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THESIS

at:

@9125)”

This is to certify that the
dissertation entitled

INVESTIGATION OF THE WALL-SHEAR-STRESS
SIGNATURE IN A BACKWARD-FACING—STEP FLOW USING
OSCILLATING HOT-WIRE SENSORS

presented by
YONGXIANG Ll

has been accepted towards fulﬁllment
of the requirements for the

PhD. degree in Mechanical Emeerirﬂ

 

 

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6/01 c'JCIRCJDateDuesz-sz

INVESTIGATION OF THE WALL-SHEAR-STRESS SIGNATURE IN A
BACKWARD-FACING—STEP FLOW USING
OSCILLATING HOT-WIRE SENSORS
By

Yongxiang Li

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

INVESTIGATION OF THE WALL-SHEAR-STRESS SIGNATURE IN A
BACKWARD-FACING—STEP FLOW USING OSCILLATING HOT-WIRE SENSORS

By

Yongxiang Li

A new high—ﬁequency oscillating-hot-wire (OHW) sensor for measurement of the
wall-shear—stress magnitude and direction was developed for the purpose of studying the
wall-shear-stress ﬁeld behind an axi-symmetric backward-facing step (BFS). The stress
direction is determined from the phase angle between the imposed and measured
oscillation velocity, and the corresponding magnitude is obtained from the low-pass
ﬁltered signal of the sensor, after removal of the modulating inﬂuence of the oscillation.

Two OHW sensors with oscillation frequency of 2.8 kHz were utilized to capture
and investigate the space-time characteristics of the wall-shear signature behind the BFS
ﬂow at three Reynolds numbers of 4300, 8700 and 13000 (based on step height). One of
the sensors was ﬁxed at approximately 5H (where, H is the step height) downstream of
the step, while the other sensor was moved to acquire data at 14 different streamwise (x)
locations ranging from 0.3H to 10H. The results agreed qualitatively with existing one-
point measurements, such as the mean/rms skin-ﬁiction distribution, forward ﬂow
probability, and power spectra, in planar BF S ﬂows. However, some fundamental
quantitative differences were found including a shorter reattachment length and a larger x
location for the peak rms wall-shear ﬂuctuations. These differences were attributed to the
axi-symmetric nature of the present geometry, transverse curvature of the step, or

differences in the measurement methods.

Consistent with the wall-pressure literature in similar type of ﬂows, the power
spectra of the ﬂuctuating wall-shear stress revealed the existence of two characteristic
frequencies: f = 0.1 and 0.65. The former is associated with low-frequency shear-layer
ﬂapping and the latter corresponds to the passage of the separated shear-layer vortex
structures. The downstream convection velocity of the vortices was inferred from the
two-point cross-correlation of the wall-shear ﬂuctuations. The resulting value was
approximately 0.45 of the ﬁ'eestream velocity, which falls below the range reported in the
wall-pressure literature of 0.5 — 0.6 of the freestream velocity, in investigations of various
separating/reattaching ﬂows in planar conﬁgurations. The lower value is apparently a
result of the deceleration of the vortices as they approach the mean reattachment point by
the stronger adverse streamwise pressure gradient associated with mean-pressure

recovery over the shorter reattachment distance in the axi-symmelric geometry.

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to my advisor Dr. Ahmed Naguib for
giving me an opportunity to work on this project and for his support and encouragement
throughout my graduate studies, for his help in my professional and personal growth, and
for his thorough review of the manuscripts of this thesis.

I would also like to thank other committee members Dr. Dean Aslam, Dr. Brage
Golding, Dr. John Foss, and Dr. Manoochehr Koochesfahani for many helpful comments
and discussions of this thesis.

I appreciate the help from my fellow graduate students. Thanks are due to Laura
Michele Hudy for the great help and contribution in designing, aligning the experimental
model and acquiring the boundary layer data. Antonius Aditj andra has been a great help
during the data acquisition. I have also had many valuable discussions with Mohamed
Daoud.

I am indebted to my father, my mother, my brother and sister for their support
through all these years. Finally, my ultimate thanks go to my wife Yuehui (Christine )
Qin for her understanding and love. Her support and encouragement was in the end what
made this dissertation possible.

This study was made possible through ﬁmding from the National Science
Foundation under grant number CTS-0116907, monitored by Dr. Michael W. Plesniak,

director of the Fluid Dynamics and Hydraulics program.

iv

TABLE OF CONTENTS

LIST OF TABLES .................................................................................................. vii
LIST OF FIGURES ............................................................................................... viii
LIST OF SYMBOLS ............................................................................................ xiii
CHAPTER 1 ................................................................................................................ 1
INTRODUCTION .................................................................................................... 1
1.1 Background ......................................................................................................... 1
1.2 Literature Review ................................................................................................ 4
1 .3 Motivation ......................................................................................................... 1 5
1.4 Objectives ......................................................................................................... 16
CHAPTER 2 .............................................................................................................. 18
EXPERIMENTAL SETUP ................................................................................ 18
2.1 Wind Tunnel ..................................................................................................... 18
2.2 Axi-symmetric Backward-facing—step Model ................................................... 20
2.3 Boundary Layer Proﬁles ................................................................................... 32
2.4 Couette Flow Calibration Facility ..................................................................... 34
CHAPTER 3 .............................................................................................................. 42
OSCILLATING-HOT—WIRE (OHW) TECHNIQUE .......................... 42

3.1 Review of Techniques for Measurement of Direction-reversing Wall-shear
Stress 42

3.2 OHW Principle .................................................................................................. 46
3.3 OHW Response ................................................................................................. 53
3.4 OHW Measurements (Phase Method and Phase Calibration) .......................... 61
3.5 Effect of Wire Height above the Wall .............................................................. 65
3.6 Effect of Wire Oscillation on the Flow ............................................................. 69
3.7 Speciﬁcation of OHW Sensors Use in the Present Experiments ...................... 74
CHAPTER 4 .............................................................................................................. 78
ONE-POINT MEASUREMENT RESULTS ............................................ 78
4.1 Boundary-layer Velocity Proﬁles at Separation ............................................... 78
4.2 One-point Wall-shear Measurements in the Separated/reattaching Flow ......... 85
4.2.1 Mean Reattachment Length ...................................................................... 85
4.2.2 Mean/rms Skin-ﬁiction Coefficient and FFP ........................................... 90

4.2.3 Probability Density Function (pdf) ......................................................... 100

4.2.4 Power Spectra ......................................................................................... 103
4.2.5 Temporal Autocorrelation Function ....................................................... 110
CHAPTER 5 ............................................................................................................ 116
TWO-POINT MEASUREMENT RESULTS ........................................ 116
5.1 Two-point Cross-correlation ........................................................................... 1 16
5.1.1 Two-point Cross-correlation with the Reference Sensor Located near x,
1 16

5.1.2 Filtered Two-point Cross-correlation with the Reference Sensor Located

near x, 127
5.1.3 Two-point Cross-correlation with the Reference Sensor Located near 0.5x,

132
5.2 Frequency Dependence of the Convection Velocities .................................... 137
5.3 Additional Discussion ..................................................................................... 146
CHAPTER 6 ............................................................................................................ 153
CONCLUSIONS AND RECOMMENDATIONS ............................... 153
6.1 Conclusions ..................................................................................................... 153
6.2 Recommendations ........................................................................................... 1 5 8
BIBLIOGRAPHY ................................................................................................. 161

vi

LIST OF TABLES

Table 3.1 Combinations of the sensors and spacers corresponding to different

measurement locations of the movable probe ........................................................... 76
Table 3.2 Measurement positions of the movable sensor ............................................... 77
Table 4.1 Parameters for the boundary layer at separation ............................................. 79
Table 4.2 Reattachment length values ............................................................................ 87

vii

LIST OF FIGURES

Figure 1.1 Backward-facing-step ﬂow structure .............................................................. 2
Figure 2.1 Schematic diagram of the wind tunnel (dimensions in mm) ......................... 19
Figure 2.2 Picture of the axi-symmetric backward-facing—step model ........................... 21
Figure 2.3 Schematic of the BF S model with dimensions (/in mm) ............................... 22
Figure 2.4 Piano-wires and boundary layer trip downstream of the model nose ........... 23
Figure 2.5 Magniﬁed view of the measurement zone showing wall-sensor covers: (a) 3D
view, (b) Cross-section view ..................................................................................... 25
Figure 2.6 Picture of the sensor cover containing two OHW sensors ............................ 26
Figure 2.7 Static pressure distribution downstream of the step at four azimuthal
locations and U00 = 15 m/s ......................................................................................... 27
Figure 2.8 Picture of the setup for checking azimuthal symmetry of the flow ............... 28
Figure 2.9 Azimuthal symmetry of the ﬂow at separation for U... = 15 m/s ................... 29
Figure 2.10 Freestrearn static pressure distribution upstream of the step ....................... 31

Figure 2.11 Typical image of the probe tip and step area used for hot-wire positioning
near the wall: (a) Wire at the initial position, (b) Wire at 0.5 mm above initial

position ................................................................................................................ , ...... 33
Figure 2.12 Picture of the Couette Flow Facility ............................................................ 36
Figure 2.13 Cross Section of the Couette Flow Facility (dimensions in mm) ................ 36
Figure 2.14 Image of the gap between the rotating and stationary discs ....................... 38
Figure 2.15 Comparison between DNS results and Couette ﬂow assumption ............... 41
Figure 3.1 Schematic diagram illustrating the OHW sensor concept ............................. 46

Figure 3.2 Schematic diagram illustrating OHW sensor response for different ﬂow
directions: (a) Flow direction is towards the right, (b) Flow direction is towards the
leﬁ ............................................................................................................................. 48

viii

Figure 3.3 Schematic of a two-layer bending piezo-element ......................................... 48

Figure 3.4 Picture of OHW sensor plug .......................................................................... 49
Figure 3.5 Cross-section view of OHW sensor plug (dimensions in mm) ..................... 50
Figure 3.6 Sketch showing wire connections to the OHW sensor .................................. 51
Figure 3.7 Video image sequences of the hot wire during oscillation: (a) Wire is straight
during oscillation, (b) Wire bends during oscillation ............................................... 55
Figure 3.8 Picture of calibration-facility adaptor plug .................................................... 56
Figure 3.9 Dimensions of adapter plug for calibration of OHW sensors (dimensions in
mm) ........................................................................................................................... 56
Figure 3.10 Magnitude calibration of the hot wire with/without oscillation ................... 57
Figure 3.11 Phase averaged hot-wire output at the oscillation frequency ....................... 58

Figure 3.12 Demonstration of the effect of the ﬂow direction on the phase of the OHW
modulation signal: (a) Modulated velocity signals, (b) OHW driving signals ......... 60

Figure 3.13 A typical direction calibration of the OHW sensor ...................................... 64
Figure 3.14 OHW Operation schematic diagram ............................................................ 65
Figure 3.15 Power spectra of three different runs for wire height y = 48 um ................. 67
Figure 3.16 Histogram of three different runs for wire height y = 48 um ....................... 68
Figure 3.17 Power spectra for wire heights y = 48, 96 and 124 um ................................ 68
Figure 3.18 Histogram for wire heights y = 48, 96 and 124 um ..................................... 69
Figure 3.19 Comparison of power spectra of OHW & conventional hot wire (HW) at
different x positions: (a) x/x, = 0.06, (b) x/x, = 0.92, (c) x/x, = 1.54 .......................... 72
Figure 3.20 Comparison of histogram of OHW & conventional hot wire (HW) at
different x positions: (a) x/x, = 0.06, (b) x/x, = 0.92, (c) x/x, = 1.54 .......................... 74
Figure 3.21 One of the combinations of the sensor cover (in mm) ................................. 75
Figure 3.22 Picture of the sensor cover during insertion in the test mode] ..................... 76

ix

Figure 4.1 Mean velocity proﬁles of the boundary layer at separation .......................... 80

Figure 4.2 Comparison of a mean velocity proﬁle from the current study to that of
turbulent and laminar boundary layers. .................................................................... 81

Figure 4.3 Inner—scaled Mean velocity proﬁles of the boundary layer at separation ..... 84

Figure 4.4 Rms velocity proﬁle of the boundary layer at separation .............................. 84
Figure 4.5 Streamwise distribution of the mean skin-friction coefﬁcient .............. 85
Figure 4.6 Streamwise distribution of forward ﬂow probability .................................... 87
Figure 4.7 Comparison of the mean skin friction coefﬁcient results with other studies:
(a) ReH< 10000, (b) Rey> 10000 ............................................................................. 92
F igurc 4.8 Minimum skin-ﬁiction coefﬁcient versus Re” .............................................. 94
Figure 4.9 Comparison of forward ﬂow probability results with other studies .............. 95

Figure 4.10 Comparison of ms skin-friction coefﬁcient results with other studies: (a)
ReH< 10000, (b) ReH> 10000 ................................................................................... 97

Figure 4.11 Wall-shear pdf at different x positions: (a) ReH= 4300, (b) ReH= 8700, (c)
Re” = 13000 ............................................................................................................ 102

Figure 4.12 Wall-shear power spectra at different x locations: (a) ReH= 4300, (b) Rey=
8700, (c) Re” = 13000 ............................................................................................. 106

Figure 4.13 Power spectra of the ﬂuctuating wall pressure at different x locations (Hudy
(2003), private communication) .............................................................................. 106

Figure 4.14 Wall-shear power spectra plotted on semil-log scale for different x locations:

(a) Rey= 4300, (b) Rey= 8700, (c) Re” = 13000 ................................................... 108
Figure 4.15 Wall-shear power spectra plotted on semi-log scale at x/xr z 0.07 ............ 109
Figure 4.16 Autocorrelation coefﬁcient of the ﬂuctuating wall-shear stress at different x

locations: (a) ReH= 4300, (b) ReH= 8700, (c) Re” = 13000 ................................... 1 12
Figure 4.17 Autocorrelation coefﬁcient of the ﬂuctuating wall-shear stress at x/xr z0.07

................................................................................................................................. 112
Figure 4.18 Contour maps of the auto-correlation coefﬁcient of the ﬂuctuating wall-

shear stress: (a) ReH= 4300, (b) ReH= 8700, (c) Re” = 13000 ............................... 1 15

Figure 5.1 Cross-correlation coefﬁcient of the ﬂuctuating wall-shear stress with the
reference sensor located near x,: (a) ReH= 4300, (b) ReH= 8700, (c) Re” = 13000120

Figure 5.2 Contour maps of the cross-correlation coefﬁcient of the ﬂuctuating wall-
shear stress with the reference sensor located near x,: (a) ReH= 4300, (b) Re”:
8700, (c) Re” = 13000 ............................................................................................. 122

Figure 5.3 Correlation-peak time delay at different x location and associated line ﬁts 125
Figure 5.4 “Local” convection velocity at different x positions ................................... 126

Figure 5.5 Local convection velocity and pressure coefﬁcient versus strearnwise
locations .................................................................................................................. 127

Figure 5.6 Cross-correlation coefﬁcient of the low-pass ﬁltered ﬂuctuating wall-shear
stress with reference sensor located near x,: (a) Re” = 4300, (b) ReH= 8700, (c) Re”
= 13000 ................................................................................................................... 130

Figure 5.7 Contour maps of the cross-correlation coefﬁcient of the low-pass ﬁltered
ﬂuctuating wall—shear stress with reference sensor located near x,: (a) Rey= 4300,
(b) ReH= 8700, (0) Re” = 13000 ............................................................................. 132

Figure 5.8 Cross-correlation coefﬁcient of the ﬂuctuating wall-shear stress with the
reference sensor located near 0.5x,: (a) Re”: 4300, (b) Re”: 8700, (c) Re” = 13000
................................................................................................................................. 135

Figure 5.9 Contour maps of the cross-correlation coefﬁcient of the ﬂuctuating wall-
shear stress with reference sensor located near 0.5x,: (a) ReH= 4300, (b) ReH= 8700,

(0) Re” = 13000 ....................................................................................................... 136
Figure 5.10 Coherence of the ﬂuctuating wall-shear stress with the reference sensor

located near x,: (a) ReH= 4300, (b) ReH= 8700, (c) Rey = 13000 .......................... 141
Figure 5.11 Streamwise dependence of the phase angle at selected frequencies: (a) Re”=

4300, (b) Re”: 8700, (c) Re” = 13000 ................................................................... 144
Figure 5.12 Convection velocities obtained from phase-angle analysis ........................ 145

Figure 5.13 Comparison of near-wall turbulence-intensity proﬁles between measurement
results and one-dimension model prediction (Devenport and Sutton (1991)) ........ 148

Figure 5.14 Comparison between wall shear and pressure power spectra .................... 150

Figure 5.15 Comparsion of ms skin-ﬁiction coefﬁcient from the unﬁltered and high-
pass-ﬁltered signals ................................................................................................. 151

xi

Figure 5.16 Percent 1’“, energy associated with vortex passage .................................... 152

IMAGES IN THIS DISSERTATION ARE PRESENTED IN COLOR.

xii

3.9..

\\

LIST OF SYMBOLS

deﬂection of the piezo-element

amplitude of wall-pressure—gradient ﬂuctuations
amplitude of traveling wave

amplitude of piezo-element driving voltage
capacitance of the piezo-element

mean skin-friction coefﬁcient

minimum mean skin-friction coefﬁcient

rms skin-friction coefﬁcient

high-pass-ﬁltered rms skin-friction coefﬁcient
meaerressure coefﬁcient

diameter of hot-wire sensor

diameter of the separated-shear-layer vortices
frequency

normalized frequency (f' = fx,./ U 00 )

gap between the two discs of the Couette-ﬂow facility
step height

half the height of the fence in fence/splitter-plate ﬂow
boundary-layer shape factor

current through the piezo-element

wavenumber

sensing length of the hot-wire sensor

xiii

’ns

m¢

Poo

Il‘l

f). .
I1172

I’. i
7272

rw

Rz’r’
I? 1.
7172

lie»

1Q611

length of the piezo-element

slope of the correlation-peak locus
slope of phase-angle plot
instantaneous pressure

reference mean pressure

mean surface pressure

power consumption of the piezo-element

power spectra
normalized power spectrum, P; = Pxx /(l/2 p U 020 )

power spectrum of 1w from the reference sensor
0
cross-spectrum of rw between the reference and movable sensors

power spectrum of 1;, from the movable sensor

radius of the rotating disc of the Couette-fow facility

radial position of the hot-wire sensor in the Couette-ﬂow facility
model radius

autocorrelation coefﬁcient

cross-correlation coefﬁcient

9.122

Reynolds number based on gap size between two discs, Re h =
v

Reynolds number based on step height, Re” = U ,0 H/v

xiv

 

Reg Reynolds number based on momentum thickness of the boundary layer at

separation, Ree = U do 6/v

t time
u instantaneous streamwise velocity
u, ﬂow velocity

um measured oscillation velocity

u, friction velocity (u, = ./ rw / p )

u' ﬂuctuating streamwise velocity
u'_v' shear Reynolds stress
U mean streamwise velocity

Uc convection velocity

U0 oscillation velocity

U 00 freestream velocity

U+ inner-scaled mean velocity (U+ =U/u ,)

ﬂuctuating normal velocity

V voltage across the piezo-element

VM forward measurement velocity of an OHW sensor

VIM backward measurement velocity of an OHW sensor

V, radial velocity component in the Couette-ﬂow facility

Va azimuthal velocity component in the Couette-ﬂow facility
Vy wall-normal velocity component in the Couette-ﬂow facility

W width of the piezo-element

XV

Twl

TwZ

streamwise coordinate (measured from the step)
mean reattachment length

Fourier transform of the reference sensor signal

conjugate of the Fourier transform of the movable sensor signal

distance normal to the wall

y in wall units (y+ = yu, /v)

spanwise coordinate

proportionality constant between Cm," and Rey“, CM", = orRef"5
boundary-layer thickness

boundary-layer displacement thickness

oscillating boundary-layer thickness, 50 = 2—V
a)

_ 1/2
, 2
normalized oscillating boundary-layer thickness, 6+ = (TI—0H1“, /p]

momentum thickness of the boundary layer at separation
time delay

dynamic viscosity

kinematic viscosity

ﬂuid density

mean wall-shear stress

wall-shear stress obtained from the reference sensor

wall-shear stress obtained from the movable sensor

xvi

¢ohw

ﬂuctuating wall-shear stress

I
phase difference of rwbetween the movable and reference sensors

phase lag between the oscillating-hot-wire output and the piezo-driving voltage
angular frequency

spanwise vorticity

wall-normal vorticity

coherence

VI I -r'tﬁ

streamwise separation distance between the two sesnors

initial phase angle

 

‘ r

rotational angular velocity

xvii

CHAPTER 1

INTRODUCTION

1.1 Background

The importance of separating and reattaching ﬂows in practical engineering
applications and research of fundamental ﬂuid mechanics is well known. These ﬂows
have been extensively studied using various experimental and numerical techniques
during the past decades. Some recent investigations on the topic include: wall-shear
measurements by Tihon et al. (2001) and Spazzini et al. (2001), wall-pressure
measurements by Hudy et al. (2003) and Lee and Sung (2002), laser Doppler velocimetry
(LDV) by Rinoie et al. (2002), particle image velocirnetry (PIV) by Kostas et al. (2002)
and Scarano et al. (1999), direct numerical simulation (DNS) by Le et al. (1997), and
large eddy simulation (LES) by Fureby (1999).

A backward-facing step (BFS) is one of the simplest geometries in this class of
ﬂows. However, the ﬂow structure downstream of the step is rather complicated. As
Shown in Figure 1.1, there are four main regions in the ﬂow ﬁeld behind the step: a
separated free shear layer, primary and secondary recirculation regions, and a
redeveloping boundary layer.

The boundary layer upstream of the step separates at the step edge due to the
discontinuity in geometry. The separated ﬂow has low-speed ﬂuid on one side and high-
speed ﬂuid on the other side. According to the Kelvin-Hehnoltz instability theory, such
velocity variation across the thin shear layer result in the formation of spanwise vortices

downstream of the step edge. The vortices convect downstream and grow in size via

pairing and entrainment and, eventually, impinge on the wall before continuing to travel
downstream. The formation and growth of the vortices has been visualized by many
researchers and, recently, detected in the PIV measurements by Scarano et a1. (1999) and

Kostas et al. (2002).

 

 

 

 

 

 

 

 

 

 

 

 

 

Separated free _ _ _
shear layer 13“”de
streamline

I Redeveloping

l/l/l/I/l/fl/ boundarVIaVCI'
/
Hﬁ /
i i \

1 —-> x
Secondary Primary Mean reattachment
recirculation recirculation point
region region

Figure 1.1 Backward-facing-step ﬂow structure

The reattachment of the separated shear layer leads to a primary recirculation
region, where the ﬂow is highly turbulent and unsteady. The reattachment mechanism is
driven by the need to balance the entrainment ﬂow into the separated shear layer. In
particular, the curvature of the streamlines associated with the reattachment process
produces an adverse pressure gradient that drives the backﬂow into the recirculation
region to provide the necessary ﬂow for this balance. The curvature of the streamlines
was eliminated in the experiments of Morris (2002) to establish a single stream shear
layer by supplying the entrainment ﬂow externally to create a zero pressure gradient in
the free stream for the separated shear layer to grow. In the back-step ﬂow, the

recirculation region is distinguished from the separating Shear layer by a dividing

streamline, which is a contour of the time-mean stream ﬁmction originating from the
limiting streamline at the step and terminating at the wall as shown in Figure 1.1.

The reattachment point is deﬁned as the position where the dividing streamline
ends on the wall. It is generally understood that instantaneously the reattachment point
moves upstream and downstream due to the ‘ﬂapping’ motion of the separated shear
layer as discussed in Eaton and Johnston (1981), Driver et al. (1987), Spazzini et al.
(2001), Lee and Sung (2002) and Hudy et al. (2003), amongst others.

Just downstream of the step, there is a secondary recirculation region or corner
eddy, which was observed, for example, from mean streamline plots in the particle-
ﬁ'acking-velocimetry measurements in a BF S ﬂow by Kasagi and Matsunaga (1995) and
PIV measurements by Kostas et al. (2002). The plots also demonstrated that the
velocities in this region of the ﬂow were very small.

Finally, a new sub-boundary-layer begins to develop downstream of the
reattachment point. J ovic (1998) investigated the turbulence structure in the redeveloping
boundary layer region in a BFS ﬂow ﬁ'om the measurements of mean static pressure,
Skin-friction and velocity. He found that the near-wall boundary layer reached an
equilibrium turbulent boundary layer by 15 step heights away from the step, and the eddy
viscosity in the outer layer recovered to the equilibrium value more than 100 step heights
downstream of the step.

Although the primary interest of the current study is the near-wall ﬂow region, the
following literature review will ﬁrst summarize the ﬂow structure behind a BFS obtained
ﬁ'om ﬂow ﬁeld measurements with emphasis on more recent studies employing

techniques that yield instantaneous spatial information such as PIV and sensor arrays.

 

 

This is done in order to provide the reader with the background necessary to draw links
between the wall-shear measurements done here and the dominant ﬂow features in
separating/reattaching ﬂows. Next, the discussion will focus on the ﬂow information
extracted from the most recent studies of the near-wall region, including those based

wall-pressure as well as wall-shear.

1.2 Literature Review

Eaton and Johnston (1981) provided a comprehensive review of the mean
reattachment length (x,), structure of the separated shear layer and turbulence data above
the wall for the BFS ﬂow from the literature available at that time. The mean
reattachment length is one of the most important parameters and has been used as a
length scale to normalize the streamwise position in correlations of the mean-pressure
coefﬁcient, skin-ﬁiction coefﬁcient and probability of ﬂow direction. The values of x,
reported in almost all experiments fall in a range which extends ﬁom 4.9 through 8.2 step
heights. According to Eaton and Johnston (1981), x, could be affected by any of the
following parameters: initial boundary-layer state (laminar or turbulent), initial boundary-
layer thickness, freestream turbulence, pressure gradient and aspect ratio (step width over
step height). Eaton and Johnston (1981) concluded that, for a fully developed turbulent
boundary layer, x, is generally 5.5-7.0 step heights, independent of the Reynolds number.
This was conﬁrmed later in the study by Armaly et al. (1983). According to the studies
by de Brederode and Bradshaw (1972), the effect of the aspect ratio could be neglected if
it is larger than 10.

The other two parameters (boundary-layer thickness and freestream turbulence)

have an adverse effect on x,, That is, x, decreases with the increasing boundary-layer
thickness and freestream turbulence. In addition to the above inﬂuences, the blockage
ratio (the area ratio of the model over the test section of a wind tunnel) should also affect
x, for test models suspended in a unifrom stream such as the axi-symmetric BFS used in
the current study (Figure 2.2 and Figure 2.3 display the model details with associated
discussion provided in section 2.2).

