is consistent with the earlier discussion that as one approaches the
wall the largest scales become more inactive, contributing less to v', and hence to v'-p'
correlation, while the smaller scales become more active, producing a "flatter" spectral
distribution where a broader range of scales is involved in generating v', and v'-p'
correlation.

Because the spectra in Figures 4.25 and 4.26 correspond to § = 2.33, which is

farther from the hotwire location than g = 2.0 but not as far as E, = 1.33, they exhibit an

intermediate behavior between that of (1),. at i = 1.33 and g = 2.0. In general the

p.
analysis of the cross-spectra between the wall-pressure and the normal-velocity
component reinforces the conclusion made in Chapter 3 that the p' fluctuations are

dominated by the vortical structures in the outer part of the boundary layer. Specifically,

the semi—logarithmic plots of the cross—spectra reveal that the wall-pressure is mainly

142

 

1)

(1/2 p U

Vp7

(D /

' 1 "1
ms = 0.05 5
y/s = 0.125
7 1 y/S = 0.25 ,
\ w; = 0.3 75 i
\ y/8 = 0.5
, y/8 =0.625 .,

 

,L . # :1,+..1,, ,, , 1 a. 1

fX/U
r (13

Figure 4.23. Cross-spectra between the normal velocity (at g = 2.05) and wall-pressure (at
g = 2.0) for y/8 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

(I)

vp7

3
(er/Uw)¢ /(1/2pU)

9

-5
2‘10 ..

y/8 = 0.05 l
y/B = 0.l25 , _
y/6 = 0.25
y/6 = 0.375
y/8 = 0.5

* y/6 = 0.625 1

er/Um

Figure 4.24. Semi-log plot of the cross-spectra between the normal velocity (at l; = 2.05)
and wall-pressure (at E, = 2.0) for y/8 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

143

w

/(1/2pU3)

va n

(D

was = 0.05
y/8 = 0.125
y/8 = 0.25

“\ y/d = 0.375 I

\ ‘ ya = 0.5

,, y/8 =70.625 ..

 

f Xr/Uao

Figure 4.25. Cross-spectra between the normal velocity (at g = 2.05) and wall-pressure (at
E,= 2.33) for y/8= 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

3,)

Vplo/(1/2 p U

(f Xr/Um) a)

y/B = 0.05

y/S = 0.125 _.
y/S = 0.25

y/S = 0.375 1
y/8 = 0.5

y/8 = 0.625

fX/U
1' CO

Figure 4.26. Semi-log plot of the cross-spectra between the normal velocity (at g = 2.05)
and wall-pressure (at I; = 2.33) at for y/8 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

144

correlated with the normal velocity in the frequency range corresponding to the passage
of the vortical structures in the outer-shear layer even at heights near to the wall at which
the normal-velocity fluctuations generated by those structures are significantly attenuated
due to the bounding-wall effect. Furthermore, the velocity illusively seems to correlate
better with p’ farther upstream, which is a direct reflection of the vortical structure being
more energetic upstream than downstream as they undergo a relaxation process while
traveling.
4.2.2. Velocity/Wall-Pressure Cross-Correlations

In this section, the relationship between the wall-pressure and velocity fields is
investigated further in terms of their space-time cross-correlation coefficient. The spatial

cross-correlation coefficient is calculated at zero time-shift (1) between the measured

velocities at g = 2.05 and the wall-pressure at all streamwise locations using the

 

 

 

 

equations:
R1,, (§,y;§.) = u (§°’y’:p(§’t) (4.1)
R.,,(§,y;§.) = V (§O,y,t:)p (é’t) (4.2)

where 5,0 is a parameter representing the streamwise location of the hotwire, E, is the
streamwise location of the microphone and the overbar refers to the time average. On the

other hand, the corresponding temporal cross-correlation coefficient is calculated using:

 

 

 

 

R.p(y,t;§1)= u'(§o,y,t+;)p'(§pt) (4.3)
Rvp(y,t; a i) : v'(§o’ y’t +;) p’(gi’t) (44)

145

where Q is a parameter indicating the streamwise location of a selected microphone.

The cross-correlation is calculated from the discrete-time data using the Fast
Fourier Transform (FFT). Data are divided into 2048 records, each containing 1024
samples, then the cross-correlation is calculated from the average of the inverse-FFT of
the products between the FFT of each velocity record with the conjugate of the
concurrent pressure record.

Color contour maps of the spatial cross-correlation coefficients between the wall-
pressure fluctuations, and the streamwise- and normal-velocity components

(Rup & Rvp ) are illustrated in the top and bottom plots in Figure 4.27, respectively. The

abscissa is the dimensionless streamwise location of the microphones and the ordinate is
the height of the hotwire normalized by the boundary-layer thickness. In interpreting the
results in Figure 4.27, the reader is reminded that at a given height, the results represent
an average spatial wall-pressure pattern associated with the passage of flow structures
past the location of the hotwire. If the structures registering at the probe were not
responsible for any p' generation, their velocity signature would have no specific

relationship to p', and hence the correlation would be zero everywhere, resulting in a

vanishing pressure pattern. For example, consideration of Rup results at y/5 = 0.4 show

that on average p' is in phase with u' directly beneath the hotwire and that the peak in the
pressure pattern is also found immediately below the velocity probe (since this is where
the highest correlation is found for the probe height considered). In contrast, an out-of-
phase pressure signature is found on average at a location that is approximately 0.5Xr

upstream of the hotwire location. The combination of the p' peak and valley forms

146

Hotwire R
location upo

y/8

 

  

1.4 1.6 1.8 ' 2 1 2.2
a Ki/Z

Figure 4.27. Contour plots of the cross-correlation coefficient at zero time shift between
the streamwise (top plot) and normal (bottom plot) velocity and the wall-pressure at the
location of the ten downstream most microphones (from E, z 1.33 to 2.33)

147

a spatial wall-pressure signature with wavelength 3.0 of approximately 0.96Xr (see Figure
4.27 for illustration).

Noting that the Rup contour lines are vertical for y/8 > 0.15, it is possible to

conclude that the dominant scale (10) of the wall-pressure signature is invariant provided
that the pressure signature is generated by the flow structures in the outer part of the
boundary layer. This scale is approximately the same as the dominant wavelength found
from the wavenumber-frequency spectrum in Chapter 3 (z 0.8-0.9X,). Lee and Sung
(2002) reported the dominant scale of wall-pressure fluctuations associated with the
large-scale vortical structure to be approximately 0.54Xr within the
separation/reattachment zone of a backward-facing step. This was slightly smaller than
that reported by Kiya & Sasaki (1985), 0.6X,. The difference between the scale found in
the present flow and the flow upstream of reattachment could be a result of the increase
in size due to merging of the vortical structures as they travel downstream.
Merging of vortices was examined in a canonical free shear layer by Ho and Huang
(1982), among others, who reported a consequent increase in size and decrease in
frequency of vortical structures.

In addition to the existence of the dominant length scale 2.0 throughout the
boundary layer, there appears to be a second, smaller, distinguishable scale near the wall

(y/S < 0.15). More specifically, two local peaks of negative correlation are found at the

lower edge of the Rup plot on either side of the main positive correlation lobe. The

streamwise spacing of the two peaks gives rise to a length scale (in; see Figure 4.27) that

is about V210. This suggests that the near-wall (y/5 < 0.15) velocity fluctuations are on

average associated with a p' signature of smaller scale than that associated with the outer-

148

shear layer motion. This hints to the existence of a smaller-scale wall-pressure-
generating mechanism that is different in nature than being the direct footprint of the
outer-shear layer structures (which are presumably associated with the XO scale). This
mechanism, however, is expected to somehow still be related to the outer-shear layer
structures since the analysis of the velocity-pressure cross spectra clearly revealed the
negligible contribution of the sub-boundary layer structures to the velocity-pressure
correlations even at the lowest y position (this was especially true when considering the u
component of the flow velocity). It is noted here that this is the first occasion in this
work that evidence of an important near-wall pressure source is identified. Subsequent
analysis in this chapter will clarify the nature of this source and its relation to the large-

scale vortices.

The Rv 0 results shown in the bottom plot of Figure 4.27 are consistent with the

p
Rupo results in revealing two characteristic streamwise length scales. These scales are

reflected in the spacing between the negative and positive correlation lobes that dominate
the correlation map. The spacing (71.0) is large and fairly uniform throughout most of the

y range, but it narrows down near the wall, as demonstrated by the labels in the figure.

The value of he obtained from the Rvpo results is somewhat shorter (~ 0.8X,) than that
estimated from the Rupo data. On the other hand, the Rvpo results seem to be “90
degrees out of phase” with the Rupo results. In particular, when Rup'o shows a strong
positive correlation, RVp0 is zero. Similarly, when RVpo has a strong positive or negative

correlation, Rupo is zero. Additionally, unlike Rupo , R magnitude attenuates

VPo

149

 

quickly with decreasing height near the wall. This is consistent with the wall attenuation
of v', referred to earlier in 4.1.2 and 4.2.1. Overall, the cross-correlation maps shown in
Figure 4.27 show good qualitative match with those presented by Lee and Sung (2002)
for the flow within the separation/reattachment zone of a backward-facing step.

The temporal cross-correlation between the streamwise and normal velocity
components and wall-pressure fluctuations at three microphone locations similar to those
selected for the cross-spectrum results earlier (2‘, = 1.33, 2.0 and 2.33) are shown in
Figures 4.28 and 4.29, respectively. Both figures depict color contour maps of the cross-
correlation results. Note that positive time difference (1) means that the occurrence of the
velocity disturbance at the hotwire lags that of the corresponding pressure signature. In

other words, positive 1 refers to the past of the pressure data, and vice versa. The Rup

plots in Figure 4.28 reflect the signature of a quasi-periodic disturbance with a
dimensionless time scale (To) of 1.8. This value is consistent with the fiequency of the
peak in the semi-log (DUI, plots shown in Figures 4.16, 4.18 and 4.20. That is, the
frequency of the cross-spectrum peak was found to be approximately 0.55 which equals

1/1.8. On the other hand, the Rvp results (Figure 4.29) depict a somewhat smaller “Co of

1.4. This value is also consistent with the somewhat higher-frequency ( fUX' z 0.7) peak

w

 

of the semi-log (Dvp results given in Figures 4.22, 4.24 and 4.26. Moreover, the results

are in agreement with the larger k0 value found from the spatial Rupo in comparison to

that found from Rvpo . It is also interesting to note that, consistent with the spatial

150

 

rU/X
<1) I"

Figure 4.28. Maps of the cross-correlation coefficient between the streamwise velocity at
i = 2.05 and wall-pressure at i = 1.33 (top plot), i = 2.0 (middle plot) and g = 2.33
(bottom plot)

151

 

r Ugo/Xr

Figure 4.29. Maps of the cross-correlation coefficient between the normal velocity at i =
2.05 and wall-pressure at Q = 1.33 (top plot), 2:; = 2.0 (middle plot) and E, = 2.33 (bottom
plot)

152

correlation observation, the temporal correlation data of RVp (Figure 4.29) do reveal a

narrowing down of the dominant time scale as the wall is approached for the two

locations that are closest to the hotwire. This narrowing of scale near the wall is not

evident in the RVp results for the microphone farthest from the hotwire (Q = 1.33). A

surprising result is the absence of a smaller time-scale signature in the temporal Rup data

near the wall for all streamwise locations. The reason for this may be clarified by
realizing that the physical interpretation of the temporal correlation results in Figures
2.28 and 2.29 is fundamentally different from that of the spatial correlation data in Figure
2.27. More specifically, unlike the latter which infer an average spatial p' pattern
associated with the velocity observations at the hotwire location, the former reveal a
temporal u' pattern associated with the pressure observations at one of three streamwise
locations. As will be clarified in 4.3, the small-scale variation in the spatial wall-pressure
pattern is in fact related to small-scale variation in v', but not u', and associated variation
in near-wall, pressure sources. Thus, since Figures 4.28 and 4.29 should be interpreted in
terms of the velocity rather than the pressure field, the appearance of the smaller near-
wall length scale in Rvp, but not Rup, results is not in contradiction with the results in
Figure 4.27.

