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This is to certify that the
thesis entitled

AMPLITUDE-MODULATED EXCITATION OF A
SEPARATED FLOW USING AN EXTERNALLY DRIVEN
HELMHOLTZ RESONATOR

presented by

Antonius Krisna Aditjandra

has been accepted towards fulﬁllment
of the requirements for the

Master of Science degree in Mechanical Engineering

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Major Professor’s Signature

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AMPLITUDE-MODULATED EXCITATION OF A SEPARATED FLOW USING AN
EXTERNALLY DRIVEN HELMHOLTZ RESONATOR

By

Antonius Krisna Aditjandra

A THESIS

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

MASTER OF SCIENCE
Department of Mechanical Engineering

2006

ABSTRACT
AMPLITUDE-MODULATED EXCITATION OF A SEPARATED FLOW USING AN
EXTERNALLY DRIVEN HELMHOLTZ RESONATOR
By

Antonius Krisna Aditjandra

A ﬂow-excitation scheme based on an externally driven Helmholtz resonator was
developed in order to force, and alter the global characteristics of, the separated ﬂow over
an axisymmetric, backward-facing step. Because the resonance frequency of the forcing
device was well above the range in which the ﬂow is receptive to excitation, low-
frequency, amplitude modulation of the resonator’s disturbance was employed to
effectively excite the separated shear layer, causing shrinkage of the separation bubble.
This approach also prevented contamination of measurements of the unsteady wall-
pressure beneath the separating/reattaching ﬂow by acoustic noise from the forcing
device.

The driven resonator was employed to force the ﬂow over a range of forcing
frequencies and amplitudes. An optimal forcing frequency, corresponding to a minimum
bubble size for a given forcing level, was found at a non-dimensional forcing frequency
of F+ 2 0.541. Additionally, unsteady-wall-pressure measurements were conducted
employing a 15-microphone sensor array. A frequency-wavenumber-spectrum analysis
of these data suggested that the spatial scale, i.e., streamwise wavenumber, of the wall-
pressure disturbance at a given forcing frequency could be predicted from the unforced
ﬂow’s frequency-wavenumber spectrum. This finding could be useful in devising

“wavenumber control” schemes for controlling flow-induced noise and vibration.

Copyﬁghtby
ANTONIUS KRISNA ADITJANDRA

2006

DEDICATION

To Papa, Mama, Paulus, and Karin,

for their everlasting devotion, compassion, and love

iv

ACKNOWLEDGEMENTS

I would like to express my gratefulness to my advisor, Dr. Ahmed Naguib, whom
I have learned so much from throughout the seven years span he had taken me as his
undergraduate research assistant and graduate student. Dr. Naguib has truly been a
wonderful and supportive advisor, teacher, and motivator that challenged his students to
be the best that they can be. Outside the academic world, Dr. Naguib has shown amiable
friendship that helped propel me through this research project. So, Dr. Naguib, thank you!

Many thanks to my thesis defense committee members, Dr. Manooch
Koochesfahani and Dr. Brian Feeny, for their constructive inputs and recommendations
of this thesis project. I would also like to acknowledge Roy Bailiff and Mike Mclean for
their help at the machine shop; Jill Bielawski, Aida Montalvo, and Elaine Bailey for their
help with numerous kinds of paperwork.

I would also like to thank my fellow graduate students from the Flow Physics and
Control Laboratory: Laura Hudy, Yongxiang Li, Mohamed Daoud, Chad Stimson, and
Barry Trosin for their assistance with the data collections and many discussions regarding
my research projects. Their friendship and camaraderie throughout the years surely did
not go unnoticed. Also, I would like to acknowledge Ke Zhang and Brendan Vidmar for
their useful feedback on my thesis. Thanks are due to TMUAL and TSFL groups for the
fun times together researching, teaching, attending MUFMECH, and socializing.

Aside from academia, I would like to thank some of my friends: Kathy Weathers,
Liz Schweitzer, Fr. Jerry Vincke, Betsy Murphy, Ken Stinson, Margaret Beahan, Megan
McCullough, The Liberato’s, and St. John’s Pastoral Team/Staff. Especially for my best

friends, I must extend my special thankfulness. Brian Preston for the great times we

spent together camping, ﬁshing, and hanging out talking about science. Elaine and Neal
Adams for the abundant support and encouragements they provided me with and the
many meals and fun family times we shared together. Stefanus Tanaya for being a cool
roommate for so many years, which of course came with the package of many late night
conversations, “Gran Turismo” challenges, and cars fixings together even when the
temperature outside was below 25°F. Fr. Mark Inglot for the prayers, encouragements,
and many glasses of excellent “vino” that went along with many philosophical
discussions. The Preston’s for their kindness and making me be apart of their family.
They have made my many years at MSU that much more fun, meaningful, and enjoyable.
Last but not least, I must thank the most important people in my life. Thank you,
Mama and Papa, for everything that you have done for me. Without the two of you, I
would not be where I am at in my life right now. The two of you have taught me how to
thrive and succeed in life. I am indebted to the two of you and I would like to especially
express my deepest gratitude, from the bottom of my heart, for your unending support
and love. I am forever grateful to have you as my parents. To my brother, Paulus, and
his family, Fanny and Sean-Oliver, thank you for all of your support and love you have
generously given to me throughout my life. Thank you for all of the long distance
encouragements you provided me with concerning life and graduate school England
surely is not close by. Finally, I would like to thank my sister, Karin, for her unwavering
love and support throughout my life. I cherished the many unforgettable moments we
shared together traveling across US and here at MSU. Also, I would like to thank Bryan,
my sister’s husband, for his support. Life really is that much better with all of you in it!

This research project was made possible through NSF grant #CTSOI 16907.

vi

TABLE OF CONTENTS

 

 

 

 

 

 

LIST OF TABLES ix
LIST OF FIGURES ....... ................ - - x
NOMENCLATURE--- xiv
1 INTRODUCTION- - -- 1
1.1 Background ............................................................................................................... 1
1.1.1 Nature of the separated ﬂow over a backward-facing step ................................ 1
1.1.2 Control of separated ﬂows ................................................................................. 3

1.2 Literature review ....................................................................................................... 5
1.2.1 Helmholtz resonators: background, concepts, and physics .............................. 11

1.3 Motivation ............................................................................................................... 15
1.4 Objectives ............................................................................................................... 16

2 EXPERIMENTAL SETUP AND METHODOLOGY - - -- 17
2.1 Experimental setup .................................................................................................. 18
2.1.1 Wind-tunnel facility ......................................................................................... 18
2.1.2 Backward-facing-step (BF S) test model and ﬂow-excitation device .............. 19
2.1.2.1 Cylindrical (axisymmetric) model ............................................................ 20
2.1.2.2 Flow excitation device: extemally-driven Helmholtz resonator ............... 22

2.1.3 lnstrumentations and their calibration .............................................................. 26
2.1.3.1 Microphones .............................................................................................. 26
2.1.3.1.] Panasonic microphone ....................................................................... 27

2.1.3.1.2 Emkay microphones ........................................................................... 29

2.1.3.2 Hotwire anemometry ................................................................................. 31

2.1.3.3 Mean-pressure measurements ................................................................... 34
2.1.3.4 Pitot tube and temperature sensor ............................................................. 35

2.1.3.5 Helmholtz resonator driving system ......................................................... 35

2.1.3.6 Data-acquisition and experiment-control hardware .................................. 40

2.2 Experimental methodology ..................................................................................... 42
2.2.] Experiments categorization and data acquisition parameters .......................... 42
2.2.2 Model alignment .............................................................................................. 43
2.2.3 Helmholtz resonator characterization .............................................................. 44
2.2.4 Mean- and unsteady-wall-pressure measurements ........................................... 49

3 CHARACTERIZATION OF THE HELMHOLTZ RESONATOR -- 51
3.1 Determination of the resonator’s acoustic and ﬂuidic response ............................. 55
3.2 Characterization of the nature of the resonator’s ﬂuidic disturbance ..................... 67
3.2.1 Background ...................................................................................................... 67
3.2.2 Flow visualization of the ﬂow ﬁeld in the vicinity of resonator’s opening ..... 72
3.2.2.1 Buoyancy effect (unforced condition) ...................................................... 73

3.2.2.2 Sine-wave forcing effect ........................................................................... 74
3.2.2.2.] Forcing amplitude representative of those used in present study ...... 75

vii

3.2.2.2.2 High intensity forcing amplitude ....................................................... 76

 

 

 

3.2.2.3 Amplitude-modulation forcing effect ....................................................... 78

3.2.3 Hotwire measurements of the resonator’s velocity ﬁeld .................................. 80

3.3 Resonator’s disturbance characteristics under modulated forcing conditions ........ 88

4 EFFECT OF FORCING ON THE SEPARATED FLOW 99
4.1 The effect of forcing on the mean-pressure distribution ....................................... 100
4.2 The unsteady-wall-pressure signature beneath the forced bubble ........................ 107
4.2.1 RMS pressure distributions ............................................................................ 107
4.2.2 Power Spectrum Density (PSD) ..................................................................... 110
4.2.3 Frequency-wavenumber spectra ..................................................................... 113

5 CONCLUSIONS AND RECOMMENDATIONS - -- 120
5.1 Conclusions ........................................................................................................... 120
5.2 Recommendations for future work ....................................................................... 124
REFERENCES - .................... -- -- - - ......... - 125
General references ...................................................................................................... 128

viii

LIST OF TABLES

Table 2.1. Data acquisition parameters for broadband and harmonic tests

ix

LIST OF FIGURES

Figure 1.1. Backward-facing step flow features ................................................................ 2
Figure 1.2. A sketch of a Helmholtz resonator during oscillation (left), and equivalent

second-order mechanical system (right) ................................................................... 12
Figure 2.1. Schematic of the wind tunnel (dimensions in meter) .................................... 18

Figure 2.2. General schematic of the BF S axisymmetric model inside the wind tunnel. 20
Figure 2.3. BFS axisymmetric model with Helmholtz resonator (dimensions in m) ...... 21
Figure 2.4. Half-sectional schematic of the Helmholtz resonator (dimensions in mm) .. 23
Figure 2.5. Helmholtz resonator assembly on the axisymmetric model .......................... 24

Figure 2.6. A schematic of one of the resonator’s support brackets: isolated bracket (left),
and assembled on model (right) (dimensions in mm) ............................................... 24

Figure 2.7. Panasonic microphone probe (dimensions in mm) ........................................ 28
Figure 2.8a-b. Photograph of Panasonic, Emkay, and hotwire sensors: (a) Panasonic
microphone mounted on an azimuthal traverse mechanism, and (b) hotwire sensor
at y = 0 (within the resolution of the measurement) above slit opening ................... 30

Figure 2.9. Hotwire probe (dimensions in mm) ............................................................... 32

Figure 2.10. A sample image used in positioning the hotwire sensor near the resonator’s

slit .............................................................................................................................. 33
Figure 2.11. A schematic showing the subwoofer location relative to the wind tunnel
intake (dimensions in meter) ..................................................................................... 36
Figure 2.12. Block diagram of the Helmholtz-driving system ........................................ 40
Figure 2.13. Streamwise distribution of Q, at four different azimuthal locations ........... 44
Figure 2.14. Location of the hotwire measurements across the slit opening ................... 48

Figure 3.1. Power spectrum of the sound pressure just outside the Helmholtz resonator’s
slit for 16 different azimuthal angles ......... p ............................................................... 56

Figure 3.2. Power spectrum of the resonator’s cavity pressure with and without (blue line)
speaker excitation. Different line colors correspond to repeated measurements. 57

Figure 3.3. Power spectrum of hotwire sensor output ..................................................... 58
Figure 3.4. Coherence plot for the input and output acoustic signals of the resonator.... 60
Figure 3.5. Coherence plot for the hotwire and Panasonic signals .................................. 61

Figure 3.6. Magnitude of the acoustic systemfunction of the Helmholtz resonator ....... 63

Figure 3.7. Phase of the acoustic system function of the Helmholtz resonator ............... 64
Figure 3.8. Magnitude of the ﬂuidic system function of the Helmholtz resonator .......... 65
Figure 3.9. Phase of the ﬂuidic system ﬁmction of the Helmholtz resonator .................. 66