To predict x, and other aspects of the back-step ﬂow, researchers have dedicated
special attention to studying the structures. Eaton and Johnston (1981) found that the
structure of the separated shear layer upstream of the reattachment zone was very similar
to a plane mixing-layer, where large-eddy motion resulted ﬁom the roll-up of the shear
layer into spanwise vortices and multiple pairings of these vortices. This picture of the
ﬂow structure could be drawn more accurately with recent advances in PIV, which yields
multi-dimensional, quantitative data of the spatial ﬂow structure. Scarano and
Riethmuller (1999) investigated the instantaneous two-dimensional velocity distribution
using digital PIV measurements of the ﬂow over a BFS at Reynolds number (based on
step height H) of 5000. The vector plot of the instantaneous velocity ﬁeld suggested that
the vortex formation started at x/H =1. Scarano et al. (1999) further analyzed their data
using a pattern recognition algorithm to characterize the spatial occurrence and size of the
spanwise vortices in and around the shear layer. From the statistical characteristics of the
Spatial occurrences, they detected clockwise vortices with vortex diameter (d.) as small as
0.12H and attributed the origin of these vortices to the Kelvin-Helmoltz instability of the
plane shear layer. The vortices approached the bottom wall at about x/x,=0.58, then

continued to travel in the streamwise direction.

Surprisingly, Scarano et al. (1999) also found signiﬁcant number of counter-
clockwise rotating vortices with size of d/H = 0.12 in the free shear layer downstream of
x/x, = 0.67. They attributed the generation of these vortices to the three-dimensional
instability and breakdown of the primary clockwise rotating vortices. The vortex
characteristics were further analyzed through application of conditional averaging
techniques to the results of the pattern recognition process. The conditionally averaged
ﬂow ﬁeld exhibited elliptic and quasi-circular streamline patterns at x/H of 1.4 and 7,
respectively, which proved the existence of the expected swirling motion associated with
the spanwise vortices. The peak vorticity at the vortex core was also found to be 3 - 5
times larger than the mean vorticity level in the region 5 < x/H < 8.

Kostas et al. (2002) performed another PIV investigation of a two-dimensional
BFS ﬂow at Reynolds ntunber, based on step height and ﬁee stream velocity, of 4660.
Their work not only Showed the measurements of velocity and vorticity in the x-y and x-z

planes, but also examined the contribution of the various length scales of the ﬂow

structure to the in-plane Reynolds stress (u'v') and turbulent kinetic energy dirstribution
within and around the separation bubble. The instantaneous vorticity ﬁeld depicted
vortical structures in both staggered (i.e. the vortices are discretely distributed) and in-
line (i.e. the vortices are next to each other like a train of vortices) conﬁgurations,
illustrating the dominance of vortices in the ﬂow and their mutual interaction.
Measurements of the Reynolds stress and turbulent kinetic energy production showed
that the vortex interactions occurred at locations of the peak values of turbulent kinetic
energy production and Reynolds stresses. Upstream of reattachment, small scales were

found to be responsible for the Reynolds stress and turbulent kinetic energy production.

On the contrary, the Reynolds stress and turbulent kinetic energy production were
predominantly affected by the large-scale structures downstream of reattachment.

Kostas et al. (2002) also used the mean vorticity distribution to identify the
separated shear layer by locating the region of concentrated negative spanwise vorticiy
(0);). In this manner, the shear layer growth could be found from the width of the contour
plot of the mean vorticity distribution in the x-y plane. As anticipated, this showed the
shear layer to be very thin immediately downstream of the step, but it continued to grow
along the streamwise direction until the reattachment position. Downstream of the mean
reattachment point, the vorticity decreased gradually with increasing x.

On the other hand, a non-negligible level of wall-normal vorticity (0),) was
detected in the x-z plane located at y = H and near the step. Similar levels of (1), could not
be found at higher and lower y locations (0.5H and 1.5H to be speciﬁc). The authors
suggested that the observed a), at y = H reﬂects the reminiscence of the near-wall
streamwise vortices associated with the boundary-layer low-speed streaks. As these
vortices are swept into the separated shear layer, they get kinked and stretched exhibiting
re—orientation of (1),, (streamwise vorticity) into (1),. Evidence in support of this scenario
was provided from the two-point spanwise correlation function of coy in the separated
shear layer at y = H and x = 2H. The correlation results indicated the existence of a
quasi-periodic structure in the spanwise direction with an average wavelength equal to
the typical spacing of the boundary layer streaks.

In addition to PIV experiments in a BF S, Furuichi et al. (2003) investigated the
spatio-temporal velocity ﬁeld of the ﬂow through an axisyrnmetric sudden expansion in a

pipe using an ultrasonic velocity proﬁler. They plotted the instantaneous reattachment

location and showed its obvious ﬂuctuation. They also attributed this ﬂuctuation to the
ﬂapping of the separated shear layer. A particularly interesting ﬁnding was that the
amplitude of the ﬂuctuation in the reattachment position was smaller for turbulent-
boundary—layer separation than for laminar or transitional boundary layer separation. The
main frequency of the ﬂapping motion was found from the power spectrum of the
streamwise ﬂuctuating velocity around the reattachment region. The peak of the power
spectrum was located at f H/Uc z 0.01 (or f x/Uw z 0.1), where Uc is the downstream

convection velocity of the vortical structures (extracted from velocity measurements) and

U ,0 is the freestream velocity upstream of the step. As have most other researchers, the

authors obtained a downstream convection velocity of 0.5U..o in the turbulent regime.
They reasoned that the ﬂapping motion resulted from the interaction between the
downstream-traveling shear-layer vortices and part of the recirculation ﬂow moving
upstream. In their wavenumber-frequency plots, up and downstream convection
velocities were observed in the separation of the transitional boundary layer. However,
only downstream convection velocity was detected for the separation of laminar or
turbulent boundary layers.

The unsteady behavior of the reattaching shear layer of the BFS ﬂow was studied
systematically by Driver et al. (1987). From the streamline plots conditioned on short,
intermediate and long reattachment cases (where the reattachment length was detected
using concurrent pulsed wire measurements that was operated as a thermal tuft), the
authors identiﬁed the amplitude of the ﬂapping motion to be less than 20% of the
thickness of the shear layer at reattachment. The ﬁ'equency of the ﬂapping motion was

investigated using wall-pressure and near-wall streamwise velocity measurements. In the

power spectra plots of the pressure and velocity, there was a peak at a low frequency of
fir/Ugo = 0.18, which was believed to be the frequency of the ﬂapping motion. However,
the energy level at this ﬁequency was very low, compared to that of the dominant
frequency in the spectrum (fx,/U.,o = 0.6), which was the same as the characteristics
ﬁequency of the vortical structures in free shear layers investigated earlier by Winant and
Browand (1974). Thus, the low frequency ﬂapping motion actually did not contribute too
much to the total energy of the ﬂow.

As Driver et a1. (1987) did, many researchers focused on the investigation of the
near-wall ﬂow structure. Two common methods for studying wall/near-wall effects of the
ﬂow structure are wall-pressure and wall-shear measurements. Knowledge of the surface
pressure ﬁeld is necessary to understand and control vibration and noise generated due to
pressure ﬂuctuations. Measurements of the wall shear, on the other hand, directly deﬁne
the reattachment point, where the mean wall-shear stress is zero. There is a wealth of
information available on one- and two-point measurements of the wall pressure in
separating/reattaching ﬂows. More recently, Hudy et al. (2003) conducted more
comprehensive measurements using wall-pressure-array measurements in a splitter plate
with a fence conﬁguration. Lee and Sung (2001, 2002) did multiple-arrayed ﬂuctuating
pressure measurements in a BFS ﬂow model. Compared with the one- and two-point
measurements, the array measurements enabled attainment of more details of the spatio
as well as temporal information of the ﬂow structure. Therefore, only these references of
wall-pressure measurements are reviewed here.

Lee and Sung (2001, I) used an array of 32 microphones to measure the pressure

ﬂuctuations in the separated and reattaching ﬂow over a BFS at Reynolds number (based

on step height) of 33000. A variety of signal processing techniques were employed in the
analysis of the array data: time-domain and spectral methods in Lee and Sung (2001, I),
unsteady wavelet analysis in Lee and Sung (2001, II) and spatial box ﬁltering and
conditional averaging in Lee and Sung (2002). Consistent with most one- and two-point
measurements, Lee and Sung (2001, I) identiﬁed two peaks in the pressure power spectra
at: fH/U... = 0.015 (or ﬁr,./U..o = 0.11) and fH/U... = 0.068 (or fx/U.o = 0.5). These
frequencies corresponded to the ﬂapping motion and passage of the large-scale vortical
structures, respectively. From their coherence and wavenumber spectra plots, they
suggested that the pressure ﬂuctuations were primarily associated with the large-scale
vortical structures and boundary-layer—like decaying modes at high frequencies. Their
later analysis in Lee and Sung (2001, H) showed that the pressure ﬂuctuations were
predominantly generated by the vortices, which were associated with two types of
signatures: global oscillation, where all pressure ﬂuctuations were in phase across the
entire separation bubble, and propagating disturbance, with a convection velocity of
0.6Uw. The two modes are synchronized with the ﬂapping motion that induced expansion
and contraction of the separation bubble. The authors provided further support for these
observations using conditional measurements of the velocity ﬁeld that are based on wall-
pressure events in Lee and Sung (2002).

Hudy et al. (2003) investigated the space-time character of the ﬂuctuating wall
pressure on a splitter-plate that was attached to and placed in the wake of a fence. The
Reynolds number based on the fence height above the splitter plate was 7900. The
majority of the data analysis was based on measurements from 28 microphones on the

centerline of the plate. The autocorrelation of the pressure ﬂuctuations implied the

10

existence of two time scales. A large time scale dominated the pressure signature from
the fence to 0.25x,. Downstream of 0.25x,, a shorter time scale prevailed. The former was
attributed to the ﬂapping motion and the latter corresponded to the Shear-layer structures.
These observations were consistent with the power spectra results. More speciﬁcally, the
peak of the power spectra obtained close to the fence was found at f(2Hj/Uw= 0.02-0.03
(i.e. ﬁts/U“,= 0.12-0.18); Farther downstream, the spectral peak shifted to a frequency of
f(2Hﬂ/U¢,= 0.1-0.15 (i.e. fir/U“,= 0.6-0.9). The convective nature of the wall-pressure
signature was clearly illustrated in their cross-correlation and phase-angle results. For x >
0.25x,, an average convection velocity of 0.57 U... in the downstream direction was found.
More interestingly, Hudy et al. (2003) also found that an upstream convection velocity of
0.21 U... was associated with low-frequency pressure ﬂuctuations in the region extending
from the fence to 0.25x,. This, combined with other results, led them to propose that the
ﬂapping motion was produced by the existence of an absolute instability of the
recirculation bubble.

Compared to wall-pressure measurements, there is little information available on
the turbulent wall-shear-stress ﬁeld in the literature. Since the near-wall ﬂow behind a
backward-facing step is highly turbulent and ﬂow reversal occurs frequently, it is
difﬁcult to conduct measurements of the wall-shear stress in this region. Although
modern techniques such as PIV and LDV are capable of capturing the unsteady direction-
reversing ﬂow information downstream of the BFS, their capability to conduct near-wall
measurements is limited or they may require provisions that are too elaborate and/or
costly to resolve the near-wall ﬂow. F emholz et al. (1996) did a comprehensive review of

techniques for measurement of the wall-shear stress in wall-bounded and separated ﬂows.

11

 

They concluded that only wall hot wires and pulsed wires were capable of measuring the
unsteady wall shear. But wall hot wires can’t be used in cases involving ﬂow reversal.
Wall pulsed wires, on the other hand, measure the wall-shear magnitude and direction,
but their frequency response is limited to tens of hertz. As a result of these difﬁculties,
only recently, information other than the rms (root mean square) and instantaneous
direction of the ﬂuctuating wall-shear in backward-facing—stcp ﬂows have become
available in Spazzini et al. (2001) and Tihon et al. (2001). A review of the pertinent
literature on mean and ﬂuctuating wall-shear measurements in BPS ﬂows follows.

The wall shear is generally presented in the following normalized form in the

literature:

 

t
C = W (1.1)
f 1/2pU020

where Cf is the skin-friction coefﬁcient, rw is the wall-shear stress, p is the ﬂuid

density. Westphal et al. (1981) measured mean and ms skin-ﬁiction coefﬁcients in a
BFS ﬂow using a wall pulsed-wire. Devenport and Sutton (1991) investigated an axi-
symmetric sudden expansion also using a wall pulsed-wire. Later, Jovic and Driver
(1995) studied the effect of Reynolds number on the mean skin ﬁiction coefﬁcient in a
BF S ﬂow using a Laser-oil Flow Interferometry (LOI). Most recently, Tihon et al. (2001)
investigated wall-shear vectors behind a BFS using a three-segment electrodiffusion
probe in a water channel, while Spazzini et al. (2001) measured the wall shear to
investigate the ﬂapping motion in a BF S ﬂow using a special wall-mounted double hot-
wire probe.

In all of the above studies, the distribution of the mean skin-friction coefﬁcient

12

'n I 5.11 “

 

was similar. The mean Cf has a small positive value from the step to a certain xo/xn the
value of which was 0.34 in Tihon et al. (2001), 0.2503 in Spazzini et al. (2001), 0.23 in
Jovic and Drivier (1995), 0.10 in Devenport and Sutton (1991), and 0.13 in Westphal et
al. (1981). Downstream of this position, the mean Cf crosses zero and becomes negative
beneath the recirculation zone from xo/x, to x/x, = 1. The peak value of negative skin
ﬁiction was located at x/x, of 0.69 in Tihon et al. (2001), 0.60 in Spazzini et al. (2001),
Jovic and Drivier (1995), and Westphal et a1. (1981) and 0.55 in Devenport and Sutton
(1991). Downstream of reattachment, the mean Cf increases ﬁ'om zero to a certain
positive value at x/x, around 1.5 in most studies. Finally, farther downstream of x/x, =1.5,
the mean Cf decays very slowly.

In the mean skin-ﬁiction distribution, there are two positions where the mean Cf
value is equal to zero. The ﬁrst one indicates the time-mean location of the separation
point at the boundary between the primary and secondary recimlation regions. The
second one corresponds to the mean reattachment location.

The magnitude of the mean Cf was found to depend on the Reynolds number.
This dependence was investigated by Jovic and Driver (1995). They found that the skin-
ﬁiction coefﬁcient magnitude decreased with increasing Reynolds number. The
minimum skin-friction coefﬁcient (Cﬂm), i.e., that corresponding to the largest negative
shear stress, was found to have a power law relationship with Reynold number with an
exponent of —1/2. This suggested that the ﬂow near the wall within the recirculation
region Was viscous-dominated or laminar-like. Tihon et al. (2001) also found a similar
relationship and supported the same conclusion.

The rms skin-friction coefﬁcient results are also similar in the literature. CM,”

13

 

exhibits very low values beneath the secondary recirculation region, but it increases
sharply to a maximum value at x/x, of about 0.78 in Tihon et al. (2001), 0.70 in Spazzini
et al. (2001), 0.70 - 0.90 in Devenport and Sutton (1991), 0.60 in Westphal et a1. (1981),
and 1.0 in Adams and Johnston (1988). Downstream of this position, the rms skin-friction
begins to decay slowly, maintaining a steady value after reattachment. Overall, it may be
stated that the peak rms skin-ﬁiction coefﬁcient is located between the location where the
mean skin-friction coefﬁcient reaches its negative peak and the mean reattachment point.

The percentage downstream/upstream ﬂow, also referred to as forward/reverse
ﬂow probability, is another quantity used to determine the reattachment position in a BFS
ﬂow. The method was used by Eaton (1980), who employed a thermal tuﬁ to detect the
wall-shear direction, Westphal et al. (1981), Tihon et al. (2001) and Spazzini et al.
(2001). The location of zero mean skin-ﬁiction coefﬁcient was found to be consistent
with that of a forward or reverse ﬂow probability of 50%, as demonstrated in the
literature. Furthermore, the dependence of the probability on the location within the
separation bubble is consistent with that of the mean skin-ﬁiction coefﬁcient. In
particular, most of the time, the ﬂow is towards the downstream direction beneath the
secondary recirculation region and in the upstream direction below the primary one.
Downstream of reattachment, in the redeveloping boundary-layer region, the downstream
ﬂow probability is larger than 50%.

Tihon et al. (2001) and Spazzini et al. (2001) obtained time-resolved information
of skin-ﬁiction measurements extending ﬁ'om the step to beyond x,. From the power
spectra of ﬂuctuating wall-shear stress, Tihon et al. (2001) found that the frequency

(ﬁr/Ugo) of the peak value in the spectra varied ﬁom 0.15 to 0.65 between x/x, =0.3 and

14

0.8. These frequency values were consistent with those obtained from wall-pressure
studies of Lee and Sung (2001, I) and Hudy et al. (2003). Spazzini et al. (2001) focused
their investigation on the low-frequency motion: i.e., the ﬂapping motion. As found in
Hudy et al. (2003), the wall-shear ﬂuctuation was also dominated by low-frequency
unsteadiness from the step to x/x, z 0.25-0.30, where the secondary recirculation zone
ended. Downstream of x/x, 5:: 0.25, the spectra were dominated by a higher frequency
component. From wavelet analysis of the measurements and ﬂow visualization, Spazzini
et al. (2001) proposed that the cause of the ﬂapping motion was a cyclic motion of

growing and breakdown of the secondary recirculation bubble.

1.3 Motivation

Although there is a wealth of information on the ﬂow structure above the wall,
and associated wall-pressure in BFS ﬂows, only a few studies of the turbulent wall-shear
stress are available. The latter are also limited to one-point measurements as seen from
the above review. Insight into the spatio-temporal character of the wall-shear-stress ﬁeld
can’t be examined ﬁom these measurements and questions pertaining to this character
remain open. For example, is the spatial and temporal behavior of the wall-shear-stress
ﬁeld consistent with the time and spatial scales of the separated Shear layer identiﬁed in
the above literature review? Do the low-frequency wall-shear-stress ﬂuctuations contain
both up and downstream propagating disturbances as found in wall-pressure
measurements of Hudy et al. (2003)? Addressing such issues is important to answer some
of the open questions concerning this classical, yet important, ﬂow.

The database to be accumulated here to answer these questions will also be

15

beneﬁcial in supporting modern development in computational ﬂuid dynamics (CF D).
CFD has developed rapidly over the past decade. Direct Numerical Simulation (DNS)
and Large Eddy Simulation (LES) are able to provide the detailed ﬂow structure
information. However, computational results need to be veriﬁed and validated with
experimental data. Spatio-temporal wall-shear measurements to verify wall inﬂuences of
computed turbulent structures are still rare, if existent at all. Therefore, the current study
contributes information that is useful for benchmarking of DNS and LES codes. The axi-
symmetric geometry of the back-step model used here was selected in part due to
suitability for comparison with computations based on periodic boundary conditions. It
is noted here also that there are very few experimental studies of BFS ﬂow in the

literature that are based on such an axi-symmetric geometry.

1.4 Objectives

The corresponding objectives for the current work are summarized as follows:

1. To develop a new hot-wire-based sensor for near-wall measurements in
unsteady direction-reversing ﬂows with frequency content up to hundreds
of hertz or a few kilohertz.

2. To develop a low-speed calibration facility for calibrating near-wall hot-
wire-based sensors.

3. To measure the wall-shear stress in an axisymmetric backward-facing-step
ﬂow at different Reynolds numbers. Information such as the mean/rms of
the wall-shear stress, ﬂow-direction probability, power spectra, etc, will be

collected and compared with the literature.

16

To conduct two-point measurements in the backward-facing-step ﬂow to
obtain spatio-temporal information of the wall-shear stress such as the
two-point space-time cross-correlation function. The spatio-temporal
information from the wall—shear will be compared with those obtained

from wall-pressure and velocity-ﬁeld measurements in the literature.

17

CHAPTER 2

EXPERIMENTAL SETUP

This chapter discusses the experimental setup and measurement techniques. The
setup consists of the wind ttmnel, axi-symmetric backward-facing—step (BF S) model and
wall-shear-stress calibration facility. Each of these components is presented in details in

sections 2.1 through 2.4.

2.] Wind Tunnel

As shown in Figure 2.1, the wind tunnel is composed of ﬁve sections: contraction,
test section, pre-diffuser, difﬁrser and axivane fan driven by a GE 15 HP DC wound
shunt motor and controlled using a GE gp-100 adjustable speed controller. Each section
is supported on wheels and could be easily moved around. The ﬂow is drawn into the
contraction section and discharges to the atmosphere at the exit of the fan section. The
total length of the tunnel is approximately 9 m and its centerline height is 1.346 m.

The duct of the contraction section is made of a composite lamination of
ﬁberglass-reinforced plastic and a rigid wood core material molded over precision
tooling. The contraction has a 6.25:1 area ratio and an inlet section that is 1.549 x 1.549
m in height and width. There is a precision aluminum honeycomb and three graduated,
high porosity screens mounted in front of the contraction to reduce the turbulence
intensity of the intake air. The freestream turbulence intensity in the test section was

found to be less than 0.45% from measurements in the velocity range of 5 -— 15 m/s.

l8

Fan Diffuser Pre-Diffuser Adj-11.591316 Test Contraction
\ Ceiling Section

”‘7‘ ’/—*-¢‘/=

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I‘_ 1 486—>~——1 829———><—1005—ﬁH——1 829—><———2096——>
Figure 2.1 Schematic diagram of the wind tunnel (dimensions in mm)

The test section is 1.829 m long, and its upstream end is bolted together with the
contraction section with a sponge ﬁlling material in-between for sealing purposes. An
adjustable ceiling that is hinged at the ﬂow entrance is used to establish zero pressure
gradient in the freestream, leaving a cross sectional ﬂow area of 609.6 x 609.6 mm at the
upstream end. Two 12.7 mm wide slots in the middle of the ceiling are used for insertion
of hot-wire and pitot-tube probes into the tunnel. The slots could also be used to install
thin plates ﬁtted with static pressure taps to monitor the pressure distribution along the
freestream ﬂow inside the tunnel.

The pre-diffuser is 990.6 mm long. The bottom wall has an angle of 6° with
respect to the horizontal direction. The upstream end of this section is attached to the test
section, while the downstream end leads to the diffuser. An air gap, covered with rubber
filling material, is left between the latter and the pre-diffuser to minimize vibration
transmission from the fan/motor. The diffuser, which is used to recover the pressure of

the exiting ﬂow, is 1.651 m long and has an angle of 588° and area ratio of 1.9. At the

19

downstream end of the diffuser, a fan-diffuser adapter converts the square cross section

of the diffuser into a circular area to match the inlet geometry of the fan.

2.2 Axi-symmetric Backward-facing-step Model

Figure 2.2 shows a picture of the axi-symmetric backward-facing-step model
supported on a traverse table. The corresponding schematic diagram of the model is
plotted in Figure 2.3 to show the dimensions of all components. The total length of the
model is 2.383 m. At its upstream end, the model is ﬁtted with an aluminum half-
spherical nose made with a radius of 63.5 mm. The 2D section is made from 127.0 mm
diameter aluminum tubing with a wall thickness of 6.35 mm. The step section is also
made from the same aluminum tubing with 101.6 mm outside diameter and 6.35 mm in
wall thickness. The difference between the tubing diameters creates a nominal step height
(H) of 12.7 mm between the rotating and step sections. Downstream of the latter, the
motor section is a thick steel tubing with 101.6 mm outside diameter and 69.9 mm inner
diameter. The bottom surface of the tubing is ﬂattened and seated onto the top of a
support bar and the two parts are attached ﬁrmly together using eight 3/8-inch screws.
Finally, the downstream end of the model is terminated with conical tail section to avoid
abrupt termination of the geometry.

The dimensions of the steel support plate are 19.05 mm thick, 482.6 mm long and
501.65 mm high. The bottom of the plate is connected to the top of the traverse table,
providing a rigid support of the model at the downstream end. At the opposite end, four
piano wires located at about 90° with respect to each other are attached to the 2D section

of the model, as shown in Figure 2.4, to prevent deﬂection due to the long overhanging

20

portion of the model. The wires pass through narrow slots in the windows of the test
section and are straightened with tension and connected to a support framework outside
the tunnel. Since the diameter of the wires is only 0.99 mm, their effect on the ﬂow is
anticipated to be negligible. This will be veriﬁed later by checking the azimuthal
symmetry of the ﬂow around the model in the ﬂow region of interest.

Tail

i\-‘Iotor Section

Step

 

Figure 2.2 Picture of the axi-symmetric backward-facing-step model

The traverse table has two angle adjustments: yaw, along the spanwise direction,
and tilt, along the vertical direction. In addition, three translation adjustments are
available in the vertical, streamwise and spanwise directions. These positioning
ﬂexibilities ensure that the BFS model is installed at the center of the wind tunnel and

aligned with the freestream velocity direction.

21

Ea as 222.8% as, 388 mam 2e .8 23883 2 8am

 

mdwmm

 

III teem ll

TI :3 1!

 

 

N132
I deh IVA. m.omm 1L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.2: G

 

 

 

 

 

062 9

mac m

22

 

Piano wires

Figure 2.4 Piano-wires and boundary layer trip downstream of the model nose

The boundary layer is tripped using sandpaper just downstream of the nose
section (see Figure 2.4) and develops to 2D (axi-symmetric) turbulent boundary layer
state at the separation point. The development length of the boundary layer was selected
by examining the study of Driver and Johnston (1990) of non-separating two- and three-
dimensional turbulent boundary layers over an axi-symmetric model. Based on this
examination, a length of about 1.2 m was found to be sufﬁcient for the transition of the
boundary layer to a turbulent state for the range of ﬂow speeds anticipated. More
speciﬁcally, the length from the tip of the half sphere nose to the edge of the Step is 1.219
In. On the other hand, the distance from the step edge to the support plate edge is 487.7
mm, or 38.4 step heights. For reference, Eaton and Johnston (1981) found the
reattachment length for a planar two-dimensional backward-facing-step ﬂow to fall in the
range of 4.9 - 8.2 step heights. As will be seen later, the largest reattachment length found

in the current measurements is approximately 4.9H. Hence, the support plate is almost

23

eight reattachment lengths, or 40H, downstream of the step.