Another important outcome of the temporal correlation analysis is the clear
convective nature of the flow structures responsible for the velocity/wall-pressure
correlation. This is evident in both Figures 4.28 and 4.29, where the correlation contour
map retains the same shape while shifting towards negative time delay with increasing

streamwise location of p' observation. Thus, one may estimate the convection velocity

153

 

of these structures by tracking the time delay ( I;

w

 

) at which a particular feature of the

f

correlation map appears in the temporal Rup results for two microphones spaced by

distance Ag. In this manner, the convection velocity would be estimated as Ag divided by
the change in the time delay. To choose a suitable feature in the correlation results, it is

noted that correlation peaks tend to be broad and difficult to pin-point with good

accuracy. Therefore, the zero-crossing of the Rvp results was selected as the feature to

track since the gradient of the correlation results (with respect to r) is substantially

sharper than that associated with the Rup correlation. To demonstrate, the Rvp

correlation results averaged over the heights y/5 = 0.375 to 0.625 are shown in Figure
4.30 for three microphone locations of g = 1.67, 1.89, and 2.11. The averaging in y is

employed to reduce any data scatter that may influence the ability to identify the zero

crossing. Moreover, this average is conducted over a y range that is centered around the

center of the outer shear layer (i.e., where —W is max., or y/5 z 0.48). Thus, the
resulting convection velocity should be that of the large-scale vortices.

To obtain an average convection velocity over the streamwise extent of the ten
most downstream microphones, the 2‘, location of each microphone is plotted versus the
normalized time delay of the zero-crossing of the corresponding RVP in Figure 4.31. The
slope of a straight-line fit through the data yields a convection velocity value of 0.82Uw.
This agrees quite well with the convection velocity values extracted in Chapter 3 using
various spatiotemporal statistical measures of the wall-pressure field, providing
additional evidence that the convection velocity of the wall-pressure signature is in fact a

direct reflection of the convection velocity of the outer-shear layer structures.

154

c’°"'."o

 

 

-1 -05 o 0.5 "1
r U /X
00 I

Figure 4.30. The average cross-correlation between the normal-velocity and wall-
pressure in the range of y/6 = 0.375 to 0.625, for i = 1.67, 1.89 and 2.11

 

 

2.6'“ T” T 7’ T477 WTTTT’ ’TTTT’T-' ”'f::f;u__l.TT'T'.’ VTi::i, _-a
0 Data
. ....___ Line fit: Uc/Uoo = 0.82
2.4 7“ — 1' 9:99-97?“- -~— - - ~~-—
N...
2.2- - ~' \ 7 iiiii
x f ,5 1 ..
1‘\.¥ . 1 Hotwrre
21. 1 \.>\.u .- ..----location .......
up
1.8' ’ _
I \‘\“\\l
l.6-~ ~ 1 ,7 ~ \\
. \{\ 1
\ ‘
1 1 1 k\\
1_4' '— “"*""' 2 I 'T‘\\"\;\‘
1
12 _ 7 ; . I ,,_' ; ., .
-0.4 -0.2 0 0.2 0.4 0.6 0.8

r Uw/Xr

Figure 4.31. Plot for extraction of the convection velocity from Rvp results

155

4.2.3. Conditionally—Averaged-Data Analysis

To gain an understanding of the type of flow structures that produce the
correlation results presented in the previous section, the conditionally averaged velocities
(< u’ > & < v' >) and wall-pressure fluctuations (< p' > are examined here); where < >
denotes the conditional, or ensemble, average of the quantity. Specifically, the interest
here is in obtaining the conditionally averaged velocity vector measured by the X-hotwire
when strong positive and negative pressure events occur at the microphone closest to the
hotwire (i = 2.0). Because there is a small offset in the streamwise direction between the
positions of the two sensors (Ag = 0.05), the pressure time series was delayed by an
amount equal to 0.05/Uc (Uc = 0.81Um) to approximate the wall-pressure information

directly below the X-wire; i.e., at 1’; location 0.05 farther downstream. Note that

 

hereafier, all pressure and velocity time series will be low-pass filtered below fUX' z 2.2

w

to focus only on the structures in the frequency range that is most significant to the
pressure fluctuations. Filtering was implemented in post processing using the "filtfilt"
function of MatLab version 6.5. This routine filters the time series in the time-forward
and -reverse directions to ensure that the filter response does not result in any phase lag
of the filtered data. Additionally, both velocity and pressure time series were filtered.

To demonstrate the influence that the filtering process has on the p'-related flow
features, the spatial and temporal cross-correlation results shown in Figures 4.27 through
4.29 were obtained again after low-pass filtering the data and the outcome is shown in
Figures 4.32 through 4.34. Comparison between the results before and after filtering

shows that with the exception of some smoothing of the contour plots and higher

156

correlation values, the overall correlation shape does not change as anticipated. Thus, the
filtering will only aid in emphasizing the flow features that are most relevant to the wall-
pressure generation process while removing the influences of the sub-boundary layer
flow structures, which are insignificant for p' generation in the current flow field as
demonstrated thus far.

The top and the bottom plots in Figure 4.35 illustrate the conditionally averaged

mean-removed velocity fields associated with positive and negative wall-pressure peaks,

w

. . . . . rU
respectrvely. The absc1ssa represents the dimensronless time (——X—) whose zero-

1'

reference is the instant of the pressure peak occurrence. A plot of the conditionally
averaged pressure < p' > normalized by pm is provided directly below each of the vector
fields for reference. The positive and negative pressure peaks where identified using the
following process: whenever the p' time series value exceeded an arbitrarily selected
threshold of 1.5p'm, above or below zero, the time of occurrence of the closest local
positive or negative peak, respectively, that exceeded the threshold was found. This
corresponded to a time offset of zero, and the average of all velocity vectors occurring at

. . . r U .
all such instants for the entire velocrty data record represents the result at X °° = 0 1n

f

 

Figure 4.35. Other positive, or negative, values of I represent the average velocity at the
corresponding time delay, or advance, from the occurrence of the peak.

The top plot in Figure 4.35 depicts that the positive wall—pressure peak coincides
with "inrush" of high-speed fluid towards the wall. The plot also illustrates that <u'> is

positive in the vicinity of the positive wall-pressure peak but starts to switch to negative

157

Hotwire
location

 

Hotwire R
location V130

0.6
0.5

0.41

y/5

0.3

0.2

0.1

   

Figure 4.32. Contour plots of the cross-correlation coefficient at zero-time shift between
the low-pass filtered streamwise (top plot) and normal (bottom plot) velocity and the
wall-pressure at the ten downstream most microphones (from g = 1.33 to 2.33)

158

 

—0.5 0
r U /X
00 I'

Figure 4.33. Maps of the cross-correlation coefficient between the low-pass filtered
streamwise velocity at E, = 2.05 and wall-pressure at 9; = 1.33 (top plot), a = 2.0 (middle
plot) and g = 2.33 (bottom plot)

159

 

GP-«u—w . -

 

   

0.5 7

   

0
rU/X
00 r

Figure 4.34. Maps of the cross-correlation coefficient between the low-pass filtered
normal velocity at g = 2.05 and wall-pressure at i = 1.33 (top plot), g = 2.0 (middle plot)
and i = 2.33 (bottom plot)

160

 

 

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mg

m} .Q\A.mv m}

76.3” 0.47 0.5

0.2 7
161

W 40.2
-pressure, <p’>, and conditionally-averaged mean-removed velocity-

2.05

vector field associated with positive (top plot) and negative (bottom plot) wall-pressure

Figure 4.35. Wall

peaks at g

values at I)?” z 0.3 and -0.5. In the flow over blunt flat plate, Kiya et al. (1982) and

f

 

Kiya & Sasaki (1985) reported positive u' fluctuation near the wall between successive
vortices. They attributed the instantaneous excess in streamwise momentum near the
wall to the penetration of the irrotational freestream fluid close to the wall in between the
vortical structures. They also associated this region of the flow with the formation of
saddle point, when viewed in a frame of reference that is translating with the convection
velocity of the vortices.

The bottom plot in Figure 4.35 shows the conditional-velocity-vector field
associated with negative wall-pressure peaks. As seen in the plot, the negative wall-
pressure fluctuations seem to coincide with negative u' fluctuation. This was also
observed by Kiya et al. (1982) and Kiya & Sasaki (1985) who also indicated that the
negative pressure coincided with the passage of a spanwise-vortex center, which is a
significant source of low pressure.

The above discussion leads to the following simple scenario for the successive
generation of positive and negative wall-pressure peaks. As the large-scale vortical
structures of the outer-shear layer travel downstream, they entrain irrotational freestream
fluid in between each pair of successive vortices. Accordingly, the accompanying
interface between the irrotational flow in the freestream and the vortical structures moves
up and down in the direction normal to the wall in synchronization with the passage of
the vortices above a certain streamwise location. Specifically, when a vortex passes over
a point of observation, it thickens the boundary layer leading to a streamwise momentum
deficit across the boundary layer and associated negative wall pressure, as found above.

On the other hand, the entrained irrotational flow between the vortices reduces the

162

boundary layer thickness and provides streamwise momentum excess. This coincides
with the saddle points up and downstream of each vortical structure and is associated
with the generation of positive wall pressure.

The x-momentum deficit and excess accompanying the passage of the vortex
center and saddle point, respectively, are examined in Figure 4.36. The figure depicts a
comparison between the mean streamwise-velocity profile and the conditionally averaged
streamwise-velocity profiles at the instant of occurrence of the positive and negative p'
peaks. The plot shows that the conditionally averaged streamwise-velocity profile
associated with the positive wall-pressure peak has higher velocity at all heights included
in the present measurements, which confirms the fact that positive pressure coincides
with x-momentum excess across the boundary layer. On the other hand, the conditionally
averaged profile synchronized with the negative wall-pressure peaks drops below the
mean-velocity profile at all heights. The above results clarify some of the features of the

spatial correlation results in Figure 4.32. In particular, the top plot in the figure shows a
vertical positive Rup correlation lobe that extends across the entire y range at a g location
of 2.05. Based on the analysis of the conditional streamwise-velocity profiles it is now

clear that positive p' is associated with positive u' across the whole y range and vice

versa, thus resulting in the vertical positive correlation lobe seen in Figure 4.32. The
corresponding feature in the Rvp correlation (bottom of Figure 4.32) is a strong
streamwise gradient of the correlation value that is associated with a switch in the
correlation sign from positive to negative. This would be consistent with the scenario

identified above that associates the pressure generation with vortex centers and saddle

points. Both of these features are characterized with a switch in the direction of the v'

163

component from the upstream side of the source to the downstream one. In the

following, the association of vortex centers and saddle points with wall-pressure peaks

will be established more convincingly.