Figure 3.10a-f. Smoke-wire images of the buoyancy-induced ﬂow under unforced
condition .................................................................................................................... 74

Figure 3.11a-l. Smoke-wire images of the ﬂow above the resonator’s slit under
sinusoidal forcing conditions at a level representative of that used to obtain data in
this study ................................................................................................................... 76

Figure 3.12a-i. Smoke-wire images of the ﬂow above the resonator’s slit under
sinusoidal forcing conditions at a level stronger than that used to obtain data in this
study .......................................................................................................................... 77

Figure 3.13a-o. Smoke-wire images of the ﬂow above the resonator’s slit under
amplitude-modulated forcing conditions at highest level of forcing ........................ 79

Figure 3.14. Phase-average of hotwire output at different x locations and y = 0 (top);
forcing signal (bottom) .............................................................................................. 83

Figure 3.15. Velocity proﬁles across the resonator’s slit opening during peak ejection
phase (top) and peak suction phase (bottom) ............................................................ 84

Figure 3.16. Conditional average of the hotwire output for different forcing levels (data
obtained at x/a' = -0.5 and y/d = 0) ............................................................................ 85

Figure 3.17. Conditional average of hotwire output at 11 different y—positions (y/d = -
0.11, 0.09, 0.29, 0.49, 0.69, 0.89, 1.09, 1.29, 1.49, 1,69, and 1.89 from top to bottom)
................................................................................................................................... 87

Figure 3.18. Sample of the driving acoustic signal at 10 Hz modulation frequency and
two forcing levels ...................................................................................................... 91

Figure 3.19. Velocity measurements at x/d = -0.5 and y/d = 0 corresponding to the
amplitude-modulated acoustic forcing shown in Figure 3.18 ................................... 92

xi

Figure 3.20. Sample of the driving acoustic signal at 18 Hz modulation frequency and
ﬁve forcing levels ...................................................................................................... 93

Figure 3.21. Velocity measurements at x/d = -0.5 and y/d = 0 corresponding to the
amplitude-modulated acoustic forcing shown in Figure 3.20 ................................... 93

Figure 3.22. Magniﬁed velocity measurements at x/d = -0.5 and y/d = 0 corresponding to
the two largest amplitude-modulated acoustic forcing shown in Figure 3.20 .......... 94

Figure 3.23. RMS of the low-frequency-disturbance velocity versus the input peak-to-
peak of the driving acoustic pressure ........................................................................ 95

Figure 3.24. A sample power spectrum of the low-frequency ﬂuidic disturbance
associated with 18 Hz modulation frequency: (a) logarithmic scale, (b) linear
ordinate scale ............................................................................................................. 96

Figure 3.25. Sample power spectra of the acoustic pressure associated with 18 Hz
modulation frequency and two lowest forcing levels: (a) 10.6 Pa peak-to-peak, (b)
24.9 Pa peak-to-peak ................................................................................................. 97

Figure 4.1. Mean-pressure distribution under no forcing condition .............................. 100

Figure 4.2. Reattachment length estimate from the present study and Li (2004) data .. 101

Figure 4.3. Forcing-frequency effect on the mean-pressure distribution beneath the
separation bubble at C ,, = 0.0690% ......................................................................... 103

Figure 4.4. Forcing-amplitude effect on mean-pressure distributions at V 2 0.541 ..... 105
Figure 4.5. Effect of forcing frequency (F) on RMS wall-pressure measurements ..... 109
Figure 4.6. Effect of forcing level (C #) on RMS wall-pressure measurements ............. 110

Figure 4.7. One-sided power spectra of the surface pressure at different streamwise
locations downstream of the step ............................................................................ 111

Figure 4.8. Streamwise distribution of the pressure-spectrum magnitude at the forcing
frequency for forcing level of C), = 0.0095% .......................................................... 113

Figure 4.9. Wall-pressure frequency-wavenumber spectrum for the unforced-ﬂow case
................................................................................................................................. 115

Figure 4.10. Wall-pressure frequency-wavenumber spectrum for the forced ﬂow (F 2
0.300 [f;_,H/U00 = 0.075] and C), = 0.0095%) ............................................................ 116

xii

Figure 4.11. Wall-pressure frequency-wavenumber spectrum for the forced flow (F 2
0.541 [ﬁH/Uo0 = 0.135] and C), = 0.0003%) ............................................................ 117

Figure 4.12. Wall-pressure frequency-wavenumber spectrum for the forced ﬂow (V 2
0.541 [feH/Ua, = 0.135] and C,, = 0.0095%) ............................................................ 118

Figure 4.13. Wall-pressure frequency-wavenumber spectrum for the unforced ﬂow,
showing straight-line ﬁt to the peak of the convective ridge .................................. 119

IMAGES IN THIS THESIS ARE PRESENTED IN COLOR

xiii

NOMENCLATURE

fc
ﬁg
fm
ﬂ

magnitude response of second order system

speed of sound at ambient temperature

P — P .
= LL39; = mean pressure coefﬁcrent
1 , U
2 ,0 air 00
Zduj rmsz . . - '
= 7;,— : momentum-based non-d1mensronal forcmg-level coefﬁcrent
CD

slit width of resonator
frequency

carrier frequency

= 2f,,, = excitation frequency
modulation frequency

sampling frequency

 

= feL : zfmxr
U00 U00

= reduced forcing frequency
step height of backward-facing step
resonator’s system-response function
discrete-frequency index

wavenumber

neck length of resonator

effective neck length of resonator

length scale characteristic of the separation bubble size

number of samples in a time series

xiv

Pa
Pap-p
1%
P:

I
P w

Poo

<q>

qmax

Red

ReH

Sid

SI]

“1‘. rms

driving acoustic pressure of the Helmholtz resonator
peak-to—peak driving acoustic pressure of the Helmholtz resonator
pressure inside the cavity of resonator

acoustic pressure imposed at the slit opening of resonator
ﬂuctuating wall-pressure

mean surface pressure at a given streamwise location

freestream pressure

= Vuz + v2 = velocity magnitude
phase-averaged q

unsteady component of q
maximum magnitude of q

time averaged component of q

radius of circular neck opening for conventional resonator

U , d
= Ji— or M = Reynolds number based on the resonator’s slit width, d
v v
U coH

 

= = Reynolds number based on the step height, H
v

surface area of the resonator’s opening

Strouhal number based on the resonator’s slit width, d
Strouhal number based on the neck length of resonator, 1
time

period of forcing signal

streamwise velocity component

RMS jet velocity at frequency of 2f,,,

XV

x,

z(t)

(00

average jet velocity during ejection phase of a driven resonator
maximum jet velocity during ejection phase of a driven resonator
freestream velocity

wall-normal velocity component

cavity volume of resonator

streamwise coordinate relative to back-step edge

mean reattachment length

wall-normal coordinate

amplitude-modulated signal

Amplitude of amplitude-modulated signal

end correction factor for estimation of the resonance frequency of the acoustical
resonator

damping ratio

power spectrum of quantity b

cross spectrum between quantities g and b
power spectrum of Q

phase angle of second-order mechanical system
coherence between quantities g and b

density of air

kinematic viscosity

= 24f: angular frequency

angular natural frequency

xvi

1 INTRODUCTION
1.1 Bac round
1.1.1 Nature of the separated ﬂow over a backward-facing step

Separated ﬂows are encountered . frequently in engineering applications.
Examples include ﬂow over wings, turbine and compressor blades, inside diffusers, and
around automobiles. The occurrence of flow separation generally, but not always, results
in adverse effects on the performance of engineering devices: reduced liﬁ, increased drag,
generation of noise and vibration, etc. Therefore, it is desirable to understand the ﬂow
physics associated with separated ﬂows for the purpose of predicting, and hence design
for minimizing, these adverse effects as well as for devising passive or active ﬂow
control schemes to eliminate or reduce the separated ﬂow regions.

A fundamental ﬂow geometry that is frequently used as a model for separated
ﬂows in the laboratory is the backward-facing step. Extensive studies have been done on
separating and reattaching ﬂows over the simple, classical geometry of a backward-
facing step using a variety of both experimental and numerical methods. This geometry
is fairly simple, however, the structure of the ﬂow downstream of the step is rather
complex and retains many of the features of the ﬂow structure encountered in
engineering applications.

Generally speaking, there are four ﬂow-regions downstream of the back step: a
separated free shear layer, primary and secondary re-circulating ﬂows, and redeveloping
boundary layer. As depicted in Figure 1.1, the shear layer separates from the edge of the
back step. The discontinuity in the geometry creates substantial velocity variations

across the separated, thin shear layer, with low-speed ﬂow on the lower side of the layer

and high-speed ﬂow on the other side. The resulting large, velocity gradient leads to the
formation of spanwise, wall-parallel vortices that originate from the well-known Kelvin-
Helmholtz instability. These vortices grow in size and convect downstream with a
convection velocity that is typically in the range of 50 - 60% of the freestream velocity.
The primary re-circulation region is caused by the need to balance the entrainment
ﬂow into the separated shear layer. In this region, the ﬂow is highly unsteady and it
“drives” the, low-speed, second re-circulating region located just downstream of the back
step. Driver et a1. (1987), Heenan and Morrison (1998), Cherry et a1. (1984) and Lee and
Sung (2001, 2002), amongst others, argue that the unsteadiness in this particular region is

due to the ‘ﬂapping’ of the shear layer.

     
 
 
 
 
 

 

 

 

 

 

 

 

Uco y
f Separated free
shear layer Dividing
streamline .
Redevelopmg
//////////(/// boundary layer
5
H a
1 /
_ /
Secondary /" \ Mean
recirculation Primary reattachment
region recirculation region point

Figure 1.1. Backward-facing step ﬂow features

The above scenario of the flow structures in the backward-facing step ﬂow
represents the generally accepted view. However, in the more recent study of Hudy et a1.
(2006) over an axisymmetric, backward-facing step, an alternate view was proposed.

Utilizing an array of wall-pressure microphones to estimate the spatio-temporal evolution

of the ﬂow structure downstream of the step through linear stochastic estimation, Hudy et
al. (2006) proposed that a wake-like ﬂow structure is dominant within the separation
bubble. In this alternative scenario, large-scale (order of step height) vortex structures
roll up from the separated shear layer at a location that is nominally half the reattachment
distance downstream from the step. Once they reach a size comparable to the step these
vortices accelerate and convect downstream. Hudy et al. (2006) attributed the difference
between this scenario, which supports observations from an earlier computational study
by Wee et al. (2002), and the traditionally accepted one to be related to the spanwise
uniformity of the mean ﬂow in the axisymmetric study. The bulk of the literature on the
backward—facing-step flow is based on a rectangular geometry, which suffers from end-

wall effects.

1.1.2 Control of separated flows

The concept of traditional boundary layer control (i.e., steady blowing or suction
through a slit opening) introduced by Prandtl in 1908, according to Greenblatt and
Wygnanski (2000), has been well studied. It was found that these traditional methods for
controlling the ﬂow work well. However, more recent research showed that a more
efﬁcient way to control separated ﬂows is by periodic excitation of the boundary layer at
separation. In 1948, Schubauer and Skramstad (1948) ﬁrst introduced this type of ﬂow
control by periodically exciting the laminar boundary layer to produce a known
instability called Tollmien-Schlichting waves (stated in Greenblatt and Wygnanski, 2000).

In periodic forcing of separated ﬂows, it is generally believed that large coherent

structures are responsible for the momentum transfer that leads to the control of the

separated ﬂow. These coherent ﬂow structures, which are generated by the ﬂow
instability to the periodic excitation, produce a signiﬁcant change in the behavior of the
separated ﬂow region. In other words, “excitation accelerates and regulates the
generation of large coherent structures, transferring high momentum ﬂuid across the
mixing layer”, according to Greenblatt and Wygnanski (2000). Amitay et al. (2001) also
suggested that “forcing of a separated flow by means of oscillatory net mass injection,
has been found to be an effective means for control of separation with substantially less
mass ﬂux than required from steady blowing.” Therefore, periodic excitation appears to
be the more effective and preferred method to control a separated ﬂow as opposed to the

traditional steady blowing/suction method.