The measurements on the model were taken primarily in the region extending
from the step to 127.0 mm (i.e. 10 step heights) farther downstream. Figure 2.5 shows a
magniﬁed view of the measurement region. Within this zone, four sensor covers are
located at the top, bottom, left and right of the cylindrical model surface. There are static
pressure taps in sensor covers numbered 2, 3, and 4 in Figure 2.5. The slot for sensor
cover 1 could be used to install two oscillating-hot—wire sensors for two-point wall-shear
measurements (as shown in Figure 2.6) or static pressure taps mirroring those in cover 3.
The static pressure taps were primarily used to verify the alignment of the model parallel
to the ﬁeestream direction.

For the sake of the alignment process, 24 static pressure taps are installed in
sensor cover 2. The ﬁrst of these is located at 4.76 m away from the edge of the step
with all taps equally spaced at an inter-sensor spacing of 4.76 mm. For the remaining
sensor covers, only eight static pressure taps are used in each cover, with the ﬁrst tap
positioned at 19.05 mm downstream from the edge of the step. The corresponding
distance between two neighboring taps is 9.52 mm. All of the signal wires and static
pressure tubes are routed inside the outer model shell, coming out from two holes at the
upstream junction between the outer shell and support plate. The cables and tubes then
run the height of the support plate, covered from the ﬂow by a shroud in front of the plate
as shown in Figure 2.2. Finally, all static-pressure tubes are connected through a 48-port
scarrivalve to a Setra Model 239 pressure transducer with a range of 0-5 inch water (i.e.

1245 Pa).

24

(a) Sensor Cover 2

 

(b) A-A Cross-section View

Sensor Cover 2 with 24
pressure taps

 
   
   

Sensor Cover 3
with 8 pressure
taps Sensor Cover 1
with OHW
sensors (or 8
pressure taps)

 

 

Sensor Cover 4 l
with 8 pressure I
taps

Figure 2.5 Magniﬁed view of the measurement zone showing wall-sensor covers: (a) 3D
view, (b) Cross-section view

25

 

Figure 2.6 Picture of the sensor cover containing two OHW sensors

The model was aligned by adjusting the tension of the piano wires and the
position of the traverse table. First, coarse alignment of the model was done using a level
gauge and tape-measure in order to position the model centerline at the center of the test
section and make it parallel to the nominal ﬂow direction. This was followed by ﬁne
alignment, which relied on the static pressure measurements from sensor covers 1- 4. The
static wall pressure along the azimuthal direction of the model should be equal at the
same streamwise (x) location if the model is perfectly aligned. Figure 2.7 shows static
pressure measurements after alignment of the model for a ﬁ‘eestream velocity of 15 m/s.
The labels of east, north, west and south in the legend refer to those measurements from
sensor covers 1 —— 4, respectively. The abscissa represents the streamwise location
downstream of the step normalized with the reattachment length (x,), the determination of
which will be discussed in details in Chapter 4. The ordinate represents the pressure

coefﬁcient Cp deﬁned as follows:

26

Cp =-—————ps'p’2 (2.1)
I/Zono

where p, is the surface static pressure along the model, p, is a reference pressure
measured at the entrance to the test section, ,0 is the air density, and U... is the freestream
velocity. The four curves in the ﬁgure collapse at most x positions. The largest
discrepancy is found at x/x, around 1 and is less than 8% of the total Cp variation. It is
believed that this discrepancy is caused by small increase in the pressure on the lower

(south) side of the model resulting from ﬂow stagnation on the Shroud.

 

 

0.15 .
-6- North

o1-e—South
“-A—East

0.05»

 

 

 

Figure 2.7 Static pressure distribution downstream of the step at four azimuthal
locations and U... = 15 m/s

To further examine the alignment of the model, the azimuthal uniformity of the
ﬂow just upstream of separation was checked using a single hot-wire probe. The sensor

was positioned at a certain height y above the wall and less than 0.5 mm upstream of the

27

step to measure the mean streamwise velocity U at different azimuthal angles ((1)) while
maintaining the same height. To rotate the probe around the centerline of the model, a
ring holder was fabricated with an inside diameter that matches the outside diameter of
the model downstream of the step. The step ring was made of two halves that could be
split and re-assembled around the model such that the ring's center coincides with that of
the model, as shown in the photograph in Figure 2.8. As also evident from the ﬁgure, the
hot-wire probe was mounted on the ring holder, which is located well downstream of the
step to avoid interference with the ﬂow at the point of measurements. By rotating the
ring in the azimuthal direction it was possible to move the probe to different azimuthal

angles.

   
   
 

: Step
V Traversing ring

Hot-wire probe

Figure 2.8 Picture of the setup for checking azimuthal symmetry of the ﬂow

Azimuthal measurements for four different heights are illustrated in Figure 2.9.

The heights are normalized by the local boundary layer momentum thickness 19

28

(determination of 19 will be discussed in section 4.1). For each height, U is plotted versus
the azimuthal angle (with zero angle corresponding to probe position on the top side of
the model) and plotted as percent of the average velocity over all angles. At the three
lower y positions, the velocity varies within i10% of the average value of the
measurements. However, since these measurements were within the boundary layer, the
error could also stem from probe-height variation caused by inaccuracies of the azimuthal
traversing method. At y/0 = 10.59, which is almost at the edge of the boundary layer, the

variation is less than 4%.

120.4 MW, ~ ,
‘e',L/9,=1'65

. —91 we = 5.;9

   

910" § E ---------- ~
I z ; I .
80 ,,-,,L i__, .4 80' , h , . 7 1,.-. ,,
0 90 18 270 360 0 90 180 270 360
11, (degree) 4) (degree)
1207 120‘ ~

if, ., : i ,. ,
—A—Wy/6= 7.94 II I —1— we = 10.59

 

o 90 180 2Io 360 0 ’9‘0 7180 270 860
4, (degree) :1 (degree)

Figure 2.9 Azimuthal symmetry of the ﬂow at separation for U... = 15 m/s

29

After alignment of the model, the ceiling of the test section was adjusted to obtain
zero pressure-gradient condition in the freestream upstream of the step. For this purpose,
two 3.2 mm thick by 12.7 mm wide plates instrumented with 16 static pressure tapes
were ﬁtted into the probe-access slots along the ceiling centerline. The plates were ﬂush
with the ceiling surface. In addition, 66 micron-thick kapton tape was used to cover any
possible gaps between the inserts and the ceiling. The location of the ﬁrst static pressure
tap (Cl) was 19 mm downstream of the tip of the model. The pressure taps C1 through
C7 were 76.2 mm ﬁ'om center to center. Because of a discontinuity in the probe-access
slot, the distance between taps C7 and C8 was 299.2 mm. The remaining taps (C8 - C15)
were also spaced 76.2 mm ﬁom center to center. The results for freestream velocities of
5, 10 and 15 m/s are given in Figure 2.10. The x locations of the pressure taps are _
normalized as follows: (x-xJ/Q where x is the distance to the tip of the model, x, is the
position of the step, and 6 is the momentum thickness of the boundary layer at 0.5 mm
upstream of the step. The two vertical broken lines included in the ﬁgure indicate the
positions of the model tip and the step ((x-xJ/Q = 0). The error bars are used to indicate i
10% variation of the measured values. The results demonstrate that a nominally zero
pressure gradient, corresponding to a constant pressure value to within 3:10%, is
established for at least 3006 upstream of the step. The initial pressure drop around the
nose of the model is presumably caused by ﬂow acceleration associated with the fairly
small model blockage of 3.4% (where blockage is deﬁned as model cross sectional area

upstream of the step divided by test section ﬂow area).

30

I _ um: 5rn/s
nor

'0.“ ' o

__ _l_ ____

oe-ooe -

-093.

 

-o.1,

.V2L_

 

 

_, L_._.-______ -11..__.._n__.-. . izz.,,x,AJL_
0'12 -500 -400 -300 -200 -100 0

o. _.- . .. .____..- . —
I

—_J_.

-0.02

-o.04I I
(ft-0.06: I
-o.oa» I

-0.1L

 

 

 

-0.12

bison—ELIOTLSM -360 Ciao—”T0

I

-o.o2I

I:

1

MI

' 0

0° -o.061

-0.08r

 

 

 

LI

-500 .460 4430 4’66“ i1—00_—_0
x-x,

19

-0.12

 

 

Figure 2.10 Freestream static pressure distribution upstream of the step

31

2.3 Boundary Layer Profiles

Boundary layer mean-velocity proﬁles were obtained at 0.5 mm upstream of the
step using a single hot-wire probe. The hot-wire probe was installed on a traversing
mechanism driven by a TMG Model 56188-01 hybrid stepper motor. The traverse system
was calibrated against a dial indicator with a 12.7 um resolution. The calibration results
Showed that the system traversed a distance of 3.3 pm per motor step, leading to a very
ﬁne control of the probe motion.

The initial position of the hot-wire probe above the surface was measured using
an imaging system. The systemconsisted of a 768 x 494 pixel Sony XC-75 CCD camera
connected to a PC computer via a National Instruments IMAQ PC1-1411 frame grabber.
A Perkin Elmer MVS-2060 strobe light sent a short duration, high intensity light pulse to
illuminate the probe tip and step area from the opposite side of the camera (as depicted in
Figure 2.11) every time when an image was acquired. The distance between the tip of the
wire and top of the model was found in pixels and then converted to m using a
calibration constant. The latter was found by moving the probe a known distance of 0.5
mm and ﬁnding the number of pixels between the probe's initial and ﬁnal locations. The
calibration constant was typically about 4-6 urn/pixel.

Once the initial position of the probe was found, it was moved up to the
freestream and calibrated against a pitot-static tube. The stagnation and static ports of the
pitot tube were connected to the high and low inputs, respectively, of a Setra Model 239
pressure transducer. Two transducers were available: one with a pressure range of 0.5
inch of water (i.e. 124.5 Pa), and the other with a higher range of 5 inch of water, or 1245

Pa. Both transducers had an output of ﬁve volts at the full-scale pressure. Thus, the

32

former transducer, with its smaller range but higher sensitivity, was used for ﬂow
velocities less than 14 m/s, while the higher range transducer was used for velocities
above 14 m/s. The ﬂow temperature was monitored by an Omega DP25-TH thermistor
indicator for use in calculation of the air density and temperature compensation of the

hot-wire response.

(a) (b)

 

 

Hot-wire prong

   

 

Figure 2.11 Typical image of the probe tip and step area used for hot-wire
positioning near the wall: (a) Wire at the initial position, (b) Wire at 0.5 mm above initial
posrtron

After calibration, the hot-wire probe was traversed back down to the initial
location near the wall. The accuracy of re-positioning of the wire was veriﬁed using a
dial gauge attached to the traversing system. The probe was then traversed away from the
wall to 33 different heights, or y positions. At each height the mean and rms (root mean
square) velocity were recorded for ten seconds at a sampling rate of 1000 samples/s. The
traversing steps followed a logarithmic distribution with smaller values in the near-wall
region where the mean-shear is high, and larger ones closer to the freestream. After the

experiment, a “post” hot-wire calibration was done to check any drift in the sensor's

33

response. If there was more than 5% error between pre and post calibration, the data were
discarded and the experiments were repeated. The results for the boundary layer proﬁles

and associated characteristics will be analyzed in section 4.1.

2.4 Couette Flow Calibration Facility

In the current study, a new hot-wire technique was used to measure the wall-shear

stress. Therefore, a facility for calibration of near-wall hot-wire sensors was required to

be deve10ped. Some of the common methods to calibrate wall-shear sensors include the
use of a ﬁrlly-developed laminar or turbulent channel ﬂow or comparison against a direct
measurement methods, such as oil-ﬁlm interferometry. Here, a Couette-ﬂow facility was
used for the calibration. This choice was motivated by the fact that the facility met certain
Special considerations that are required for successful measurements of the velocity very
close to the wall (within 100 pm) in a separating/reattaching ﬂow environment. First, the
calibration range must extend down to velocity, or shear-stress, values that are very close
to zero. A Simple extrapolation of King’s law-type calibration to lower velocity values is
not valid since the heat transfer mode from the wire switches ﬁom forced, where King’s
law is applicable, to natural convection as the velocity approaches zero (in addition to the

increasing importance of heat conduction to the prongs near zero velocity). Second, the

calibration facility should have provisions to reverse the ﬂow direction in order to verify

and document the Wire’s directional sensitivity. Third, since ultimately the measurement

of interest is 1..., one needs to know the wall-shear—stress value at different calibration

velocities.

The idea for the Couette ﬂow calibration facility was ﬁrst discussed by Khoo et

34

al. (1998). The facility consists of two discs: one of the discs is stationary and the other is
spinning. Ifthe gap between the discs is small enough, the azimuthal ﬂow velocity varies
linearly across the gap; that is, establishing a Couette ﬂow ﬁeld. The ﬂow direction could
also be reversed easily by switching the rotational direction of the spinning disc.
A similar device was designed and constructed in this study as demonstrated in
Figure 2.12. The picture gives an overview of the device. Figure 2.13 shows a cross
section of the facility along with relevant dimensions. The rotating disc is made of
aluminum and the stationary disc is fabricated from plexiglass. Both discs have a
diameter of 317.5 mm. The rotating disc is connected to a steel drive shaft seated in a
bearing assembly. Two NTN high-precision angular contact bearings (model number:
7006CDB/GNP4) at the right end of the bearing assembly are ﬁxed in position in back-
to-back arrangement by trapping the inner and outer races of the bearings using shoulders
or nuts. A deep groove ball bearing (model number: 6204—2RSl) is ﬂoated at the other
end of the bearing assembly by ﬁxing its inner, but not outer, race. The purpose of the
ﬂoating arrangement is to allow the shaft to deform due to thermal expansion. A timing
pulley (model number: Browning 40XLBO37) at the end of the shaft was connected
through a timing belt to another pulley (model number: Browning 20XLBOB7) on the
shaft of a 12/24-volt DC motor (model number: Dayton 4Z144). The pulley on the main
shaft is twice as large as the pulley on the motor. In this fashion, it was possible to run the

rotating disc at a speed as low as 18 rpm to calibrate the sensors at very low speeds.

35

Rotating

  
 
    

Disc
Bearing ISDtlitéonary
Assembly

I \SDCHI

('1 ‘Sensor

Figure 2.12 Picture of the Couette Flow Facility

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rotating Stationary
Disc Spring-loaded
support screw
Bearing
Assembly
Timing Belt \
f7 T
g ~ Supporting
190.5 bracket
L I _ T
_- . ~ 476.3"* ' *

Figure 2.13 Cross Section of the Couette Flow Facility (dimensions in nun)

36

The Couette facility motor is driven by a Sorensen DCR 40-13B DC power
supply. Experimentation with the motor-driving unit showed that when driven by the DC
power supply, the motor has lower running noise and less vibration than when operated
using a SCR DC motor controller. However, the disadvantage of a DC power supply is
that there is no over-load protection of the motor. This was remedied by monitoring the
Sorensen power supply's current meter to ensure that the current drawn by the motor does
not exceed the rated value. At all speeds, the current was less than 80% of the full load
current of the motor. Finally, the rotational speed of the motor was measured using an
Omron EE-SBSV photo-detector switch. The sensor output was switched off whenever a
piece of black tape adhered to the rotating disc passed in front of the sensor. This resulted
in output of one 5V pulse per revolution from the switch. During calibration, the pulses
were sampled, counted and divided by the sampling time to yield the average rotational
speed of the disc.

Proper-rotation and vibration of the rotating disk was checked by a Kaman KD-
2300 hall-effect displacement sensor, the sensitivity of which was 0.5 pm per milli-volt,
that measured the distance from the surface of the disk to the tip of the sensor within the
measurement range (005 mm). The sensor was installed in the facility's supporting
bracket on a three—dimensional traversing mechanism and adjusted to be perpendicular to
the rotating disc with the sensor tip about 0.25-mm away. The maximum variation
measured in the distance between the disc surface and the sensor tip was 27 pm over the
range of rotational speeds employed here. This variation represented less than 5% of the
600 um gap between the two plates. The latter was set using feeler gauges while

ensuring parallelism of the two plates by adjusting three spring-loaded support screws

37

(see Figure 2.13). The resulting uniformity of the gap was veriﬁed during rotation using
high-magniﬁcation video imaging of the gap. A sample frame from the video recording
is given in Figure 2.14. Examination of different frames of the video demonstrated that

the gap variation was less than 37 um.

Rotating
/ disc

Stationary
disc

 

 

Gap

Figure 2.14 Image of the gap between the rotating and stationary discs

Khoo et al. (1998) calibrated near-wall hot-wires successfully in a similar type of
facility as that used here and veriﬁed their operation in fully-developed laminar and
turbulent channel ﬂow. The same authors also did a Direct Numerical Simulation (DNS)
to solve for the ﬂow ﬁeld between the two discs and compared the DNS results to the
Couette ﬂow assumption. These results are displayed in Figure 2.15, where three plots
showing the radial (V, , azimuthal (Va) and wall-normal (Vy) velocity proﬁles across the
gap are given. The symbols in the plots depict DNS data while the solid lines represent
asymptotic solution obtained by Stewartson (1953) for the ﬂow between two inﬁnitely

wide discs and small Reynolds number based on the gap size:

38

52-112
V

 

Re}, = (2.2)

where Q is the rotational angular velocity, It is the gap between the two discs and v is the
kinematic viscosity. Also included in the ﬁgure is a dotted line indicating the typical.
height above the wall for the sensors used here during calibration.

A few important conclusions may be drawn from Figure 2.15. First, the
agreement between the DNS and analytic solution for the radial and azimuthal velocities
is good for all Reynolds numbers providing conﬁdence in both solutions. For the wall-
normal velocity, the agreement between the two approaches is limited to Re h of 3 or
less. However, the latter velocity component is more than three orders of magnitude
smaller than the other two components, and therefore its inﬂuence on the hot-wire
calibration can be ignored. A second important conclusion is the fact that V9 is in fact

found to vary linearly across the gap (middle plot) as expected for Re In of 4.5 or smaller.

This provides conﬁdence in that the Couette-ﬂow assumption made here is valid for the
entire calibration range since the largest Re h value encountered during calibration is
equal to 4.7.

For the purpose of the calibration, the hot-wire sensor was installed in a 63.5 mm-
diarneter wall plug whose top surface was ﬂush with the inside surface of the stationary
disc. The center of the plug, where the sensor's center is also located, was at a radial

position (r...) of 120 m. If the wire height above the wall is denoted by y, then the

velocity ( V9) and wall shear (I...) seen by the wire are given by:

Q-rw

 

V0 = y- (2.3)

39

dV .Q-r
Twzﬂ dyg=y hw

(2.4)
where u is the dynamic viscosity. The facility was typically operated from 0 - 1800 rpm,
leading to wall-shear—stress variation within the range of 0 to 0.66 Pa.

Prior to calibration, the wire height was determined using a high-magniﬁcation
video system. The wall plug was installed on an ULTRAlignTM traversing mechanism
(model number: Newport M-462) with a resolution of 2 um. Using focused strobe light
illumination, the image of the wire and surface of the wall plug was viewed on a video
monitor. The wire position in the image was ﬁrst marked on the monitor. The wire
height above the wall was then determined ﬁ'om the micrometer reading of the traverse of
the translation distance necessary to traverse the plug surface within the view of the
microscope to the marked location of the wire.

The calibration velocity/ wall-shear values were selected to cover their actual
range in the BFS ﬂow. Speciﬁcally, the minimum wall shear is zero in this case, while
the maximum mean-skin friction-coefﬁcient is never above 0.003 in the literature; e.g.,
Jovic and Driver (1995) and Spazzini et al. (2001), which corresponds to a wall-Shear
value of 0.42 Pa at 15 m/s, the highest velocity of interest. Two separate curve ﬁts were
used: polynomial ﬁt for the low velocity /wall-shear values (about 0 —l .2 m/s or 0 — 0.21
Pa) where natural convection effects may be important, and King's law ﬁt for high
velocity /wall-shear values (about 1.2 — 3.8 m/s or 0.21 -— 0.66 Pa). There were about ten
points in each calibration range to ensure reliability of the curve ﬁt. Also, each calibration
data point was obtained from averaging the wall-shear magnitude and hot-wire voltage

over at least 50 rotations of the rotating disc in order to "average out" the effect of any

background noise such as that caused by motor vibration.

4O

Vr/(w-r)

 

 

     

 

 

 

 

 

 

 

0.06 f
. Symbols: numerical results: 9 Reh = 0.5
0.04 1' '. Line: series solutions ‘ I Re}, : 1.0
A RE}, I 1.5
0'02 . F . x Reh : 3.0
0 e ‘ etc Reh :45 ,
o R) : 6.0
-0.02 1 ‘h
-0.04 4
-0.06
0.0 0.2 0.4 0.6 0.8 1.0
y/h
V9 my - r)

 

 

 

 

0.0 0.2 0.4 0.6 0
y/h

V}, /(a) - r)
6.0E-5
4.0E-5
2.0E-5
0.0E+0
-2.0E-5
-4.0E-5
-6.0E-5
-8.0E-5

 

 

 

 

 

0.0 0.2 0.4 0.6 0.8 1.0
y/h

Figure 2.15 Comparison between DNS results and Couette ﬂow assumption
(Khoo et al. (1998)

41

 

 

 

CHAPTER 3
OSCILLATING-HOT-WIRE (OHW) TECHNIQUE

The main measurements in this thesis were conducted using a new oscillating-hot-
wire (OHW) sensor developed in the Flow Physics and Control Laboratory at Michigan
State University for the purpose of this work. The sensor is able to capture the near-wall
velocity/wall shear signature of unsteady direction-reversing turbulent ﬂows. The concept
and validation of the operation of a prototypical OHW sensor suitable for use in ﬂows
containing unsteady frequencies of the order of tens of hertz have been demonstrated in
Li and Naguib (2003). However, near-wall ﬂows behind a BFS involve signiﬁcant ﬂow
unsteadiness occurring at frequencies of the order of hundreds of Hz to a few kHz. Thus,
oscillation frequencies of the order of a few kilohertz are desired for successful operation
of OHW sensors. The development of such an OHW sensor is described in this chapter.
Section 3.1 reviews existing techniques for measuring the direction-reversing wall shear
in air ﬂows. Section 3.2 introduces the basic principles of the OHW sensor, while the

operation and characteristics of the sensor are presented in Sections 3.3 through 3.5.

3.1 Review of Techniques for Measurement of Direction-reversing Wall-Shear
Stress

The near-wall ﬂow behind a backward-facing step is highly turbulent and ﬂow
reversal occurs frequently. Therefore, it is difﬁcult to conduct measurements, especially,
of the wall-shear stress, in this region. The conventional wall-hot-wire sensors ﬁequently

used for wall-shear measurements would not work here for their inability to discern the

42

shear direction. A brief review of the techniques that are able to measure direction-
reversing wall Shear follows.

Hanratty and Campbell (1983) reviewed six principal methods to measure the
local wall shear and found only the direct method (i.e. ﬂoating element) and sublayer
fence were able to determine the direction as well as magnitude of the wall-shear stress.
Femholz et al. (1996) updated the review of the ﬂoating element and sublayer fence and
also added two more methods for reverse wall-shear measurements: oil-ﬁlm
interferometry and wall pulsed wires.

The ﬂoating-element is suspended by beams, with some gap between the element
and surrounding walls. The wall shear acting on the surface of the element causes beam
deﬂection that could be measured to obtain the force (i.e. wall shear) acting on it. But the
requirement of a relatively large surface area for the small shear-stress values typically
encountered in low-speed ﬂows limits its usage. More recent studies attempted to resolve
this issue through the development of microfabricated ﬂoating elements with extremely
small area ( for example, Padmanabhan (1997) and Schmidt et al. (1988) ), but sensors of
this type with acceptable signal-to-noise ratio, sensitivity and reliability are yet to be
demonstrated.

The sublayer fence is mounted on the wall, with the height of the fence less than
ﬁve wall units to ensure that the fence is immersed within the viscous sublayer. The
measured pressure difference between the front and back of the fence is used to obtain
the wall-shear stress after appropriate calibration. However, the fence technique is still
limited to primarily measuring the mean skin friction coefﬁcient. It is unclear if the

technique could be adapted for time-resolved measurements.

43

The oil-ﬁhn interferometry is another technique to directly measure the direction-
reversing wall shear. An extremely thin oil ﬁlm is spread on the solid wall and
illuminated using a laser beam, or some other monochromatic light source. Interference
of the reﬂected light ﬁom the top and bottom of the ﬁlm is used to measure the ﬁlm
thickness, which is related to the wall shear acting on the ﬁlm through well known
analytic equations. Although this is probably the most accurate method available today
for mean skin ﬁiction sensing, it can't be used for unsteady measurements because the
ﬁhn doesn’t respond quickly enough to instantaneous wall-shear-stress variation.

Pulsed-wire anemometry was ﬁrst developed by Bradbury and Castro (1971) for
velocity measurements in separated ﬂows. A Similar approach for measuring wall-shear
stress was later developed by Westphal et al. (1981). In pulsed anemometry, the probe
consists of a central heating wire surrounded by upstream and downstream cold wires.
The central wire, which is typically oriented at 90 degrees with respect to the cold wires,
is heated periodically. The velocity is measured from the time of ﬂight of the hot tracer
ﬂuid from the central wire to the downstream or upstream wires, for forward or reverse
velocities, respectively.

There are several difﬁculties and limitations of pulsed anemometry. First, to
avoid thermal diffusion effects, the sensing volume size is typically no better than one to
two millimeters on the side, limiting the spatial resolution of the sensor. This sensor
separation also limits the frequency response to tens, or a few hundred hertz. Second, in
ﬂows with large velocity gradients, such as near walls, the measurements need to be
corrected as discussed recently by Schober et al. (1998). Finally, pulsed wires require

elaborate and careful calibration that renders them inconvenient for applications

involving array measurements.

A different variation on pulsed anemometry was introduced by Downing (1972)
who used three parallel wires to measure the velocity in a one-dimensional pulsating ﬂow
in a pipe. In this approach, the central wire was operated as a conventional constant-
temperature sensor and used to measure the magnitude of the velocity. The ﬂow
direction was deduced by incorporating the two outside wires, which were located
upstream and downstream of the central wire, in opposite legs of a Wheatstone bridge to
form a thermal tuft similar to that used by Eaton et al. (1979). Although this method
overcomes some of the disadvantages of the time of ﬂight technique, the ﬁ'equency
bandwidth remains limited because of the separation of the thermal tuft sensors and their
thermal inertia.