0.7' 7 ., a, 7:7 7 7I7 7 *2i ’7' "':7.0
A Positivepeak ‘ ‘ 7':
V NegativePeak; v ‘
0.6179 rMeanprofilei ; .~ ~ . v'.? 7 7 16.0
1 V .A
1 V CA
0.51 ~ ' 1 , 7 7.7-..‘77 ' ~15.0
.V .A .
V .A
‘ V .A 1
04.7 :- 0.; 114.0
Q A 5
>1 '7 ..A >\
0.31 *V O A , w , 13.0
V O A
V O A
V O A ‘
1 ‘ 'V.O AA
0.1" 7 , ”V ‘7‘. 111.0
x r ‘ 1 1 1
V
(61.7 7 7 7 .1 7 7 . 1K. #7. -77 ..‘0
.5 0.6 0.7 0.8 0.9 1

u/U0°

Figure 4.36. Comparison between the mean streamwise-velocity profile and the
conditionally-averaged streamwise-velocity profiles associated with positive and negative

wall-pressure peaks
Although Figure 4.35 illustrates the main features of the conditional velocity field
and associated wall-pressure signature, it does not show the real physical picture of the
flow structures. To obtain such a picture, the velocity vector field has to be viewed from
a coordinate frame that moves with the convection velocity of the flow structures (Fiedler
1988). Fielder (1988) investigated this issue extensively and referred to the vector-field
representation in Figrre 4.35 as triple-decomposition, which is obtained by looking at the

mean-removed velocity vector field in laboratory coordinates. However, he argued that

164

in order to obtain the velocity vector field that represents the real physical picture of the
flow structures, one has to do a double-decomposition of the flow by viewing the total
velocity vector field (mean + fluctuation) relative to an observer moving with the
convection velocity of the flow structures. The top and the bottom plots of Figure 4.37
show the same results as in Figure 4.35 after adding the mean velocity and viewing the
result in a frame of reference that is moving with a convection speed of 0.81Uoo. The top
plot of the figure shows clear evidence of a saddle point that is synchronized with the
positive wall-pressure peak directly below it, while the bottom plot provides a clear
physical picture of a vortical structure with a center located directly above the negative
wall-pressure peak. This provides a strong support for the central role played by the
vortex structures and saddle points in the scenario of wall-pressure generation outlined
earlier. Note that strictly speaking, because the abscissa in Figure 4.37 represents time
rather than space, the results are indicative of the passage of the vortex structures and
saddle points past the point of observation of the pressure, rather than the actual spatial
structure of these features. However, as discussed in Chapter 3, it is believed that Taylor
hypothesis of frozen turbulence holds fairly well for the large-scale vortices, and hence
the results in Figure 4.37 are likely to represent well the actual spatial structure. A more
quantitative assessment of this statement is provided in 4.3.
4.3. Stochastic Estimation

The above analysis has led to the identification of the quasi-periodic passage of
the outer-shear layer vortices and saddle points resulting from their mutual interaction as

the primary source of wall-pressure generation. Generally, the identification is based on

165

 

 

                             

'— 6.75

 

77
77.7.7. 7 m
7 7- 7 .
77,1 . 7 _
7,, . 77
7,7,7”, §
77/47” \3

7,

.=‘ €/
:4

I
7‘

.17"—

{/1
"'

     

_" I

73

Ebb?

735 7.

0.4

0.2

7 0.1

 

 

L011 7 -03 7—012

-0.5

_27 7.

0.1

tU/X

 

0.3 0.4 0.5

0.2

 

—o.2’ '

—o'.3

—o.4

71.5

and conditionally-averaged velocity-vector field

Figure 4.37. Wall-pressure, <p'>,

-pressure peaks, viewed

associated with positive (top plot) and negative (bottom plot) wall

in a frame of reference moving with 0.81Um

166

time-averaged statistical information, which does not reveal an in-depth picture of the
variation in the characteristics of the individual flow structures or the nature of the
mechanism(s) leading to the wall-pressure generation. Hence, in this section an analysis
is conducted using the stochastic estimation of the flow field based on the instantaneous
p' signature across the ten most downstream microphones. As explained below, the
procedure yields an estimate of the "quasi-instantaneous" velocity over the full y range of
the measurements. Subsequent evolution in time of this estimate yields a picture of the
quasi-instantaneous flow structures traveling past the streamwise location of the hotwire,
which can be examined to reveal some of the variability in the individual flow structures
and associated wall-pressure-producing mechanisms.
4.3.1. Principle and Equations

Application of the stochastic estimation technique for identification of
organized, or coherent, motion in turbulent flows was first proposed by Adrian (1975)
and then fiirther refined by him and his coworkers (e.g. Tung and Adrian 1980). Since
then, applications of stochastic estimation have expanded to range from the identification
of the conditional eddies of isotropic turbulence, Adrian (1979) and Adrian & Moin
(1988), to the extraction of the spatio-temporal flow field characteristics of turbulent
boundary layer structures, Guezennec (1989) and Choi & Guezennec (1990).
Classically, the stochastic estimation technique estimates any variable at one location in
the flow from a known variable at some other location. However, and as pointed out in
the objectives section, stochastic estimation is used in the present work to estimate the

velocity field above the wall from its wall-pressure signature. In particular, the technique

seeks to get the estimated velocity ufi'sfij, + 'r‘,t + r) from a known wall-pressure p'(‘r‘o ,t);

167

where f is the position vector of the pressure-observation point, the components of

0
which are x0, 0 and 20, in the streamwise, wall-normal and Spanwise direction,
respectively, f (x-xo, y, z-zo) and r are the spatial-offset vector and time—shift between the
estimated velocity and wall-pressure event, respectively, subscript i is the tensor index
notation specifying the velocity component and subscript s is used to denote the

stochastic estimate. Note that “is“; +"r’,t+r) and p'(fo,t) are the mean-removed

values. The former is obtained from a Taylor-series expansion in terms of the latter as

follows:
ug,$(fo + f, t + r) = Ai('r‘o + it) p'(?0,t) + Bi('r‘o + it) p'2('fo,t) + - -- (4.5)
where, A, (f0 + f, r) and BIGo + 'f, r) are the estimation coefficients for the linear and

quadratic terms in the expansion, respectively. Note that both coefficients depend on the

location of observation of the event (to) but not its time of occurrence (t). This is

because the flow field considered is generally inhomogeneous and hence statistical

properties of the flow in the vicinity of a particular event location (1;) will differ from

those at another location. On the other hand, the flow is stationary in time, and therefore
its statistical properties are independent of the time of occurrence of the event.

The primary goal of the stochastic estimation procedure is to obtain the
coefficients A, and B,, so that the velocity field can be estimated from equation 4.5. If
only the linear term in the equation is included in the estimation, it is known as Linear
Stochastic Estimation (LSE). However, if the first two terms are included, the estimation
is a Quadratic Stochastic Estimation (QSE). Note that both LSE and QSE can be

implemented using wall-pressure conditions from multiple points in space

168

 

simultaneously, yielding a "multi-point" estimation. The multi-point estimate is
generally more effective in capturing the "true" nature of the flow structures, and it is the
one of primary focus here given the availability of multi-point information from the
pressure-array data. The multi-point estimate will only be of the LSE type because of the
complexity involved in obtaining a multi-point QSE. However, the results will be
compared to those from single-point LSE and QSE. Derivation of the equations used to
obtain the estimation coefficients for all of these cases is given below.

I. Single-Point LSE

As mentioned earlier, the single-point LSE (denoted by SL hereafter), provides a

linear estimation of the conditional flow filed, u;(?0 +'f,t + r) from the wall-pressure,

p'(fo,t). For brevity, the following analysis will only consider the x~component of the

velocity, but the resulting equations are the same for the y-component. The estimated u'

is calculated from the following equation:

u'.,(fo + it + T) : Au,lin(-fo + it) p'(ifi) (4-6)
where, Au, 1m is the coefficient of the linear estimate of u'. This is determined such that
the long-time mean squared error between the velocity and its estimate, e("fO +f) , is

minimized. This is the maximum likehood estimate, assuming a Gaussian distribution of
the error about the estimate. Explicitly, at a specific location within the flow domain the

mean squared error is:

 

e(f0 +r) =(u',(ro +f,t+r)—u'('fo +f,t+r))2

(4.7)
: (Au. lin p'(ro ) tC) _ U'Cfo 'l' f, t ‘l" 1))2

 

where, the overbar denotes time averaging. The value of Aqun that minimizes the error

may be found then by requiring:

169

 

 

 

de d ' ,_ ,1 _
dA =dA KAN.p(r.,t)-U(r.+r,t+t))’l=0 (4.8)

u. lin u. lin

Exchanging the order of the derivative and time integration operations:

 

d
dA

 

[(AuJin FIG), 0 ‘ “'60 + it + T))2l

u. lin

.—. 205...“. p'2(fo, t) — u'(f0 + i‘,t + r) x p'(Fo,t)) (4.9)
= (Au.lin p'2 (to, t) — u'('r; + 'r’, t + t)p'(i';,, t))= 0

 

 

Equation 4.9 is satisfied if

 

 

_ u'(fo + 131 + r) p'(fo,t) _ u'(ro + ‘r‘,t + r) p'(ro,t)

Au.lin — ,2 _ :2 -
p (r..t) pm..(r.,t)

 

 

 

(4.10)

Inspection of Equation 4.10 shows that Au,“ equals the ratio between the cross-
correlation of u' and p', at a time delay corresponding to the delay between the estimate
and the event, and the square of the root-mean-square, or second PDF moment, of the
wall-pressure. Equations 4.6 and 4.10 yield the final form of the equations employed for

SL estimation:

 

 

 

 

u;=["(r°+r.’.t+..t)p(r°’°]p' & v;=Iv(r.+r',2t+_r)p(r..t)Ip. (4.11)
pm(1;,,t) prms(ro’t)
II. Single-Point QSE

Similar to the single-point LSE, SL, the single-point QSE will be denoted by SQ.

Based on equation 4.5, the main equation for obtaining the SQ estimate is given by:
u'S(fo + 'r',t + r) = Au,quad ('r‘o + 'r‘, r) p'(‘r’o, t) + Bu(‘r‘o + 'r‘, r) p'2 (1;, t) (4.12)

where, Amquad and Bu are the unknown coefficients of the first- and second-order terms
for the SQ method. Similar to SL, those coefficients are determined by a least square

error minimization process:

170

 

e (to + r) = (AWad p'(ro, t) + Bup'2(ro,t) — u'(i"o + r, t + o)’ (4.13)

By taking the derivative of Equation 4.13 with respect to Amquad and Bu individually and

equating the result of each to zero, the error is minimized in a least-square sense:

 

 

 

 

dACrlr:uad = dquuad [(Au quad p’(izo ’ t) + Bup’2 (f0, 0 — u’(-f0 + f’t + 1))2]
= d [(All quad p'(‘g, t) + Bup'2 (to, t) — u'(f0 + f, t + 10):] (4.14)
dAu.quad ‘

 

= ZlA...q..d 13' +3.1)” -U')>< p' = (AM... 137’— +3.? -U'—p’)=0

 

 

548—6: : 21%[(Au‘quad p'(fo9t) + Bup'2(i:o’t) - u'(i:o +f,t+t))z:l
d y - l "’ ' ‘ -.
236-— (AILQuadp(roat)+Bup2(ro’t)—u(r0+r’t+1))2] (4'15)

 

= 2(1chl p'(fo,t) + Bup'2(r;,,t) — u'('fo + 'f,t+ 1))x p'2(ro,t)
= (Au.quad p'3('r;,,t) + Bup"’(fo,t) — u'(fo + ?,t+'c) p'2(‘r‘°,t))= 0

 

 

 

Solving Equations 4.14 and 4.15 for Au, quad and B, gives:

 

 

 

B _ p'2(?o, t) u'(f0 + it + 1:) p'2 (f0, t) — p'3('r'o, t) u'('r; + it + r)p'(fo, t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

u (4.16)
(Memo)? —p'“<r..t) p'2<'r;,.t)
u'(fo + r, t + I) p'(fo,t) — Bup'3(fo,t)
Au. quad : '2 _ (4.17)
p (ro’t)
Similarly, the coefficients for the v-component estimation are given by:
r2 - r - —- r2 - _ r3 - r - - r -
B. = p (r.,.t)V(ro +r,t+r)p (ran) p (r..t)V(r., +r,t+t)p(r.,t) (418)
(final —p"(r..t) pea-0,1)
I - - p - _ ,3 _.
Amquad = V(ro+r9t+T)p(ro’t) va (lb’t) (419)

p"(?.,.t)

171

III. Multi-Point LSE

Multi-point LSE, or ML, is essentially based on a weighted linear combination of
wall-pressure events at multiple points in space. In the present study ML uses ten points
on the wall, corresponding to the ten downstream most microphones. In this case, the
relation between the estimated velocity and wall-pressure can be stated as:

u's = Aul p: + A“2 p'2 + Am, p; +---+ Aulo p:o (4.20)
where, Au], Au2,...Au10 are the unknown coefficients of the estimation. Note that
microphone #1 is located at g = 1.33 and microphone #10 is the most downstream one (Q
= 2.33). Similar to the procedure of SL and SQ, those coefficients are determined by

minimizing the mean squared error, e, between the estimate and the actual velocity:

 

e:-(1Aul pl+Au2 p; +Au3 p;+”.+AuIO 13:0.”1'1')2 (4.21)
To obtain Au], the derivative of Equation 4.21 with respect to Au] is set to zero, leading

to:

 

de
dA =dAul

ul

d[(Au1p1+Au2 p2+A113 133+. +Au10p10—u u)2]

 

 

=d—-A [(A..p.+A p’.+A 132+ +A....pI.-U')l (4'22)

=2(A.. p1 +A.. p2 +A.. p2 +---+A...p1.. -U')><p’=0

 

Rearranging equation 4.22 and applying the same procedure to the rest of the

microphones:

(A..p’p'. +A .zipp'fiA r'p'=.+-~+A...pip10) 317i
(A..p_'.p'.+A p_'p'.+A p'p".+-~+A...p2pi..)=u_p. (4.23)

(A... $17. + A... piop'. + A... piop'. + - - - + A... plopiol= U'pI.