1.2 Literature review

There are various means by which periodic excitation can be achieved. Since the
primary focus of the present investigation is to develop such a method for the control of
an axisymmetric, backward-facing-step ﬂow, a brief review of methods for separated-
ﬂow control is given here, along with highlights of the inﬂuence of oscillatory control on
the ﬂow behavior. It is to be noted that the review is only intended to outline some of the
popular ways for achieving the control through representative studies rather than
providing a comprehensive list of all possible control approaches and in-depth analysis of
the resulting ﬂow behavior.

In general, there are two types of ﬂow control: passive and active. In passive
control, no actuator is required, and hence this type of control does not require external
energy input. On the other hand, active control requires the use of an actuator that is
driven utilizing external energy input. An example of a passive, yet unsteady, control of
a separated ﬂow may be found in the study done by Urzynicok and Femholz (2002).
These investigators employed a Helmholtz resonator that was driven to oscillate at
resonance by the natural unsteadiness of the ﬂow above the resonator’s opening, and
hence the unsteady control input to the ﬂow did not require external energy. Urzynicok
and Femholz (2002) utilized the passive resonator to demonstrate separation control in
diffuser and airfoil ﬂows. Though the approach of the passive-driven resonator is
attractive, there are no means by which the device operation could be adjusted to changes
in the Reynolds number and other ﬂow conditions. Speciﬁcally, since the resonance
frequency of the excitation device is not guaranteed to match the natural, or optimal-

control, frequency of the ﬂow, the control is most effective only over a narrow range of

ﬂow parameters. It is also notable here that Urzynicok and Femholz (2002) found that
the most effective location for controlling the separated ﬂow was to locate the resonator
opening as near to the point of separation as possible.

Amitay et al. (2001) gave an example of actively controlling the separated ﬂow
over an unconventional-proﬁle symmetric airfoil. Eight piezoceramic disks (four disks
for each of two actuators located side-by-side) were used to drive synthetic (zero net
mass ﬂux) jets that were placed along the spanwise dimension of the airfoil. The piezo
disks were placed inside a cavity behind the jet exit, and thus, the actuators may be
classiﬁed as internally excited devices. Amitay et al. (2001) implemented the ﬂow
control utilizing high and low actuation frequencies. The former frequency, F” = 10.6
(where F is the frequency normalized using a length characteristic of the streamwise
scale of the separated ﬂow and the freestream velocity; the corresponding physical
ﬁequency is f = 740 Hz), was more than an order of magnitude larger than the natural
shedding frequency of the airfoil, which was F+ 1: 0.7 (f2 50 Hz). The low frequency of
F+ = 1.5 (f = 110 Hz) was selected to be within the receptivity range of the separated
shear layer (i.e., on the order of the natural shedding frequency of the airfoil). The low
actuation frequency was achieved by using the entire volume of the actuator cavity,
highlighting the difﬁculty to design an actuator that has the capability of operating at the
low-frequency range (where the ﬂow is most unstable) when geometrical constraints are
present.

Amitay et a1. (2001) found that without the synthetic jet control, the ﬂow
separated at the leading edge of the airfoil at angles of attack beyond 5°. However, the

ﬂow became completely attached for angles of attack up to (and partially beyond) 17.5o

with the application of the synthetic jet control. The control was achieved through low
actuation frequencies that were within the receptivity band of the separated shear layer

rather than the high actuation frequencies. The Reynolds number range used for their

study was between 3.1x105 and 7.25 x105 , based on the chord length.

Extending the work of Amitay et al. (2001), Glezer et a1. (2005) emphasized the
aspects of low- and high-frequency actuation for ﬂow control. They systematically
varied the actuation frequency, corresponding to ﬁve different Strouhal numbers (F) of
0.7, 1.1, 2.05, 3.3, and 10. They argued that low-frequency (order of the natural shedding
frequency of vortex structures from the separated-ﬂow zone) actuation produces time-
dependent ﬂow disturbances, whereas the high actuation frequency results in a basically
time-invariant ﬂow modiﬁcation. In the former case, they found that the control led to
persistency of the vortices beyond the trailing edge of the airfoil. They suggested that
this was due to the effective coupling of disturbances to the ﬂow at the excitation (also
structure formation) frequency that was close to the natural shedding frequency of the
airfoil. This means that the low-frequency control produced highly organized structures
that advect downstream.

As the Strouhal number was increased to 3.3, Glezer et al. (2005) showed that the
shear-layer vortices became smaller and “somewhat smeared”, indicating a deterioration
in phase-locking between the ﬂow structures and the actuation. This led to the
conclusion that the actuation becomes less effective as the actuation frequency was
increased beyond a certain optimal value. However, Glezer et al. (2005) made a case that
when the actuation frequency was increased to an even higher value (i.e., Strouhal

number of 10), the ﬂow would remain attached to the suction surface of the airfoil with

no “evidence of organized, phase-coherent vorticity concentrations.” Therefore, the
separated ﬂow was well suppressed at adequately high (an order of 10 higher than the
natural shedding frequency) actuation frequency.

Chun and Sung (1996) provided another example of active, oscillatory, control of
a separated, turbulent, shear layer. For their investigation, they utilized a classical
backward-facing-step geometry over a ﬂat plate. At the point of separation, sinusoidal
velocity disturbances were generated through a slit opening at the edge of the step that
ran the full spanwise length of the back step. The main driver of this internally driven
actuator was a subwoofer that was mounted to one of the walls of a cavity placed behind
the slit. Actuation of the subwoofer led to the generation of disturbances that excited the
main ﬂow through the slit opening at the separation edge.

Chun and Sung (1996) demonstrated that the effect on the ﬂow of local forcing at

the separation edge was signiﬁcant. The Reynolds number range used in their study was

varied in the range 1.3x 10" S Re” S 3.3x 104 (where Re” is the Reynolds number based
on step height), while the actuation frequency was varied between 25 Hz and 5 kHz.
Their ﬁndings indicated that as the forcing amplitude was increased, the reattachment
length became shorter. This trend became non-linear, showing signs of saturation,
beyond a certain, critical, forcing level. Furthermore, they found that the most effective
forcing frequency for reducing the reattachment length was close to that of the vortex
shedding frequency of the unforced ﬂow (i.e., natural frequency of the separated ﬂow),
which was St” = 0.275 (where St}, is the Strouhal number based on step height and
freestream velocity). However, when the excitation frequency was increased in the range

St” 2 0.8, the reattachment length became independent of the excitation frequency. In

fact, in this frequency range, the reattachment length became larger than that of the
unforced case. Thus, the mechanism for reducing the reattachment length seemed to be
more sensitive to the excitation frequency than the forcing amplitude.

In a later study, Chun et al. (1999) utilized the same experimental parameters and
conﬁguration as Chun and Sung (1996) in an attempt to understand the effect of
spanwise-varying, local forcing on the turbulent separated ﬂow over the back-step. The
spanwise variation in the forcing was achieved by blocking the control-jet’s slit opening
at the step edge at periodically-placed locations along the span of the slit. The most
effective forcing frequency, St” = 0.27, was found to be the same for all spanwise
wavenumber employed in the forcing. This frequency value was the same as found in
their earlier investigation utilizing spanwise-uniform forcing.

In 1998, Chun and Sung (1998) undertook a visualization study of a, locally
forced, laminar, separated ﬂow over a backward-facing step in a water channel. The
Reynolds number (Rey) used in their experiment of 1200 was substantially lower than the
other studies from the same group (cited above). The local forcing was of the form of an
oscillatory jet driven by a scotch-yoke mechanism. The scotch-yoke mechanism, which
was connected to a piston and cylinder system, could produce different forcing
frequencies and amplitudes, leading to the establishment of a periodic jet through a slit at
the edge of the back step. In the experiment, dye was used for visualization and LDV
was utilized to measure the velocity components in the re-circulating-ﬂow region. The
range of forcing frequencies, in terms of Strouhal number (Sty) based on the step height,

used in the study was 0.305 S St” S 0.955.

The ﬁndings of Chun and Sung (1998) showed that the optimum Strouhal number
for shortening the reattachment length was 0.477. This value was higher than Chun and
Sung (1996) and Chun et al. (1999) ﬁndings; however, the Chun and Sung (1998) study
was for the case of a laminar rather than the turbulent separation examined in the latter
studies. Moreover, this value was in the neighborhood of the natural-ﬂow-instability
frequency (Sty = 0.4) found by Roos and Kegelman (1986) in their backward-facing-step,
ﬂow study of laminar, separated shear layers. Chun and Sung (1998) also found that the
reattachment length increases with increasing Strouhal number beyond the optimum-
control Strouhal number, which was consistent with their previous assertion made in
Chun and Sung (1996). Moreover, they claimed that the secondary re-circulation ﬂow
located just downstream of the back step disappeared all together when the optimum
forcing frequency was applied.

In a review paper, Greenblatt and Wygnanski (2000) provided many examples to
demonstrate the importance of oscillatory addition of momentum to produce effective
hydrodynamic interaction between the actuators and the ﬂow in active ﬂow control.
They believed that large, coherent structures, introduced by the oscillatory control, are
responsible for enhancement in the momentum transfer across the separated ﬂow, leading
to reduction in the length of the separated ﬂow. They also mentioned that periodic ﬂow
oscillations could be achieved by “speakers/acoustic drivers, pistons, oscillatory-ﬂow
valve-system, piezoelectric-based diaphragms two-dimensionally, or surface mounted
mechanical actuators such as ribbons, ﬂiperons, or piezo-based benders.”

In contrast to a later study done by Glezer et al. (2005), which was brieﬂy

summarized above, who showed that higher reduced frequency (i.e., F order of 10 or

10

higher) could be effective for controlling the separated ﬂow, Greenblatt and Wygnanski
(2000) suggested that the optimum reduced frequency, F, for the majority of the tested

cases ranged between 0.3 and 4.

1.2.1 Helmholtz resonators: background, concepts, and physics

For reasons that will become clear in section 1.3, an externally driven Helmholtz
resonator was selected as the actuating mechanism for the present study. However,
before getting into the motivation of the present investigation, it is helpful to brieﬂy
summarize the origin, theory, design, and physics of a Helmholtz resonator.

The idea of a Helmholtz resonator was ﬁrst introduced and theorized in 1860 by
Hermann von Helmholtz. This acoustic resonator was later named aﬁer Helmholtz and
employed in the mid of 19th century for sound ampliﬁcation of particular frequencies
from a complex sound ﬁeld. Moreover, this idea was applied, quite universally, as a
sound-damping system in jet engines, train stations, and many other sound-damping
applications. A Helmholtz resonator, also known as acoustic absorber, is basically a rigid
container (or cavity) containing ﬂuid (e.g., air) with an opening (or “neck”). A driven
Helmholtz may be modeled using a simple, single-degree-of-freedom, second-order,
spring-mass-damping system. The ﬂuid that occupies the volume within the neck of the
resonator acts as the mass, the compressible air inside the cavity behaves as spring
stiffness, and viscous dissipation as well as sound radiation from the resonator provides

damping. A schematic of a typical Helmholtz resonator is shown in Figure 1.2.

11

Oscillatory
ﬂuidic mass

Oscillatory
Spring motion \

   

Figure 1.2. A sketch of a Helmholtz resonator during oscillation (left), and
equivalent second-order mechanical system (right)

The driving source of oscillations in a resonator can be either ﬂuidic or acoustic.
In response, the driven Helmholtz resonator produces oscillations of the mass of air
contained within the neck of the resonator. During oscillation, when the mass is pushed
slightly inside the cavity, the ﬂuid within the rigid container will be compressed.
Consequently, the cavity pressure will rise and force the mass to move in the opposite
direction. When the mass motion reverses, the pressure inside the cavity will eventually
rarify, opposing the outward motion of the mass. The resonance frequency of the
oscillation, which can be derived from analogy with the mechanical, second-order system,
is a function of the geometry of the cavity, neck, and lip/oriﬁce, in addition to the speed

of sound (e.g., see Kinsler et al., 1982), as given by the following equation:

(1.1)

where coo is the angular resonance frequency, c is the speed of sound at ambient
temperature, S is the neck cross-sectional area, V is the cavity volume, and l' is the
effective neck length (given by 1' = l + 8, where l is the neck length and 5 is an end
correction accounting for the additional mass motion at the outer and inner ends of the
neck).