Most recently, Spazzini et al. (1999) constructed a direction-sensitive wall-shear
sensor using two S-um diameter tungsten hot wires that were mounted on top of a small
l-mm diameter cavity while being separated by a small gap in the direction parallel to the
main ﬂow direction. The sensor was able to detect the ﬂow direction since the output of
whichever wire was in the downstream direction was affected by being exposed to the
thermal wake of the other wire. Hence, by measuring the signal from both wires
simultaneously it was possible to measure the shear-stress magnitude and direction, after
appropriate calibration. Although the sensor of Spazzini et al. (1999) has better spatial
and temporal resolution than that achieved by earlier methods, it is fundamentally based
on thermal wake effects like pulsed anemometry and variations on it. Therefore, the

temporal response of the sensor is limited by the separation of the two hot wires, rather

than by the much faster bridge circuit response.

45

3.2 OHW Principle

From the above review, it is evident that a wall-shear-stress sensor with fast
frequency response for use in direction-reversing ﬂows is still not available today. Li and
Naguib (2003) demonstrated a prototype of an oscillating hot-wire sensor for
measurements in unsteady direction-reversing ﬂows. As shown in Figure 3.1, the hot-
wire prongs are attached to an oscillating base driven by a sinusoidal voltage. Ideally, the
oscillating frequency is much higher than the unsteady ﬂow frequency. Measurements of
the velocity are then acquired at the oscillation center, where the oscillation velocity is
the largest. In this manner, two measurements are acquired during one cycle of
oscillation: one when the wire moves to the right, referred to as the backward

measurement, and the other one, or forward measurement, when the wire moves towards

the left.
Direction-Reversing Flow
”f
_ Hot Wire
Prong 5
Oscillating support @
UoSin (Zizfot) Wall
Side View End View

Figure 3.1 Schematic diagram illustrating the OHW sensor concept

In Figure 3.2(a), the ﬂow direction is towards the right resulting in an effective
cooling velocity equal to the Stunmation of the oscillation velocity (U0) and ﬂow velocity
(Hf) for the forward measurement since the wire moves against the ﬂow. On the contrary,

for the backward measurements, since the wire moves with the ﬂow, the effective cooling

46

velocity of the sensor is the difference between the two velocities. In this case, the
forward is greater than the backward measurement. However, if the direction of ﬂow
velocity reverses as shown in Figure 3.2(b), the opposite scenario takes place, where the
backward is greater than the forward measurement. Thus, the ﬂow direction could be
determined by comparing the two measurements. On the other hand, the magnitude of
the ﬂow velocity/wall shear could be obtained from the measurement with the wire
moving against the ﬂow. Since only one velocity magnitude and direction is obtained
during one cycle of oscillation, the oscillation frequency of the OHW sensor is also the
sampling frequency of the measurements. The prototype sensor used by Li and Naguib
(2003) was oscillated at a frequency of 490 Hz and U, of 0.5 m/s using a 63.5 mm-long
by 31.8 mm-wide by 0.58 mm-thick piezoelectric cantilever oscillator. Hence, it is
suitable only for use in ﬂows containing unsteady frequencies of the order of tens of Hz.
A great effort has been taken to build a higher frequency and smaller size OHW
sensors for measurement at higher frequencies. In order to reach this goal, characteristics
of the piezoelectric oscillator has to be understood ﬁrst. Figure 3.3 shows a two-layer
piezo-element clamped at the left end. The length and width of the element are L and W,
respectively. The deﬂection (Ad) caused by the driving voltage (V) is proportional to L2
and V, and the resonant frequency of the element is proportional to l/LZ. Thus, in order
to increase the resonant frequency, the element length has to be shortened. But that would
sacriﬁce the oscillation amplitude. Theoretically, the amplitude of oscillation velocity
should not change since it is proportional to the product of the oscillation frequency and
amplitude. However, in practice, a small drop in U, was observed since the added mass at

the tip of the element, for example, ﬁ'om the prongs and epoxy, had larger effect on

47

smaller piezo—elements. Although the oscillation amplitude could be increased by a
higher driving voltage, the voltage is generally limited by the characteristics of the

element to below 120 volts.

(a) Flow direction is towards the right

Flow Flow
‘2’ “I :5 “1
%, we,
Vfwd=luf|'Uo wad=qu|+Uo

(b) Flow direction is towards the left

Flow Flow

:1 14f (1: “f
%, w.
Vfwd=lqu+Uo wad =|uf|-U0

Figure 3.2 Schematic diagram illustrating OHW sensor response for different ﬂow
directions: (a) Flow direction is towards the right, (b) Flow direction is towards the left

 

Figure 3.3 Schematic of a two-layer bending piezo-element

48

Figure 3.4 shows a photograph of the OHW sensor developed for use with the
axisymmetric BFS model. The main dimensions are given in Figure 3.5. The sensor is
mounted in a 7.62 mm-wide wall plug with a curved top surface with a radius of 50.8 mm
to match the curvature of the outer shell behind the step of the test model. The prongs of
the hot-wire are attached using epoxy on one end of a 12.7 mm-long by 6.35 mm-wide by
0.58 mm-thick piezoelectric beam, protruding through 0.51 mm-diameter holes in a
plexi-galss insert in the wall plug. The surfaces of the piezo-element are coated with a
thin layer of nickel so that it is electrically conductive for supplying the driving voltage to
the piezo-element. To isolate the hot-wire signal from the driving signal, a thin layer of
3M electric tape is placed between the prongs and piezo-element as shown in Figure 3.6.
At the opposite end of the prongs, the piezoelectric beam is rigidly ﬁxed using a clamp
made out of phenolic to electrically isolate the piezo element’s electrodes from the plug.
In turn, the clamp is supported on two dowel pins that are force ﬁtted to the wall-plug
body. The pins provide rigid support to the piezo element in the direction of oscillation
while allowing the element’s position to be adjusted in the wall—normal direction to place
the hot wire at heights (y) up to 5 mm above the wall. Once at the desired location, two
set screws are used to ﬁx the sensor’s position.

Dowel pin Sensor holder

Set screw

 

Piezo-element Sensor clamp

Figure 3.4 Picture of OHW sensor plug

49

Piezo-element

Hot-wire

Sensor holder

    

 

 

 

Sensor clamp

' F, < r 7.62
< 33.0 F
e - . 59-24 - Side View
End View

Figure 3.5 Cross-section view of OHW sensor plug (dimensions in mm)

Details of the wiring connections of the OHW sensor are illustrated in Figure 3.6.
One end of stripped gauge 32 wires is soldered to the bottom end of the prongs. The other
end of the wires is soldered onto a printed circuit board that is adhered using epoxy to the
bottom of the piezo—element, with a thin layer of self—adhesive copper ﬁlm sandwiched
between the board and element. The adhesive side of the copper ﬁlm is in contact with
the PC board, while the other side of the ﬁlm, which is electrically conductive, touches
the piezo-element's electode to provide the connection with the piezo-driving signal. On
the other hand, connection to the hot-wire anemometer circuit is made using two 76 mm
long 28 AWG hook-up wires, which are also soldered onto the PC board to make
electrical connection. The use of the PC board to transfer the electrical connections to the
prongs prevents the outside cabling from interfering or inﬂuencing the motion of the free-
end of the pizo element. For the same reason, the piezo-driving cables are soldered onto
the copper ﬁlms instead of the piezo-element. This prevents any distortion of the

oscillation of the piezo-element because of "cable tugging", which would have adverse

50

effects on the performance of the sensor. More speciﬁcally, the element oscillates at a
high frequency but with very small oscillation amplitude. For example, at ﬁ'equency of
4800 hertz, the amplitude is only of the order of the wire diameter, making it susceptible
to any minor interference. In earlier-version OHW sensors that did not incorporate
similar cable-separation provisions, wiggling the hot-wire or piezo-driving cables

distorted the sensor's output signal.

Hot-wire
/

  
   
 
  
    
     

Prongs

Piezo-
element

Hot-wire
Cable

Sensor
Clamp

Sensor/

Clamp

Copper
Piezo-driving Film
cable

Figure 3.6 Sketch showing wire connections to the OHW sensor

An Agilent 33120A function generator is used to supply the piezo-driving signal.
The output of the function generator is ampliﬁed with a P1000 Haﬂer audio ampliﬁer
before feeding the power-ampliﬁed signal to the secondary side of a 1:186 step-down
transformer to increase the voltage amplitude of the signal. In this fashion, the voltage

amplitude of the sinusoidal driving Signal could be adjusted up to 180 V peak to peak.

51

The choice of the driving circuitry is determined from the power required to drive the
piezo- element, which is estimated as follows: the current 1' through the piezo-element at
time t is

d V

i(t) =C7t- (3.1)

where V is the voltage across the piezo-element and C is the capacitance of the piezo-
element. Since V is a sinusoidal wave with an amplitude A, then

V(t) = Av sin(27rﬂ + (D0) (3.2)
where f is the frequency of the sine wave and (Do is the initial phase angle at t = 0. The
power P is given by

P=imxvm=7r-f-Av2-C (3.3)

For a capacitance C of 6 nf, a typical value for the piezo-element, even for as high
of a voltage as 120 V and frequency of 5000 Hz, the power consumption is only 1.36
watts. This is well within the power capacity of both the power ampliﬁer and transformer
(110 and 4.7 watts, respectively).

The sensing element of the OHW sensor is a 3.75 rim-diameter tungsten wire. The
two ends are electroplated with copper and soldered onto the probe prongs. The sensing
length (l); i.e., unplated portion, of the wire is 0.8—1.2 mm and is centered in the middle
of the 4 mm-gap between the prongs. The resulting I/d ratio (where d is the wire
diameter) of the wire is greater than 200.

In implementing the OHW sensor, one concern is that vibrations of the tungsten
wire could be excited by the imposed wire motion. Such vibration will not only result in

measurement inaccuracy but also in shortening the life of the wire. In order to verify that

52

the wire did not vibrate during oscillation, a microscope composed of Sony XC-75 CCD
camera and VZMTM Model 1000 lens, with a magniﬁcation factor that is adjustable in the
range 2.5X - 10X, was used to visualize the oscillation of the wire. The resulting image
resolution was estimated to be 1.68 pm. That is, the system is capable of detecting
oscillation amplitudes of the order of the wire diameter. An MVS-2060 ﬁber-optic strobe
light that can be ﬂashed in the frequency range of 0-60 hertz was used to illuminate the
wire during oscillation in order to provide recording of an alias of the oscillation
frequency so that the high frequency oscillation could be observed visually. Figure 3.7
shows a sample sequence of images captured during the sensor oscillation. In Figure
3.7(a), the wire clearly remains straight, moving like a rigid beam. Figure 3.7(b), on the
other hand, depicts a case where substantial wire vibration was seen at an oscillation
frequency of 4000. The wire oscillation is excited because the oscillation frequency of the
piezo-element is close to the resonant frequency of the wire, which depends on the
residual stress of the wire and the plated and unplated length of the wire. The problem

could be easily remedied by changing the oscillation frequency or replacing the wire.

3.3 OHW Response

Although typical hot-wire sensors are known to have wide bandwidth of 20 — 40
kHz, evidence of substantial reduction in the dynamic response of the hot wires near zero
mean-ﬂow velocity exists in the literature. Elger and Adams (1987) used a calibration
facility that created an oscillating, zero-mean ﬂow by a reciprocal motion of a piston
driven by a loudspeaker in a cylinder. In their experiments, the ﬂow oscillates at 25,

45.2and 70 hertz. And the oscillation velocity amplitude is in the range 0.22 — 5.60 m/s.

53

The dynamic Nusselt number was calculated from the measured anemometer output
voltage. The mean value of the dynamic Nusselt number corresponding to zero-mean
ﬂow increased with ﬂuid motion amplitude, indicating that the heat transfer mode is not
purely natural convection. This suggested that a dynamic calibration is needed for the
oscillating ﬂow measurement. But the authors also concluded that the peak ﬂow velocity
of an oscillating ﬂow could be accurately obtained from a steady calibration. Perry and
Morrison (1971) also found that the dynamic calibration method was more accurate and
consistent than the conventional procedure for measurement of absolute turbulence
intensities.

Therefore, with the use of kilohertz OHW, it is important to calibrate and check
the hot wire ability to respond to the imposed oscillation in a controlled environment
where the mean velocity could be adjusted to very small value and the direction of the
ﬂow could be reversed. The Couette ﬂow facility mentioned in Section 2.4 served this
purpose. An adapter plug as shown in Figure 3.8 is used to adapt the curved top surface
of the OHW sensor plug into a ﬂat surface to match the shape of the inner surface of the
stationary disc of the Couette ﬂow facility. Figure 3.9 shows the main dimensions of the
plug. The OHW sensor plug is inserted into the slot of the adapter and aligned with
reference marks as Shown in Figure 3.8 to make the top surface of the sensor plug ﬂush
with that of the adapter. Once aligned properly, the sensor plug is ﬁxed with two set
screws ﬁom opposing sides. Two pieces of 66-um thick kapton tape were used to cover

any discontinuity between the sensor and adapter plugs at the edges of the slot.

54

(a) Wire is straight during oscillation (dash line is used as a staight-line reference)

 

Figure 3.7 Video image sequences of the hot wire during oscillation: (a) Wire is straight
during oscillation, (b) Wire bends during oscillation

55

OHW sensor Kapton tape

Reference

mark Plug

Set screw

 

     

 
    
       

R5080
’. , . . \
$2.84 I ‘ . ' , . ' §\\\‘
3 evenly ’ ' I N
spaced ' *I 4.76
holes 7.62 14.28

Front View Side View

Figure 3.9 Dimensions of adapter plug for calibration of OHW sensors (dimensions in

mm)

To examine the response of the OHW sensor, it was ﬁrst desired to verify that the

imposed oscillation does not inﬂuence the wire calibration; i.e., the oscillation merely
results in linear amplitude modulation of the sensor output. To this end, the wire was
ﬁrst kept stationary and calibrated over the velocity range of 0.1 - 4 m/s. Subsequently,

the calibration was repeated when the OHW was driven by i 30 Volt ac signal while

56

varying the frequency from 1.1 to 4.8 kHz. Figure 3.10 shows the hot-wire calibration
obtained without wire oscillation (solid line), compared to that obtained from the mean
hot-wire voltage for wire oscillation frequencies of 1.1, 2, 3 and 4 kHz. The no-
oscillation calibration extends down to a velocity of 0.1 m/s. On the other hand, data
obtained during oscillation are shown for a velocity range that is larger than 0.3 m/s. In
this range, a good agreement is evident between the results with/without oscillation. This
veriﬁes that the high-frequency small-amplitude oscillation of the wire does not result in
an anomalous or non-linear behavior that may affect the Wire’s calibration or its accuracy

in retrieving the velocity magnitude.

 

 

0.95I

STAT
f=1100 Hz
f=2000 Hz
f=3000 Hz
f=4000 Hz

 

’50” 0.9

   

I
I
I

Irf+

 

 

 

0.85»-

 

 

 

0.75»
I J
0.7l i 1 1 4'
0 1 2 3 4 5
U(m/s)

Figure 3.10 Magnitude calibration of the hot wire with/without oscillation

To examine the unsteady response of the wire to the imposed oscillations at

different frequencies and ﬂow velocities, the mean-removed hot-wire output was

57

acquired and phase averaged with respect to the OHW driving voltage. The phase
average aids in removing the effect of any background disturbances related to motor
vibration inﬂuences or other sources of noise. A portion of the resulting signal is shown
in Figure 3.11 for an oscillation frequency of 2 kHz and different mean velocities in the
range 0.34 — 3.8 m/s. The reduced dynamic-response ability of the wire with decreasing
speed is evident from the data as the amplitude of the measured oscillation velocity (um)

is attenuated and an increased phase lag is observed with the reduction in the mean ﬂow

 
  
  
  
 
 
  
  
  
  

velocity.
0.1514 ..
. U=0.34m/s
uo,m I u=o,53m/s
0.1~ U=0.8lm/s
(m/s) « U=l.20m/s
I

U=l.62 m/s
. . U=2.03 m/S
‘ . U=2.49 m/s

‘~ U=2.9l m/s

7 U=3.36 m/s
U=3.80 m/s

0.05‘»

-o.05:7

-o.1I,V‘

-o.15.

'o' ' 7 "19” W ’26 T .30-.. " 40
SampleNumber

Figure 3.1 1 Phase averaged hot-wire output at the oscillation frequency
For ﬂow speeds larger than 1 m/s, um is fairly constant. However, when the ﬂow

velocity drops below 1 m/s, larger attenuation is found. The reduction close to zero ﬂow

velocity is presumably due to shifting of the primary heat transfer mode of the wire from

58

forced to natural convection and increasing importance of heat transfer through the
prongs by conduction, leading to a larger thermal time constant. Although the Constant
Temperature Anemometer (CTA) circuit compensates for the sensor’s thermal response,
the compensation generally works well for ﬂow velocities larger than roughly 1 m/s
where the thermal time constant apparently does not vary signiﬁcantly. Below 1 m/s
there seems to be an appreciable change in the time constant, for which the CTA does not
adapt.

The reduction in the dynamic response should not cause a problem in operating
the OHW sensors as long as the amplitude attenuation is not too large for the Sinusoidal
modulation of the wire output to be detectable at very low velocities. In Figure 3.12, the
modulation Signal at the oscillation frequency was detectable at ﬂow velocities as low as
0.03 m/s.

In addition to amplitude attenuation, as the ﬂow velocity decreases, a phase-angle
delay is observed between the imposed and measured oscillation. The delay becomes
progressively larger with decreasing ﬂow velocity. Similar effects on the phase response
of hot wires were also made by Egler and Adams (1989) and Antonini et al. (1976). They
explained the reason for the phase lag was because the temperature distribution around
the wire could not follow the wire oscillation. In Figure 3.11, the biggest phase lag is less
than 90°, which is found at the lowest ﬂow speed. It is well known that the thermal
response of the hot-wire sensor could be modeled as a ﬁrst-order dynamical system. Even

as the oscillation frequency approaches inﬁnity, the phase lag reaches only —90°.

59

(a) Modulated velocity signals

 

 

—e~ Positive

 

 

I I
~ _._ Negative r
C‘F‘i‘ﬁif‘ i CJMCC'
u 0 6 .. or" n.) , “XII“ : C
. ............ u ———————————— a ..................... 1:...pr ........... (gen—I
N,‘ I ‘ g; Cir". ,l/./‘ .‘ ‘., ,1 Q :2 h /. '\. a J _
(m/s) . - 1,. . U— 0 55 m/s
at - _ 1. . ~ ‘ o

O 5 t- . . 13:72. _ .. ...... .o 3 ..{‘.’29,.. . “of.“ I .. ﬂ” .'. ... ..

I 1:78 ' ~‘l . . “k¥;7f;7 . . T F “ﬂi'ﬁfrk ‘.

 

W ..
‘e ,..k ' 9‘
‘~- ‘A "‘
0.2 -~~ 3'81: ----,
lCWC sat-v“ t

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

 

.........

.4—0—0 44‘. ‘
I \‘ 9,3..332; ' ’0 .. > 4 I.» 1““ _ 0" 0‘ _ “e.
., - e “a e ‘e-" U — 0 03 m/
0' .~::»3'~«”~4:..E.~i*’9‘*~e - . -' ~‘ *“*’“""ff~»-'«-«.:j - en ‘6’ “’1‘ ” e" W’ "ﬁr ' S
. ‘trf'ﬂ - - ‘yo‘e‘ ‘

(b) OHW driving signals

0 . 8 ______.__

e d I

(V)

   

 

Sample Number

m_ﬁ__ _.__~_._ _
,.._

 

—-:::+— POSIIIVG I
_._ Negative

‘——- v”

 

 

 

 

 

Sample Number

Figure 3.12 Demonstration of the effect of the ﬂow direction on the phase of the OHW
modulation signal: (a) Modulated velocity Signals, (b) OHW driving signals

60

3.4 OHW Measurements (Phase Method and Phase Calibration)

In Section 3.2, it was demonstrated how the ﬂow direction information is
extracted from comparison of the forward and backward measurements. However, as
seen from the previous section, the dynamic response of the OHW sensor leads to a phase
shift at low ﬂow velocities, which further results in smaller difference between the
forward and backward measurements. This may be seen from Figure 3.11: the two dotted
lines indicate the oscillation phase corresponding to the forward and backward
measurements, respectively. When the ﬂow velocity is U = 0.34 m/s, the difference
between the two measurements is only about 0.08 m/s. In a real ﬂow situation, this small
difference may be easily masked by unsteady ﬂow-velocity variation. Moreover, because
of the amplitude attenuation of the OHW signal at low velocity, it is not possible to
obtain the exact magnitude of the ﬂow velocity by subtracting the oscillation velocity
from the measured cooling velocity as outlined in section 3.2.

To overcome the above problems, a different method was used to recover the ﬂow
information from the OHW time series. In particular, it can be shown as follows that the
phase difference between the wire output and the piezo-driving voltage, hereafter denoted
by mm at the oscillation frequency experiences a 180° jump when the ﬂow switches
direction. In Figure 3.1, the instantaneous effective cooling velocity of the wire is: luf+
Unsin(27y'ot)| (The absolute value is used here because a hot-wire sensor can’t determine
the ﬂow direction.) If the ﬂow velocity is positive, the measured velocity is W +
Uosin(279’0t). On the other hand, the measured velocity is l-uf + U,sin(279€,t)| = luji -
Uosin(279’0t) if the ﬂow direction switches. There is 1800 phase difference between the

former and latter measured velocities.

61

To demonstrate the concept, the OHW sensor was used to take measurements at
different velocities under clockwise as well as counter—clockwise rotation of the rotating
disc in the Couette ﬂow facility. The direction of ﬂow velocity in this facility depends on
the rotational direction. Two different rotational directions produce two opposite
directional ﬂows. The results are plotted in Figure 3.12 for the velocities of U = 0.03, 0.2
and 0.55 m/s. For each speed, two data sets are plotted (open circles and dots)
representing two opposing ﬂow directions. For orientation purposes, the bottom portion
of Figure 3.12 contains a plot of the driving voltage of the OHW. The data clearly
demonstrate the reversal of phase of the wire sinusoidal modulation signal as the ﬂow
changes direction for all ﬂow speeds. It is noted here that for the lowest ﬂow velocity
and negative direction, small (about 0.01 m/S) negative values of the sensor output are
seen in Figure 3.12. Since this is the “raw” output of the sensor before demodulation and
extraction of direction information, no negative values should be observed. The
discrepancy is of the order of the accuracy of the calibration process in the vicinity of
zero velocity.

From Figure 3.11, it is evident that in a given ﬂow direction, the phase of the
oscillation in the hot-wire output changes with decreasing velocity. Because this phase
change does not exceed 90°, it does not create a problem in direction determination via
the phase method. In other words, in the absence of phase lag, rial... would jump from,
say, 0° for positive to 180° for negative ﬂow velocity. This would be the same in the
actual situation at high ﬂow velocities where no limitation on the Wire’s dynamic
response has been observed. On the other hand, at very low velocities, or wall shear

stress, values, Ito)... would jump from 90 to 270 instead of 0 to 180 degrees because of the

62

phase lag. Albeit this change, positive and negative ﬂow directions could still be
distinguished without ambiguity since in one direction e50)", falls in the range 0 — 90
degrees, which does not overlap with its range in the opposite direction (180° - 270°). A
problem would have been encountered if the Wire’s phase lag at low Speeds exceeds 180
degrees, resulting in an overlap in Ito}... values in the positive and negative directions.
However, as mentioned before, the wire’s phase lag is limited to less than 90°.

Itch... is unique for each OHW sensor because it depends on the dynamic response
of the piezo-element as well as that of operation of the anemometer and the hot-wire
sensor at different velocities. Therefore, Ito)... response must be veriﬁed for individual
sensors by calibrating against the shear stress in the positive and negative directions. An
example of such a calibration for a wall-shear OHW sensor is shown in Figure 3.13. For
reference, the ideal ¢0hw response, exhibiting 180° phase jump with ﬂow reversal, is also
illustrated in the ﬁgure using a broken line. It is clear that 410;... values in one direction do
not overlap with those in the other, conﬁrming that the direction can be determined
without ambiguity. In implementation, ¢ohw is found from calculation of the cross
spectrum between the piezo-driving signal and the hot-wire output at the oscillation
frequency for every cycle of oscillation. The process of detecting the ﬂow direction is
simpliﬁed by realizing that Ito}... values in the positive shear direction correspond to a
negative real component of the cross spectrum, and vice versa. Thus, for every cycle of
oscillation, the Sign of the real part of the cross spectrum is used to determine the ﬂow
direction. Finally, it should be noted that it is believed that the deviation between the
ideal and actual ¢ohw behavior (see Figure 3.13) is primarily caused by the limitation of

the wire dynamic response near zero velocity, as discussed above. In addition, it is also

63

believed that at these low velocities, low noise levels at the oscillation frequency
(resulting from capacitive coupling of the driving signal to the hot wire prongs) may
inﬂuence the imposed modulation of the wire output. Understanding of the latter requires
additional investigation, which was not done here since the observed at»... response

indicate that the problem does not result in ambiguity in shear direction determination.

 

200;? " I f ——.

 

150

100»

 

 

 

 

 

 

1w (Pa)

Figure 3.13 A typical direction calibration of the OHW sensor

To determine the magnitude of the velocity from OHW measurements, the sensor
output was low-pass ﬁltered to remove the sinusoidal modulation due to wire oscillation.
The cut-off frequency of the ﬁlter was set at half the oscillation frequency. Since the
latter is also the sampling frequency of the shear information, the ﬁlter also prevents
aliasing due to subsequent downsampling of the data. The low-pass ﬁltered signal is then
converted to wall-shear stress using the magnitude calibration. Figure 3.14 provides a

block diagram summarizing the overall operation of the OHW sensor.

64

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low-Pass Filter
I: E
: Ind
f,/2! K _.
U
@ Constant E
Temperature Phase
f0/2: IM Detection UrDirection
I (Cross
Spectrum)
- - H' -P F'lt INWVI
Ojglilatw” 1gb ass 1 er To Oscillating
river Oscillation Driving Signal Support

 

 

 

Figure 3.14 OHW Operation schematic diagram

3.5 Effect of Wire Height above the Wall

For a wall hot-wire sensor to measure wall-shear stress, the wire must be located
within the viscous sub-layer; i.e., y+ < 5 (or y < 5v/u,). Referring to the measurements by
Tihon et al. (2001), Spazzini er al. (2001), and Jovic and Driver (1995), the maximum
reverse and forward skin-friction coefﬁcient is less than 0.004 at the highest Reynolds
number. For a given step height, the highest Reynolds number corresponds to the largest
wall shear and friction velocity, and therefore the largest y+ for the same distance (y).
Consequently, the largest wire height above the wall in viscous units is actually found at
the highest Reynolds number. Combining U... = 15 m/s, which corresponds to the largest
Reynolds number in the present study, with Cf = 0.004, the maximum wire height was
estimated to be 112 pm. This provided a guide for initial placement of the wire above the

wall since the ﬁiction velocity was not known a priori.