172

The equation set 4.23 can be reorganized using matrix notation as follows:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P—I—r I I fi —I—T- — 1 F I I 1
mm mm mm 13213.2 A“. up.
p'2pi p21); p’2p2 p'2p12 A“, u’p'2
T7 .7—I “I—, I—2 I—i (4-24)
1321). 132192 p.132 p.132. A..2 = up.
p121)? p121); p121); 19121312 Aw v'pi2
and then the coefficients can be calculated from:
A“, pip: pip; pip; pipio U'pI
A“, p213: p'2p’2 p'2p2 p'2p12 U’p’2
‘77 TT T? 2—7 T7 (4-25)
A2 = p21). mm 13.1). p2p22 up2
_Am p121)? piop; p121); p12p12 _U'p12J
Similarly the coefficients for ML estimation of v' are given by:
A” pip? pip; pip; pip?o ' V'pi
Av, p21)? p’2p; p213; p'2p12 V’pi
I 2 ‘fi fi —,7 (426)

A2 = 5:15: 32172 132192 p2p.2 m

A p121)? pfop; 1312132 p121); m

_ VIO_ _ _l — d

 

 

 

 

 

 

The inverse matrix in equations 4.25 and 4.26 (first matrix on the right hand side)
is calculated from the wall-pressure measurements alone and it contains information on
the two-point correlation among the data from all microphones. The second matrix on
the right hand side of the equations represents the cross-correlation between the velocity
component to be estimated and the wall-pressure at all microphones. The values of these

cross-correlations at different heights are represented in the cross-correlation maps shown

173

 

in Figure 4.33. Note, though, that the values in the figure are normalized by the
corresponding root-mean-square values of the velocity and pressure; i.e., they represent a
correlation coeflicient.

It is interesting to note that the estimation of the conditional velocity vector field,
using any of the three stochastic estimation methods, is obtained from unconditional
statistics, which are the p'-p’, u'-p' and v'-p' two-point correlations. By comparing the
conventional conditional average with stochastic estimation, it is found that the bulk
processing in stochastic estimation, involving the calculation of the stochastic estimation
coefficients, is conducted only once no matter what the number of conditions to be
investigated is. This makes the method substantially more efficient than the conventional
conditional average when dealing with large two- or three-dimensional databases
(Guezennec 1989); though this gain in efficiency may be at the expense of some
reduction in accuracy if the estimation does not converge to the conditional average using
a linear or a quadratic estimation.

4.3.2. Comparison Between ML, SL and SQ

In order to assess their capability of estimating the velocity field, the results of
three different estimation methods ML, SL and SQ are compared in this section. The
assessment is based on correlation of the stochastically-estimated velocity vector field
with that of the measured velocity. In addition, the auto-spectra of the measured
streamwise- and normal-velocity are compared with those of their stochastically-
estimated counterpart. Moreover, the effect of the microphone location on the estimation
is examined for the cases involving single-point estimation. In this examination, the

upstream (at E, = 2.0) and downstream (at g = 2.11) microphones closest to the hotwire

174

location are used, and will be denoted hereafier by subscripts 1 and 2, respectively.
Specifically, SL1 and SL2 refer to the single-point LSE based on the wall-pressure data at
E, = 2.0 and E, = 2.11, respectively, while SQ2 denotes the single-point QSE based on the
data from the microphone located at 3; = 2.11.

To compare the different estimation methods, the correlation coefficient between
the measured streamwise- (Cu) and normal-velocity (CV) time series and the
corresponding stochastically estimated time series are evaluated at all heights for the four
different estimation methods (ML, SL1, SL2 and SQ2). The results are shown in the top
and bottom plots in Figure 4.38, respectively. In general, the distributions depict the
streamwise- and normal-velocity estimated using ML to be better correlated with the
measured values than those obtained using single—point estimation. On the other hand,
comparison among the single-point methods reveals that whereas SL1 provides the best
estimation results of u' (i.e., highest Cu value), the corresponding v' estimation is poor
across the boundary layer. This is caused by the low v'-p' correlation for the pressure
data obtained at 5 = 2.0, as seen from Figure 4.32. By utilizing the microphone at E, =
2.11 in the estimation (SL2 results), where the v'-p' correlation is higher, a significant
improvement in the single-point estimation of v' is found, as reflected in the increase in
Cv (see Figure 4.38, bottom). The enhancement in v' estimation is accompanied by a
small deterioration in u' estimation, because of a decrease in the u'-p' correlation
associated with the microphone at Q = 2.11. These results highlight one of the main
disadvantages of single-point estimation relative to that based on multiple points. More

specifically, single-point estimations exhibit sensitivity to the location of observation of

175

0.5 :—

O.4 *

0.3-

0.2-

0.1,

0.5«

0.4.. ~

0.3-~~

0.2

0.1—

0 ML
9 ‘ v SL1
' SL
00 . 1 o. ,
00‘...0 v'. 9.. 'SQZ
V VV 7' i

0.1 0.2 03 0.4 0.5-“0.6 0.7 0.8
VS

.... b.00.00099..;... ..
o
. ifillifilfi“f" . ‘.
o f ___________
I
i 22222
. v
. ,2 VV VVVU_'VVVV
VYV VV'v V ‘
v 7'
'7'

0.1 0.2” i 03 "0.4 0T5 0.6 0.7 0.8

Figure 4.38. Correlation coefficient between the measured and stochastically-estimated
streamwise- (top plot) and normal- (bottom plot) velocity using ML, SL1, SL2 and SQ2

176

the event, and hence in order to obtain a good estimation one must select a location where
a high u'-p' and v'-p' correlations are maintained simultaneously. In contrast, in ML, the
estimations from all points of observations are superposed such that p' information at
locations with high velocity-pressure correlations automatically compensate for
information at points where the correlation is low.
Another point of interest in relation to the data shown in Figure 4.38 is the comparison
between the SL2 and SQ2 results. As evident, both methods provide an equally good
estimation, yielding similar Cu and CV values across the flow. This suggests
that the inclusion of the quadratic term does not provide substantial improvement in the
estimation for the current flow. Therefore, it was not worth a while to pursue the
substantially more complicated multi-point QSE in this study.

To examine the spectral agreement between the measured and stochastically-

estimated streamwise- (u’,) and normal-velocity (v',), a comparison between their spectra

is presented in Figure 4.39 and 4.40, respectively, for y/8 = 0.05, 0.125, 0.25, 0.375, 0.5

 

 

 

and 0.625. The abscissa and ordinate represent the dimensionless frequency ( fUX' ) and
the dimensionless velocity spectrum ( U3“ and UV; ), respectively. In general, the

spectra in Figures 4.39 and 4.40 show that the spectra of the estimated velocities
qualitatively posses the same shape as the actual spectra but with an attenuation in the

energy content. The least attenuation is found in the ML case (particularly within the

f X . .
frequency range corresponding to the peak pressure fluctuations, —U—'- z 0.7), glvmg

w

177

additional evidence for the ability of the multi-point estimation to provide a better
estimation of the velocity field.

The above analysis, although highlighting the advantage of ML versus SL
estimation, does not shed light on the accuracy of the estimation in calculating the
conditional average of the velocity. In order to assess the accuracy of the estimates, one
must compare the estimated values to those obtained from conditional averaging. For the
ML estimation, it is practically impossible to achieve such a comparison. To elaborate,
the reader is reminded that the ML estimation at a given point in time corresponds to the
average velocity associated with the specific pressure pattern "seen" by the ten
downstream most microphones. In order to obtain the conditional average corresponding
to this pattern, one must search the wall-pressure data records for a large number of
“exactly" similar pressure patterns with the same pressure values seen by each of the ten
microphones, which would be quite difficult to achieve. Therefore, the estimation
accuracy will only be assessed here for single-point estimation, with the understanding
that the accuracy of the ML estimation is expected to be at least the same, if not better.

In order to examine the accuracy of the single-point estimates, the conditionally-
averaged (<u'> & <v'>) and the estimated streamwise (us) and normal (vs) velocities
using SL2 and SQ2 are calculated for different values of the pressure condition (p'). In
particular, <u'> and <v'> are obtained by averaging u' and v', respectively, at times when
the wall-pressure value falls within a window of 05me centered around the p' condition
value. By repeating this process for different values of p' that cover the whole range of

wall-pressure fluctuations, the dependence of <u'> and <v'> on p' can be examined.

178

-2
10, - ,
, y/8 =0.05
-4 \
10 1
-2 ”511222.?
10 ML
1 SL'
SL2
SQ2
10'8'_l 2. 40
10 10
-2
10 1 — — .
y/8=0.25
-4
10 1 ,,,,,,
-6,
10 .2
-81
10 3] 2 z
10 10
-2
10 ; . I
‘ w5=o5

uu (I)

(D

10' ;

1
10’84 2 w . ;
10" 100

fX/U
r 00

 

 

L 1'1 1.-.‘

-2
10 1 ~ W . 1
‘ -  : y/8 =0.125

 

'4' 2 1
10 1 .\ _ _ _
1 ‘ // ”‘1‘.
10‘9” \.__\ - 1
.\ 1
_ 1 ; 1
10 8 -1 2_ A» 0 W 1
10 10 10
-2
10 1 We- . ..
1 I y/8 =0.375 1
: /
1 / 1‘
-4

 

i "i \
-6 *\
10 1 *‘2 x 1
. 1“. \-\\ 1
1 Q“ 1
-8 1 j .
10 "* 1 “ 0‘ ’t‘”*”‘“'-’ 1
10 10 10
-2
10 . *** . ' '
i y/8 = 0.625
-4 ' ’ ‘ .\
10 1 / r \ ************
-- \1
/ “i\“\.\\ \’\‘ J
6 1
10 r " \ \
\. \
10’8 2 .. . _ I
1

10- 100 101
fX/U
r 00

Figure 4.39. Comparison between the measured and stochastically-estimated streamwise-
velocity spectra using ML, SL1, SL2 and SQ2 for y/8 = 0.05, 0.125, 0.25, 0.375, 0.5 and

0.625

179

 

10- .- .__. . _. . --.. - 1._7. "1"":"1 10-2" '_ ' "7" ' 'Tr'v 1
y/Szogs . ML~ . y/5=0.125 1

 

10' 1- ~ 10'21-____-...._ as
y/5=0.25. f 1 y/5=0.375{ 1

/

10 1 ’ 7. 7 ’ ' ’ j 10 ’ ‘/” ,. ’ _ jguj " \ ’’’’’’ 1
1 /' , 1 \ 1

-3 i -8 1
10 l 0 I 10 -l 0 I

10' 10 10 10 10 10

t WW1- . . ......-, 10'21._W . .. .
‘ y/a=o.625,

',-"" > 7 7L - b l/‘//’.."' .-M‘r.\~"~.\_

.4 /V " ' 1 \‘\ - fl \\
10 ~ , : . 1 10 -~ , - ~ ~ x
1 . , , - I J, ,V K
1 / 1, ' 1’ \ , - / "4 . \