The above equation (also known as the Rayleigh equation) generally works well
for estimating the resonance frequency provided that the wavelength of sound is much
larger than any of the dimensions of the resonator; the cavity dimensions are of the same
order in all three spatial directions, and the neck is centered above the cavity. Chanaud
(1994) utilized lumped-parameter models in conjunction with solutions to the wave
equation to derive analytical expressions for the resonance frequency of resonators with
asymmetrically placed oriﬁces, asymmetric cavities, and extreme geometric shapes.

Chanaud (1994) discovered that resonance-frequency calculations obtained from
the Rayleigh equation worked well for thin oriﬁces that are centered on cubical cavities
(referred to as “symmetric condition”). Chanaud (1994) then concluded that the Rayleigh
equation has strong limitations on its validity with extreme changes in the resonator’s
geometry from the symmetric condition.

Another important aspect of Helmholtz resonators relates to their behavior when
driven at high-intensity sound levels. In this regard, Ingard and Labate (1950), Ingard
and Ising (1967), and Lebedeva (1980) presented signiﬁcant results concerning acoustic
streaming and acoustic non-linearity in the vicinity of the resonator’s oriﬁce. Especially
interesting is the research done by Ingard and Labate (1950), where smoke-particle

visualization of the acoustical streaming (steady ﬂow) phenomena in the vicinity of

13

oriﬁces with different geometrical parameters were recorded. Ingard and Labate (1950)
classiﬁed the streaming ﬂow patterns into four distinct regions of ﬂow based on the
particle velocity in the oriﬁce. These four regimes will be discussed in detail in section
3.2. Here, it is noted that the Ingard and Labate (1950) study clearly demonstrated the
existence of steady ﬂow patterns that are produced in the vicinity of the resonator’s
oriﬁce as a result of driving the actuator at high intensity of sound. Such steady ﬂow
pattern cannot be modeled with the simple mechanical system shown in Figure 1.2,
which can only account for linear effects. However, as will become clear in Chapter 3,
this streaming ﬂow is the key for the success of the ﬂow excitation device developed in

this study.

14

1.3 Motivation

Numerous studies have previously been done on control of the separated ﬂow
over a classical backward-facing-step geometry. Of these, only one study (Liu et al.,
2005) employed wall-pressure sensor array for the investigation of the space-time
characteristics of the surface-pressure ﬁeld beneath an excited separating/reattaching
ﬂow downstream of “two-dimensional” (rectangular) backward-facing step. Moreover,
in all of the existing investigations, in a back-step or other separated-ﬂow geometry, the
actuation device produces substantial acoustic noise at the same frequency as that of
actuation. Consequently, noise from the control device and the wall-pressure signature of
the excited ﬂow structures are both picked up by the wall microphones, and cannot be
separated by frequency ﬁltering. Therefore, there is a need for an actuator with an

acoustic signature that does not overlap with the excitation frequency of the ﬂow.

15

1.4 Objectives

The speciﬁc objectives of the present study may be summarized as follows:

1. To develop an actuator/actuation-scheme (speciﬁcally, an externally-driven
Helmholtz resonator) that is suitable for studies of the surface pressure beneath
oscillatory-controlled ﬂows. The actuator is required to produce ﬂow
disturbances at the edge of the back step of an axisymmetric model that are
capable of altering the global (overall) characteristics of the separation bubble.
Moreover, disturbances to the ﬂow from the actuation device should be at a
frequency that is different from any acoustic noise emanating from the device.

2. To characterize the operation of the driven Helmholtz resonator using
measurements of the unsteady pressure at the neck and within the cavity of the
resonator, and velocity measurements as well as visualization of the ﬂow in the
vicinity of the neck.

3. To characterize the overall effect of forcing by the Helmholtz resonator on the
separation bubble through mean-wall-pressure measurements.

4. To conduct a preliminary investigation of the wall-pressure ﬂuctuations beneath
the separation bubble under natural and forced conditions utilizing wall-pressure-
array measurements. In particular, to be examined are measurements of the RMS

pressure ﬂuctuations, power spectra, and frequency-wavenumber spectra.

16

2 EXPERIMENTAL SETUP AND METHODOLOGY

This chapter provides description of the experiments, measuring instruments, and
test procedure employed in the present study. The experiments conducted may be
classiﬁed into two main categories based on their goal. The ﬁrst type of experiments
targeted the characterization of the acoustic and ﬂuidic response of the ﬂow excitation
device (i.e., the extemally-driven Helmholtz resonator). The second type of tests aimed
to characterize the inﬂuence of the excitation device on the wall-pressure ﬁeld beneath
the separating/reattaching shear layer. This characterization includes both the mean- and
ﬂuctuating-pressure ﬁeld. To conduct both types of experiments, three measurement
techniques were used. More speciﬁcally, microphones were used for measurement of
acoustic and hydrodynamic pressure ﬂuctuations; static-pressure taps, coupled with a
pressure scanner and high-sensitivity pressure transducer, were employed to capture the
mean-pressure ﬁeld beneath the separated ﬂow; and hotwires provided the ﬂow-velocity
data for capturing the ﬂuidic response of the resonator. In the following chapter, these
techniques along with the experimental procedure are described in detail. This is done

following the description of the facilities and test model.

17

2.1 Experimental setup

The experimental setup includes the wind-tunnel facility, axisymmetric test mode]
with the backward-facing-step (BF S), and all of the instruments used during the

experiments, along with their calibration processes.

2.1.1 Wind-tunnel facility
The wind-tunnel facility used here consists of ﬁve parts: contraction, test section,
pre-diﬁ‘user, diffuser, and axial fan. A 15 HP DC wound shunt GE motor with adjustable

speed controller drives the fan. A schematic of the wind tunnel may be seen in Figure 2.1.

Adjustable

. . Test
ce111n .
l g 83°90" Pre-diffuser lefuser F?”

- |
__._1:>_-_- -_- ._t-T -_ ....... q.._- -.__[

Contraction

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Flow

. !
/ I I 1.346
|—— 8.096 1.829 1.005 L— 1.889 —-+-— 1.486 —~ l

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\‘

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1. Schematic of the wind tunnel (dimensions in meter)

The total length of the wind tunnel is slightly more than 8 meters and the test-section-
centerline height is nominally 1.35 meters from the ﬂoor. The contraction has 6.25:1
area ratio with the cross section at the inlet being 1.524 m X 1.524 m in width and height.
Precision honeycomb and variable-porosity screens are mounted at the front end of the
contraction to reduce the turbulence intensity of the airﬂow in the test section to less than

0.5% at freestream velocities up to 15 m/s.

18

The test section has an inlet area of 0.6096 x 0.6096 m2 with an adjustable ceiling
to achieve a nominally zero, pressure gradient in the freestream. The ceiling also has two
slots, along its centerline, with 12.7 mm width to allow insertion of instrumentations (e.g.,
hotwire traversing mechanism, Pitot tube, etc.). The side and lower walls of the test
section consist of acrylic windows inserted into sanded-plywood panels that are coated
with transparent polyurethane to ensure smoothness of the surface. Four acrylic windows
supported by aluminum frames, two on each of the sides of the test section, are attached
using hinges and clamps to the tunnel to allow for easy access to the test model inside the
tunneL

Similar to the test section, the pre-diffuser also has an adjustable ceiling to avoid
any discontinuity in the tunnel walls between the test section and the pre-diffuser section.
Except for this adjustable ceiling, the walls of the pre-diffuser diverge at an angle of 6°.
Additionally, the ﬂoor has a 50 mm wide slot that is used as a pass-through for the test
model’s support plate. The downstream end of the pre-diffuser is then connected to the
diffuser directly. The diffuser has an expansion angle of 588° and an inlet-to-outlet area

ratio of 1:19 to recover the pressure drop across the test section.

2.1.2 Backward-facing-step (BFS) test model and ﬂow-excitation device

The BFS model has an axisymmetric geometry and is supported by a heavy
traverse table that allows for adjustment of the yaw and pitch angles as well as the height,
and lateral position of the model. The centerline of the model is aligned with the center
of the inlet area of the test section. The length of the model extends from 2 0.3 m

downstream of the test section entrance to near the downstream end of the pre-diffuser

l9

section, as seen in Figure 2.2. In the following, more speciﬁc details are provided
concerning the geometry of the model and the Helmholtz resonator employed to excite

the separated ﬂow.

 

 

 

 

 

 

 

 

15>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l
l _ 1 — _1TH'— _
/‘ '\ " ' lnnnnn
Jr . .
l I \\ 2%
/ / Helmholtz I .. g
Test section BFS model resonator

Hollow shroud for instrumentation cables and

static-pressure tubing feed through 7 Q

Adjustable
support frame

 

 

 

 

 

 

\\ \ \\\\\\\ \\ \

Figure 2.2. General schematic of the BFS axisymmetric model inside the wind
tunnel
2.1.2.1 Cylindrical (axisymmetric) model
The axisymmetric model has a total length of 2.38 m (see Figure 2.3) and is
mostly made of aluminum tube with wall thickness of 6.35 mm. The nose of the model
has a half-sphere shape and is made of aluminum as well. Downstream of the nose and
upstream of the step, the model has a diameter of 124.4 mm. Downstream of the step, the

diameter decreases to 100.0 mm, creating a step height, H, of 12.2 mm. The step is at the

20

downstream end of an acrylic component, or module, that forms the step and contains the
Helmholtz resonator (description of this module is given in the next subsection, 2.1.2.2).
Piano wires with nominally 1 mm in diameter are used to support the front end of
the model to avoid the “cantilever deﬂection” associated with the long overhang of the
model. These wires are located nominally 0.10 m downstream of the tip of the nose and
pass through narrow slots in the side and bottom acrylic windows of the test section near
its entrance. Outside the test section, the piano wires are connected to a supporting
framework with fair amount of tension. Li (2004) found the wires to have no interfering
effect on the separated ﬂow. This was demonstrated through hotwire measurements

around the perimeter of the model just upstream of the step.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Helmholtz Steel
Flow C Sand paper resonator support
R 0.0688 /l {—0 0.1244 / {-0 0.1000
1 1“/—— ‘ — 47 " 1 —
1 + 0.3604
_. 1.1633 =44 0.5433 —" =1 -
~ 2.3885 '—

 

 

 

 

Figure 2.3. BFS axisymmetric model with Helmholtz resonator (dimensions in 111)

Three aluminum-oxide, wet-cut, 120 grit, sand paper strips were adhered to the
model surface at the upstream end (just downstream of the nose) of the model to hasten
the transition of the boundary layer to a turbulent state. Finally, the tail end of the model
has a conical shape to ensure that there is no sudden changes in geometry that might

create undesirable disturbance to the overall ﬂow structures around the model. For

21

additional details about the model geometry, its internal construction, etc., the reader is

referred to Chapter 8.2 (appendix) of Hudy (2005).

2.1.2.2 Flow excitation device: externally-driven Helmholtz rescmtol

To form the resonator’s cavity volume, a 3.175 mm-thick acrylic cylindrical shell
and a 6.35 mm-thick acrylic cover were assembled around the model just downstream of
the existing step in the model as shown in Figure 2.4. The streamwise location of the
cover was adjusted, using gage shims, to maintain a uniform slit opening, d, of 0.5 mm
around the model’s perimeter. The corresponding slit height (or resonator’s neck length,
l) was 3.175 mm. Because it is well established in the literature (e.g., Greenblatt and
Wygnanski, 2000, and Chun and Sung, 1996, 1998) that the most effective unsteady
forcing of a separated ﬂow is achieved very near the point of separation, the resonator’s
slit was made to coincide with the back—step edge. This was accomplished by the
creation of a 45° chamfer in the cover, on the downstream side of the slit as depicted in

Figure 2.4.