65

In order to experimentally check proper positioning of the wire above the wall,
measurements were conducted with the height of the sensor set to three different values:
48, 96 and 124 um. The x location for the measurements was chosen to be well beyond
the expected x, position (x = 9821-!) such that t... has recovered near its largest value after
reattachment. The freestream velocity was set to 15 m/s, corresponding to Reynolds
number based on step height of Re” = 13000. At each height, the experiment was
repeated for three times with the wind tunnel turned off in between runs. This was done
in order to compare the agreement between the data for different wire heights against the
measurement repeatability from a given height. The data were acquired for one minute
for each run. No wire oscillation was imposed during this set of experiments since the
measurement location is well downstream of the expected reattachment position, and
hence there is no directional ambiguity of the ﬂow. The work of Tihon et al. (2001) and
Spazzini et al. (2001) provide supporting evidence that the ﬂow at the chosen
measurement location is always in the downstream direction. More importantly, the
ﬂow-direction—probability results from the current study, given later in 4.2.4, verify that
the ﬂow is in fact always in the downstream direction at x = 9.82 H.

Figure 3.15 and Figure 3.16 show the power spectra and normalized histograms,
respectively, of the wall-shear stress measured from three repeated runs at y = 48 pm.
The normalization of the histogram refers to the fact that the sum of the histogram values
equals unity. The data show that the power spectra exhibit a high degree of repeatability.
On the other had, some scatter is observed in the vicinity of the histogram peak.
However, the mean values of the wall-shear stress obtained from the histograms are

0.277, 0.279 and 0.282 Pa, respectively. The differences in these values are less than

66

1.8%. This small difference and the repeatability of the spectra demonstrate the precision
of the current measurement procedure.

Figure 3.17 and Figure 3.18 display the power spectra and normalized histograms
for data acquired at three different wire heights, respectively. The data corresponding to
the three different cases collapse well in both plots. The mean values of the wall-shear
stress calculated from the histograms are 0.277, 0.276 and 0.275 Pa. The small variation
of the measurements with the wire height indicates that all three positions are within the
viscous sublayer, where the velocity is linearly proportional to the wall shear stress. In
fact, if the highest observed '1:w value of 0.282 Pa is used, one wall unit will be 31.3 urn.
The corresponding wire heights (y+) are equal to 1.5, 3.1 and 4.0; i.e., all positions are
within the viscous sublayer. Based on this, the wire heights were set at 82 and 97 pm for

the two OHW sensors used in the two-point measurements.

 

.Lg

 

I

 

1O _.4 —A7.—¥ . _A.~._._.~ ~_‘._‘_A_I.' ,,.. _..._.. 'h-CL #4."; _..-_L i. 2'. —A—. ~—.‘__‘._A_ _L..#1

1o 10 1o3 10
f(Hz)

4

Figure 3.15 Power spectra of three different runs for wire height y = 48 um

67

 

 

 

 

 

 

 

 

r (Hz)

Figure 3.17 Power spectra for wire heights y = 48, 96 and 124 um

68

 

 

I‘ +h=48pm I
+h=96um I
—- h=124umI

0.3;»

MI

._ ,_. ._._......

Histogram
O
L. .O
9!- - N

.0
A

0.05’

 

 

 

Figure 3.18 Histogram for wire heights y = 48, 96 and.124 urn

3.6 Effect of Wire Oscillation on the Flow

Although highly unlikely, it seemed appropriate to check whether the oscillation
of OHW sensors disturbs the unsteady direction-reversing ﬂow or not. For this purpose,
data obtained at ReH=13000 and x/x, = 0.06, 0.92 and 1.54, i.e. before, close to and after
the mean reattachment point, were utilized. At each location, measurements were
conducted when the oscillation was turned on as well as off. The former case
corresponded to an OHW sensor with the oscillation frequency of 2.8 kHz, and the latter
represented a conventional hot-wire sensor. The premise of the test is based on the fact
that notwithstanding its directional blindness, the conventional hot wire output voltage
should provide the correct magnitude of the wall shear stress. Therefore, the low-pass-
ﬁltered voltage signal of the OHW, which also provides the magnitude information of the

wall shear, should yield the same voltage output magnitude as the conventional hot wire.

69

If the OHW operation does disturb the ﬂow, then one would expect that the frequency
spectrum, or some other statistical measure, of the low-pass-ﬁltered OHW voltage would
deviate from that obtained with the stationary wire. I

Figure 3.l9(a), (b) and (c) show a comparison between the power spectra of the
raw voltage data of the oscillating- and conventional-hot-wire sensors, labeled as “HW”
and “OHW” in the legend. A low-pass ﬁlter with a cut-off frequency of 1.4 kHz, or half
of the oscillation frequency, was also applied to the OHW signal. Since the sampling
frequency of the OHW sensor is also the oscillation frequency, this ﬁltering eliminates
any possible aliasing in the OHW signal. The spectra obtained ﬁ'om the ﬁltered signal are
also included in the plots in Figure 3.19. Inspection of the ﬁgure clearly shows that
within the pass-band of the ﬁltered signal, the voltage spectrum of the OHW and HW
sensors agree quite well for all three measurement locations. The imposed oscillation on
the OHW sensor is also evident from the harmonic peak at f = 2.8 kHz. Therefore, one
may conclude that the oscillating hot wire operation does not generate any ﬂow anomaly.
It is also noteworthy that the energy level of the oscillation signal is at least two orders of
magnitude larger than that of the ﬂow at the oscillation ﬁ'equency. This yields a strong
signal-to-"noise" ratio, which is necessary for successful determination of the ﬂow
direction.

Another important observation in Figure 3.19 is that the “ﬂow” energy contained
in frequencies above 1 kHz is only very small component of the total energy. This may
be seen ﬁ'om the fact that the Spectrum level at frequencies higher than 1 kHz is two
orders of magnitude or more lower than the spectral peak value. Hence, the selected

oscillation frequency should be high enough to resolve the wall-shear signature of the

70

most energetic ﬂow structures.

(a)%r = 0.06

.4

 

10 :--l,-._---- . ~
. o HW
.5 g — OHW
10 I + OHW Filtered
.5
Z
$107 I
(210“3 ..
$103 .
104°:
.11:
10 .-
10°

 

10"»F ,W - .-..H, ﬁzz .mvﬂzuz ..,-. ﬁzz..- 7. .. _..E.I

I O HW I

10.5 ' — OHW I

2‘ + OHW Filtered 3.
.5 3
NA‘IO I
2, 2
E - .
2107
g I

‘21043 i

a) if 7

g _ : :

8‘109

10'”;

. l

-11I -

10 . ----n---------_--------- w

10° 0 102 103 10‘
f(Hz)

 

 

 

10“ {E e We ___.r__._ a -.j
I O HW ;
«6 l — OHW .-
10 E + OHW Filtered I
-6 7‘ .
«“10 I E
2 E 3
gm”
«(310“3 i .
gl
£10 I
104°?» 1
10'11I i _r A l A lint , --.M.__+ HEMI
10° 101 102 103 10‘

f (Hz)

Figure 3.19 Comparison of power spectra of OHW & conventional hot wire (HW) at
different x positions: (a) x/x, = 0.06, (b) x/x, = 0.92, (c) x/x, = 1.54

Further comparison between the oscillating and stationary hot wire measurements
was conducted by comparing the normalized histogram results obtained in both cases. As
evident in Figure 3.20(a), (b) and (c), the “OHW” and “HW” histograms are consistent
with each other, at the same x locations used for the spectral comparison in Figure 3.19.
This suggests that the two measurement methods produce the same statistics of the
voltage, and hence shear-stress, magnitude.

In summary, the above comparison demonstrates the ability of the OHW to
operate successfully without disturbing the ﬂow ﬁeld. Moreover, the selected oscillation

frequency appears to be high enough to capture the majority of the energy in the ﬂow.

72

—i— OHW Filtered

 

 

if —e— HW
—l— OHW Filtered

,,, . ,_ ﬁ—r

Histogram

 

73

 

 

Y—e—HWY

 

—+— OHW Filtered E
0.5 4
0.4»
E
a
30.3
E
I
0.2»

.0
_a.

 

 

 

 

0.9

(9.75 3
E (V)

Figure 3.20 Comparison of histogram of OHW & conventional hot wire (HW) at
different x positions: (a) x/x, = 0.06, (b) x/x, = 0.92, (c) x/x, = 1.54

3.7 Speciﬁcation of OHW Sensors Use in the Present Experiments
Two OHW sensors (OHW 1 & 2), similar to that pictured in Figure 3.4, were

developed for the current study. The height of the wire above the wall for OHW] and
OHW2 was set to 97 and 82 pm, respectively. The length of the sensing portion of
OHWl sensor was 1.259 mm, and its cold resistance was 6.01 D at room temperature of
22.7 °C. Similarly, the active sensing length of OHW2 sensor was 1.0 mm with cold
resistance of 5.26 0 at room temperature of 22.8 °C. The overheat ratio of the two

sensors was 1.6 and the oscillation frequency was set to 2800 hertz.
As shown in Figure 3.21, the two sensors were assembled together with spacers to

form a sensor cover ﬁtted into the slot behind the BFS (see Figure 3.22). One of the

74

sensors was used as a reference sensor by keeping its x location ﬁxed for all
measurements. The other one could be mounted at seven different x locations on one side
of the reference sensor by using different-width spacers. By ﬂipping the plug 180
degrees, seven additional positions could be easily obtained. Table 3.1 lists the width and
arrangement of the spacers and sensors corresponding to the seven positions of the
movable sensor. In the table, M and R are used to denote the movable and reference
sensors, respectively, and the width of each element in mm is given in brackets. Table 3.2
lists the measurement locations for the movable sensor normalized by the step height.

The corresponding reference sensor location is 5.17H.

OHW sensor OHW sensor
(movable) (ref) Sensor cover

1L

 

Spacer

 

 

59.69

 

 

 

 

127.0

 

 

 

Figure 3.21 One of the combinations of the sensor cover (in mm)

75

OHW] OHW2 Sensor cover

 

Figure 3.22 Picture of the sensor cover during insertion in the test model

Table 3.1 Combinations of the sensors and spacers corresponding to different
measurement locations of the movable probe

 

Combinations for the sensor plug

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Spacer 1 Mgﬁazblle Spacer 2 Rigegce Spacer 3
1 M (7.62) Spacer (52.07) R (7.62) Spacer (59.69)
2 Spacer (13.97) M (7.62) Spacer (38.10) R (7.62) Spacer (59.69)
3 Spacer (21.59) M (7.62) Spacer (30.48) R (7.62) Spacer (59.69)
4 Spacer (29.21) M (7.62) Spacer (22.86) R (7.62) Spacer (59.69)
5 Spacer (36.83) M (7.62) Spacer (15.24) R (7.62) Spacer (59.69)
6 Spacer (44.45) M (7.62) Spacer (7.62) R (7.62) Spacer (59.69)
7 Spacer (52.07) M (7.62) R (7.62) Spacer (59.69)

 

76

 

Table 3.2 Measurement positions of the movable sensor

 

 

 

 

 

Position 1 2 3 4 5 6 7
x/H 0.31 1.45 2.07 2.69 3.31 3.93 4.55

Position 8 9 10 11 12 13 14
x/H 5.79 6.40 7.02 7.64 8.26 8.88 10.02

 

 

 

 

 

 

 

 

77

 

CHAPTER 4

ONE-POINT MEASUREMENT RESULTS

This chapter contains two main sections: one on the boundary-layer velocity
proﬁles at separation, and the other containing a discussion of the single-point
characteristics of the wall-shear stress beneath the separating/reattaching ﬂow. In the
second section, wall-shear results pertinent to the reattachment length, mean/rms skin-
friction coefﬁcient, probability density function, power spectra, and autocorrelation are

discussed and compared with the literature.

4.1 Boundary-layer Velocity Proﬁles at Separation

The boundary-layer characteristics at the point of separation have signiﬁcant
effects on the separating/reattaching ﬂow over a BFS, as demonstrated by Eaton and
Johnston (1981), Armaly et al. (1983) and Westphal and Johnston (1984). For example,
the boundary-layer thickness weakly affects the reattachment length, while the boundary
layer state (turbulent or laminar) has a signiﬁcant effect on the reattachment length. In the
current study, the mean and ms streamwise velocity proﬁles (u) were documented at a
location that was 0.5 mm (or approximately 15-20% of the boundary layer momentum
thickness for all three Reynolds number investigated) upstream of the step. Figure 4.1
displays the mean velocity proﬁles at three different velocities: 5, 10 and 15 m/s. The

distance above the wall (y) plotted on the ordinate is normalized by the momentum

thickness (0), which is deﬁned as follows:

78

_ My). -2312.
i9— [EDT/:[l U00] dy (4.1)

where u(y) is the mean velocity at position y. The value of 6? was found by numerical
integration of the measured velocities using the trapezoidal rule (also known as Newton-
Cotes formula). The outcome is shown for all cases in Table 4.1. The corresponding
Reynolds numbers based on 6 are also listed in the table, along with the boundary-layer

thickness (6), displacement thickness (6') and shape factor (HI; = 576).

Table 4.1 Parameters for the boundary layer at separation

 

 

 

 

 

 

U..(m/s) 5 10 15
5(mm) 28.5 24.5 21.6
6' (mm) 4.03 3.54 3.05
0(mm) 3.12 2.77 2.40
H]; 1.29 1.28 1.27
Reg 1139 2057 2739

 

 

 

 

 

 

As seen from Table 4.1, the Reg range of the boundary layer is on the lower end
required for the establishment of a turbulent state. The lowest known Reg value of a
turbulent boundary layer is 670, which was obtained in the ﬁrst Direct Numerical
Simulation of a turbulent boundary layer by Spalart (1988). In the laboratory, Naguib
(1992) studied turbulent boundary layers that spanned a Reg range starting from
approximately 1500.

The boundary layer proﬁles shown in Figure 4.1 have large velocity magnitudes

in the near-wall region (more than 40% of the freestream velocity at the lowest y position

79

of 0.355 mm). This is reminiscent of the 'fuller' nature of turbulent velocity proﬁles,
where the more effective momentum transport across the layer, brought about by the
turbulent eddies, causes penetration of the high-momentum ﬂuid deep into the boundary
layer. A comparison between one of the current proﬁles with a Blausius and turbulent
velocity proﬁles at a similar Reg is shown in Figure 4.2. The data demonstrate the

turbulent character of the separating boundary layer in this study.

 

aoI—u- * ~- - E
Q l _e.Re=1139
\ 25I 129-2057
A i —1— e6-

_A_Ree=2739

20+

15}

l

10E

5:»

of __ 022"” 04

 

Figure 4.1 Mean velocity proﬁles of the boundary layer at separation

The turbulent nature of the separating boundary layer may also be put on
quantitative basis by considering the boundary layer shape factor (H12). Hg is a useful
quantity to describe the shape of velocity proﬁle. For example, in a zero-pressure-
gradient turbulent boundary layer, the shape factor is given as 1.3 in Schlichting (1979)
and 1.31 in Morris (2002). For a laminar boundary layer, H12 is 2.59, which is obtained

from the Blasius solution. The shape factor values for the boundary layers established in

80

this study are nearly equal to those ﬁ‘om Schlichting ( 1979) and Morris (2002) (see Table
4.1). They are also in good agreement with those in some of the BFS ﬂow studies: 1.29 in

Chun and Sung (1996) and 1.26 in Baton (1980), as examples.

301-——-~ ~—.————— --___x-_ W“ I _____
I -e.- Ree=2739 .
Q 25; O Reef2953, Nagu1b(1992) I
>5 . "‘— Blasms J
zoI
15$

 

 

U/Um

Figure 4.2 Comparison of a mean velocity proﬁle from the current study to that of
turbulent and laminar boundary layers.

Conventionally, the friction velocity (u, = 41,, / p ) is used to non-
dimensionalize the mean velocity (U+ =U/u,) and the inner, or viscous, length scale to

normalize y ( y+ = yur /u) in plots of the boundary-layer proﬁles. A 'canonical', or
'fully-developed', turbulent boundary layer is known to posses the so called 'logarithmic

region' where the mean-velocity varies with height according to: U + = 2.44 ln(y+ ) + 5.0.

The constants in the last equation are determined empirically. Generally speaking, there

is a good deal of scatter in the values reported in the literature, with 2.44 and 5.0 being

81

the most commonly cited ones. The primary difﬁculty in obtaining the constants of the
'log law' is the ability to measure the wall-shear stress, or ﬁiction velocity, accurately.
One typical, albeit inaccurate method, to determine ur is known as the Clauser ﬁt. In this
procedure, the log law is assumed to have a known value for the slope or intercept, which
leads to the value of ur by comparing to the slope or intercept of a least-squares line ﬁt to
the logarithmic portion of the mean velocity proﬁle. However, the accuracy of the
method is generally no better than 10 - 15% because of uncertainty in determining the
limits of the log region. Recently, Osterlund et a1. (1999) conducted careful
measurements of the wall-shear stress in a turbulent boundary layer over a wide Reynolds
number range using oil-ﬁlrn interferometry. Their results showed that appropriate
implementation of the Clauser-ﬁt requires the ﬁt to start above y+ = 200. This would
leave too little data to obtain a reliable ﬁt at Reynolds numbers comparable to those
investigated here, and therefore this method was not employed.

No attempt was made to measure the wall-shear stress at the point of separation
directly, since the main focus of the current study is the separated region downstream of
the step rather than the separating boundary layer. In fact, Morris (2002) showed that in
geometrical separation of turbulent boundary layers, a 'sub-shear layer' controls the
dynamics and subsequent development of the ﬂow downstream of separation. Only the
near-wall vorticity of the boundary layer participates in the initial instability of the sub-
shear layer. An effective momentum thickness of ten wall units was found to
characterize the width of the region signiﬁcant to the separating-shear-layer instability.
This is well below the log region, and hence the speciﬁc constants of the velocity proﬁle

are not expected to have important bearing on the separated ﬂow. However, for the

82

purpose of qualitatively examining the behavior of the current boundary layer proﬁles
under the traditional scaling, the friction velocity was estimated using the empirical
correlation of Ludwieg and Tillmanm (1949) for the skin-friction coefﬁcient for an

equilibrium turbulent boundary layer (see Hinze (1959), page 486):
C f = 0.246x 10‘0-678” 12 Ref-263 (4.2)

The mean velocity proﬁles are normalized with the ﬁiction velocity obtained

using Equation 4.2 and u7 = ‘/(CfU3, ) / 2 . The results are plotted on a semi-logarithmic

scale in Figure 4.3. All proﬁles seem to collapse fairly well using the traditional inner
scaling, with the exception of data near the edge of the boundary layer where the classical
Reynolds-number-dependent deviation from the log law (know as the 'law of the wake')

is observed. The data also exhibit a 'log-law—like' behavior from approximately y+ = 30
to 300. The best ﬁt to these data is u+ = 2.141n(y+) + 5.65 , which yield constants that are

about 13% off ﬁom 2.44 and 5.0. The deviation may be attributed to a number of reasons,
including the error in determining the friction velocity, the transverse curvature effects of
the axi-symmetric shape used in the current study, or the development length of the
boundary layer, which is shorter than that employed in most studies aimed at establishing
equilibrium turbulent boundary layers.

Finally, the streamwise rms velocity proﬁles are shown in Figure 4.4 for all
Reynolds numbers. The data exhibit the classical behavior of a turbulent boundary layer
with the largest rms value located very close to the wall. Collectively, the results
discussed in this section indicate that the separating boundary layer is turbulent. It is

uncertain, however, whether it is in what is considered in the literature to be a 'canonical'

83

state. As discussed above, this issue is not of central signiﬁcance to the objectives of the

current study.

 

 

To Ree=1139
+ Ree=2057

201 A Ree=2739
— Fit

     

 

10E . l
5» \ I
u+ =2.14ln(y+)+5.65 1
01-0 . ...2 22.2...2222i 2222222222...” 2___2.. 2.2.2.... 222.23 222—22- .. 2.2.22.2 4
10 10 10 10 10
y+

Figure 4.3 Inner-scaled Mean velocity proﬁles of the boundary layer at separation

   
    

30, __. 2 2. 2 2 2 2 22 , 2 22
m . —e- Ree=1139
R 25} _+_ Reo=2057

I .4. Ree=2739

 

 

..k. . 2

0 " 0.02 0.04 “6.06“ ."

urine/U00
Figure 4.4 Rms velocity proﬁle of the boundary layer at separation

84

4.2 One-point Wall-shear Measurements in the Separated/reattaching Flow
4.2.1 Mean Reattachment Length

At the mean reattachment location (x,), the mean wall-shear stress equals zero.
Thus, the streamwise distribution of the mean skin-friction coefﬁcient (Cf) could be used
to directly ﬁnd x,. Using the results from the movable sensor at different streamwise
locations over the range x = 0.3OH to 9.84H (the speciﬁc positions are speciﬁed in Table
4.2 section 3.7), Figure 4.5 was obtained for three Reynolds numbers of Re”: 4300, 8700,
and 13000. Note that the plot also contains data from the reference sensor at x = 5.07H.
The vertical axis represents the mean skin-ﬁiction coefﬁcient, and the horizontal axis
indicates the corresponding normalized x location. The data for all Reynolds numbers

have the same trend.

 

 

 

 

 

-3
5x10? f 4 22.22.22.222
—9— ReH = 4300 t .
4:2. ReH = 8700 c = ;
3,_,_ ReH=13000 ,, ’ I
2. 0 '
0‘1
0 e"
-1. »
-2.
-31.. 22. . 2 2 222 _222 2
0 1 5 6 7 8 9 10

 

x/H

Figure 4.5 Streamwise distribution of the mean skin-friction coefﬁcient

85

The mean skin-friction distribution in Figure 4.5 displays two positions where Cf
crosses zero. The ﬁrst position close to the step deﬁnes the location of the mean
secondary separation point or corner eddy length as discussed above. The position, where
Cf: 0 at the farther downstream location, gives the mean reattachment length. The values
of x, for the Reynolds numbers investigated are tabulated in Table 4.2.

Another method to ﬁnd the mean reattachment point is to use the distribution of
forward/reverse ﬂow probability, as demonstrated in Baton (1980), Westphal and
Johnston (1984), Spazzini et al. (2001) and Tihon et al. (2001). For example, a forward
ﬂow probability (hereafter referred to as FFP) of 100% means the ﬂow is always towards
the downstream direction. On the contrary, FFP = 0% means the ﬂow is always towards
the upstream direction. At the mean reattachment point, FFP = 50%. Figure 4.6 displays
three curves representing the FFP distribution for all three Reynolds numbers, which
exhibit the same trend. Consistent with the distribution of the mean skin-ﬁiction
coefﬁcient, there are also two positions where F FP = 50%. The farther downstream one
indicates the mean reattachment location, the values for which are listed in Table 4.2.

As depicted in Table 4.2, the difference in determining the reattachment length
using the Cf or FFP is less than 1.5%. The reattachment length slightly increases with
rising Reynolds number over the range of investigation. The same trend was also found
by Spazzini et al. (2001), where Rey was varied from 3500 through 16000. The increase
could be attributed to the variation of the boundary-layer thickness. Both Eaton and
Johnston (1981) and Adams and Johnston (1988) found that the reattachment length
increased slightly with decreasing boundary-layer thickness. Since the Reynolds number

is increased by increasing the freestream velocity in the present study, the thickness of

86

the boundary layer at separation does in fact decrease with increasing Reynolds number

(the speciﬁc values of 6 may be found by referring to Table 4.1).

100I— *

  
  

 

FFP (%)
0|
9

 

25 —e— Re=4300
I + Re=8700
I —l—- Re=13000

0! — 22 — 2 — 4—
0 1 2 3 4 5 6 7 8 9 10

Figure 4.6 Streamwise distribution of forward ﬂow probability

Table 4.2 Reattachment length values

 

 

 

 

x/I-I
xr/H x,/H .
ReH = = o Drfference (%)
(Cf 0) (FFP 50 /o) Mean
4300 4.26 4.33 1.43 4.30
8700 4.60 4.67 1.45 4.64
13000 4.84 4.88 1.00 4.86

 

 

 

 

 

 

 

Another important observation is that the reattachment length (x,/H) reported here
is smaller than that in the bulk of the literature; for example, x/H = 5.39 at Re” = 5100

(Spazzini et a1. (2001)), 5.1 at Re” = 4800 (Tihon et al. (2001)), 6.00 at Re” = 5000

87

(Jovic and Driver (1995)) and also smaller than the minimum x, (4.9 H) reported in the
review by Eaton and Johnston (1981). But in all of these studies, either a 'two-
dimensional' planar BFS geometry or sudden expansion of a pipe was used. Although the
ﬂow in the sudden expansion is also axi-symmetric, the geometry has concave curvature,
and, the boundary layer characteristics at separation may be different. Therefore, it is
possible that the shorter reattachment length found here is caused by the geometry
difference. Kim et al. (1996) also found that the reattachment length in laminar
boundary-layer separation over an axi-symmetric BFS, composed of an elliptic-cone-
shaped obstruction placed in front of a cylinder tube, was always shorter than that of the
planar BF S ﬂow. They attributed this phenomenon to the effect of transverse convex
curvature.

To examine the reasonableness of this hypothesis further, the reattachment
mechanism needs to be scrutinized. As argued by Chapman et al. (1958) and Eaton
(1980), the reattachment physics could be attributed to the mass balance between the
ﬂuid re-entrained into the shear layer and the pressure-driven backﬂow. For example, if
there is more pressure-driven backﬂow, the reattachment length will increase so that the
shear layer has a longer length to entrain the backﬂow, and the pressure gradient
decreases leading to a reduction in the backﬂow. On the contrary, increasing entrainment
rate to the shear layer will decrease the reattachment length, which has been veriﬁed in
the control experiments of the backward-facing step ﬂow; for example, Chun et al.
( 1999) and Lai et al. (2002).