, \ l/ ‘-_ ‘
‘ ‘ ‘ ‘3 // / 1 ‘1

VV w

1 O 1 7777777 I I \‘ \X‘ ' 1 0 L 7' ’ ’ " ‘ ‘11 \\
. . \\ a ,

"i l w . a 10-811, . O a 1
10 10 10 10 10 10

fX/U fX/U
r oo 1' m

Figure 4.40. Comparison between the measured and stochastically-estimated normal-
velocity spectra using ML, SL1, SL2 and SQ2 for y/S = 0.05, 0.125, 0.25, 0.375, 0.5 and
0.625

180

 

 

 

Figures 4.41 and 4.42 depict a comparison between <u'> and <v'>, and us and vS
calculated using the linear and quadratic estimation methods (SL2 and SQ2, respectively)
as a function of the wall-pressure condition at a = 2.11 for y/B = 0.05, 0.125, 0.25, 0.375,
0.5 and 0.625. In Figure 4.41, the ordinate represents <u'>/urms and u_,/urms while the
abscissa is p'/pms. It should be noted that <u’> is calculated for a wall-pressure range of

i2.25p,ms, which is found to contain more than 90% of the observed wall-pressure values.
Generally, Figure 4.41 shows the attenuation in both the conditionally-averaged and
estimated velocities; i.e., the largest positive or negative value of <u'> and us is
approximately 0.5ums. More importantly, the figure shows that, for a wall-pressure range
of iprms, the values of us calculated using SL2 and SQ2 agree well with those of <u'> for
all heights, and that SQ2 does not provide any improvement in the accuracy of the
estimate. On the other hand, when the magnitude of p' exceeds or drops below pm, a
deviation is seen between the linear estimate and the conditional average. This deviation,
however, does not exceed 20% of um, which indicates that within the entire range of
pressure conditions, the SL estimation accuracy is better than 80% of um.

The deviation between the linear estimate and the conditional average when lp'l >
prms seems to be odd; i.e., the estimate overpredicts the average for p' > prms and
underpredicts it for p' < -pm,. This odd deviation, which may be accounted for by the
inclusion of a cubic term in the estimation, seems to increase as the height decreases. In
other words, the results suggest that including a cubic term in the estimation should
improve the accuracy of the results beyond the 80% obtained when truncating the
estimation after the linear term. The inclusion of this higher order term will not be

pursued here, however.

181

0.6

air.

I y/s =4 0.125 W

   

0.4-- , ~ , .
‘9 04
0.2 ~+ .
O-- ..
i
'0.2 ’ fi'

V 0 CondAve

-O.4 ‘ ' 2 -- — Linear
"" Quadratic
”0.6 — ; ' " ' 1 i W 1 ' ‘.:
-3 -2 -1 O 1 2 3

 

 

   

0.6 r -. .. _c_____._
y/5 = 0.625
0.4““ w; a ,,,,,,, i
I
0.2 ~-
0 a _
-0.2-
-o.4
-0.6 __ ; _.__ L a . 1
-3 -2 -1 O 1 2 3
137me

Figure 4.41. Comparison of the conditionally-averaged and stochastically-estimated
streamwise-velocity using SL2 and SQ2 for y/S = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

182

O Cond. Ave.
0.4 " ' * * ‘ " — Linear l
1 1 I" Quadratic
02$- 1. __ _ _, *fTi—‘T— i

   

 

 

 
   
  

 

-3 -2 -1 0 1 2 3
0'6 o.6.~— Y— ~—
0.4‘§‘" 0'4iu6-
0.2 0,2!- ————————————— a
0" 0i“ ________
.02 -0,2-- ;
-0.4 _Oo4l. .‘ , ..
, y/6=O.3:75 ,
‘0.6 t _O.6 L‘Ann_,_idi_ _ ; ‘- nfi . :
-3 -3 -2 -1 o 1 2 3

O.6-
0.4lL—-—-E—— -. ......
0.2l ........... f

I
1 ‘ ] i
1 1 1

 

-o.21~-~-—~

04;
l y/5=O.625 .

-06 '~- - , 1 -- r~ u: 1 L. 1 , ;
-3 -2 -1 0 1 2 3

 

137me

Figure 4.42. Comparison of the conditionally-averaged and stochastically-estimated
normal-velocity using SL2 and SQ2 for y/5 = 0.05, 0.125, 0.25, 0.375, 0.5 and 0.625

183

 

 

Conclusions concerning the accuracy of the normal-velocity estimation may be
drawn from Figure 4.42. The figure shows that, at the lowest y location (y/S = 0.05), SL2
is capable of representing <v'> with good accuracy. For higher y locations, the three data
sets show good agreement within the range: lp'l < iprms. Similar to the observations of
Figure 4.41, Figure 4.42 depicts that the quadratic term does not include more
information and, in contrast, the linear estimation seems to more accurately estimate <v'>.
Although the need for a cubic term is not as clear in the normal-velocity results as it is in
the case of the streamwise velocity, Figure 4.42 reveals that the need for such a term is
strongest at y/8 = 0.3 75 (where the v' fluctuations are strongest).

In summary, the comparison of the velocities estimated using ML, SL2 and SQ2
with the instantaneous and conditionally averaged ones reveals that the quadratic term
does not provide more information to the estimated results. Moreover, the multi-point
estimation (ML) was found to represent the conditional velocity field better than single-
point-based estimation, with an accuracy that is projected to be larger than 80% of the
RMS velocity for both velocity components and at all heights examined here. Therefore,
in the remainder of this thesis only ML results will be employed.

4.3.3. Stochastic Estimation Results and Wall-Pressure Sources

Figures 4.43 depicts the ML velocity vector field, along with the associated wall-
pressure time series at E, = 2.0 (corresponding to the microphone location immediately
upstream of the hotwire) for three consecutive time windows plotted in three plots, one
above the other. The vector fields are viewed in a frame of reference translating with a
velocity of 0.81Uco in the downstream direction. Also, in the velocity vector field

windows, two scales are shown for the ordinate: one giving the y location normalized by

184

the boundary layer thickness (displayed on the left side of the plot), and the other (shown
on the right side) provides y/h values. For the wall-pressure plots, the ordinate is the
fluctuating wall-pressure normalized by pm. The corresponding abscissa, which is
common for the velocity vector field and wall-pressure plots, represents the non-
dimensional time.

It should be noted that r values displayed in the plot are folded (i.e., they increase
in the time-backward direction) relative to an arbitrarily selected time offset (to); i.e., r = -
(t-to); t < to. By changing to, one can display different time windows from the data
records. On the other hand, the time reversing operation orders the data in such a way
that the progression of information with increasing time is from right to left. This
facilitates viewing of the vector field in a frame of reference where increasing 1 may be
interpreted as increasing downstream distance by invoking Taylor's hypothesis of frozen
turbulence. The intent here is not to claim that the vector fields displayed in Figure 4.43
represent the instantaneous spatial structure, but rather that the time evolution of the
velocity vector field is consistent with the passage of the flow features shown in the
figure. The degree to which the actual spatial structure of the flow features resembles
that depicted in the vector plots as they sweep past the hotwire is actually quite
reasonable as discussed in the following paragraph.

The stochastically-estimated data, shown in Figure 4.43, may be used to represent
the spatial velocity field using Taylor’s hypothesis of frozen turbulent eddies (Taylor
1938), which provides an approximation to obtain spatial information from temporal data

and is, strictly speaking, only applicable to statistically stationary, homogenous, flows.

185

 

Lumely (1965) showed that the hypothesis gives adequate accuracy if Bini<< 1; where
U

C

Uc is the mean convection velocity of the flow structures. Altemately, the hypothesis is
reasonable over a time window (T) that is much smaller than an eddy "tum-over time"
( E/urms; é’ being a characteristic length scale of the dominant eddies). That is, there is
very little time for the eddy to evolve and change state during T. This implies: T <<
é/urms. The dominant length scale of the structures related to the wall-pressure generation
may be estimated from the wavenumber-frequency spectral peak in Figure 3.17 as
approximately 6 = Xr (kx z 1 /X,). On the other hand, the u,ms plot in Figure 4.4 shows

that urms is of the order of O.1U.,o (note lower um,s values improve the applicability of the
hypothesis). Thus, if the reader wishes, it appears reasonable to interpret the structural
features (not the vector field as a whole) in Figure 4.43 as being spatial structures over a
time window T Um/Xr << 10; i.e., T Ugo/Xr z 1. For guidance, this time window is
demonstrated in the vector plots in Figure 4.43. Additional support for the
reasonableness of Taylor hypothesis in such interpretation is provided by the practically
invariant pressure fluctuation level in the streamwise direction in the vicinity of the
hotwire location (see Cpo in Figure 3.2), and the constant convection velocity associated
with all frequencies and time scales of the wall-pressure field as found earlier from the
wavenumber-frequency analysis.

Overall, the velocity-vector field snap shots displayed in Figure 4.43 are
dominated by quasi-periodic vortical structures. These vortices are large in scale and

have cores (as identified visually) that appear mostly within a small y range in the

186

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L .. \QxMhlwwfl L. .
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= 2.0 (bottom of each plot) and the associated

Figure 4.43. Wall-pressure time series at g

stochastically-estimated velocity vector field (top of each plot) using ML for three

consecutive time windows, viewed in a frame of reference translating with velocity of

0.81Uw (largest vector = 0.25Um)

187

neighborhood of y/5 z 0.5. One can also depict saddle points inbetween the vortices,
associated with the interaction between successive vortical structures. These
observations are consistent with the picture drawn above flom the previous statistical
analyses. However, the stochastically estimated velocity fields based on the
instantaneous spatial wall-pressure footprint reveal the richness in variability in the
characteristics of the flow structures: something that can't be discerned flom the earlier
results. This variability includes structure scales, locations, temporal (and hence spatial)
spacing, and interactions (including variability in saddle point locations and apparent
vortex merging). Some observations concerning these characteristics are discussed in the
following paragraph.

In the first (top-most) window in Figure 4.43, four successive vortical structures,
the centers of which are marked using "x", are evident. Note that at this stage, a vortex is
identified whenever the vector field locally reflects a circulatory flow pattern around a
zero-velocity center. Of course, the appearance of such pattern is highly sensitive to the
selection of the translation velocity of the flame of reference. The dependency of the
vortex identification on the velocity of the flame of reference may be seen in Figure 4.44.
The figure depicts that varying the velocity of the flame of reference by iO.OSU.,o greatly
affects the visual identification of the locations of vortical structure centers. Therefore,
the observations discussed here require substantiation by some other flame-independent
measure, which is done immediately following this discussion. The non-dimensional
temporal spacing between the three right-most vortices in the top vector plot of Figure

4.43 is approximately 1.3, which is equal to the dominant time scale identified earlier in

188

 

 

 

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GOP plot),

(bottom plot); largest vector = 0.25Uw (note that the "x"

estimated velocity vector field using ML viewed in
189

and "0" locations are fixed for all three plots to help identify the variability in the vortex

flames of reference translating with three different velocities of 0.76Uao
and saddle-point locations, respectively)

Figure 4.44. The stochastically-
0.81Uoo (middle plot) and 0.86Uuo

the autocorrelation of the wall-pressure and the cross-correlation between the normal-
velocity and wall-pressure, while that between the two left-most vortices seems to be
shorter (approximately equal to 1.0). In the middle window, three vortices are also
captured over the same time flame. These vortices, also, appear not to be separated

uniformly; i.e., their centers are located at time instants of 4.7, 6 and 6.6. Therefore, the

'ch
X

f

 

left-most and middle vortices are = 1.3 apart while the separation between the

middle and right-most ones, which appear to be undergoing a vortex-merging process, is
0.6, reflecting the variability in the temporal separation between successive vortices, and
hence the quasi-periodic nature of the vortical structures. Finally, the bottom-most
window in the vector-field sequence depicts three consecutive small vortices that are
possibly undergoing a merging process, in addition to a separate fourth vortex at the right
side of the window. The reader is cautioned that the identification of vortex merging
flom a single snap shot of the vector field is somewhat speculative since space-time
velocity information, which is not available in the present study, is required to examine
the time evolution of the vortices and ascertain that a merging process is in fact taking
place. In-between the vortex structures, saddle, or stagnation, points (marked by "0")
that result flom mutual-vortex interactions are observed. Some of the saddle points are
observed as close to the wall as y/5 z 0.13, signifying the fairly deep penetration into the
boundary layer of the high-speed momentum fluid flom the fleestream associated with
these saddles.