22

Flow Outer shell

 

 

 

 

 

 

 

 

 

, l
/ // / // // / // I '
/ // / / // / / j
‘} / / / / 1
\ \ \
\ \ \ V \ \ f\ \
\ \.
\ \ \\ \ \\ . . \
\ t \ \ |\ \
R 50.000 / .' /
Cavity volume ll Slit opening Cover

Figure 2.4. Half-sectional schematic of the Helmholtz resonator (dimensions in mm)

To mount the resonator on the model, it was necessary to make both the shell and
the cover out of two halves that were aligned using dowel pins and were assembled
around the model as depicted in Figure 2.5. Once assembled, the two halves were held
together using a 63.5 um-thick Kapton (Polyimide) ﬁlm tape with silicone adhesive.
Four acrylic brackets were utilized to support the shell portion of the resonator (Figure
2.6). These brackets occupied a very small fraction of the cavity volume in order to
minimize their inﬂuence on the device’s resonance frequency. After considering the four

support brackets, the cavity volume was found to be nominally 202,304 mm}.

23

   
  

Axisymmetric

Dowel pins

   
 
 

Outer shell

 

 

Acrylic bracket
L7 9.525/-—
9.525
./4
8837 l
L 3.175 541]
1 1.»
‘ 50.800

 

 

 

 

Figure 2.6. A schematic of one of the resonator’s support brackets: isolated bracket
(left), and assembled on model (right) (dimensions in mm)

24

To predict the resonance frequency of the resonator, Equation 1.1 is used, in connection

with an end correction (a) that is given by (e.g., see Kinsler et al., 1982):

6 = 1.7 r (outer end ﬂanged) (2.1a)

6 = 151' (outer end unﬂanged) (2.1b)

where r is the radius of the neck opening (taken here as the radius of a circle with the
same area as that of the current resonator’s slit). The Helmholtz resonance ﬁ‘equency
was calculated to be 425 Hz and 447 Hz when employing Equations 2.1a and 2.1b
respectively for the end correction. These values are quite far from the actual resonance
frequency of the device of 657 Hz (see section 3.1). This is not too surprising since
Equations 2.1a and 2.1b work well for symmetric cavity geometry (similar dimensions in
all three dimensions) with centered circular neck. The present resonator’s geometry
deviates substantially from this simple geometry, with its approximately rectangular
(when unwrapped), high-aspect—ratio (width/height) cavity, and off-center, rectangular
neck. Ingard (1953) derived analytical expressions for the end correction of the neck
length of various resonators’ geometrical shapes.

Of these shapes, the square cavity with rectangular neck provided the closest
approximation to the geometry of the present resonator (although this does not account
for the high-aspect-ratio of the cavity and the eccentricity of the neck). Calculation of the
resonant frequency based on the results of Ingard (1953) for the square cavity yields a
resonant frequency of 585 Hz. Although this new prediction remains signiﬁcantly far

from the true frequency (about 11% deviation), it is substantially better than that based on

25

the simplest resonator geometry (equations 2.1). It is suspected that if the high aspect
ratio of the resonator and eccentricity of the neck were taken into account, the proper
frequency value would be predicted. However, no analytical solutions that account for
these factors were found.

During operation, external sound waves at the resonance frequency were used to
excite the resonator, as will be described in section 2.1.3.5. In response, large oscillation
of the air mass within the neck of the device was produced, creating an energetic
unsteady air jet in the immediate vicinity of the point of separation. The coupling of this
jet to the ﬂow ﬁeld is what ultimately leads to the manipulation, or control, of the

separation bubble and associated wall-pressure ﬁeld.

2.1.3 Instrumentations and their calibration

The next several subsections provide a description of the instrumentation used for
the experiments and their calibration processes. The instrumentation include
microphones, hotwire anemometry, static-pressure-measurement system, Pitot tube and
temperature sensors, along with other hardware such as the subwoofer/speaker utilized in

driving the Helmholtz resonator, and the data acquisition hardware.

2.1.3.1 Microphones

 

There are three common types of microphones: capacitive, electret, and
piezoresistive (micro-fabricated) microphones. Capacitive and electret type microphones
have a wide frequency response (a typical bandwidth is 3 Hz — 70 kHz for 'A" capacitive

microphone), and very good sensitivity. They are capable of detecting pressure

26

ﬂuctuation levels that are well below one millionth of standard atmospheric pressure. On
the other hand, the piezoresistive type has a frequency response that is comparable with
that of capacitive and electret microphones on the high-frequency end, but it can extend
to DC (zero frequency) on the low-frequency end. The piezoresistive sensors also tend to
have sensitivity that is one or two orders of magnitude lower than a comparable size
capacitive and electret microphones. Therefore, piezoresistive microphones are generally
suitable for measurements of high-level pressure ﬂuctuations and when information on
the mean pressure (zero frequency) is desired.

For low-speed ﬂows, such as the case in the present study, the wall-pressure
ﬂuctuations are on the order of fraction of a Pascal. This very low level of pressure
ﬂuctuations eliminates the possibility of utilizing piezoresistive sensors because of their
relativity low sensitivity. Additionally, to achieve high spatial resolution and to minimize
the cost of the microphones, Emkay (model FG-3629-P16) electret microphones were
utilized for measuring the surface pressure. These sensors also have a very compact
package size that allowed them to be integrated into test model within the available space.

In addition to the Emkay microphones, a stand-alone electret Panasonic
microphone was used to sense the sound pressure imposed on the Helmholtz resonator.
A summary of the characteristics of both types of microphones and other relevant

information is provided below.
2. 1.3.1. 1 Panasonic microphone

The Panasonic microphone used here is an omnidirectional back electret

condenser microphone (model P9925-ND). It has a nominal manufacturer sensitivity of

27

—35 3: 4 dB (relative to 1 V/Pa; i.e., nominally 18 mV/Pa) for a frequency range of 20 to
20,000 Hz. The overall dimensions of this microphone cartridge are as follows: outside
diameter is 6.0 mm and length is 3.4 mm. The sensing diaphragm of the microphone is
exposed to the sound ﬁeld through a 2 mm-diameter sensing hole covered with a thin,
sponge—type material. This microphone cartridge is mounted in a homemade PVC and
brass tubing to provide means for holding the microphone (Figure 2.7). The resulting
probe length is 104 mm. The microphone cartridge is slip ﬁtted into the PVC tubing and
made ﬂush at the tube opening. The outside of the PVC tubing was machined such that
the probe would ﬁt into the mounting hole of a Larson-Davis microphone calibrator
(model CAL 200) for determining the sensitivity of the Panasonic microphone on daily
basis. The brass tube is press ﬁtted into the PVC on the opposite end of the microphone

and held together by epoxy around its circumference to ensure rigid construction.

Microphone PVC (2) 13.00
cartridge ‘2) 7-00

,'.‘ "/1 ‘/ / ’ E

‘\.Y\“\.x

Brass tubing Wires
(2) 7.13

    

¢ . I

d— 9.50 —>l-I— 11.00 =!= 83.50

Figure 2.7. Panasonic microphone probe (dimensions in mm)

 

 

 

 

 

—-4

 

 

To operate the microphone, it was connected to one of the ports of a 16-channel
homemade, electrical circuit. Details of the circuit, which is powered by a 9 V, DC

power supply, is provided in Daoud (2004). The output signal of the microphone was

28

connected to one of 16 channels of a National Instrument A/D board (model NI 6024B)
through a two-channel, Larson-Davis signal conditioner (model 2200 C) to remove any
DC offset from the output signal. The microphone was calibrated using the Larson-Davis
precision acoustic calibrator at 114 dB and 1000 Hz. Typically, the resulting sensitivity

of the microphone was found to be 14.8 mV/Pa or —36.6 dB (relative to 1 V/Pa).

2. 1. 3. I. 2 Emkay microphones

In the study by Hudy (2005), a 32-sensor array of Emkay (model F G-3629-P16)
microphones was constructed and integrated into the axisymmetric, back-step model in
order to measure the space-time signature of the wall-pressure ﬂuctuations downstream
of the step. Each of the microphones has a nominal manufacturer sensitivity of —53 i 3
dB (relative to 10 V/Pa; i.e., nominally 22.4 mV/Pa) for a frequency range of 100 to
100,000 Hz. The actual sensitivity value for the individual microphones was obtained by
Hudy (2005), where these microphones were bench calibrated using a B&K (model
4226), multifunctional calibrator in the frequency range between 20 and 15,000 Hz. The
outside diameter of the microphone cartridge is nominally 2.6 mm. The sensing
diaphragm of the microphone is exposed to the sound ﬁeld through a 0.77 mm-diameter
sensing hole.

The present investigation utilized only part (15 sensors) of the same microphone
array for capturing the unsteady wall-pressure ﬁeld downstream of the resonator. The
remaining microphones were not utilized for the same purpose since they were either
contained within the volume of the resonator, or located downstream of the measurement

extent of interest. However, one of the microphones contained within the resonator’s

29

cavity was used to measure the sound pressure within the resonator, and hence determine
the resonator’s acoustic response. Figure 2.8a provides a photograph showing the

placement of the microphones relative to the resonator.

Panasonic microphone
traverse holder Traverse 1' mg Support bracket Hotwire sensor

Helmholtz '

.V resonator
Panasomc _, .. ,
microphone
Kapton tape
Panasonic
111ic1‘ophonc

\ limkay

microphone

 

Figure 2.8a-b. Photograph of Panasonic, Emkay, and hotwire sensors: (a)
Panasonic microphone mounted on an azimuthal traverse mechanism, and (b)
hotwire sensor at y = 0 (within the resolution of the measurement) above slit
opening
Note that the ﬁrst, most upstream, microphone utilized in the measurements of the
ﬂow-induced pressure ﬂuctuations is located 11.73 mm downstream of the resonator
cover’s sharp oriﬁce (i.e., the step). The rest of the microphones have a distance of 4.8
mm centerline-to-centerline between each other. This microphone array was used to
sense the unsteady pressure induced by the ﬂow structure downstream of the separation
point.

30

2.1.3.2 Hotwire anemometg

A single, hotwire probe was used to measure the velocity within the oscillatory jet
produced by the forced resonator. The hotwire sensor was made from Tungsten wire
with a diameter of 3.75 pm. The wire was electroplated in a copper-sulphate solution to
deﬁne a sensing length (un-plated portion) of 0.78 mm. This provides a wire length-to-
diameter ratio of more than 200:1. The plated wire is soldered to the tip of two stainless
steel prongs of the hotwire probe, which is nominally 300 mm long and has an outside
diameter of 5.56 mm. In turn, the prongs are soldered to lead wires that are threaded
through the length of the probe body and terminated with a SMA — coaxial connector for
coupling to the anemometer circuit. A schematic of the hotwire probe may be seen in
Figure 2.9.