It is conjectured here that because of the axi-symmetric geometry of the BFS, the

vortical structures at separation are substantially more coherent in the transverse

88

(azimtuthal) direction than in the planar geometry, and more capable of sustaining higher
entrainment rates than in the planar geometry, leading to a shorter reattachment length.
In the planar geometry, the transverse coherence, or two-dimensionality, of the vortices is
inﬂuenced by side-wall effects directly, as well as indirectly, through the imperfect two-
dimensionality of the mean ﬂow itself. The coherence (and energy) of the vortices may
be enhanced through artiﬁcial excitation of the ﬂow. Chun and Sung (1996) employed
local forcing to control the turbulent ﬂow separating over a planar BF S. The
reattachment length was reduced from 7.8H to 5.0H. It was shown that the forcing
enhanced the shear-layer growth rate and produced roll-up of a large vortex at separation,
leading to higher entrainment rate and shorter reattachment length. In a similar manner,
one could imagine a highly coherent vortex ring shedding from the transverse convex
curvature at the step and growing quickly to a large size through pairing interactions,
affecting a higher entrainment rate.

Another factor to consider in examining x, results is the boundary-layer-thickness
to step-height ratio. In the current study at Re” =13000, Reg = 2739, (SH =1.72 and x,/H =
4.88. This may be compared with two well-documented experiments in a planar BFS
ﬂow: Re” = 33000, R39 = 1340, (WI =0.41 and x,/H =7.8 for Chun and Sung (1996) and
Re” = 39455, Reg = 852, 5’H =0.23 and x/H =7.95 for Eaton (1980). The boundary layer
in all studies is turbulent. The current 8’11 is the biggest, so the reattachment should be

the smallest.

89

4.2.2 Mean/ms Skin-friction Coefﬁcient and FFP

The mean skin-ﬁiction coefﬁcient distribution is re-plotted in Figure 4.7 after
normalization of x using x,. Data from the literature are also added to the ﬁgure for the
purpose of comparison. The measurements by Spazzini et al. (2001), Tihon et al. (2001),
and Jovic and Driver (1995) were obtained using a "double" hot-wire type skin-friction
probe, an electro-difﬁrsion technique and laser-oil-interferometry (LOI) technique,
respectively. The ﬁrst two methods infer the shear stress from heat/mass transfer ﬁom the
sensor and require calibration, while the last one measures the shear stress directly and
does not require calibration. LOI is the most accurate wall-shear measurement technique
known today, but it can only provide mean-shear measurements.

The mean skin-friction plots in Figure 4.7 are divided into two parts: (a) for Re” <
10000 and (b) for Re” >10000, in order to show the data points clearly. Despite
differences in the experiments, such as geometry and Reynolds number, all data sets
exhibit the same trend. From the separation point to x/x, z 0.33, the value of the mean
skin ﬁiction is very small and slightly above zero. This region is referred to as the
secondary recirculation zone, or comer eddy. Kim et al. (1996) studied the axi-symmetric
separation and reattachment of a laminar boundary layer and were able to identify the
comer eddy only when the diameter ratio of the up and downstream cylinders at the step
was less than 2.0. In the current study, the diameter ratio is 1.25, and hence the ﬁnding of

the comer eddy here is consistent with the conclusions of Kim et al. (1996).

90

(a) Re” < 10000

 

 

ReH =4300

ReH =8700

ReH =3500 (Spazzini et al. (2001))
Re” =5100 (Spazzini et al. (2001))
Re” =3860 (Tihon et al. (2001))
Re... =8840 (Tihon et al. (2001))
Re” =5000 (Jovic & Driver (1995))
Re” =6800 (Jovic & Driver (1995))

DxV+DO>O

91

(b) Rey > 10000

 

 

 

 

 

 

 

‘60 0:5 1 “1.5 i if 2 2.5
x/xr

ReH =13000

ReH =10000 (Spazzini et al. (2001))
Re" =16000 (Spazzini et al. (2001))
Re” =11900 (T ihon et al. (2001))
Re” =10400 (Jovic & Driver (1995))

XVDOO

Figure 4.7 Comparison of the mean skin friction coefficient results with other studies:
(a) ReH< 10000, (b) ReH> 10000

From x/x, z 0.33 — 1.0; i.e., within the primary recirculation region, the mean Cf is
negative and the maximum "reverse" Cf is at x/x, z 0.68. Downstream of the
reattachment, the mean Cf rises slowly to a steady value at x/x, z 2. The magnitude of
mean Cf in Tihon et al. (2001) is generally higher than all other measurements. Hence,

these data are not used for comparisons concerning C; magnitude. The data at R811

=13000 agree well with those at Re” = 10400 (Jovic & Driver (1995)), and Re” =10000

92

(Spazzini et al. (2001)) except for the region around the maximum negative shear, where
the present data are closer to those from Spazzini et al. (2001) than the others. Similarly,
the data at Rey = 8700 collapse on those at Re” = 5100 (Spazzini et al. (2001)) and Re”
=6800 (Jovic & Driver (1995)) except for the region around the maximum shear reversal
(x/x, = 0.68). The magnitude of Cf from the current data is somewhat smaller than that
obtained from the other studies around x/x, = 0.68. Finally, at the lowest Reynolds
number of Re” = 4300, the results are consistent with Rey = 3500 data (Spazzini et al.
(2001)) upstream of reattachment. However, farther downstream, the data at Rey = 4300
are obviously larger than those at Re” = 3500 (Spazzini et al. (2001)). The discrepancy
might be caused by the differences in the geometry and/or the separating boundary layer
state. In Spazzini et al. (2001), the step has planar geometry and the ﬂow upstream of the
point of separation is a two-dimensional fully-developed channel ﬂow.

Figure 4.8 depicts the magnitude of the minimum skin-ﬁiction coefﬁcient (Cﬂm),
which is the maximum skin-friction coefﬁcient of the backﬂow, against the Reynolds
number. More data points from J ovic and Driver (1996) are included in the ﬁgure for the
purpose of comparison. All data points collapse fairly well onto a straight line with —1/2
slope when plotted using logarithmic scale. This kind of power-law relationship between
CM”. and Re” was also found in Devenport and Sutton (1991), Jovic and Driver (1996)
and Tihon et al. (2001). Tihon et al. (2001) included data ﬁ'om Jovic and Driver (1996),
Westphal et al. (1987) and Devenport and Sutton (1991). They concluded that the ﬂow
near the wall in the primary recirculation region is viscous-dominated or laminar-like,
where the local skin-friction coefﬁcient is inversely proportional to square root of the

Reynolds number.

93

 

 

 

 

 

 

 

 

 

 

 

l

5.103»- I

2‘."
”l 0 present l
l A Jovic & Driver (1996) .
— -1/2 slope
10“ 2 2 . . 2 2 . 2 2 2 2 . . . .
103 104 105

ReH
Figure 4.8 Minimum skin-friction coefficient versus Re”

Despite differences in various experiments, it is of interest to note that a universal
empirical correlation between Cm». and the Reynolds number provides a fairly good
agreement with all data sets. Tihon et al. (2001) investigated the skin-friction coefﬁcient
inside a water channel using a two-dimensional BFS geometry, J ovic and Driver (1996)
studied the air ﬂow over a two-dimensional BFS geometry while the current study is
concerned with the air ﬂow over an axi-symmetric backward-facing step. Figure 4.9
shows a comparison with published results for the forward ﬂow probability. All of the
measurements show FFP value that is close to 100% just downstream of the step. This
value decreases rapidly to 50% at x/x, z 0.33, which marks the boundary between the
comer eddy and primary recirculation region. Downstream of this position, the FFP

decreases to 0% followed by an increase to 50% again at x/x, z 1, or the reattachment

position. The FFP continues to rise until it reaches 100% around x/x, z 1.33. The x/x,

94

location corresponding to 50% F F P at the point separating the primary and secondary
recirculation zones, becomes smaller with increasing Reynolds number. This suggests
that the size of the secondary recirculation zone becomes smaller relative to the primary

recirculation zone as the Reynolds number increases.

 

 

 

 

: 6
80 (3+ 8‘
9 g Q
g 60» v ‘6 8
s. E
u. 40. 9; j
20' $27.-
l t ‘
05” 0 1 ” ”1.5 ‘ 2 25
Mr
0 Re” =4300
‘ Rep-1 =8700
0 Re” =13000
o ReH =3500,Spazzini et al. (2001)
A ReH =5100,Spazzini at al. (2001)
v Re" =10000,Spazzini at al. (2001)
0 Re“ =16000,Spazzini et al. (2001)
+ Rel. =3860,Tihon et al. (2001)

Figure 4.9 Comparison of forward ﬂow probability results with other studies

The observations from Figure 4.7 and Figure 4.9 are quite consistent. Positive
values of the mean skin-friction coefﬁcient correspond to FFP > 50% (i.e., most of the

time the ﬂow is in the downstream direction), while negative Cf is found at locations

95

where FFP < 50%. At FF P = 50%, sz 0: thus the consistent agreement found earlier in
determining x, from F FF and Cf(see Table 4.2). Finally, it is interesting to note that the
location of the maximum reverse shear (x/x, z 0.68) is different from the location of FFP
z 0 (x/x, = 0.57).

Figure 4.10(a) and (b) display the rms skin-friction coefﬁcient from the current
measurements and those from the selected studies. The trend of C;’ is the same for all
measurements. It increases monotonically with increasing x, reaching a maximum in the

vicinity of the mean reattachment point, followed by a slow decay thereafter.

(a) Re” < 10000

 

 

 

 

x 10‘3
5 , + . - 4 1
+++ l
l + 1
4 ++
1 ++ +
3I + +++
~ ... I + o +++++ l
1 ‘ ‘ 8 O I
+ “ ‘AAAAA A I
l
O 2 2 2 l
0 0 5 1 1.5 2 2 5
x/x
,
0 Re“ =4300
A ReH =8700
0 Re... =3500,Spazzini et al. (2001)
A ReH =5100,Spazzini et al. (2001)
+ ReH =3860,Tihon et al. (2001)

96

(b) Re” > 10000

 

 

-3
5 x_1_0__ f *7
4.
3 I.

{r

 

 

Re" =13000

ReH =10000,Spazzini et al. (2001)
Re“ =16000,Spazzini et al. (2001)
Re” =37300, Westphal et al. (1981)
Re“ =36000, Adam & Johnston (1988)

XUDOO

Figure 4.10 Comparison of rms skin-friction coefﬁcient results with other studies: (a)
ReH< 10000, (b) Rey> 10000

Qualitatively speaking, the Cf’ results bear similarity to that associated with the

rms wall-pressure ﬂuctuation in Lee and Sung (2001) and Hudy (2001), where the rms
pressure rises with increasing downstream distance till x/x, z 1 and continues to decay

aﬁerwards. This suggests that the driving mechanism of wall-shear ﬂuctuations is linked
to that of wall-pressure ﬂuctuations. Such a link would be consistent with the conclusions

of Devenport and Sutton (1991) that the ﬂuctuations of the streamwise velocity near the

97

wall were driven by pressure gradient ﬂuctuations imposed by the separated shear layer.
One component of this pressure gradient is that associated with the low-frequency
ﬂapping of the shear layer. The other is that generated by the momentary passage of the
vortical structures above the point of 1;. observation.

The detailed characteristics of the shear-layer vortices were examined recently by
Kostas et al. (2002) using PIV measurements within the separated ﬂow region. They also
proposed that small-scale vortical structures were responsible for Reynolds stress and
turbulent kinetic energy upstream of reattachment while large-scale structures were
dominant downstream of reattachment. Moreover, Scarano et al. (1999) found that the
vortex formation started from x/x, z 0.17, and that the vortices approached the bottom
wall at x/x, z 0.58 in their PIV investigation of a BFS ﬂow. Owing to their smaller size
and larger distance above the wall, the smaller vortices found near separation presumably
have less of an effect on the wall-shear than the larger ones, farther downstream. The
gradual increase in C;' with x/x, is, thus, partly a manifestation of the increase in energy
of the passing vortices and their movement closer to the wall.

Near the step, the very small skin-friction ﬂuctuation could be caused by the
unsteadiness of the corner eddy, which is associated with the ﬂapping of the shear layer.
Downstream of the reattachment, Cf’ decays slowly due to the lack of free shear layer
activity there. The energized vortical structures from the separated shear layer decay and
diffuse as they travel downstream.

Comparing the magnitude of Cf' from the vaious investigations,, the data from
Tihon et al. (2001) are higher than others, as was the case with the mean Cf results (see

Figure 4.10). Although the data from Westphal et al. ( 1981) and Adam and Johnston

98

(1988) are close to those from Spazzini et al. (2001), they were obtained at much higher
Reynolds number. Except near x,, the current rms skin-friction coefﬁcient values at Rey =
4300 fall between the data at Re” =3500 and 5100 from Spazzini et al. (2001). At Re” =
8700 and 13000, the current data are smaller than those at Re” =10000 and Re” =16000,
respectively. However, the reader is reminded of the axi-symmetric geometry of the back
step used here, which may be the cause of some of the observed differences.
Additionally, with the exception of the data from Tihon et al. (2001), all other
investigations employed one or more variation of thermal-wake tagging by a heated wire
to detect the ﬂow direction. The present investigation is the ﬁrst to employ ﬂow
direction detection that is not based on thermal-wake detection.

It is also interesting to examine the location of the peak rms skin-ﬁiction
coefﬁcient. The data from Westphal et al. (1981) and Spazzini et al. (2001) peak at x/x, z
0.72, Tihon et al. (2001) at x/x, z 0.76, Adam and Johnston (1988) at x/x, 211.25, while the
current results exhibit the peak at x/x, = 1. It is well documented that the peak rms wall
pressure occurs slightly upstream of the reattachment point in two-dimensional BF S ﬂow
and splitter-plate ﬂow. Driver er al. (1987) found the peak rms wall-pressure at x/xr z
0.91, Heenan and Morrsion (1998) at 0.91, Lee and Sung (2001) at 0.93, and Hudy et al.
(2003) at 0.92. Due to the limited number of measurement locations, there is no sensor
data available from x/x, = 0.9 — 1.0 in the current study. It is possible that the maximum
rms skin-ﬁiction coefﬁcient found here could be located at x/x, = 0.9 —1.0, which would

be consistent with the wall-pressure measurements.

99

4.2.3 Probability Density Function (pdf)

The normalized pdfs of the ﬂuctuating wall-shear stress are shown in Figure 4.11
(a), (b) and (c) for all Reynolds numbers. Consistent with the mean/rms skin friction
coefﬁcient, at x/x, = 0.07 and Re” = 4300, the values are very close to zero. Deviations
from the mean are very small, signifying a very low level of ﬂuctuations. The shape of
the distribution is generally similar to those at the higher Reynolds numbers. At x/x, =
0.6, which is close to the position of maximum wall shear in the reverse direction, most
of the data have negative values and the distribution spreads over a wider range,

corresponding to large ﬂuctuation level.

(a) Re” = 4300

3000 . 400 A

 

 

 

 

 

 

 

 

 

 

 

 

XIxr=0'07 j
I 300
2000»
'3 200
1000i
100 j-
o 2_ 2 2 _ ___22 2l
-0.01 O 0.01 -0 01
300 2 400
300»
200
100 ,
0 222 2
-0.01

 

100

(b) Re” = 8700

 

 

 

 

 

101

 

 

 

 

 

 

 

0.01

(c) Re” =13000

 

 

 

 

 

 

 

 

 

 

 

 

 

4000 2. 1000 -— 2 2 — ~—— —
x/xr=0.06 ' I x/xr=0.54
3000 1 - _
“g 2000 I 500I I
1000 l .
. ___l ol I I
49.01 0 0.01 -0.01 0 0 01
1000 r—---—---— 1 800 .. .
I VX5032 x/xr=1.54
' I 600 I
R 5°°I I 400 I
I
. I 200.; j ,
49.01 0 0.01 49.01 0 A 0.01
of of

Figure 4.11 Wall-shear pdf at different x positions: (a) ReH= 4300, (b) ReH= 8700, (c)

Re” = 13000

At locations near x/x, =1, i.e., the mean reattachment location, the histogram

exhibits a bi-modal character. Such a histogram is characteristic of quasi-periodic, or

limit-cycle, type signals. The “bi-modality” is related to the movement of the

instantaneous point of reattachment up/downstream of x z x,, and may be indicative that

such movement is the result of a limit cycle resulting from global instability of the ﬂow.

Hudy et al. (2003) suggested that the ﬂapping of the shear layer, which would cause the

reattachment point movement, is driven by an absolute instability of the primary re-

circulation bubble, leading to self-sustained, or limit-cycle, expansion/contraction of the

102

separation bubble. The bi-modal character of the wall-shear histogram is in agreement
with this scenario. Finally, downstream of the mean reattachment location, x/x, = 1.54-
1.75, most of the measurements are positive, which is in agreement with the fully

attached ﬂow well beyond x,, as found earlier from the FFP results.

4.2.4 Power Spectra

The power spectra of the ﬂuctuating wall-shear stress at different locations are
plotted in Figure 4.12 (a), (b) and (c) for Reynolds numbers of 4300, 8700, and 13000,
respectively. The original time series was divided into small records (4096 points per
record). The spectra (Pu) were obtained by averaging the spectra of these records as

follows:

M5

X(f)*C0nj(X(f))
1 (4.3)
mez

 

Pxx(f) = "

where f is frequency, m is number of record, N is number of points in each record, and
X0) is Fourier transformation of data record x(t).
To avoid clutter of the plot, each spectrum was shifted up by one decade relative

to that one x location upstream of it. The spectra and ﬁequency are normalized as

 

follows:
4- Pxx
Pxx = ___—1 2 (4.4)
IEPUiI
:1- f-x,
= 4.5
f U0, ( )

103

Two primary observations may be made from Figure 4.12. First, for all Reynolds
numbers, the spectrum value decays with increasing frequency at all x locations. This is
fundamentally different ﬁ'om wall-pressure spectra, which exhibit a spectrum "peak"
switch with increasing x. To demonstrate, wall-pressure spectra measured by Hudy et al.
(2003) on the same model and Re” = 8700 are shown in Figure 4.13. Unlike rw' spectra,
the peak of the wall-pressure spectrum switches from being within the low-ﬁequency
range near separation to a frequency of f z 0.5 for x/x, > 0.9. This discrepancy will be
discussed in the following chapter (section 5.3) when presenting a uniﬁed picture that
accounts for both one- and two-point observations.

Although no 239' spectrum peak is found around f z 0.5 in Figure 4.12, there is a
clear "ﬁlling" of the spectrum in the vicinity of this frequency as x increases. This may
be identiﬁed by the decrease in the spectrum slope in the middle range of frequencies as
identiﬁed in the ﬁgure. For x approximately less than 0.3x,, a slope of —4/3 is found. At
larger x locations, the slope becomes —2/3. Also at locations sufﬁciently far downstream,
x/x, > 0.8, the spectral slope at the high-frequency end exhibits a -5/3 slope. This is
indicative of the establishment of an inertial subrange, presumably associated with the re-
development of the boundary layer after reattachment.

The rise in the spectrum energy in the middle range of frequencies as x
approaches xr (around f z 0.5) may be seen more clearly by plotting the spectra on a

semi-log plot, as seen in Figure 4.14. For these plots, the horizontal axis is log(f"'), and
the vertical axis is linear and represents the product of P; x f. In this fashion, the

geometrical area under the spectra represents the energy level. For example, the area

under the spectra around x/xr = 0.07 is smallest, which corresponds to very small energy

104

-

i.e., very small ﬂuctuation of a. On the contrary, the area under the spectra around the

reattachment position is largest, where the ﬂuctuation is highest.

(a) Re” = 4300

-6
10 222 .22 222222222

 

1o“3 ’-

-10

10

42
.O_§ 1o 2

 

-14

10

~16

10

 

 

-1a

10 9.2... 2. 222 ......._k_.22k2._ ___22 2. ___“- hux-._~._2 2;. _A2 A*L.C~A b—b ___—___.L .a.__._4._¢ A_._r

10'2

(b) Re” = 8700

.5

 

10

10" -

-10

10

 

.12
.032 10 2

  

 

.— x/x, = 0.57 _.
<— x/x, = 0.31

10

10'

(c) Re” =13000

 

 

  

 

 

 

   
 

 

 

 

10.6 IF— ‘—'_'—'“”“*_'ﬁ aﬁ h}
10'8 } --------------
M ‘ --------- : .213
1o“°~ \ <-x/x,= 1.67 .
. - (3.3—Vb: 1.17
§ 1042'
'm " \ <—x/x,=0.79 “
10-14” ““““““““ «X/xr=0.54 A
<—x/x,=0.29 i
10'“? .:
18 <-x/x,=0.06 j
10 . ....l W W: . ALA
10'2 10'1 10° 101 102
f.

Figure 4.12 Wall-shear power spectra at different x locations: (a) ReH= 4300, (b) Re”:
8700, (c) Rey = 13000

 

 

 

 
 

 

 

(DPP
109 -
”Rs. 4— x/x,z 2.3
106 .
4— x/x,z 1.8
103 - "2.x {— x/xrz 1.3 1
"""""""""""""""""" <— x/x,z 0.9
00 _ -.""-\..~.‘ ‘
1 p - 4— x/x,z 0.6 'f-
a l... . __ﬁg/M;
10'2 10’l 10° 101 102

Figure 4.13 Power spectra of the ﬂuctuating wall pressure at different x locations
(Hudy (2003), private communication)

106

(a) Re” = 4300

 

 

 

“'1
E

I

4

I

5x 10“3
{ ,
‘ . — x/xr=0.07
, .-°-'°': --- x/xr=0.33 r
4' '° -.-. x/xr=0.61
:: ...... xlxr=0.90
33»
. a: ‘ '
Q 2f r.,\—l°V-‘ k;
I. 3'. :_
I “. °-.
.1 9‘. 3
1% I" ‘3.
i \3. ,.
oL—d ""' ""‘
10'2 10'1 10° 101

(b) Re” = 8700

 

FT

— xlx =0.07
{:3 -..- xlx =o.31
-.-. x/x =o.57
...... x/xr=0.83

q

n-I

 

 

 

 

. C
,,«- $5.: 1
.n 3 °.
. I"! ‘
j
... .’
0.. I
.. .’o ~"-\’AA”‘
"‘ v ‘ "\
‘
:— T "“ “w_-__ _
-1 0 1
10 10 10

107

(c) Rey =13000

 

 

.9
2.5§_1_0__,_,_,,,,,, , - ,t._-s-__w,-._- , V!
_ — xlxr=0.06 i
:,‘: --.. xlxr=0.29 r
2 gigs-z, -.-. x/xr=0.54 J
.5‘ 3"- ...... x/xr=0.79 "
15‘ t3 ‘
'u— . i s a.
X

1 .’ 5' \ t
o I” F” W ‘ #702
Figure 4.14 Wall-shear power spectra plotted on semil-log scale for different x locations:
(a) ReH= 4300, (b) ReH= 8700, (0) Re” = 13000

At x/x, z 0.07, most contribution to the total energy of the ﬂuctuating components
of the wall-shear stress comes from low frequencies for all three Reynolds numbers. This
may be seen more clearly from Figure 4.15, where a different vertical scale is used to
magnify the view of the spectra at x/x, = 0.07. Ignoring the harmonic peak at f z 1 and
the lowest Reynolds number, which is believed to be associated with noise, the energy
content is concentrated around a frequency of f z 0.1. A peak at f z 0.16 was found in
the pressure measurements of Hudy et al. (2003) upstream of O.25x,. Spazzini et al.
(2001) also detected a peak at f z 0.1 in the wall-shear spectra measured upstream of
O.28x,. For the locations downstream of x/x, z 0.30, the build-up of energy up to the
reattachment location clearly falls in a frequency range that is centered around an f value

0.65 (see Figure 4.14). This frequency range encompasses the passage frequency of the

108

shear layer vortices, as reported in the literature (see next paragraph for speciﬁc details).
Thus, it is apparent that the increased level of wall-shear ﬂuctuations is a manifestation of
the increased energy and proximity of these vortices as they approach the reattachment
point.

The two frequencies of f = 0.1 and 0.65 were also identiﬁed in the limited wall-
shear literature available. For example, Spazzini et al. (2001) found the low and high '
frequencies to be f = 0.08 and 1.0, respectively. More abundant observations of these
frequencies are available from wall-pressure measurements. Peak frequencies of f =
0.12 — 0.18 and 0.6 - 0.9, were found by Hudy et al. (2003) in a fence/splitter-plate
model, f' = 0.11 and 0.50, by Lee and Sung (2001) in BPS flow, and f = 0.18 and 0.6, by

Driver et al. (1987) in BFS flow.

 

—e— YRef:4300ﬁ 1‘
—A— Re=8700
—+— Re=13000 4'

 
  
  

 

xf

3r
.0}?

 

 

 

 

10'2 10'1 10° 101 102
f'

Figure 4.15 Wall-shear power spectra plotted on semi-log scale at x/xr z 0.07

109

4.2.5 Temporal Autocorrelation Function

The autocorrelation coefﬁcient (R (1’) obtained from single-point measurements is
generally used to identify the dominant time scales of the signal of interest. R H is
essentially the inverse Fourier Transform of the power spectrum. In the summary, Rf,»

may be calculated using the following equation:

 

1ti.(t)rl.(t - 9) (4.6)

Rr‘r'(9) z , (t)2
1w

 

 

where the overbar denotes time averaging, Sis the time delay, and tw'(t) is the ﬂuctuating
wall-shear time series. Figure 4.16 (a), (b) and (c) display RH for three Reynolds
numbers at x/x, z 0.07, 0.7, 1.3 and 1.8. These positions correspond to those immediately
after the step, in the vicinity of x, and farther downstream of x,, respectively. The general
trends in the data are the same for all three Reynolds numbers. As expected, R” equals 1
at 9 = 0 and decreases monotonically with the increase in the time delay. At x/x, z 0.07,
Rf’f has a long tail and approaches a zero value at SUw/xm i 15 for Re” = 4300 and i 8
for Ken = 8700 and 13000 (see Figure 4.17), while it decreases much more quickly and
drops to zero at SUm/x, z i 2 at the rest of the locations. The longer correlation time
means that larger scale (or lower frequency) motion dominates the ﬂuctuation of the wall
shear. Therefore, it may be concluded that low-frequency disturbances are dominant
immediately after the step and higher-frequency ones are more dominant farther
downstream. This result is consistent with the power spectrum analysis in the previous
section, where it was found that most of the energy was contained at the low-frequency

end of the spectrum at x/x, z 0.07, while most of the contribution to the total energy came

ﬁ'om higher frequencies downstream of x/x, z 0.3.