Overall, the depicted vector field corroborates the idea that the wall-pressure
generation process is in fact dominated by the passage of quasi-periodic vortices and their

mutual interaction. As the vortex cores convect past a point on the wall, they produce

190

 

negative pressure, followed by positive-pressure generation associated with the passage
of the saddle points. This scenario, which was suggested earlier flom examination of the
conditional—average of the velocity vector field in 4.2.3, can be further examined using
the plots of the simultaneous velocity vector field and wall-pressure signature presented
in Figure 4.43. Generally, the three time windows in the figure confirm that the negative

and positive p' peaks are synchronized with the passage of the vortex cores and saddle

points, respectively. Practically all time instants of observing vortex cores (I Use = 0.2,
X

f

 

1.15, 2.3, 3.4, 4.7, 6.0, 6.6, 7.7, 8.35, 8.7 & 9.9) coincide with the occurrence of a
negative peak or local minimum in the wall-pressure. There is a small time offset
between the observation of the negative p' peak at the microphone, and the passage of the
vortex center past the hotwire. This offset is due to the fact that the microphone is
located slightly (Aé = 0.05) upstream of the hotwire, and hence the p' peak occurs slightly

ahead of the passage of the vortex core past the hotwire. On the other hand, positive wall—

pressure peaks evidently coincide with the occurrence of saddle points (e.g. I U» = 0.6,

 

f

1.6, 3.0, 4.0, 5.4, 6.2, 7.3, 8.0, 8.5, 9.2 & 10.2).

The aforementioned flow features are consistent with the study of Kiya et al.
(1982), who used a discrete-vortex model to simulate the flee-shear-layer flow above a
separation bubble formed by the flow past the nose of a thick splitter plate. Their results
show negative pressure peaks in the vicinity of vortical structures and positive peaks
when the vortices are absent. In addition, the above observations are consistent with the
recent study of Naguib and Koochesfahani (2003) in which it was shown that for a vortex

ring interacting with a flat wall, the primary source of negative pressure was associated

191

 

 

with the high vorticity in the vortex core. On the other hand, positive-pressure generation
was dominated by the straining motion caused by the interaction of the vortex with the
wall or with other secondary vortices that are produced flom such interaction.

The above analysis presumes that a coordinate-flame translation velocity of
0.81U.O is appropriate based on the convection velocity information obtained earlier flom
statistical analysis of the pressure and velocity information. Based on the observations of
Figure 4.44, the reasonableness of identifying certain flow structures as vortices needs to
be verified here, and therefore we examine the vorticity field associated with the
estimated velocity field. It should be understood that the existence of regions of high
vorticity does not necessarily imply the existence of a vortex (e.g., Blasius boundary
layer contains vorticity but no vortices). Nevertheless, vortex cores are expected to be
associated with localized high-level vorticity. Hence, the association of the vortex
structures identified above with locally high vorticity at the core is required to justify the
above discussion. It is also noted here that unlike the velocity vector field, vorticity is

independent of the selection of the coordinate-flame translation velocity.

 

The spanwise vorticity, (02,, = ——$‘ , is calculated flom the estimated

velocity vector field using a central-finite-difference method and plotted using a color
contour map that is superimposed on top of the corresponding velocity vector field as
shown in Figure 4.45. In order to concentrate only on regions with significant
vorticity, the vorticity contours are shown for values that exceed 50% of the highest
vorticity in the figure. Figure 4.45 reveals that the vortex cores identified earlier are in

fact associated with locally high values of vorticity. In particular, the top-most window

192

 

.)L;

\\I \\\\x

\ 11\1IlI///J-

\KL...,V,..n

 

IU/X

Figure 4.45. Vorticity and associated stochastically-estimated velocity vector field at
three consecutive time windows, viewed in a frame of reference translating with 0.81Um

193

depicts four main high-vorticity cores, which coincide with the occurrence of the
identified cores of vortical structures and their associated wall-pressure negative peaks.
However, the middle window depicts only two spots of locally high negative vorticity,
but significantly stronger than those found in the first window, which represent the left-

most vortex and the two right-most merging vortices. Similarly in the bottom-most

window, the existence of a broad (7.5 S. Tum S 9.3) and concentrated (Tux- z 9.9)
X X

f I’

 

 

regions of high negative vorticity seem to correspond well to the three merging vortices
and the right-most vortex identified earlier in Figure 4.43. The three windows also show
a thin but extended layer of negative vorticity concentration near the wall. This layer is
clearly associated with the high mean velocity gradient near the wall, modulated by the
passage of the outer-shear layer vortical structures. The above suggests the
appropriateness of the selected flame of reference for viewing the structure of the flow.
It should also be noted that in this flame of reference, the identified vortex cores are
observed in the vicinity of y/5 z 0.48, which agrees well with the identified outer-shear-
layer center based on the peak shear Reynolds stress.

To attain a deeper understanding of the relationship between the wall-pressure
signature underneath the investigated boundary layer and the pressure-generating flow
structure, the distribution of the turbulent wall-pressure flow sources of the
stochastically-estimated, or quasi-instantaneous, velocity field are investigated. As
discussed in Chapter 1, Poisson’s equation governs the pressure fluctuations in
incompressible turbulent flows. For the flow considered here, which possesses a

dominant mean shearing component, du/dy, the Poisson's equation may be simplified to:

194

 

 

LV2P'=_zfld_V_u;j
p dy dx ’

“3.1 = q (4.27)
where u is the mean velocity in the streamwise direction, q is the spatial distribution of
wall-pressure sources, and the prime indicates a fluctuating (mean-removed) quantity.
As described in 1.1.2, the first term on the right hand side of equation (4.27) is called the
linear pressure source, while the second term is known as the nonlinear one.

In many investigations of the wall-pressure in turbulent wall-bounded flows, only

the linear term is considered because of the obvious simplicity in measuring, analyzing,

and modeling the linear relative to the non-linear term. This simplification is typically

justified because of the large value of :iTu in comparison to that associated with the
y

turbulent-velocity gradients comprising the non-linear term. However, DNS
computations by Kim (1989) and Chang et al. (1999) enabled calculation of both the
linear and non-linear terms in fully-developed turbulent channel flow. Both studies did
indicate that the non-linear contribution to the wall-pressure was at least as important as
the linear one. The first indirect experimental verification of this conclusion was later
provided by Naguib et al. (2001) who showed that for the stochastic estimation of the
flow field based on single point wall-pressure events to converge to the conditional
average, a quadratic stochastic estimation must be used. Considering an analysis based
on the source terms of Poisson's equation, Naguib et al. found that the improvement to
the stochastic estimation via inclusion of the quadratic or higher-order terms was
attributed to the importance of the non-linear sources of the wall pressure.

The reasonableness of neglecting the non-linear source terms for the current flow

may be substantiated using an order of magnitude analysis. Referring to the mean-

195

 

flfljfil‘fl .

 

!' r‘ Ilu.m.r .-
‘ r

velocity profile in Figure 4.36, it is seen that gil— is ahnost uniform in the outer part of the
y

flow (y larger than approximately 0.155) and is approximately equal to 0.3Uoo/8. As
found earlier in the discussion of the spatial cross-correlation results in 4.2.2, the
dominant length scale of the pressure-generating structures is of the order of Xr for y/6 >
0.15. Thus, the ratio of mean-velocity gradient to any of the turbulent velocity gradients
is of the order of (0.3 Ugo/8)/(O.1 Ugo/X1); note that the estimate of the velocity scale of the
turbulent fluctuations as 0.1U00 is quite reasonable based on the um, and vrms profiles (see
Figures 4.3 through 4.6). This yields 3X,/8 z 7. Closer, to the wall (y/8 < 0.2), where the
du/dy is of the order of Ugo/6 (within the measurement range) and a shorter length scale of
)1.- z Xr/2 are found, the ratio becomes approximately 10. Thus, it is evident that the
mean-velocity gradient across the layer is in fact substantially higher than the turbulent
velocity gradient, providing some support of the dominance of the linear source of the
wall pressure. It should be noted that this conclusion is not in conflict with the
significance of the non-linear source terms found in canonical wall-bounded flow by Kim
(1989), Chang et al. ( 1999) and Naguib et al. (2001), since the current flow is clearly a
non-equilibrium one and the dominant p' sources are those originating in the separating
shear layer and hence they bear no relation at all to the sources in the equilibrium wall-
bounded flows.

Based on the above discussion, the following analysis of p' sources will only
consider the first term on the right-hand-side of equation 4.27, which is the linear source,
or qL. The reader is reminded, though, that the analysis based on Figures 4.41 and 4.42

showed that one should include a cubic term in the stochastic estimation for a full

196

 

account of the conditional average, which provides a hint that the non-linear pressure
source terms may not be totally negligible, based on the results of Naguib et al. (2001).
Therefore, the analysis of the linear source alone should be viewed as an analysis of the
leading-order term of the pressure-generating sources. Future studies, considering full
3D stochastic estimation of the flow can then quantitatively assess the relative importance
of the linear and non-linear sources.

The time evolution of the y distribution of the dimensionless linear pressure

sources (qL X,5/U.o2) for the same three time windows considered in Figure 4.43 is shown

I

I O O u v D O C
in Figures 4.46. Note that because qL IS the product of d—d— , it IS normalized by
y x

U

d) w

?7(— , where 5 and X, are the length scales in the y and x directions, respectively.

The source distribution is shown using three color contour plots of qL calculated flom the
stochastically-estimated velocity field. The associated wall-pressure time series is also
provided in the figure below each of the contour plots. Note that in order to compute

dv'

 

, the temporal variation of v' is converted to spatial variation using Taylor’s

hypothesis in conjunction with the local mean velocity over a very narrow time window
(corresponding to the duration of acquisition of three successive data points that are used
for implementation of a central finite-difference method to calculate the derivative. Note
that this window width is 1/60th of the period T shown in Figure 4.43, over which Taylor

hypothesis was estimated to work reasonably well).

197

 

 

7 7 7:5 '3 8.5 7 '9" 9.5 7 10 16.5
I U /X
00 1'

Figure 4.46. Pseudo-instantaneous linear pressure source (qL) flooded contour maps and
associated wall-pressure for three consecutive time windows

198

Inspection of Figure 4.46 shows that whenever a substantial negative pressure
peak is observed, negative pressure sources are found directly above the peak. Although
the sources are distributed across the entire y range, they exhibit two high-concentration
spots, or localized peaks: one far from the wall in the outer-shear layer, near y/6 = 0.48,
where the Reynolds—shear-stress peak was found earlier, and the other close to the wall
(y/5 < 0.2). In a similar fashion, substantial positive pressure peaks are also associated
with positive sources that are located directly above the positive pressure peak, with the
highest sources also found in the vicinity of the outer-shear-layer center, and below y/5 =
0.2.