Constant Temperature Anemometer (CTA) from TSI (model 1750) that was
powered by a 15-Volts, Elenco Precision, variable, power supply (model XP — 581) was
used to operate the single-hotwire sensor. The resistance of the hotwire was typically
4.80 Q and the overheat ratio was set to 1.62. The output of the CTA was split into two
paths: one that was connected directly to one of the analog-input channels in a National
Instrument A/D board (model NI 6024); the other one was routed through a two-channel,
Larson-Davis signal conditioner (model 2200 C) to remove the mean voltage and amplify
the ﬂuctuating signal using a gain of 10. This mean-removed/ampliﬁed signal allowed
better digitization resolution of the unsteady velocity information. In the processing of
the hotwire signal, the unconditioned voltage time series was averaged to yield the mean
voltage. The mean-removed/ampliﬁed one was divided by the gain factor then added to

the mean voltage to retrieve a digitized record of the full hotwire output. Note that the

31

cut-off frequency of the high-pass ﬁlter of the Larson-Davis signal conditioner was 1.6
Hz. This is well below any of the frequencies associated with the measured velocity

ﬂuctuations, and hence the signal conditioning process should not lead to the loss of any

relevant information.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

’ I I“ . Stainless
// ‘5.
,1 ‘X steel prong
// \
t \\
1’ .1. ‘1‘ 4-times
l . .
11 3 60 O 78 : magmﬁcatron
(I
t 1 T {I
t 1
r \ ‘ It Copper
1’ ‘\ // plated wrre Hotwire
\
If \ / probe
I \\ x/
If \ K JJJJJJJ ¢’/
1 .- x’
1’ .- , 4’ 1 I
l/I" \ ‘\ I l l 1 * I I /
1 ,’ 1 1
1 l x 1 0 5.56 , ,
\ J/x
‘ ~~' ‘ t 1 |
l l
—‘ 80.00 = = 300.00 ———I

 

 

 

 

Figure 2.9. Hotwire probe (dimensions in mm)

During the experiments, the hotwire probe was held by aluminum mounting
bracket that was connected to a traversing mechanism via a tube that passed through the
slot in the ceiling of the test section. This traversing mechanism was driven by TMG,
Hybrid Stepper Motor (model 5618 S — 0105) coupled with TMG, MS 2.0 series, quad-
step controller. The latter was driven by an 8-bit digital I/O interface integrated into the

NI 6024E multi-purpose A/D card. The whole assembly of the stepper motor unit was

32

located outside the test section. The resolution of the stepper motor/traversing-
mechanism system, which was calibrated using a 12.7 um resolution Starrett dial
indicator (model 25-631), was found to be 0.824 1.1m per motor step (where one full
revolution of motor shaft corresponds to 800 steps). To position the hotwire sensor
accurately relative to the resonator’s slit, a digital imaging system was utilized. The
system consisted of a Sony (model XC-75) Charge Couple Device (CCD) video camera
connected to a PC-driven, National Instruments, IMAQ frame grabber (model PC1-1411).
A Nikkor 500 mm (model NH — 27) lens was utilized for capturing images of the hotwire
tip near the resonator opening. A Steady Lite, high-intensity, lighting manufactured by
Stocker & Yale, Inc. (model 13 Plus Lite Mite® Series) is used to illuminate the imaging
region of the hotwire tip and the lip of the resonator. A sample of a captured image is

depicted in Figure 2.10.

 

Figure 2.10. A sample image used in positioning the hotwire sensor near the
resonator’s slit

33

2.1.3.3 Mean-pressure measurements

Static-pressure-taps coupled with an appropriate pressure transducer were used
for measuring the mean-pressure distribution beneath the separating/reattaching ﬂow.
There are four arrays of static pressure taps: one on the north (top) side of the model, two
on the east and west sides, and one on the south side. The array on the north side consists
of 27 pressure taps exposed to the ﬂow while the other three sides have 7 pressure taps
each. The north static pressure taps lay along side the Emkay microphone array at the
same spatial distance of 4.8 mm from centerline-to-centerline with the ﬁrst pressure tap
located 11.73 mm downstream of the sharp edge of the resonator’s cover.

The static-pressure taps were connected to polyurethane pressure tubing that were
threaded inside the axisymmetric model and came out through a half-rounded, hollow
shroud attached to the vertical steel support plate of the model. The other ends of the
tubing were connected to 48 input ports on a rotary, solenoid-driven, pressure scanner
(model 48D9 — 1346 from Scanivalve). The rotary solenoid was driven by a homemade
solenoid-actuation circuit that was, in turn, controlled using one of two analog-output
channels available from the NI-6024E general-purpose, data-acquisition board. The
analog channel was used to generate a 0 — 5 Volts transition to step the scanner from one
static-pressure port to the next. The selected port connects to the output port of the
scanner, which is ultimately connected to a low-range (0 — 12.7 mm H20, or 0 — 124.7 Pa)
Setra pressure transducer (model 239 — XW 211). The transducer’s output is linear with

maximum output of 5 Volts at full-scale pressure.

34

2.1.3.4 Pitot tube and temperature sensor

A 5/16 -inch diameter Pitot tube manufactured by Dwyer Instrument Inc. (model
160-12) was used for measuring the freestream velocity in the test section. The device
was inserted into the test section through theqslot in the ceiling with tube’s static-pressure
ports located nominally 20 mm upstream of the step. The pressure ports of the Pitot tube
were connected to either a 0 — 13.3 Pa, or 0 — 133.3 Pa, Baratron pressure transducer
(model 223-B) depending on the ﬂow velocity. In either case, the output of the Baratron
pressure transducer is in the range of 0 — 1 Volt. The Pitot tube measurements were used
to set the freestream velocity to the desired value, and monitor its stability during the
experiments. In addition, the tube was used to calibrate the single hotwire sensor.

The temperature of the air in the test section was also measured in order to
calculate the air density for Pitot tube velocity measurements as well as for correction of
the hotwire signal for temperature effects. The temperature sensor employed was
manufactured by Omega (model DP-25-TH), which provides an analog output signal for
sampling by the computer. In addition, the sensor has an LED panel that gives
instantaneous readouts for monitoring by the operator during the experiments. The

sensor was of the thermistor type and had a sensitivity of 0.1 Volt/°C.

2.1.3.5 Helmholtz resonator driving system
As discussed previously, the Helmholtz resonator was driven by imposing a sound
ﬁeld external to the device. This was accomplished using a 0.381 m diameter, 450-Watts,

Eminence LLC subwoofer (model Kappa 15) with 0.3496 m bafﬂe-hole diameter. The

35

subwoofer was located at the centerline of the contraction, nominally 2.3 m upstream of
the intake (see Figure 2.11).

A two-channel, 110-Watts/channel, power ampliﬁer manufactured by Haﬂer
(model Trans-ana P 1000) was used to drive the loudspeaker. The two channels of the
ampliﬁer were “strapped” together; i.e., connected in parallel, to operate the speaker at
power levels up to 220 Watts. The input signal to the ampliﬁer was provided using two
Hewlett Packard function generators (model 33120A), the output of which was summed
together. The adoption of two function generators enabled the generation of amplitude-

modulated excitation of the Helmholtz resonator.

 

 

 

 

 

 

 

Subwoofer Contraction
intake
\ 9) 0.385 \
Git: Flow :
1"- 0.156 1.346
~ 8.880 ._

 

 

 

 

 

K\\\\\\\\\\\\\\\\\\\\\\\\\\\D

Figure 2.11. A schematic showing the subwoofer location relative to the wind tunnel
intake (dimensions in meter)

The need for employing amplitude-modulated, rather than pure-harmonic

excitation stems from the mismatch between the resonance frequency of the forcing

device (630 Hz: see section 2.2.3) and the frequency range of the inherent ﬂow instability

(i.e., the range in which the unsteady excitation affects the ﬂow). In particular, the latter

36

is more than an order of magnitude lower than the former. Since, it was not possible to
employ a Helmholtz resonator with sufﬁciently low resonance frequency to match that
desired by the ﬂow (the resonator’s volume would be prohibitively large), seeding of
low-frequency disturbances into the ﬂow was accomplished through low-frequency
amplitude modulation of the high-frequency excitation signal of the device. As will
become clear in Chapters 3 and 4, the success of this strategy is related to the
establishment of a “streaming” (i.e., steady) velocity component at the resonator’s
opening when exciting the resonator with high-intensity harmonic sound. Subsequent
modulation of the harmonic sound causes this steady ﬂow to oscillate at the low
frequency of modulation, thus coupling effectively to the ﬂow. Equally signiﬁcant is the
fact that the generated low-frequency ﬂuidic disturbance is at much lower frequency than
the acoustic signal driving the resonator. Therefore, the ﬂow disturbance occurs at a
frequency that is different from the acoustic noise produced by the excitation scheme.
Although it is possible to generate an amplitude-modulated signal from a single
function generator, there are some subtle issues relating to the acoustic response of the
wind tunnel that led to the use of two function generators. More speciﬁcally, mixing (i.e.,
multiplication) of two sinusoidal signals was used in the present study to form the

amplitude-modulated signal. Such a signal, say z(t), is given by the following equation:

z(t) = Zsin(27gfmt)sin(27gfct) (2.2)

where t is time, Z is the signal amplitude, fm and fc are the modulation and carrier

frequencies respectively, and no phase difference is assumed between the two sinusoids

37

because of its insigniﬁcance to the present analysis. Employing trigonometry, Equation

2.2 can be rewritten as:

z(t) = %[cos 27r(fc - ﬁn)! — cos 27r(fc + me (2.3)

Equation 2.3 shows that a harmonic signal with frequency fc that is modulated with
frequency fm can be created by adding two harmonic signals with frequencies (f,- —f,,,) and
(fc + fm). However, the amplitude of the two signals must be equal.

If a single function generator is used to generate a modulated signal such as that
given by Equation 2.2, this would be equivalent to seeding the wind tunnel with two
sound waves at frequencies of (72 -— fm) and (f, + f,,,) and equal amplitude. However,
because of the dependence of the tunnel acoustic response on frequency, the two waves
may be preferentially ampliﬁed and/or attenuated as they propagate through the tunnel.
Furthermore, reﬂection of the waves from the laboratory walls creates a standing-wave
component within the tunnel, leading to spatial dependence of the sound wave amplitude,
which in turn is dependent on the wave frequency. The net result of these effects is that,
at the location of the resonator slit, the two seeded waves may not have equal amplitude.
In the extreme case, when one is substantially attenuated relative to the other, a pure
harmonic, rather than amplitude-modulated, excitation of the Helmholtz device is
produced. In other more moderate situations, where the two signals are of comparable
but not equal magnitude, modulation is achieved but not with the desired strength. As

described above, the ability to excite the ﬂow relies on the modulation signal, and

38

therefore the ability to control the modulated-forcing-component strength properly is
important.

The employment of two function generators to produce the amplitude-modulated
signal remedies the problem described above by allowing independent adjustment of the
amplitude of each of the sound wave components. In this case, one of the function
generators is used to produce a harmonic signal at a frequency of (f; —f,,,) while the other
one produces the other harmonic component with frequency (fc + fm). The two signals are
then added together using a summation circuit before they are fed to the power ampliﬁer.
To set the amplitude of each of the two signals, the Panasonic microphone is placed at the
resonator’s opening to monitor the incident sound wave. The amplitude of one of the
waves is kept constant and that of the other is adjusted. This is done while monitoring
the microphone output signal on the oscilloscope until a good-quality amplitude-
modulation signal is observed. Subsequent adjustment to the forcing-signal strength at
the given carrier and modulation signal frequencies is accomplished by equally
amplifying the two signals.

Figure 2.12 provides a block diagram of the entire driving system. Note that the
combined signals were sampled by the A/D board in various experiments to provide a
reference for the forcing-signal phase. In addition, the signal is fed into a homemade
low-pass ﬁlter powered by a couple of Imation, AC/DC adaptor power supplies (model
JOD 48U—08). The low-pass ﬁlter was utilized only when broadband sound was
generated for obtaining the frequency response of the resonator. In this case, a white
noise signal was fed from one of the function generators to the power ampliﬁer. The

ﬁlter helped to limit the bandwidth of the white noise to that desired (to avoid subsequent

39

aliasing of the digitized data records). Finally, certain characterization tests of the
resonator required the use of pure harmonic signals. In this case, only a single function

generator was employed as the source of the driving-signal.

 

 

HP 33120A HP 33120A

Function generator I I Function generator

Summation circuit

 

 

 

 

 

 

 

 

 

 

 

’ A/D board

 

 

 

 

 

I
Low-pass ﬁlter

1

Power ampliﬁer

1

Subwoofer

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.12. Block diagram of the Helmholtz-driving system

2.1.3.6 Data-acquisition and experiment-control hardware

As brieﬂy discussed in the previous subsections, there are two different computer-
based-data—acquisition boards used for the experiments. The main data acquisition board
is a PC-based National Instrument A/D board (model NI 6024E). This board is equipped
with 16 single-ended analog-input channels and has a maximum sampling frequency of
12,500 Hz per channel if all 'channels are consumed. Because the channels were

multiplexed, a systematic error in the sampled data could occur due to an average inter-

40

channel time delay. However, this error is relatively small, 5 us, and has negligible effect
on the measurements compared to the ﬂow convective time scale, 318 us, over a distance
equal to sensor (i.e., microphone) spacing. The board was used for various
measurements that included velocity, pressure, and temperature. In addition, the board
was also used for driving the stepper motor unit connected to the hotwire traversing
mechanism.