110

(a) Re” = 4300

 

 

    
   

Rfr’
-37 1577-4732771) '2' it, is ’8
SUOJx,
(b) ReH=8700
1'7 7 #Ti #5 7'7 7 7"—
Rff ‘ — X/Xr=0 07

...j. x/xr=0.70

03} -.-. $0421.36 "
1 x/xr=§1.88 ‘

0.6) ~
1

0.4)
i

0.2)

111

(c) ReH= 13000

Rr’r’

 

-8 -6 41’ -i2 707772 4 6" 3
SUa/x,

Figure 4.16 Autocorrelation coefﬁcient of the ﬂuctuating wall-shear stress at different x
locations: (a) ReH= 4300, (b) Re”: 8700, (c) Re” = 13000

1: 7e 7‘ #¥ n.7, a 7,, 7 a

Rh: — ReH=4326 ,
l ...... ReH=8653

08" " ' " ' ' ;.‘.‘.'ReHi12979 ‘

 

46‘ ‘7 7 :11)” ‘7 "_o—TV ' To" W 2‘0
SUq/x,
Figure 4.17 Autocorrelation coefficient of the ﬂuctuating wall-shear stress at x/xr
z0.07

112

Contour plots of the autocorrelation coefﬁcient at all x locations are displayed in
Figure 4.18 (a) — (c) to show a ﬁller picture of the variation of Rf,» in space. The
horizontal axis represents the streamwise distance from the step normalized with x,, and
the vertical axis gives the non-dimensionalized time delay 9U../x,_ The color indicates
values of RH. The contour line with Rh: value of 0.1 (labeled as R0,; in Figure 4.18) is
used arbitrarily here to mark the width, or integral scale, of the autocorrelation. At the
most upstream location, the width of R” extends approximately from SUw/x, = -5 to 5.
As x increases, the correlation width decreases fairly rapidly till x/x, z 0.5. Farther
downstream, the correlation remains practically unchanged. These variations in the
integral time scale of the wall-shear stress are consistent with those ﬁom the wall-
pressure measurements by Hudy et al. (2003). These authors found that the ﬂuctuating
wall-pressure autocorrelation was wide at x < 0.25x,., but it narrowed-down quickly over
the range x z 0.25 - 0.5x,, remaining relatively unchanged downstream of that position.

Unlike the wall-pressure autocorrelation ﬁmction, R” does not posses a region
with appreciable negative correlation. The wall-shear autocorrelation approaches zero
with increasing time delay, oscillating with very small amplitude around zero. In contrast,
the wall-pressure autocorrelation of Hudy et al. (2003) crosses zero, reaching a negative
peak of -0.1 to -0.2, before oscillating around zero. The negative correlation peak is
inferred to be a manifestation of the quasi-periodic nature of the separated shear-layer
vortical structures, which dominate the wall-pressure signature in the vicinity of
reattachment. Although these vortices are also expected to have a signiﬁcant impact on
the wall-shear signature as well, it is believed that the inﬂuence of viscous damping on

the near wall ﬂow attenuates this signature; hence, obscuring the negative correlation

113

peak. A more detailed comparison of wall-shear and wall-pressure results is given in

section 5.3.

(a) Re” = 4300

   
 

 

 

 

10 .1
SUw/x, 08 R1411
5
-, ,:0.6
0
.0.4
_5 ﬂ ‘ 0.2
0
—10
x/x,
(b) ReH= 8700
10 .1
SUa/x, -
0 8 Rev
5
20.6
0
0.4

 

 

 

-lO

x/x,

114

(c) Re” = 13000
10

  

S Ugo/x,

 

 

x/x,

Figure 4.18 Contour maps of the auto-correlation coefﬁcient of the ﬂuctuating wall-
shear stress: (a) Re”: 4300, (b) Rey: 8700, (c) ReH = 13000

115

CHAPTER 5

TWO-POINT MEASUREMENT RESULTS

In this chapter, analysis of the space-time characteristics of the wall-shear stress
in the BF S ﬂow is presented. Speciﬁcally, the results ﬁ'om the two-point cross-
correlation and coherence between the time-series obtained ﬁom the reference and
movable OHW sensors are examined. Subsequently, convection velocities of the
dominant ﬂow structures are obtained through the use of two methods: one in the time
domain (based on the cross correlation) and the other in the frequency domain (based on
the phase of the cross spectrum). Since there is no literature involving two-point
measurements of the wall shear in separating/reattaching ﬂows, the current data are

exclusively compared with those from wall-pressure measurements.

5.1 Two-point Cross-correlation
5.1.1 Two-point Cross-correlation with the Reference Sensor Located near x,

The spatial correlation function is a useful tool for identifying the length scales of
the turbulent structures. In the current study, two OHW sensors separated from each other
in the streamwise direction were used to acquire the wall-shear stress simultaneously for

the purpose of obtaining the two-point correlation function. In particular, the two-point

cross-correlation coefﬁcient (Rt. t. ) is deﬁned as
l 2

 

 

Rea, (9.xm;xo) = 7”’("°”)‘”2(x’""‘9) (5.1)
l

\lx'w1(x0,t)2 Jam", ,t)2

 

 

116

where 1“,; and m represent time series of the wall-shear stress obtained from the
reference and movable sensors, respectively, and S is the time delay.
Negative/positive 9 values correspond to advancing/delaying 1“,; relative to 1w] in

calculation of R . . .
1"172

In the present study, one sensor (hereafter referred to as the reference sensor) was
placed at a ﬁxed position (xmf) and the other sensor (hereafter termed the movable sensor)
was moved along the streamwise direction. Two cases were tested. One set xref at 5.07H,
i.e., in the vicinity of the reattachment position. The corresponding xrq/x, is 1.18, 1.09
and 1.04 for Rey= 4300, 8700 and 13000, respectively. The other placed x"; at 2.28H,
i.e., in the middle of the reattachment zone. The corresponding xm/x, = 0.47, 0.44 and
0.42, in order of increasing Reynolds number. For the former reference position, the
movable sensor locations extended from the step to x/x, z 2.0. For the latter reference
sensor position, the movable sensor was placed at locations extending from downstream
of the reference sensor to x/x, z 1.0.

Figure 5.1 shows the two-point cross-correlation coefﬁcient for all Reynolds
numbers and selected movable sensor positions with the reference sensor close to the
reattachment point. In the ﬁgure, the abscissa represents the normalized time delay, the
ordinate indicates the cross-correlation coefﬁcient, and the different lines correspond to
different movable sensor positions. The black, red and blue color lines correspond to
movable sensor positions that are upstream of the reference sensor; that is, x/x, = 0.76,
0.90 and 1.04 in Figure 5.1 (a), x/x, = 0.70, 0.83 and 0.96 in Figure 5.1 (b), and x/x, =
0.67, 0.79 and 0.92 in Figure 5.1 (c). On the contrary, the cyan, yellow and green color

lines are used to represent the cross-correlation results for movable sensor positions

117

 

downstream of the reference sensor. A positive time offset means that the feature giving
rise to the correlation between the two sensors registers at the movable sensor earlier in
time before registering at the reference sensor, and vice versa.

The largest cross-correlation coefﬁcient is found at the closest spacing between
the two sensors and decreases with the increasing spacing. Upstream of the reference

sensor, the main peak of Rt. 1:. occurs at smaller time lead with decreasing distance
1 2

between the reference and movable sensors. Similarly, downstream of the reference

sensor, the main peak of RT.1" takes place at a time lag that increases with increasing
1 2

spacing between the two sensors. This is a manifestation of a downstream-traveling

disturbance.

The shape of the Rt- T. curves at the ﬁrst two upstream positions is different from
1 2

the rest. In addition to the main peak of the cross-correlation, these curves exhibit a
second peak of the same order of magnitude as the main peak but at negative time delay.
These secondary peaks are marked by open circles in Figures 5.1 (a) - (c). It is
interesting to note that the time delay corresponding to the secondary peak is consistent
with a downstream-propagating disturbance at Re” = 4300, a stationary disturbance at
Re” = 8700 and an upstream-propagating disturbance at Reg =13000. The signiﬁcance of
this secondary peak and the reasons why the convective characteristics of the disturbance

it represents change with increasing Reynolds number are not known.

118

(a) ReH = 4300

 

(b) ReH = 8700

’1':

 

119

(c) ReH = 13000

 

 

 

 

 

 

 

 

 

 

 

 

0.25177 3 .
r}. ! —; x/xr'-f0.67
12 0.2'1 "is ”4:919-
. -.L. x/xr:}o.92 .
Q15 . f .-.-...z.X/Xr¥_1.29.h
.."H' ‘3 x/xr=1.79 .
0.1 ' i f '_.; \‘ x/xr=9 09
0.05
0
-o.o5~ - j --------- —
-o.1

SUa/x,

Figure 5.1 Cross-correlation coefﬁcient of the ﬂuctuating wall-shear stress with
the reference sensor located near x,: (a) Re”: 4300, (b) ReH= 8700, (c) Re” = 13000

Figure 5.2 (a) — (c) show contour plots of the two-point cross-correlation results at
all x measurement locations. At the reference sensor position, the results are the same as
the auto-correlation with a peak value of 1.0 at zero time delay. Note that the color scale
used to represent the correlation value only spans the range — 0.1 through 0.5 to make the
lower cross-correlation values visible. The most obvious feature of the correlation maps
is inclined-lobe-shaped contours with negative slope. This pattern of contour lines is
consistent with that produced by a downstream-propagating wall-shear disturbance.
Furthermore, the slope of a locus of the correlation peaks along the center of the lobes
(highlighted with broken line in the ﬁgures) is inversely proportional to the mean

convection velocity of the disturbance.

120

(a) Re“ = 4300

SUm/x,

qr,

 

 

 

(b) ReH = 8700

9 U m/x,

7172

 

 

 

x/x,

121

(0) Reg = 13000

SUx/x, . .

0.4

 

 

   

x/x,

Figure 5.2 Contour maps of the cross-correlation coefﬁcient of the ﬂuctuating wall-
shear stress with the reference sensor located near x,: (a) Re”: 4300, (b) Re”: 8700, (c)
Re” = 13000

At the lowest Reynolds number, the lobe-shaped region over which most of the
correlation exists extends from about 0.7x, to beyond 2x, in the streamwise direction and
from time delay of - 2SUw/x, to + ZSUm/xr. As the Reynolds number increases, the area
of the lobe appears to become somewhat larger. This apparent, but small, increase in the
lobe size is caused by the fact that the reference sensor position decreases ﬁom 1.18x, to
1.04x, as the Reynolds number increases from 4300 to 13000. As a result of this
upstream movement in the non-dimensional location of the reference sensor, the entire
correlation contour plot shifts in the upstream direction; thus extending to x/x, locations
below that found at the lowest Reynolds number. On the other hand, the increase of the

temporal extent of the lobe area reﬂects the slight increase in the width of the auto-

122

correlation function, or normalized integral time scale, in the vicinity of x/x, = 1 with
increasing Reynolds number (see Figure 4.18).

Another observation from the contour plots is that the extent of the lobe in the
streamwise direction is not symmetric about the reference position. The correlation
distance is shorter upstream than downstream of the reference position. No signiﬁcant
correlation is found upstream of 0.6 - 0.7x,. On the contrary, downstream of the reference
position, the wall-shear ﬂuctuations are still correlated with the reference signal as far as
2x" and possibly farther. It is not surprising to see this long extending correlation
downstream of the reference point. It is well known that the coherent vortical structures
are sustained over fairly long distance beyond the reattachment point. (see Kostas et al.
(2002), Lee and Sung (2002)). However, the shorter correlation distance on the upstream
side is somewhat different from the wall-pressure results, where, for example a
correlation coefﬁcient of 0.1 extends to x/x, z 0.5 and in addition, there is substantial
negative correlation even further upstream, in comparison with x/x, z 0.9 in the wall-
shear case. Because the shear layer vortices are initially small and not too energetic near
the separation point, their inﬂuence on the wall pressure is generally not felt close to the
step. However, as they travel downstream, the vortices grow in size, move closer to the
wall, and become more energetic and begin to inﬂuence the wall-pressure signature
downstream of x/x, z 0.25 (Hudy et al.(2003)). For reasons that will be discussed in
section 5.3, it is believed that the wall-shear signature of the vortices is even weaker than
the wall-pressure one. Hence, they would need to travel farther downstream before
inﬂuencing the wall-shear stress: a suggestion that is consistent with the short correlation

width upstream of the reference probe location.

123

 

One way to obtain the convection velocity of the wall-shear disturbances
associated with the vortex passage is from the slope of the peak locus of the cross-
correlation contour plot. Figure 5.3 contains a plot of the correlation-peak time delay at
different locations of the movable sensor for all Reynolds numbers. The horizontal axis
represents the streamwise position of the movable sensor and the vertical axis gives the
corresponding time delay of the correlation peak. The symbols display the experimental
data, which are used to arrive at a linear least-square ﬁt to obtain the convection velocity
from the slope of the ﬁt. Note that this approach assumes that the convection velocity
does not vary with x, since the slope of a straight line is the same at all x locations. It is
noted, though, that the data do exhibit some deviation ﬁ'om the straight line, suggesting
that an x-dependent convection velocity may be more appropriate. The slope (ms) of the
line may be related to the mean convection velocity as follows:

ﬂea; (5.2)
U.no ms

The resulting mean convection velocities are 047,045 and 0.46Uao in order of increasing
Reynolds number. Compared with the wall-pressure measurements, the downstream
convection velocity is 0.57U... in Hudy (2001), 0.6U,o in Lee and Sung (2001), 0.5 —
0.6U,o in Heenan and Morrison (1998), 0.5U0° in Cherry et al. (1984), and 0.5U,o in Kiya

and Sasaki (1983). All of these values, which are obtained in various conﬁgurations
(fence-with-splitter-plate, thick splitter plate and BFS), consistently fall in the range of
0.5 - 0.6Uw.

In addition to the mean convection velocity (averaged over all scales of motion

and x locations), the local convection velocity may be estimated ﬁ'om the inverse of the

124

slope of the correlation peak locus (broken line in Figure 5.2). Figure 5.4 shows the local
convection velocities for all positions. The observed values decrease as the vortical

structures approach the reattachment point, then increase again.

 
    

 

 

 

 

1.5 -- M——-—r , , —~—— .. _Tm- -__.____ __
SUm/x, A 0 Re” = 4326 1
1 A ReH = 8653 j
+ ReH = 12979 .
0.5: _ ﬁt 1
i
0..
-0.5L
-1. l
4.5) 1
-2. _:
'2'35 "i T" "'1‘? "—3 ' 2.5

Figure 5.3 Correlation-peak time delay at different x location and associated line ﬁts

The deceleration of the vortices as they approach the separation point may be
clariﬁed through examination of the mean streamwise pressure gradient. Figure 5.5
displays the local convection velocities (He) and mean-pressure coefficient (Cp) in the
same plot. The ordinate scale on the left side represents Uc normalized by U00, and that on
the right side indicates Q, values. In the range x/x, z 0.5 — 1.0, the recovery of the mean
pressure results in the establishment of an adverse pressure gradient. It is believed that
this pressure gradient results in the deceleration of the vortices as they approach the
reattachment point. The stronger this pressure gradient is, the stronger the deceleration

effect. Since x, for the current ﬂow is shorter than that found in other studies of planar

125

 

BFS ﬂows, the pressure recovery occurs over smaller distance, leading to stronger
adverse pressure gradient. This apparently results in the lower convection velocity as

mentioned earlier.

1. , L , . i____.. -_.__.____ _ _--. ..

 

—— o ReH = 4326 ;
Uc/U” A ReH = 8653 1
08.l ... ReH =12979 ,;
0.6 J
+Al' Q
+A 0 °" 91: o
0.4;L 400 1" o 1
”also «mo 1
0.2--

Figure 5.4 “Local” convection velocity at different x positions

The reason for the acceleration of the vortices downstream of reattachment
(Figure 5.5) is not clear. However, the observation is consistent with that from the wall-
pressure measurements of F arabee and Casarella (1986) which shows the convection
velocity to continue to increase downstream of reattachment up to an asymptotic value of
approximately 0.7 Ugo. This may be partly caused by the continued increase in the size of
the vortices associated with increase in the thickness of the attached shear layer with
increasing x. As a result, the vortex center moves farther away from the wall, where the

local mean velocity is larger and the decelerating-inﬂuence of the image vortex becomes

126

 

smaller. The latter presumably also decreases monotonically with increasing x, since the
energy of the vortices, and hence their ability to induce velocity in the Biot-Sevarat sense,

continues to decrease as they travel downstream.

 

Uc/ Um

 

 

 

 

-0.2

 

Figure 5.5 Local convection velocity and pressure coefﬁcient versus streamwise
locations

5.1.2 Filtered Two-point Cross-correlation with the Reference Sensor Located
near x,

In section 4.2.3, the two frequencies of f z 0.1 and f' z 0.65 were detected in the
power spectra of Tw'. These frequencies are associated with the global ﬂapping motion of
the shear layer and the passage of the vortical structures, respectively. As just found in
section 5.1.1, the latter convects downstream at a mean velocity of 0.45 - 0.47Uw, It
seems interesting to see how the cross-correlation shape changes if the high frequency

component is ﬁltered out. To this end, a low-pass ﬁlter with a cut-off frequency f = 0.25

127

 

was used to ﬁlter the wall-shear data from both OHW sensors. The cross-correlation of
the ﬁltered signals is displayed in Figure 5.6 (a) — (c). Compared to the results in Figure
5.1, the ﬁrst observation from Figure 5.6 is that the correlation-coefﬁcient curves are
smoother and, at large time delays, ﬂuctuate around the zero correlation value with larger
amplitude and period. The second ﬁnding is that the values of the correlation coefﬁcient
are larger; that is, the signals are more coherent. The reason for the above observations
could be attributed to the removal of the effect of the high-frequency (small-scale) eddies.
Thus, the correlation is higher over longer distances because of the remaining large-scale
structure.

It is interesting to note that, albeit inﬂuences of the ﬁltering to remove the vortex
passage effects, the cross-correlation shape still reﬂects the dominance of downstream-
moving disturbance, as clearly illustrated in the contour plots of Figure 5.7(a) — (c). The
inclined lobe structure with negative slope is eminent in the plots. However, the contours
are more organized and the size of the lobes is also larger than that in Figure 5.2,
consistent with the removal of smaller-scale inﬂuences by ﬁltering.

As mentioned above, the primary motivation for the ﬁltered-signal analysis is to
check the nature of the global ﬂapping motion with frequency f' z 0.1 by removing the
higher-frequency inﬂuences of the energetic vortical structures. Kiya and Sasaki (1985)
proposed a model for the mechanism of surface-velocity ﬂuctuation: a traveling wave
with downstream convection velocity for the large-scale vortical structures and a standing
wave with a node near the middle of reattachment zone for the low-frequency ﬂapping
motion. This standing wave was also observed by Hudy et al (2003). However, the

contour plots in Figure 5.7 are still dominated by the convection motion. Possible

128

 

reasoning for this is as follows: although the passage frequency of the dominant vortices
is f z 0.6, a fairly broad range of frequencies is associated with the passage of vortices of
different sizes. This is reﬂected in the fact that the energy in the spectral plots in Figure
4.14 in the vicinity of f z 0.65 is contained in a broad, rather than narrow (harmonic),
peak. It appears from the results in this section that the inﬂuence of vortex passage
extends down to frequencies that are below the cut-off frequency of the ﬁlter. Kiya and
Sasaki (1983) also found that very large-scale vortices were shed with frequencies less
than f' z 0.2. Therefore, the wall-shear signature of the ﬂapping motion in the vicinity of

x, might still be masked by that of the vortices even for f less than 0.25.

(a) ReH = 4300

 

 

0-4 " *7 "ﬁr—+1
Rm; 1 9",. § —1— x/xr=0.76
‘ ~"‘;’; i, ‘ --- x/xr=0.90
0,3» ~ ,' i\ ‘- -. .xlxrt.04
‘ ‘ :.’ i 3. \_ -... x/xr:1.46
,- ,- -, 3 --- x/xri1.89

t)... .. x/xr=z.u.5 1

 

 

 

 

 

 

 

 

 

129

(b) R6“ = 8700

0.4 l I . 1 LT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rn}; I 1 S —- X/Xr=:o-70
- ' 1 .-, i --- x/x =‘o.83
: ; I \ I'\ : r
0.3‘ i I. I .‘ -l-I XIV -0.95
‘ 3i -’ t '\ ..
'- i! -‘ ." -.-- X/Xr-:1.36
O ql . ! I. ‘\ ‘. --- X/xr:'1.88
-“ I “'5‘; ‘.‘ “ .. x/xr==z.1z
Ii 3 .A‘ ti‘ "
I '.. I ‘ ' \ . l
1 «I 1".- U \ \ |
- 0'... . " 1‘. '- ‘
0.1 ’ .' ’ )' I I. \I
41‘! i/ ‘/-— \
.- I - '- \ . ...
F”“‘°.: ----- ’l I' ’4’ ‘. '\ $¥K--:‘\.
'- --.'>.—’ 3 ' ‘1’ /'\._ 0' --’°"".X§, 3&3?!
0.9;..— AAAAAA ...... w _ 32.... ' 7
—o 1* i 3 #1
I-8 -6 -4 -2 0 2 4 6 8
SUao/Xr
(0) R6“ = 13000
0.4ﬁ AF .
Rriryz 1 — X/Xr:=0.67 l
--é- x/xr==0.79 ;
0.3 -. . x’x“0.92 ‘
---- X/Xr==1.29
--- x/xr==1.79
0.2% .. X/Xr==2.UZ
0.1
“E‘~ s. .- a
0.” -"'.I ";;:".*‘¢‘.:;.
_ i ‘ i
0"18 -6 -4 -2 O 2 4 6 8

Figure 5.6 Cross-correlation coefﬁcient of the low—pass ﬁltered ﬂuctuating wall-shear
stress with reference sensor located near x,: (a) Re” = 4300, (b) ReH= 8700, (c) Re” =
13000

130

(a) Re“ = 4300

.0.5

SUa/x, a R .t
,‘ e 0.4 {"1

0.3

"II—“P

0.2

 

 

   

x/xr

(b) Re“ = 8700
4
SUw/x, . .

2

-2

 

 

 

131

(c) Re“ = 13000

SUaz/x,

rlrz

 

 

 

0.5 1 1.5 2
x/x,

Figure 5.7 Contour maps of the cross-correlation coefﬁcient of the low-pass ﬁltered
ﬂuctuating wall—shear stress with reference sensor located near x,: (a) ReH= 4300, (b) Re”
= 8700, (c) Re” = 13000

5.1.3 Two-point Cross-correlation with the Reference Sensor Located near 0.5x,

Since the wall-shear stress at the previous reference position did not show
appreciable cross-correlation with that from the movable sensor upstream of 0.7x,, a
second reference-sensor position around 0.5x, was employed. Speciﬁcally, the new
reference location was 0.47, 0.44 and 0.42x, in order of increasing Re”. The movable
sensor was placed at ﬁve x locations extending from the reference position to the
reattachment point.

Figure 5.8 shows the cross-correlation coefﬁcient between the two sensors for all
ﬁve locations of the movable sensor (represented by the different-type lines, as depicted

in the legend). The largest cross-correlation coefﬁcient is about 0.1 when the movable

132

sensor is placed closest to the reference probe. This value is smaller than that obtained at
the same sensor separation but with the reference probe located near the reattachment
position. As shown in Figure 5.1, the latter is about 0.15 — 0.2 depending on Reynolds
number. By increasing the sensor separation to the second smallest one (x = 0.76x, at
Rey = 4300, etc.), the correlation-peak value drops substantially to 0.05. Additional
increase in the sensor separation appears to have little effect on the magnitude of the
correlation.

The most prominent feature in the cross-correlation results in Figure 5.8 is a
sinusoidal-like signature that shiﬁs towards negative time delay with increasing x
location. The signature, which is depicted for all Reynolds numbers, is evident for all but
the most downstream position. The time scale of the disturbances responsible for the
generation of this signature may be estimated as twice the time delay between the
negative and positive peaks of the signature, which is ZUw/xr. The corresponding
frequency is f' z 0.5, which is approximately the same as that of vortex passage. This
suggests that the wall-shear inﬂuence of the shear layer vortices is detectable as early as
near 0.5x, (the approximate location of the reference probe), which is farther upstream
than implied by the correlation results with the reference probe located near reattachment.

Finally, Figure 5.9 displays contour plots of the cross-correlation coefﬁcient
relative to the new reference location. The downstream convective character of the wall-
shear signature is evident from the negative-inclined broken lines tracing the positive and
negative correlation peaks of the sinusoidal-like signature. It is interesting to recall here

that the negative correlation values were absent in the results based on the reference

133

 

probe position near x,. It is also worth noting that the negative correlation-peak value

becomes progressively smaller with increasing Reynolds number.

(a) Re“ = 4300

 

 

 
    

 

Rtir} ‘ ‘ 'f- X/Xr-':‘O.61
0.15: , ,, , - -- i ‘ --i- ”#0154
, : -.t. x/xr:0.90
0_1 .-..-.,X/XrT-f.1.04..a
.. .. x/xr=1.18
0.05

 

 

 

 

 

 

 

(b) ReH = 8700

R,,

:er

 

(c) ReH = 13000

 

S U aa/x,

Figure 5.8 Cross-correlation coefﬁcient of the ﬂuctuating wall-shear stress with the
reference sensor located near 0.5x,: (a) ReH= 4300, (b) Re”: 8700, (0) Re” = 13000
(a) Re“ = 4300

(a) Re” = 4300

 

 

 

 

x/x,

135

(b) Re” = 8700

SUm/x,

 

 

 

 

x/x,
(0) Re“ = 13000

SUa/x, . .
0.4 R’l’z

 

 

   

0.5 0.6 . 0.7 0.8 0.9 1

x/x,

Figure 5.9 Contour maps of the cross-correlation coefﬁcient of the ﬂuctuating wall-
shear stress with reference sensor located near 0.5x,: (a) ReH= 4300, (b) ReH= 8700, (0)
Re” = 13000

136

The difference in the correlation-function shape for the two different x choices of
the reference probe location warrants closer examination. When the reference probe is
located near x,, it has ahnost equal probability of being up or downstream of the
instantaneous reattachment location, depending on the phase of the ﬂapping cycle. This
results in switching of the sign of the shear signature in a manner that is clearly not
related to the vortex passage. Hence, it is not too surprising that this effect would cause
“de-correlation” of the vortex-passage signature, which in turn smears out the negative
correlation peak. This also would be in agreement with the disappearance of the
sinusoidal-like signature in the cross-correlation in Figure 5.8 at the most downstream

location, which is in the vicinity of x,.