The localized peaks of negative sources found in the outer-shear layer are located
at the height of the vortex cores identified earlier in the vector plot in Figure 4.43.
Comparison of this figure to Figure 4.46 shows good coincidence between these localized
negative-pressure sources and the vortex-core locations. Positive-pressure sources are
found immediately before and after the negative ones, but at the same height. This
provides strong evidence for the ideas presented thus far that the passage of vortex cores

past a point on the wall is the leading contributor to p' generation in the current flow. As

I

. dv . . . . .
the vortex travels it creates strong d— in the v1c1mty of the core leading to pressure
x

I

. . . . du dv' dv . . . .
generation v1a the linear mechanism (or -d_d— ). The a—vanation associated With the
u x

x
vortex passage is such that it creates a negative source at the core and positive ones

upstream/downstream of the core. The alteration in the pressure source sign flom

I

positive to negative and back again to positive may be depicted by considering -d—v—
x

199

variation associated with the conditional velocity field obtained earlier for negative

pressure peak in Figure 4.37. Figure 4.47 shows a plot of v' normalized by U00 versus

I

TUoo/X, extracted flom Figure 4.37 at different heights. The associated fldx: normalized

by Ugo/X, is also shown in the figure. The succession of positive/negative sources

I

. . . . . . v
assomated With the vortex passage 18 evrdent flom the alternatmg Sign of d—.
x
As y decreases below the vortex center, the source strength becomes weaker since

du . . . . . .
d— 15 practically uniform in the outer-shear layer and v' attenuates w1th decreasmg
y

height (see v,ms profiles and spectra in Figures 4.6 & 4.12). However, below y/5 z 0.2,
the mean-velocity gradient starts to increase substantially because of the no-slip
condition. As a result, the creation of another significant source of pressure fluctuations
is found near the wall. Physically, this source may be thought of as one that results flom
the modulation of the near-wall flow by the vortex passage; i.e., one that is the result of

the indirect influence of the vortices. Unlike the outer-shear-layer source, this source is

I

associated with weak 33% disturbances, the ability of which to produce p' is amplified by

the strong near-wall shear.

It is interesting to note that whereas, localized peaks of qL are always found below
y/5 = 0.2, whenever there are localized peaks near y/5 = 0.48, the opposite is not true.
That is, at certain instances, there are localized peaks near the wall that are not associated
with localized peaks away flom the wall. This gives rise to a pressure source near the

wall that is associated with a time scale smaller than, about half, that characteristic of the

200

 

 

 
   