The other data acquisition hardware is a PC-based, National Instruments, IMAQ
frame grabber (model PC1-1411). This frame grabber was mainly used for capturing
close-up images of the hotwire in order to position it with good accuracy relative to the
resonator’s opening. The board is capable of capturing video image frames in real time,

converting them into digital format using an 8-bit, ﬂash, analog-to-digital converter.

41

2.2 Experimental methodology

This section includes a categorization of the experiments conducted along with a
summary of the data acquisition parameters, followed by a description of the procedure

of those experiments.

2.2.1 Experiments categorization and data acquisition parameters

The experiments may be categorized into three different topics: test-model
alignment, Helmholtz resonator characterization, and mean- and unsteady-wall-pressure
measurements for examination of the ﬂow-forcing effect on the surface-pressure ﬁeld
beneath the separated/reattaching ﬂow.

In all the experiments, two-different sets of data-acquisition parameters were
utilized depending on whether time-resolved information or only statistical quantities are
to be obtained. In the former case, the sampling frequency, f,, was set to 8,000 Hz and
the total number of samples, N, was 262,144 or 2'8. The sampling frequency was well
above the largest frequency (nominally 1 kHz) where any signiﬁcant signal ﬂuctuation
existed in order to avoid any aliasing during the digitization of data. In addition, the
number of points was large enough to attain proper resolution of the lowest frequency of
interest, while providing sufﬁcient number of data sub-records for reducing scatter in the
calculation of frequency spectra.

The above parameters were used for all except the mean-pressure measurements,
where f, was set to 1,500 Hz and N was 15,000 samples. This rendered a lO-second
average of the pressure, which was long enough to obtain a stable average at all static-tap

locations. The A/D board input range used was either $5 or 250.5 Volts, depending on the

42

signal level. When coupled with the board’s resolution (12-bit), the larger range provides
a voltage resolution of 2.44 mV. The lower range reduces this value by an order of
magnitude, which was particularly important for measurements of the mean-pressure and

the resonator’s unsteady-jet velocity at low levels of excitation.

2.2.2 Model alignment

The test model was aligned parallel to the freestream direction prior to installing
the resonator on the model. This was done by adjusting the model’s support frame and
the tension of the four piano wires at the upstream end of the model. Initially, the model
surface was set parallel to the sidewalls and ﬂoor of the test section using tape measure,
ruler angle, and level. This was regarded as coarse adjustment with additional ﬁne-
tuning of the model’s alignment conducted based on the mean-pressure distribution
around the model.

As described earlier in section 2.1.3.3, there are four, static-pressure-tap arrays
located downstream of the step. Azimuthally, these arrays are positioned on the top and
bottom (indicated as the north and south here), and both sides (east and west) of the
model. A plot of the streamwise distribution of the mean-pressure coefﬁcient (Cp)
obtained from the four arrays may be seen in Figure 2.13. Note that the streamwise

coordinate is normalized by the step height (H) and that C p is given by the equation:

Cp = $192; (2.4)
7 1001'er

43

where P... is the freestream pressure, P, is the pressure measured at each of the pressure

taps surrounding the model, pm, is the density of air, and U,o is the freestream velocity.

—9— North @— Semi-é;— East —e;West 1

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.13. Streamwise distribution of CI, at four different azimuthal locations

As seen from Figure 2.13, the mean-pressure distributions at all four azimuthal
locations seem to collapse up to nominally x/H = 3. Downstream of this location the
largest deviation is found near x/H = 4. This deviation (relative to an average of all four
values at a given x position) is about 10% of the overall pressure variation within the

separation bubble.

2.2.3 Helmholtz resonator characterization

Both broadband (white noise) and harmonic sound waves were used to
characterize the Helmholtz resonator. The broadband signal was utilized in tests directed
towards obtaining the frequency response of the resonator. The main goal of these tests

was to identify the resonance frequency of the device, and to verify that the behavior of

44

the device may be approximated by that of a second-order system (as expected from a
Helmholtz resonator). In addition, these experiments were also employed to verify the
azimuthal uniformity of the incident sound ﬁeld. On the other hand, the harmonic test
signals were primarily employed to determine the ﬂuidic response of the resonator. That
is, these tests rendered information on the strength of the unsteady jet produced by the
resonator at different locations away from the slit, and for different forcing levels. In the
following, more detailed description of each of the tests is provided. Note that all the
tests were conducted without ﬂow in the tunnel.

For the broadband tests, the Panasonic microphone was placed in the immediate
vicinity of the resonator’s slit, as illustrated earlier in Figures 2.8a-b, in order to measure
the acoustic input that drives the resonator. The Panasonic microphone was mounted on
a traverse ring that was attached to the model and allowed for measurement of the input
sound at different azimuthal angles while maintaining the same location relative to the
resonator’s oriﬁce. The traverse ring is made of two aluminum halves held together by
two dowel pins and tape. This ring was assembled around the model such that the center
of the ring coincides with the center of the axisymmetric model. The tests were
performed at sixteen different azimuthal angles, starting from 22.5°, relative to the top of
the model, to 360.0° in 225° increments.

To measure the response (output) of the resonator to the input sound, the pressure
inside the resonator’s cavity was acquired using the Emkay microphone embedded inside
the resonator (see section 2.1.3.1.2). This rendered the acoustic response of the device.
Additionally, the hotwire was located slightly (y = 0.034 mm; corresponding to 2 pixels

in the image captured by the CCD camera used for the hotwire positioning) above the slit

45

opening, where for the remainder of the discussion will be referred as y = 0 (within the
resolution of the measurement), and at the slit opening’s center line in order to capture
the ﬂuidic output of the device (i.e., the velocity oscillations associated with the air mass
movement within the neck of the resonator). Figure 2.8b provides a photograph showing
the placement of the hotwire relative to the resonator. To position the hotwire above the
resonator, images of the hotwire at different locations were captured. First, the hotwire
was positioned as close as possible to the resonator’s slit. Then the hotwire probe was
traversed upward in the wall-normal direction at known increments. Concurrently, an
image was captured for every hotwire position and a corresponding dial indicator
(manufactured by Starrett — model 25-631) reading of the traversing system movement
was recorded. The captured images were then processed using MATLAB software to
obtain the number of pixels that the hotwire moved for each increment and correlate this
with the actual distance traveled by the hotwire in order to arrive at the imaging scale
factor. After averaging for all incremental movements, this factor (which is also the
measurement resolution) was found to be nominally 17.2 uni/pixel.

For the broadband tests, four channels of data were acquired: two for the hotwire
signal (unconditioned and mean-removed/ampliﬁed), one for the Panasonic microphone,
and one for the Emkay microphone located inside the resonator. Before activation of the
sound ﬁeld, measurements of the background noise were conducted. This was done in
order to verify that, once the sound was activated, the signal-to-noise ratio was
sufﬁciently high to render the response of the device. As will be seen in Chapter 3, that

this was in fact true for the entire frequency range of interest when determining the

46

acoustic response of the resonator. On the other hand, the ﬂuidic response could only be
obtained within a narrow frequency band around the resonance frequency of the device.

A similar setup was utilized for the harmonic tests. In this case, however, the
Panasonic microphone’s azimuthal location was ﬁxed at an angle of 180° relative to the
top of the model (see Figure 2.8b). Also, in addition to sampling four channels of data as
described above, the sound source signal from the function generators was captured to
provide a reference for the forcing signal phase. Table 2.1 summarizes all data
acquisition parameters employed for both the broadband and harmonic tests. Note that
all harmonic tests were conducted at a frequency of 630 Hz. This corresponds to the
frequency at which the unsteady resonator’s jet was the strongest for a given sound input.
It is interesting to point out that this frequency was somewhat smaller than the resonance
frequency of the resonator (which was 657 Hz). The reason for this difference will

become clear in Chapter 3.

Table 2.1. Data acquisition parameters for broadband and harmonic tests

 

 

 

 

 

 

 

 

 

 

 

 

Channel Channel descriptions Broadband test Harmonic test
Ch. 10 Panasonic microphone d: 0.5 Volts d: 0.5 Volts
Ch. 11 Function generator Shorted i 5.0 Volts
Ch. 12 Unconditioned hotwire :t 5.0 Volts 3: 5.0 Volts
Ch. 13 Ampliﬁed hotwire Extjmoalngihltgf 10 5111:3615 gZihltgf 10
Ch. 14 Emkay microphone d: 0.5 Volts i 0.5 Volts

Number of samples, N 262,144 262,144
Sampling frequency, f, 8000 Hz 8000 Hz

 

 

47

When utilizing a pure harmonic signal to drive the resonator, three different tests
were conducted. First, the location of the hotwire was kept at y = 0 above the slit opening
and data was acquired for seven streamwise locations spanning a distance of slightly
more than the slit width (see Figure 2.14 for demonstration of the location of the
measurements relative to the slit). This test was used to identify the location within the
slit where the velocity ﬂuctuations are the largest. Subsequently, a second test was
conducted where the hotwire was positioned at this location and the forcing level
(imposed sound amplitude) was changed. In particular, ﬁve different levels of forcing
were considered. These forcing levels corresponded to a sound-pressure-level (SPL) of

98.1 dB, 102.6 dB, 107.3 dB, 109.3 dB, and 111.7 dB based on the RMS pressure.

 
  
  
  

Location of measurements: / ’
y = 0 above slit opening ’
//
12-times ’7
ll

. . /,
magn1ﬁcat1on /,
/

X

/ 1

/
, 1
/ 1

Outer shell

   

 

 

Figure 2.14. Location of the hotwire measurements across the slit opening

48

Finally, a third test was conducted where the sound level was kept at the highest
value and the hotwire was used for measurements at different locations in the wall-
norrnal (y) direction. Starting with an initial location of y = -0.055 mm relative to the
surface of the slit opening, the hotwire was traversed in increments of 0.10 mm up to

0.945 mm above the surface of the slit opening.

2.2.4 Mean- and unsteady-wall-pressure measurements

Mean-pressure measurements were undertaken for both unforced and forced ﬂow
conditions. The unforced, or natural, case was done ﬁrst to produce a benchmark.
Subsequently, the mean-pressure distribution was acquired when forcing the ﬂow at ﬁve
different modulation frequencies (i.e., 10 Hz, 15 Hz, 18 Hz, 20 Hz, and 25 Hz) with a
carrier frequency, fc, of 630 Hz. As will become clear in Chapter 4, since 18 Hz
produced the strongest effect on the ﬂow, the mean-pressure distribution was also
measured for different forcing levels for this particular frequency to observe the effect of
forcing strength on the mean-pressure distribution. During the measurements, the
freestream dynamic pressure was also acquired with each static-tap measurement to
account for any small variation in the tunnel velocity when calculating the mean-pressure
coefﬁcient. All of data were acquired at an average freestream velocity, U00, of 3.25 m/s.

While the mean-pressure information is useful in examining if the ﬂow excitation
is pronounced enough to produce a global effect on the average ﬂow behavior (i.e.,
affecting the entire separation bubble behavior and not merely the shear layer), the
unsteady surface-pressure measurements would yield information that is related to the

alteration of the turbulent structures. The energetic ﬂow structures leave a wall-pressure

49

imprint as they advect downstream. Therefore, alteration of the ﬂow structures behavior
via the forcing should also lead to corresponding change in the space-time characteristics
of the wall-pressure. To investigate the latter, the ﬁﬁeen-Emkay-microphone array
described in section 2.1.3.1.2 was used for measuring the instantaneous wall-pressure
signature downstream of the resonator.

Because microphones are sensitive to the hydrodynamic pressure ﬂuctuations (i.e.,
that produced by the ﬂow structures) as well as the acoustic pressure ﬂuctuations
produced by speaker driving the Helmholtz resonator, it was desired to document the
latter by conducting measurements for all the forcing conditions without ﬂow.
Subsequently, these measurements were also repeated for a freestream velocity of 3.25
m/s.