5.2 Frequency Dependence of the Convection Velocities
The cross-correlation function analyzed in section 5.1 is the result of the non-

discriminant inﬂuence of all scales of motion. The cross-spectrum (Pt't' ), on the other
1 2

hand, may be used to identify the frequencies corresponding to the turbulent motions
responsible for the correlation of the turbulent wall-shear stress at two different points.

P. . is calculated as follows:
Tl"2

P. . (10:2!)1 mm; (f) (5.3)

TITZ

where f is the frequency, X1. is the Fourier transform of the reference sensor signal,
1

and X :' is the conjugate of the Fourier transform of the movable sensor signal.

P .1. (f) is generally a complex number composed of phase angle and magnitude. The
T1 2

137

 

angle of Pro 1.2 (f) indicates the average phase difference between the two signals at
1
frequency f. On the other hand, the magnitude of Pr. 1.2 (f) normalized as follows
1

deﬁnes the coherence (1‘1. 1' ) between the two signals:
1 2

 

 

 

5162“"
11'; (f )= (5-4)
12 Ptir'l(f)xpt'21'2(f)

where Pt- I- and P1. 1' are the auto power spectra of the wall-shear ﬂuctuations obtained
1 l 2 2

from the reference and movable sensors, respectively.

Figure 5.10 shows the coherence plots for all three Reynolds numbers and the
reference sensor position near the mean reattachment point. The location of the movable
sensor corresponding to each plot is speciﬁed in the legend. The selected positions, for
which data are shown, are the same as those in the cross-correlation plots of Figure 5.1
and Figure 5.6. The ﬁrst three positions are upstream of the reference position while the
last three are on its downstream side. In Figure 5.10, the abscissa indicates the
normalized frequency (f') and the ordinate gives the coherence. Consistent with the cross-
correlation results, the smaller the separation distance between the two sensors, the larger
the coherence.

At the Re” = 4300 and the most upstream and downstream locations, the
coherence is practically zero for all ﬁ'equencies. At the remaining four positions, a
distinguishable but small coherence value is depicted for frequencies below f z 1. This

frequency range encompasses the wall-shear—stress inﬂuences of the energetic vortical

structures (f' z 0.65) and the global ﬂapping motion (/ z 0.1). At Re” = 8700, the

138

 

coherence results are qualitatively similar but the magnitude is a little larger than that at
Re” = 4300 for the same separation distance. At Re” = 13000, the value of the coherence
increases further. Even at the most up and downstream positions, some coherence is now
observed at low frequencies. The increased coherence at these locations far away from
the reference sensor suggests an increase in the level of the low-frequency (large-scale)

ﬂuctuations with increasing Reynolds number.

(a) ReH = 4300

 

 

 

 

 

 

 

 

 

 

 

 

139

 

(b) Re“ = 8700

’l‘:

,. 7 ..V-
l

0.3L

 

 

 

 

 

 

0.2;

 

 

1O

10

140

 

(c) Re“ = 13000

 

r——————— —- "—w——— —— ——«———.‘

F. . . *
'1': 03 - x/xr=o.67 *: 0.3 ‘ XIX =0.79
0.2% 4' 0-2?
0.1 ;

 

 

 

 

 

 

0.3L
0.23»

0.1} « . . \
Ol____ . r , ___) Ot__,, __,,__”H, __4,_ .

10’2 10

f f'

   

 

Figure 5.10 Coherence of the ﬂuctuating wall-shear stress with the reference sensor
located near x,: (a) ReH= 4300, (b) ReH= 8700, (c) Rey = 13000

The coherence level of the wall-shear ﬂuctuation is smaller than that of wall-
pressure ﬂuctuation for the same separation distance. Again, this is attributed to the effect
of local small structures and viscous dissipation on the turbulent wall-shear stress.
However, it is still clear from Figure 5.10 that the coherence level at f < l is
distinguishable from that associated with the uncorrelated motion at high frequencies.
Since the coherence at a given frequency is a measure of the degree of phase locking
between two signals at that frequency, it seems possible to calculate the phase difference
between the signals obtained from the two shear sensors with some reliability for

normalized frequencies less than 1.

141

The phase information is of interest as it can be used to determine the convection
velocity at speciﬁc frequencies. To understand this, consider a traveling wave equation in
the form F ( x,t ) = AT sin( 27r( ft i kx)); where Aris the amplitude, f and k are frequency
and wavenumber, respectively, and “ i ” sign indicates up- or downstream propagation.
The wavenumber is related to the convection velocity by k = f/Uc and the phase
difference (‘1’) of the traveling wave at two different x locations is given by 2nkAx (Ax
being the difference between the x locations). Thus, the convection velocity Uc =
2anx/ 4) for a traveling wave, and if one is to plot 4) (which is obtainable from the cross
spectrum) versus the separation distance between the movable and reference sensors, the
slope of the phase line at any desired frequency could be used to determine the
corresponding convection velocity.

Figure 5.11(a) — (c) display plots of the phase versus x location of the movable
probe for frequencies of f z 0.06, 0.1, 0.2, 0.4, 0.6, and 0.8 and all three Reynolds
numbers. Different symbols represent the phase calculated from the cross-spectrum at
the different frequencies. A least-squares line ﬁt of 45 -vs-x data is obtained at each
frequency and displayed using a solid line in the ﬁgure. If the slope of the line is m¢, the

convection velocity is obtained from:

Uc 24f (5.5)

 

Note that the phase at the reference sensor location is zero and that phase jumps
of more than 21: between successive data points have been removed using unwrapping

algorithms. Also phase data are only shown for the x range extending from 0.8 to 1.8x,

where coherence values distinguishable above the background noise are found. As

142

 

l‘i

shown in Figure 5.11, the data exhibit less scatter at f z 0.06, 0.1 and 0.2 because of the
higher level of coherence at low frequencies. The ﬁtting line with positive slope indicates
downstream convection of the wall-shear disturbances for all ﬁ'equencies. In agreement
with wall-pressure results, no upstream convection was found because the analysis is
limited to x > O.8x,. The upstream convection at f z 0.12 was observed by Hudy et al.
(2003) from wall-pressure measurements upstream of O.5x,. This was obtained ﬁom

phase results relative to a sensor positioned immediately downstream of the separation

point (x/x, = 0.023).

(a) Re“ = 4300

 

8 ___ _n ,_ k ___ﬂ* _
WW9 i o f‘=0.06
i A f-=o.12 { ‘
6r + f‘=0.18
<> r=o.41 ’ ,
42» D f‘=0.59 o
i <1 f‘=0.82 , 4
— ﬁt > ’ 9
2t 4’ O
”...—— ,/ *3
'2i 5,4
I z
.4

143

 

(b) ReH = 87001

¢(rad) 8T

 

(c) Re” = 13000

    
 

8 Ah, ,. 7? izm 7_
¢(rad) 0 f'=0.04
6 A f'=o.11
i + f‘=0.20
‘ o f‘=0.40 ‘
4. D f"=0.60 i
<1 f*=0.80
2i — m
i
0i
l
4 <1

x/x,

Figure 5.11 Streamwise dependence of the phase angle at selected frequencies:
(a) Rey= 4300, (b) ReH= 8700, (c) Re” = 13000

144

The convection velocities calculated from Equation 5.5 are plotted versus the
normalized frequency for all Reynolds numbers in Figure 5.12. The average values of the
convection velocities are 0.47, 0.45 and 0.43 Ugo in order of increasing Reynolds number,
which are consistent with the results obtained from the cross—correlation. The broken line
in the ﬁgure represents an equation relating the convection velocities to the frequency
ﬁom the frequency-wavenumber spectrum of the wall-pressure measurements by Hudy et

al. (2003) on the same model, which is given by:

I

UC f (5.6)

U00 —1.69f‘+o.15

 

    

 

 

1r“*"‘ , #ﬁ*#~.,_:kha
A ReH= 8700
08* 0 Re": . a:
.. i=__f___ ‘
1.69 f‘ + 0.15

0.6»
Pi ,
0.2 i
ol — — 4 +— ——~——~I

 

 

‘ 0.2 0.4 0.62 0.8 i

Figure 5.12 Convection velocities obtained from phase-angle analysis

The error bar at each point is estimated using one standard deviation of the
calculated convection velocities from Equation 5.5. This is obtained from the standard

deviation associate with the estimate of the slope of the line ﬁt to the data in Figure 5.3

145

 

caused by the scatter in the data around the line (see Doebelin (1990), page 57).
Notwithstanding the data scatter around the broken line, the calculated convection
velocities are in general agreement with the trend predicted from wall-pressure
measurements. Additionally, it is seen that the convection velocity value increases with
increasing ﬁequency. This suggests that the high frequency (small scale) structures
travel at a faster speed. Also note that the results are independent of the Reynolds

number.

5.3 Additional Discussion

In this section, an attempt is made to systematically compare wall-shear with
wall-pressure measurements, and to examine the mechanism(s) causing differences and
similarities between them based on the one- and two-point results from the present study.
In both wall-shear and wall-pressure (Hudy (2003)) spectra plots, there are two dominant
frequencies: f' z 0.1 immediately after the step and f“ z 0.65 farther downstream. The
former is associated with the ﬂapping of the separated shear layer, and the latter
corresponds to quasi-periodic passage of the shear-layer vortical structures.

One of the features associated with the ﬂapping motion is the displacement of the
instantaneous reattachment position. This motion leads to variation of the dividing-
streamline curvature; thus inducing the pressure ﬂuctuations. Devenport & Sutton (1991)
proposed that the near-wall streamwise velocity ﬂuctuations are related to the streamwise
gradient of such shear-layer-imposed pressure disturbances through a one-dimensional
model. The model is based on uni-directional simpliﬁcation of the x momentum equation.

More speciﬁcally, for an inﬁnitesimally small distance above the wall, the streamwise

146

 

velocity (u) is assumed to be the dominant component and its gradient along the y
direction is assumed to be much greater than that along the x and 2 directions. The

momentum equation can then be simpliﬁed to:

2
ea. 16mm.

(5.7)

Devenport and Sutton (1991) assumed a sinusoidal pressure-gradient ﬂuctuation of the

fo ° 1 6p = A em" , with frequency 0) and amplitude Ap. For such a harmonic

disturbance, Equation 5.7 produced the following solution of the instantaneous velocity

ﬁeld:

A __2’.
u = —p sin(wt) —e 50 sin[a)t __y_] (5.8)
0) 60

where 60 = "-21 is the oscillating boundary layer thickness. Devenport and Sutton
(0

(1991) compared the predicted near-wall turbulence-intensity proﬁles using Equation 5.8

to experimental data in a sudden expansion in a pipe to verify the reasonableness of the
a [—7 ”2

one-dimensional model. As shown in Figure 5.13, when 8+ (6+ = —0[ r; /p] ) is 8,
v

corresponding to f = 0.18 near the step and f = 0.74 near the reattachment point, the
predictions are in good agreement with the measurements. It is noticed that these two
frequencies are close to the two dominant frequency values identiﬁed in the current

study.

147

 

 

 

 

J:-
I’le IVVV'IVVI'YrerVIII‘IIVI'vIII'VIII

 

 

 

._a

TDATUM FLOW ‘ FLOW WITH CENTERBODY ‘

‘ D x/xr=.16 <> x/xr=.96 ‘ ‘7 x/xr=.181 o x/xr=.77

i x x/x,=.32 o x/xr=1.12 l - x/x,=.362 a x/x,=.9o71
G x/xr= .498 A x/xr= 1,043‘

A x/xr — .48 x x/xr = 1.28
+ x/xr = .64 El x/xr = 1.6 , o X/xr = .634
V x/xr = .8 A x/xr = 1.95

Figure 5.13 Comparison of near-wall turbulence—intensity proﬁles between measurement
results and one-dimension model prediction (Devenport and Sutton (1991))

 

Since the one-dimensional model appears to be reasonable, Equation 5.8 was used
to predict the corresponding wall-shear stress associated with a pure harmonic pressure-

gradient disturbance as follows:

611 Ap . 7!
w t = u— = —._ an — 5.9
r () 6y y=0 0 sm( + 4) ( )

148

 

A
The corresponding ms of the wall-shear stress is p , which indicates that the wall-

420a)

shear ﬂuctuation is directly proportional to the amplitude, while being inversely

 

proportional to the square root of the frequency of the wall-pressure—gradient ﬂuctuations.
Thus, for equal-magnitude pressure-gradient disturbances, the induced shear-stress
ﬂuctuations would be attenuated more at higher frequencies. This may explain the
qualitative difference between the wall-pressure and wall-shear spectra observed earlier
in section 4.2.4.

The difference in the spectral shape is more evident if both spectra are plotted on
the same graph. Figure 5.14 depicts such a graph, showing the power spectra of the wall
shear and pressure for two similar positions at Re”: 13000. Each spectrum is normalized
by its own peak value to ﬁt the two spectra on the same graph. It is clear that the shape
of the wall-shear spectrum is different from that of the wall-pressure one. The wall-shear
spectrum decays with increasing frequency, while that of the wall-pressure exhibits a
broad peak at f z 0.4. It is suggested here that a similar peak is not found in the wall-
shear spectrum due to the inverse-square-root attenuation resulting from the Davenport
and Sutton model, which results in damping of higher frequency 1w ﬂuctuations relative
to lower ﬁ'equency ones.

Notwithstanding the attenuation of high-frequency 7w ﬂuctuations, the shear-layer
vortices appear to be responsible for most of the 2"“, activity. One evidence of that is the
two-point correlation results in section 5.1, which demonstrated the downstream
convective character of the wall-shear ﬂuctuations. Such a convective ﬁeld is clearly

associated with the vortical structures and not the shear layer ﬂapping. Furthermore,

149

 

 

even though a low-pass ﬁlter with cut-off frequency of f = 0.25 was used to remove the
inﬂuence of the passing vortices on the cross-correlation, the results still reﬂected the
eminence of downstream-traveling disturbances. The power spectra analysis in section
4.2.4 also demonstrated that the ampliﬁcation of the wall-shear rms value with increasing
x primarily corresponded to a rise in the spectral energy within the frequency band
associated with vortex passage. To underscore this observation further, a high-pass ﬁlter
with cut-off frequency of f = 0.25 was used to remove the inﬂuences of the low-
ﬁ'equency ﬂapping motion on the wall-shear stress signal. As discussed above, the
energy within the pass band of the ﬁlter does not account for all contribution due to
vortex passage since some of this contribution comes from frequencies below f of 0.25.
Thus, the rms calculation from the high-pass—ﬁltered signal is likely to underestimate the

total contribution of vortex passage to the wall-shear ﬂuctuations.

 

— X/Xr=1.67 (tw')
...... XIXr=1.80 (pw') i

J"
l

 

Normalize power spectra
A
o

 

 

 

 

10 A
10° 10

Figure 5.14 Comparison between wall shear and pressure power spectra

150

 

Figure 5.15 displays a comparison between the ﬁltered and unﬁltered rms values
of the ﬂuctuating skin-friction coefﬁcient, i.e.; C;' and thf, respectively, while Figure
5.16 plots the percentage of the high-frequency contribution to the total z’W energy. The
percentage increases continuously till x/x, = 0.54 and then remains practically constant
with a value of 65 - 70%. This conﬁrms that the r“, disturbances associated with the
vortices are dominant downstream of x/xr = 0.54, which is the streamwise domain in

which the cross-correlation measurements were conducted.

x10-

lr—'—-- e-r — --‘-T——-———— r -—

C . —e— Yunﬁlterd l
f + ﬁltered

0.8-

 

 

 

0.6

0.4;

 

 

Figure 5.15 Comparsion of rms skin-ﬁiction coefﬁcient from the unﬁltered and high-
pass-ﬁltered signals

151

 

 

 

 

 

 

80
S :
E7 60:
s3: 1

4oi r;

20 « T‘—
1
0L__ _ _ _ ___—L 1 ._ ._.._.;.__ ___J _ ___!
0 0.5 1 1.5 2 2.5
x/x,
Figure 5.16 Percent z"w energy associated with vortex passage L”

152

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions
I. Development of a new wall-shear sensor

A high-frequency oscillating-hot-wire (OHW) sensing technique for measurement
of the ﬂow-velocity/wall-shear-stress magnitude and direction was developed for the ﬁrst
time in the current work. The mechanism for direction determination by the sensor is
based on a 180-degrees phase jump of the modulation signal, resulting from wire
oscillation, when the ﬂow switches direction. On the other hand, the shear-stress
magnitude could be obtained in a manner similar to a conventional hot wire, after
ﬁltering the output signal to remove the modulated portion.

Magnitude and direction calibration of the OHW sensor was conducted in a
Couette-ﬂow facility that was especially designed and constructed for the purposes of the
present study. The facility provided a wall-shear calibration range of 0 — 0.7 Pa. In this
range, a good agreement was found between the magnitude-calibration results
with/without oscillation of the wire. This veriﬁed that the high-frequency oscillation of
the wire did no result in an anomalous or non-linear behavior that may affect the wire’s
calibration or its accuracy in retrieving the shear-stress magnitude.

It was also found that decreasing the ﬂow velocity (shear stress) resulted in
amplitude reduction of the sensor’s modulation signal at the oscillation ﬁ'equency. This
reduction was associated with increased phase lag between the imposed and measured

oscillation velocity of the OHW sensor. Both amplitude and phase effects did not

153

 

 

 

inﬂuence the measured shear-stress magnitude, the determination of which was based on _
the ﬁltered, modulation-free, output of the sensor. On the other hand, the phase lag
resulted in deviation of the direction response of the wire from the ideal one outlined
above. However, since this phase delay did not exceed 90 degrees, it did not cause the
modulation signal phase values to overlap for opposite shear directions. Thus, the ﬂow
direction could still be determined without ambiguity, aﬁer appropriate direction-
response calibration in the Couette-ﬂow facility to determine the range of modulation-

signal phase corresponding to each ﬂow direction.

11. Reattachment length in the axi-symmetric backward-facing-stepﬂow

The wall-shear signature behind an axi-symmetric BFS ﬂow was investigated
using the new OHW sensors while oscillating at a frequency of 2.8 kHz. Measurements
of the wall-shear stress were conducted at different streamwise locations downstream of
the step for three Reynolds numbers (based on step height) of 4300, 8700 and 13000.
The reattachment length was determined using two methods: one based on the mean skin-
friction coefﬁcient and the other on forward ﬂow probability. The difference between the
results ﬁ'om the two methods was less than 1.5%. The speciﬁc x, values varied from
4.3H to 4.86H with increasing Reynolds number. These values are smaller than those
reported in the literature for planar BFS ﬂows. It is hypothesized that the reduction in the
reattachment length is caused by enhanced entrainment of the separated shear-layer in the

axisymmetric case and/or transverse curvature of the test model used here (i.e., 6/R ratio

effect; 5 is the boundary layer thickness at separation and R is the model radius).

154

 

HI. Mean shear-stress characteristics

The streamwise distribution of the mean skin-ﬁiction coefﬁcient (Cf) was found
to be qualitatively consistent with existing literature on wall-shear measurements in BF S
ﬂows, including those that are based on direct measurements using oil—ﬁlm
interferometry. The value of Cf is positive but very small in the region extending from
the separation point to x/xr z 0.33. From x/x, z 0.33 — 1.0, Cf is negative, with the
maximum “reverse” Cf (Cﬂm) located at x/x, z 0.68. A universal empirical correlation of
the form Cﬂm = aReH'o‘s, which was suggested in earlier studies, provided good
representation of data obtained here and those from the direct shear measurements of
Jovic and Driver (1996). Downstream of the reattachment position, the mean skin-

ﬁiction coefﬁcient slowly rises, reaching a steady value at x/x, z 2.

IV. Turbulent shear-stress characteristics

The rms skin-friction coefﬁcient (C 'f) increased monotonically with increasing x, ‘
reaching a maximum in the vicinity of the mean reattachment location, followed by a
slow decay thereafter. Examination of the power spectra revealed that the rise in C 'f with
increasing streamwise distance from separation to reattachment is concentrated around
the “middle” range of ﬁequencies that is associated with the passage of shear-layer
vortices. The identiﬁcation of vortex passage frequencies was achieved from
examination of wall-pressure literature. In addition to the inﬂuence of the passing eddies

at an average frequency of f =fx,/Uco z 0.65, shear-stress unsteadiness at a lower

characteristic ﬁ'equency of f z 0.1 was found to be dominant very close to the step. This

155

 

signature was attributed to quasi-steady modulation of the mean shear stress by low-
frequency “ﬂapping” of the shear layer.

Consistent with the above observations from the power spectrum results, an
autocorrelation analysis revealed a large integral time scale immediately downstream of
the step. This integral scale decreased sharply in the region extending ﬁ'om 0.25 — 0.5x,,
but remained practically unchanged with further increase in streamwise distance.
Overall, observations from both the power spectrum and autocorrelation of the wall-shear

stress were qualitatively consistent with those documented in the wall-pressure literature.

V. T wo-point characteristics of the turbulent shear-stress

A unique feature of the present work is the compilation of what is believed to be
the ﬁrst two-point database of the wall-shear-stress ﬁeld in a BFS ﬂow. This was
accomplished using one ﬁxed, or reference, OHW sensor at xre/x, = 5.0711 and another
movable sensor that was located at 14 different x locations ranging ﬁom 0.3H to 10H.
The compiled data set was used to obtain the two-point space-time cross-correlation
function of the wall-shear stress. For all Reynolds numbers investigated, the shape of the
cross-correlation function reﬂected the dominant inﬂuence of the downstream-traveling
shear-layer vortices on the wall-shear-stress ﬁeld in the vicinity of the reattachment
location.

The downstream convection velocity of the vortical structures was obtained from

the cross-correlation results. The resulting value, which represents an average over all
time scales and x locations, was approximately 0.45 Um. A “local” determination of the

convection velocity revealed that the vortical structures decelerated as they approached

156

 

the reattachment point. This deceleration was attributed to the mean adverse pressure
gradient resulting from the pressure recovery in the range x/x, z 0.5 — 1.0. Because of the
shorter x, found here relative to other studies of planar BFS ﬂows, it is believed that the
adverse pressure gradient is stronger, leading to a more pronounced deceleration effect.
This apparently results in a lower convection velocity in comparison to values reported in
the literature (0.5 — 0.6 Uco from wall-pressure data in various geometries).

Additional information on the convection velocity was obtained through the use
of phase analysis. This analysis yielded the dependence of the convection velocity on
ﬁ'equency. In agreement with wall-pressure results from data obtained on the same
model (Hudy (2003)), the convection velocity exhibited a rise in magnitude with
increasing frequency. However, the wall-shear-based phase data experienced a good deal
of scatter, particularly at higher frequencies, because of the generally low values of the
associated coherence (i.e., low degree of phase locking between the measurements

obtained at two different locations).

VI. Some diﬂ'erences between the wall-shear and wall-pressure signatures

Although there were a number of features found to be common between the wall-
shear-stress ﬁeld measured here and the wall-pressure in BFS and related ﬂows in
general, some differences were also observed. One difference related to the “spectral-
peak switching” from the lower frequency corresponding to shear-layer ﬂapping to the
higher frequency associated with vortex passage as x increases beyond the reattachment
location. This peak switching, which was found in pressure data measured on the same

model and at similar Reynolds numbers (Hudy (2003)), was not manifested in the wall-

157

 

shear spectra. This is believed to be caused by the stronger attenuation of higher-
frequency near-wall velocity (and hence wall-shear stress) ﬂuctuations. The attenuation
was clariﬁed through the use of a uni-directional ﬂow model that was found by
Davenport and Sutton (1991) to provide a good description of near-wall velocity
ﬂuctuations in separating/reattaching ﬂows.

Another difference that was noted between wall-pressure and wall-shear
measurements is that pertaining to the values of the two-point correlation and coherence.
Generally, the correlation and coherence values decayed more rapidly in the wall-shear
case with increasing spacing between the reference and movable sensors. This was
attributed to the frequent change in shear-stress direction in the vicinity of the
reattachment point, where the reference sensor was located. Because a wall-pressure
sensor located at the same position would not be inﬂuenced by the near-wall ﬂow-
direction change, it should not suffer from the de-correlating effects of local direction

changes.

6.2 Recommendations

Notwithstanding the successful realization and use of the new OHW sensors in
this study, some extra work is required to ﬁne-tune their operation. In particular, the
issue of noise induced from coupling of the piezo-driving signal to the hot-wire sensor
requires some attention. As discussed in Chapter 3, this noise is partly responsible in the
deviation of the direction-response of the sensor from the ideal ISO-degree phase jump.
For the 2.8 kHz wire-oscillation frequency used here, the deviation was too small to

inﬂuence the shear-direction determination. However, at an oscillation ﬁequency of 5

158

 

kHz, one of the .two sensors utilized here could operate properly, while the other one
exhibited strong noise coupling ﬁom the driving signal that completely obscured the
direction-response of the wire. Thus, there is a clear need to understand the nature of the
coupling between the driving and sensing signals, the dependence of this coupling on the
oscillation frequency and other geometrical/physical parameters of the sensor, and the
best ways to minimize/eliminate this coupling. This should result in the ability to
construct sensors that can operate at tens of kilohertz to resolve even wider bandwidths
ﬂows.

Another direction in the development of the OHW sensors is to reduce their size
and construct them in array conﬁgurations. Actually, a MEMS-based OHW wall-shear
sensor has already been developed by the author for this purpose. More than one hundred
such sensors were produced on a 100-mm diameter silicon wafer. This work veriﬁed the
feasibility of microfabricating the sensors. However, it is anticipated that development of
a few generations of these MEMS sensors will be required before the realization of a
robust sensor and sensor arrays that can be used for detailed measurements such as those
conducted here.

Finally, it is recommended that some of the current measurements be repeated
using smaller sensors, or even sensor arrays, to capture the cross-correlation at smaller
sensor spacing than that realized here (0.15x,). This is motivated by the fact that the
value of the cross-correlation drops signiﬁcantly over the smallest spacing used here,
suggesting that the inﬂuenCe of small-scale disturbances on the correlation could not be
captured (e.g., the curvature of the correlation at zero separation, or Taylor microscale).

It will also be interesting to examine the wall-shear signature in the region extending

159

 

from the step to 0.25x,, where Hudy et al. (2003) found an upstream convection velocity
of wall-pressure ﬂuctuations. Because of the sensor-size limitation, only one point of
wall-shear measurements in this region could be realized in the current study. Thus, the

use of smaller sensors with smaller separation is needed to overcome this limitation.

160

 

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