--------

-
~~~~

-0025 P i; 17> -
d<v>ldx -

-o.05 , 4
0.025 m L

(X/Uoo)

 

 

 

<v’>/U & 0.1 *d<\l>/dx

 

 

LII
’ l
l
I
I
1
i
Ll
l
|
l
l
l
l
a
l
t
1
l
l
1
1
|
|
I
|
|
|
I
1
l
|
J
l
l
1
|
l
"I
|
l
i
.J

 

 

-0.05'-- 1 f ‘: :-____L... ‘1 # 1 L

-0.5 on -o.3 .02 .0.1 o 0.1 0.2 0.3 0.4 05

Figure 4.47. The conditionally-averaged normal velocity and its streamwise gradient

associated with negative wall-pressure peak for y/6= 0.05, 0.125, 0.25, 0.375, 0.5 and
0.625

201

vortex-vortex spacing away flom the wall. Consequently, pressure sources near the wall
are expected to generate wall-pressure signatures with both the smaller and larger
characteristic scales. This observation is consistent with the spatial cross-correlation
analysis presented earlier in the discussion associated with Figure 4.27.

The importance of the various flow sources of pressure to the wall-pressure
generation depends not only on the strength of the source, but also on the distance
between the source and the wall. The larger this distance is, the weaker is the source
contribution to p'. This may be seen flom the solution of equation (4.27), which is a
volume convolution integration of weighted pressure sources over the flow domain given

by (e.g., Kim 1989):

p'(x,y,z,t) = q(X.,Y.1Z.,t) dV (4.28)

p/21t _mflx -x,)2 +(y- y,)2 ‘*'(Z"Z.)2 ’

 

where x,, y, and 25 are the coordinates of the pressure source, and x, y and z are the
coordinates of the point in the flow of where the solution of p’ is sought. Thus, the
denominator represents the distance between the source and a point on the wall in the
case of the wall pressure. Moreover, the observed p' value is an integration, or
superposition, of all sources within the domain. Thus, in order to assess the true
significance of the sources identified in the discussion of Figure 4.46 above, the source
strength must be weighted, i.e., divided by, its distance (r) to the point of observation of
the pressure. This was done (accounting for the source height above the wall and the x
offset between the hotwire location and that of the microphone for which p' data are
shown below the source-distribution maps) and the outcome is shown in Figure 4.48.

The figure shows that the importance of the direct pressure-source located at the height of

202

 

 

   

2 2

- - q /r(X8 /U )
a; 01 **' // H J” , ‘r 4* \.\ ' w L. *RT’ ’ \ ”j L r w
a ./ 1. L : \«1
1

LL1

 

1 - . ‘ 2 2
‘ ;..~~».~ . 1 q/r(X6/Uw)
84E 0, r 1“, 11,.» 1 ..»’ if M L l/IV L r
a ‘ LL
_5 L L L: L . L
7 7.5 8 8.5 9 9.5 10 10.5

1 11,01,

Figure 4.48. Pseudo-instantaneous linear pressure source ((1%) flooded contour maps

and associated wall-pressure for three consecutive time windows

203

 

 

the vortex centers is greatly diminished, because of the larger distance to the wall.
Therefore, as far as the wall-pressure signature is concerned, the strongest pressure
fluctuations are generated flom the near-wall sources (y/6 < 0.2). This is quite surprising
given that all indications flom the pressure and pressure-velocity statistical analysis
undoubtedly tie the wall-pressure to the outer-shear—layer vortices. However, the reader
is reminded that the near-wall source is in fact an indirect consequence of the passage of
the large-scale vortices, and if the outer-shear layer were to somehow disappear, the near-
wall source would not exist. Moreover, the weaker outer-shear-layer sources of the wall
pressure are distributed over a substantially larger height than the stronger near-wall
sources. As seen flom the convolution-integral (equation 4.28), the net pressure
magnitude is an integration over the whole flow volume. Hence, a weaker source that is
extended over large volume may be as important as one that is very strong, but also
highly localized. Therefore, the results in Figure 4.48 do not necessarily indicate that the
direct influence of the vortices, i.e., outer-shear-layer sources, is negligible.

The linear source analysis presented above is based on the observation of typical,
but short, time records of the entire data set. To examine the consistency of the
conclusions made with the entire database, the RMS of the source strength (qms) was

calculated for different heights flom the stochastically-estimated velocity and plotted in

Figure 4.49. The abscissa is the RMS of the pressure source normalized by 951%

1'

while the ordinate is the height above the wall (y) normalized by the boundary-layer
thickness (5). The figure shows the profiles of q,ms calculated flom both the measured
and estimated velocity fields. Comparison between the two data sets shows a very good

agreement in the qualitative trend of the data with, expected, attenuation of the results

204

 

 

based on the stochastic estimation relative to those based on the actual velocity.
Consistent with the above analysis, the source RMS profile clearly depicts two localized

peaks of large pressure generation: the first one is at y/S = 0.48, which is the same height

of the maximum — u'v' . The second peak is implied flom the rise in q,ms magnitude
below y/B = 0.2. Since v' goes to zero at the wall, q,,,,, must decay to zero at y = 0. Thus

the near-wall qm, peak is located somewhere between the lowest y value shown in Figure

 

 

4.46 and the wall.
0.7 i’ L . ., - -;r
. . Linear
O o Data-Linear
06. . . . L . L L- _. -.
o
o1
o
05 .. L LLLL.L
.1
o
o
0.4- ‘ - ~ . '0 I' '
s. -.
0.3 H 1 - o e e »- 1 *******
o
1 o
, . -
024,, ~ 0 ———————— - ------ , a »1 ,1 ,,,,,
o
7 o
o
0.1 -* . ~ * 0 ~
0
0‘. . LL L . LLL . L l LLLLL LL. _.
O 0.1 0.2 0.3 0.4 0.5 0.6 0.7
2
qms (XrB/Uoo)

Figure 4.49. RMS profile of the linear pressure source calculated flom stochastically-
estimated and measured velocity time series

The results shown in Figure 4.49 do not account for the distance between the

source and the wall. Figure 4.50 shows q,ms afier proper weighting and normalization, by

205

w CD

U . . .
373(— , of the source strength to account for its location relative to the wall-pressure

observation point. The outcome reflects a practically uniform distribution of source
strength above y/S z 0.2. As discussed above, this distribution is substantially weaker
than the largest value found near the wall (by about a factor of five). However, the

corresponding y range of the uniform source is also five or more times that of the near-

 

wall source. Hence, the direct influence of the vortices may in fact be of equal

importance to their indirect influence in generation of the wall pressure.

 

 

 

 

0.7 i» : : *W" 7 : ”"iaT—‘TFI : 'fi: :21 fir:
‘ :"1 Linear
.0 o Data-Linear .
0.6 ' 70f r
9
o
- ' . Lo - -
0.5 — ~ 7 1.1 ‘ o ; 1 1 1 7777777 -, 1
o
o
0 4 O
c
Q L o
>‘ L o 1
0 3 ~ {i1 - — t ~ ~ 7777777777777777777777777777777 1
O i
o 1
- .1 e ‘ 1 ‘ ; '
0.21- e1 7‘» ,,,,,,,,,,,,,,
' o - -
o
. 1
0.1 ' ' * O
711 .
0 _LLLL L- LLL L 1 l - i 3 L _ .-
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

2 2
qrms Xr 5 /(r U00)

Figure 4.50. RMS profile of the linear weighted pressure source calculated flom
stochastically-estimated and measured velocity time series

206

I E‘l'ibyil .
.. ...-3qu I . I
r. .1)..... r

5. CONCLUSIONS AND RECOMMENDATION FOR FUTURE WORK

5.1. Conclusions

In this investigation, a 16-microphone array has been constructed and used to
acquire a wall-pressure database simultaneously with X-hotwire time series in a non-
equilibrium boundary layer originating flom a separating/reattaching shear layer. Data
were acquired over a non-dimensional streamwise (5,) range extending flom 0.67 to 2.33.
Characterization of the wall-pressure data alone showed that the wall-pressure
fluctuations were dominated by large-scale downstream-traveling disturbances that were
linked to the vortical structures generated upstream in the separated shear layer.
Notwithstanding this dominance, the p' signature of these structures decayed with
increasing downstream distance as the vortices underwent a relaxation process, gradually
loosing their energy and associated p' signature.

In contrast, the contribution of eddies generated due to the boundary-layer-like
shear near the wall became more significant with increasing g. This could be deduced
flom the increase in the energy at high flequencies in the wall-pressure spectra, which
resulted flom the growth of the inner sub-boundary layer near the wall. The
dimensionless flequency of the most energetic wall-pressure fluctuations was found to be
approximately fX,/U,,o = 0.7, which agreed well with the passage flequency of the
separated-shear—layer vortices reported in studies concerned with the
separating/reattaching flow upstream of the flow region examined here. In addition,
wavenumber-flequency—spectrum results showed that pressure signatures of all
wavenumbers and frequencies were associated with flow disturbances that travel

downstream with the same convection velocity of approximately 0.81Uoo, which was

207

 

 

about 10% U00 higher than that reported by Farabee and Casarella (1986). The lower
value in the latter study is attributed to the influence of the wake region of the separating
turbulent boundary layer. Such a region exists in the study of Farabee and Casarella but
not in the current study where a laminar boundary layer exists at separation.

The flow velocity was measured within the non-equilibrium boundary layer flow
(at 2; = 2.05), simultaneously with wall-pressure data flom the ten downstream-most
microphones (é = 1.33 to 2.33). Those measurements were used to examine the velocity-
field characteristics of the non-equilibrium boundary layer and the relationship between
the flow structures and associated wall-pressure field.

The wall-normal profile of the mean streamwise velocity deviated from that of a

canonical equilibrium turbulent boundary layer and its log-law, consistent with the earlier

studies by Ruderich and Femholz (1986), Farabee and Casarella (1986), Castro and Epik
(1998) and Song and Eaton (2002). In addition, the present data showed that the non-
equilibn'um boundary layer did not possess a wake region, which could be related to the

laminar separating boundary layer. On the other hand, the boundary layer at E,=2.05

exhibited flee-shear-like characteristics, associated with the existence of v,,,,, and -TlTV_'
peaks in the outer layer at y/S = 0.38 and 0.48, respectively, which were related to the

energetic large-scale vortical structures that were generated and energized upstream in

the separating free-shear layer. Unlike v,ms and —W , the nuns profile exhibited a flat
region of fluctuations in the range of 0.02 < y/S < 0.44.
The dimensionless auto-spectra of the strearnwise— and normal-velocity ((DUU, (DW)

revealed the increasing importance with increasing downstream distance of the small-

scale turbulence associated with the sub-boundary layer near to the wall. The former

208

 

 

revealed the existence of a more than a decade wide f '5’ 3 range, implying the existence of

an inertial sub-range, while the latter depicted a peak similar to that of the wall pressure

I'

 

at a flequency of f z 0.7. The cross-spectrum between the strearnwise- and normal-

w

velocity ((1),...) reflected a pronounced increase in Reynolds stresses in the high-flequency
end of the spectrum near the wall. On the other hand, the data also showed that the large-
scale structures were active in the outer layer, but became inactive near the wall.

The velocity-pressure cross spectra provided additional evidence that p’
generation was dominated by the vortical structures in the outer part of the boundary
layer, exhibiting a peak in the flequency range corresponding to the passage of the
vortical structures for all heights within the boundary layer. On the other hand, the
velocity-pressure spectral information also depicted growth in the sub-boundary layer
contribution to p' with increase in i. However, this contribution remained negligible
relative to that of the vortical structures over the streamwise extent covered in the present
study.

Using u'-p' and v'-p' cross-correlation, the size of the flow structures in the outer
part of the boundary layer (X,,) was estimated to be approximately 0.9-1.0 X,, which
agreed reasonably with the dominant wavelength found flom the wavenumber-flequency
spectrum of the wall—pressure data and was about 30-40% higher than that reported in the
literature of separating/reattaching flows. It is believed that the reason for this deviation
is the increase in size of the vortices as they travel from the separation/reattachment zone
to the developing boundary layer region where the current data are obtained. Such size

increase may be associated with diffusion and vortex-merging effects. Near the wall, a

209

 

 

second, smaller length scale of about V210 was also identified flom the velocity-pressure
correlation data. Later analysis of the wall-pressure flow sources revealed that this scale
was associated with pressure-generation due to the modulation of the strong near-wall
shear by the passage of the vortices. Moreover, when the cross-correlation results were
coupled with the conditionally averaged wall-pressure and velocity-field, they revealed
that positive and negative wall-pressure peaks coincide with "inrush" of high-speed fluid
towards the wall and the passage of spanwise-vortex centers, respectively. The high-
speed inrush was found in between two successive vortices and was associated with the
formation of a saddle-point positive-pressure source, when viewed in a flame of
reference traveling with the 0.81Uao. The succession of vortex centers and saddle points
resulted in the quasi-periodic character of p' reflected in all statistical measures of the
wall-pressure obtained here.

Finally, multi-point Linear Stochastic Estimation of the flow field based on
instantaneous wall-pressure information, while confirming the dominance of wall-
pressure generation by the passage of the shear layer vortical structures and their mutual
interaction, provided more insight into the quasi-instantaneous variability of the
characteristics of these structures; e. g., their wall-normal location, scales, streamwise
spacing, etc. More importantly, when the stochastic estimation results were coupled with
analysis of the leading-order term of the wall-pressure sources flom Poisson's equation,
two important wall-pressure-generating mechanisms were revealed. The first was
associated with strong dv'/dx disturbances that are produced by the passage of the high-
vorticity vortex cores in the outer part of the flow. This produced a uniformly distributed

wall-pressure source in the region y/8 > 0.2. Closer to the wall, a stronger, more

210

 

 

localized source was found to result flom coupling of weak dv'/dx disturbances resulting
flom the vortex passage with the strong mean-flow shear due to du/dy. Although this
source was stronger in magnitude than the outer-shear layer source, because of its
proximity to the wall and the strength of du/dy, it occupied a much smaller flaction of the
boundary layer, and hence integration of all sources across the boundary layer is expected
to result in practically equal importance of the outer and near-wall mechanisms for
generating the wall pressure. It is important to realize that while it is likely that the
former mechanism could be modeled by the wall-pressure field associated with quasi-
periodic passing of vortices embedded in uniform inviscid flow, the latter mechanism
requires proper account of viscous effects near the wall and associated mean shear.

5.2. Future Work

The following is a list of items suggested as a possible extension/refinement of
the present study:

1. As demonstrated herein, the stochastically-estimated velocity based on the
instantaneous wall-pressure information has a spectrum that is similar in shape
but attenuated in magnitude in comparison to the actual velocity spectrum. A
possible idea to remedy the attenuation in the estimate is to use time-delayed, as
opposed to instantaneous, wall-pressure information to estimate the velocity. The
time delay would be different for each sensor in the microphone array and taken
equal to that corresponding to the maximum correlation between the pressure
measured by the sensor and the velocity at the location of the estimate. To a
certain extent, this approach is similar to that employed in phased, or beam-

forrning, microphone arrays which are used in aeroacoustic applications to

211

 

 

identify flow sources of noise. By phase-shifting and adding signals from the
different microphones, the "net" array output can be made to respond to noise
generated flom a particular location in the flow, while canceling out contribution
flom other locations. It is recommended that such a "beam-forming" Stochastic
Estimation be attempted on the current data set, and the results compared to the
instantaneous velocity.

Conducting two-dimensional multi-point velocity measurements simultaneously
with wall-pressure measurements to obtain the spatial characteristics of the flow
field in the investigated zone. The velocity measurements could be conducted
using Particle Image Velocimetry, or by traversing an X—hotwire over a two-
dimensional spatial grid in x and y, instead of y only as done here.

Examine the space-time characteristics of the stochastically-estimated velocity
field based on the results flom the simultaneous measurements described in item 2
above.

Conduct an experiment using different X-wire configurations to measure u', v' and
w' over a three-dimensional grid above the microphone array. This will enable
calculations of the pressure-velocity correlation for all three components of the
velocity and in three dimensions, yielding a stochastic estimation of the full three-
dimensional flow structures. Subsequent analysis of the velocity field will not
only allow understanding of the full three-dimensional nature of the pressure
sources, but also assessing the importance of the non-linear sources of pressure,

which is an issue that was raised but not addressed in the current work.

212

 

 

A. Derivation of Equation (2.6)

From the X-wire probe geometry shown in Figure 2.22, the effective velocities of

HW #1 and HW #2 (U61 and Ucz) may be given as follow:
Uc,=uF,2(90—a,)—VF,2((1,) (A.l)
Uc2 =uF22(90—a,)+vF22(a2) (A.2)
Equation (A.1) can be solved for v to give:

V _ u F12(9O_al)—Ucl

 

 

 

 

 

 

 

 

 

(A.3)
F12 ((11)
plugging the above equation into equation (A.2):
2
Uc, = u F5 (90 — a,)+[“ F' (902 ‘1') U“ J F; (01,) (A4)
F1 (011)
F12 (“1) UeZ + F22 ((12) Ucl = (A 5)
u(F22(90—a2)F12(a,)+ Ff(90—a,) F22(a2))
and,
2 2
u = 2 Uez Fl2 ((1.1)-l- Uzcl F201;) (A.6)
F1 (90-(11) F2 (0'2)+F1 ((1,) F2 (90—(12)
Finally, v can be obtained by solving equations (A3) and (A6) together:
v _ LU“ F301,)Ff(90-(t,)+Ue2 E2(a,)F,2(90—a,) _ ] 1 (A 7)
E2(90—a,)F22(a2)+E2(a1)F22(90—a2) " 11,201,)
( Ue, F,2(a,) 135(90-0191» 11,, Flz(a,)F,2(90—a,) 1
V: _Uel F22(a2)E2(90_a1)-Ue1F12(a1)F22(90-a2) (A.8)
112190—11.) F5111.) 11.211110111211110 F.2(90-a.) F5111.)
K J
2 2
v=[ 2Uc2F1(902—a,)—U:,F2(90 11,) J (A9)
F1 (90—(11) F2 ((12) +F1 ((1,) F2 (90-(12)

213

 

B. Data Acquisition Settings

Table B.1 shows a list of the signal connections to the A/D converter, and
corresponding input range and gain used with each data acquisition channel. The first
channel (channel #0) was connected to the pressure transducer that measured the static
pressure upstream of the model to determine Um, while the fourth channel was connected
to the output of the temperature sensor used to measure the flow temperature for
calculation of the air density and temperature-correction of the hotwire output. Because
it was important to acquire the mean voltage of the hotwire (HW) output, which ranged
from 1 to 2 Volt, as well as resolve the substantially-smaller fluctuating voltage, the
output of the each of the wires was acquired using two channels. The first channel,
which was set with large input range (to accommodate the large mean voltage value) and
low gain, was used to calculate the mean of the HW output, while the second channel,
which was set with small input range and high gain, was employed to acquire the mean-
removed HW output voltage, which was obtained using a high-pass filter, of a cut-off
flequency of 0.2 Hz, that was built in a Larson & Davis (L&D) preamplifier (model
2200C). Subsequently, the time series of HW #1 and HW #2 were obtained by adding
the mean voltage of channel #1 and channel #2 to the voltage time series of channel #4
and channel #5, respectively. Finally, the last ten A/D channels were connected to the
last ten microphones (flom microphone # 7 to #16).

Data were acquired with a total sampling flequency of 200,000 sample/s, which
corresponded to 12,500 sample/s per channel. 221 (2,097,152) samples per channel were

acquired at each of the twenty six hotwire locations: flom y = 4 to 54 mm in increments

214

 

 

of 2 mm. A LabView program was used to operate the data acquisition system,

automatically traversing the HW probe, and acquiring/saving data flom the 16 channels.

 

1.111. B. {Tonga-11.111.115.111}

 

 

 

 

 

 

 

 

 

 

 

 

A/D Channel # Input Sigal Input range [V] Gain
Channel # 0 Pressure Tansducer :1: 5.0 1.0
Channel# 1 HW#l signal i 5.0 1.0
Clnnnel# 2 HW#2 signal :1: 5.0 1.0
Channel # 3 Temperature Sensor :1: 0.5 10.0
Channel # 4 HPF HW#l signal :1: 0.5 10.0
Channel # 5 HPF HW#2 signal 3: 0.5 10.0
Channel # 6 Most usptream microphone (Mic #7)

“m : : i 0.5 10.0
p Chamel # 15 Mostldownstream microphone (Mic #16)

 

 

 

 

 

 

 

215

References

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Princeton, NJ, 1977, pp. 323.

Adrian, R. J ., “Conditional Eddies in Isotropic Turbulence,” Physics of Fluids,
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Adrian, R. J. and Moin, P., “Stochastic Estimation of Organized Turbulent
Structure: Homogeneous Shear Flow,” Journal of Fluid Mechanics, Vol. 190, pp. 531,
1988.

Adrian, R. J ., Moin, P., and Moser, R. D., “Stochastic Estimation of Conditional
Eddies in Channel Flow,” Report CTR-S87, Center of Turbulence Research, NASA
Ames/Stanford, 1987.

Arie, M. and Rouse, H., “Experiments on Two-Dimensional Flow Over a Normal
Wall,” Journal of Fluid Mechanics, Vol. 1, pp. 129-141, 1956.

Blake, W. K., “Mechanics of Flow-Induced Sound and Vibration”, Applied
Mathematics and Mechanics, Academics, Orlando, FL, 1986.

Bonnet, J. P., Cole, D., Delville, J ., Glauser, M. N. and Ukeiley, L., “Stochastic
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