The forcing conditions for which microphones data were compiled corresponded
to a carrier frequency of 630 Hz, two modulated frequencies of 10 Hz and 18 Hz, and two
different forcing levels. Note that because the dynamic range of the microphones is
limited to about 130 to 140 dB SPL, it was not possible to conduct the measurements at
the “higher” forcing levels as the output of the microphones would “clip”. In general, the
speaker-generated sound dominated the microphone measurements. However, because
of the amplitude-modulated excitation, the sound and hydrodynamic signals were in two
widely separated frequency bands. Speciﬁcally, the sound signal was conﬁned to the two
frequencies (fc—fm) and (fc + fm), while the hydrodynamic signal was at 2f," (and not f", as
will become clear in Chapter 3). Since fc is more than an order of magnitude larger than
fm, it was possible to effectively separate the acoustic and turbulent components using

frequency-domain ﬁlters. The latter was implemented during the post processing of data.

50

3 CHARACTERIZATION OF THE HELMHOLTZ RESONATOR

This chapter contains three sections that provide details of the data analysis, and
discussion of the results pertinent to characterizing the Helmholtz resonator. The ﬁrst
section contains material related to the acoustic and ﬂuidic response of the resonator.
This is followed by qualitative and quantitative characterization of the resonator’s ﬂuidic
disturbance under pure-harmonic forcing conditions in the second section. Finally, the
third section contains further study of the resonator’s ﬂuidic disturbance under
modulated-forcing conditions.

In order to characterize the Helmholtz resonator, several analyses were conducted.
These are summarized here in order to familiarize the reader with the details of the signal
processing prior to presentation of the results. First, the Power Spectrum Density (PSD)
of various discrete-time measurement signals was obtained by means of the Fast Fourier
Transform (FFT). PSD yields information about the distribution of the “energy” of the
signal over different frequencies, allowing one to identify the dominant frequencies in a
signal (i.e., frequencies at which large ﬂuctuations take place). The one-sided PSD for a
signal b(n), where n is the discrete-time index (or sample number), is given by the

following equation:

 

 

B(k)xB*(k) 11:05
_ 1v2 ’ ’2
¢bb(k)_ 2xB(k)><B*(k) .k_12 91—1 (3.1)
N2 9 _ 9 9'”,

where, ¢bb is the discrete PSD, B(k) is the FF T of b(n), B*(k) is the complex conjugate of

B(k), k is the discrete-frequency index, and N is the number of data points in the time

51

series. Note that to obtain the physical frequency value corresponding to a particular
value of k, the latter is multiplied by the spectrum resolution Af = fs/N (where f, is the
sampling frequency). It is evident from Equation 3.1 then that the largest frequency in
the one-sided spectrum is the Nyquist frequency, or ﬁ/2.

A second quantity of interest is the cross-spectrum. Speciﬁcally, this quantity is
useful in obtaining the response of a system from measurements of its input and output,
as will become clear later. This was utilized for characterization of the forcing-device
response from measurements of the Panasonic microphone’s signal, or resonator’s input,
while employing the Emkay microphone and hotwire to capture the acoustic and ﬂuidic,
respectively, disturbance as the system’s output. Unlike the power spectrum analysis that
helps ﬁnd the dominant frequencies in a signal, the cross-spectrum analysis offers
information regarding the correlation between two different signals for different
frequencies. The one-sided, discrete-frequency, cross-spectrum was calculated from the
equation below for two signals, b and g, where deﬁnitions of various symbols are similar

to that given earlier for Equation 3.1.

 

 

G(k)xB*(k) k=0ﬁ
- N2 ’ 9 2
¢gb(k)" 2XG(k)xB*(k) .k212 ﬂ_1 (3.2)
N2 , 7 3'",

The magnitude and phase responses, respectively, of the Helmholtz resonator,
were obtained from the cross- and auto-spectrum results as follows (note that here b

indicates the sound input to the resonator; i.e., that measured by the Panasonic

52

microphone, and g is the acoustic or ﬂuidic output; i.e., that captured by the Emkay

microphone and hotwire respectively):

_| ¢g,,(k) |

J k —
I ( )l .(¢bb(k)—¢bb_background(k))’

(3.3)

 

11g, 00
(¢bb (k) _ ¢bb_background (k))

 

ZJ (k) = angle (3.4)

where J(k) is the discrete-frequency, resonator’s system function, and subscript
background indicates the background noise measured by the Panasonic microphone
without any deliberate sound excitation of the device.

Subtraction of the spectrum of the background noise from the input-sound
spectrum helps to remove noise inﬂuences (including electronic noise as well) on the
input signal when estimating the system, or transfer function. Noise inﬂuences on the
output signal are accounted for by utilizing the cross-spectrum in the numerator of J
instead of the auto spectrum of the output signal. These provisions help in improving the
quality of the estimate. However, this does not alleviate the need to have a good signal-
to-noise ratio at all frequencies of interest. Two measures were utilized here to gauge the
signal-to-noise quality of the measured input and output signals. First, the auto spectrum
of each signal with and without imposed sound ﬁeld were compared to ensure that the

latter was well below the former. Second, the coherence between the input and output

53

signals was checked to examine the phase-locking between b and g. The coherence (Q)

was obtained from the cross-spectrum using:

11gb1k>|

(3.5)
\/¢,,,,(k)>< ¢gg(k)

ngUf) =

 

Note that since the resonator is a deterministic system, the coherence between b and g
should be equal to unity across all frequencies. Deviations from this value are due to
phase variation associated with random noise. The associated reduction in 1781, depends
on the relative magnitude of the random and deterministic signal components. Therefore,
for the purpose of transfer function determination, a high value of coherence is desired.
In the present investigation, a coherence threshold of 0.7, or 70%, was adopted to gauge
the quality of the measurements at different frequencies.

Note that in calculating the power and cross-spectrum, the 262,144 points time
series were sub-divided into 512 records of 512 points each. The spectra obtained from
each of these records were then averaged yielding a frequency resolution of 15.625 Hz

and random uncertainty of 4.4%.

54

3.1 Determination of the resonator’s acoustic and fluidic response

The acoustic response of the resonator refers to the pressure ﬂuctuations produced
inside the resonator’s cavity (pc) in response to the acoustic pressure imposed on the
outside of the slit, or neck (1),). On the other hand, the ﬂuidic response pertains to the
unsteady ﬂuid velocity at the slit opening of the resonator produced by the imposed
sound.

To determine the response of the resonator over the entire frequency range of
interest, white-noise sound input was utilized. This enabled excitation of all frequencies
at once. The bandwidth of the white noise was limited to less than 4 kHz (the Nyquist
frequency of the data acquisition) by feeding the function generator output through a low-
pass ﬁlter before feeding it to the speaker’s power ampliﬁer. Figure 3.1 shows the power
spectrum of the generated sound-ﬁeld pressure just outside the resonator’s slit. The
different curves correspond to measurements conducted at sixteen different angles around
the perimeter of the slit. Also included in the ﬁgure is the power spectrum obtained with
the speaker turned off. This spectrum represents the background noise level, and
therefore is denoted by “BG” on the spectrum plot. Also note that the results are plotted
for frequencies up to 1 kHz only since higher frequencies are not of interest.

Inspection of Figure 3.1 shows that, overall, the imposed sound ﬁeld at the lip of
the resonator is azimuthally uniform up to a frequency of about 700 Hz (some narrow-
band exceptions to this uniformity are found at nominal frequencies of 310 Hz, 565 Hz,
and 670 Hz). The spectra depict multiple peaks that presumably correspond to
resonances of the wind-tunnel test section, with the highest peak found at nominally 630

Hz. This frequency then corresponds to the largest achievable magnitude of the driving

55

pressure of the Helmholtz resonator for a given input-signal-amplitude to the speaker.
Finally, consideration of the background-noise spectrum shows that, except at 360 Hz
and 720 Hz, the imposed sound-pressure spectrum is more than one order of magnitude

higher than the background noise. Therefore, the measured signal has an acceptable

signal-to-noise ratio for utilization in determining the resonator’s response.

10'

10‘

10

11,“, (Pa’)

10'

10

 

.+90.o°
,+112.5°

+2700"

‘ +3150

+225“
+ 450°
+675

135.0°
+1575
+1500°
5: ‘ 202.5°
+2250
-:»*- 247.5°

292.5°

+3375

: +3600
1 —-BG
1000

 

 

 

1 l l l l l 1

300

 

200 600
Frequency (Hz)

Figure 3.1. Power spectrum of the sound pressure just outside the Helmholtz
resonator’s slit for 16 different azimuthal angles
A similar plot to that shown in Figure 3.1 is given in Figure 3.2 for the pressure
inside the cavity (pc), which was obtained using the Emkay microphone. The data from
the Emkay microphone were acquired simultaneously with the data from the Panasonic
microphone. Therefore, there is one Emkay time series for each azimuthal position of the
Panasonic microphone. However, since the Emkay microphone position did not change
during the acquisition, the different-color data sets shown in Figure 3.2 may be used as an

indication of the repeatability of the measurements.

56

Similar to the results in Figure 3.1, the highest spectrum peak is found to be at
nominally 630 Hz for the cavity pressure. However, it is important to realize that this
frequency does not necessarily correspond to the resonance frequency of the resonator.
The highest peak observed in Figure 3.2 indicates that the strongest cavity-pressure
ﬂuctuations for a given input to the speaker are found at 630 Hz. Hence, this frequency
corresponds to the resonance frequency of the entire system consisting of the power
ampliﬁer, speaker, wind tunnel and resonator. On the other hand, the resonance
frequency of the resonator should correspond to the largest pressure ﬂuctuation amplitude
inside the resonator’s cavity for a given sound-pressure amplitude imposed on top of the
slit. Lastly, inspection of the background-noise spectrum (blue line in Figure 3.2) clearly

shows that the measured sound pressure is one or more orders of magnitude larger than

the background noise.

 

 

 

 

 

,_ L,_.\ p... / \/ _
10 \NWYAy/Vi
10.7 F 1 1 1 1 1 1 1 '3
200 300 400 500 600 700 800 900 1000
Frequency (Hz)

Figure 3.2. Power spectrum of the resonator’s cavity pressure with and without
(blue line) speaker excitation. Different line colors correspond to repeated
measurements.

57

The ﬂuidic response of the Helmholtz resonator is represented using the one-sided,
power-spectrum plot shown in Figure 3.3. For all spectra shown, the hotwire was located
at y = 0 (within the resolution of the measurement) relative to the slit opening of the
resonator. It is noted that the representation of the data here is in volts (i.e., without
conversion of the hotvvire’s output voltage to velocity). No calibration was conducted for
this test since it does not inﬂuence the outcome of the analysis, which primarily aims at
identifying the Helmholtz resonance frequency. Similar to Figure 3.2, the different color-
coded curves correspond to repeated runs of the measurements, and the blue line (see

arrow) denotes the background noise.

 

 

 

   

 

 

10" ~ -
Va.) ' - :a‘
tr> I“ Background
9' 1. . noise
.' ' £le 4
\ , i I / Z... _‘_q_
1040 1 1 1 1 1 1 1
200 300 400 500 600 700 800 900 1000
Frequency (Hz)

Figure 3.3. Power spectrum of hotwire sensor output

Several peaks can be identiﬁed in the spectra in Figure 3.3. The largest two peaks
are found at nominal frequencies of 300 Hz and 630 Hz. However, the former frequency

is clearly dominated by the background noise (this point will be reinforced using the

58

coherence analysis below). On the other hand, the peak at 630 Hz is more than an order
of magnitude larger than the background noise, and hence this peak corresponds to actual
velocity ﬂuctuations produced by the coupling of the speaker-driven pressure ﬂuctuations
with the resonator. Unlike the pressure measurements, the velocity data seem to have
large signal-to—noise ratio only within a narrow frequency band of 50 - 100 Hz that is in
the vicinity of 630 Hz. Consequently, the ﬂuidic response of the Helmholtz resonator can
only be determined reliably within this band. The results suggest that the resonator is
highly inefﬁcient in generating velocity disturbances at frequencies that depart
appreciably from the resonance frequency of the system.

As discussed earlier, coherence analysis was used to examine the phase-locking,
and hence the signal-to-random-noise quality, between the Panasonic input signal, on one
hand, and the Emkay and hotwire output signals, on the other. Based on Equation 3.5
above, the coherence could be calculated, and an arbitrary threshold of 0.7 or 70% was
chosen to distinguish between good and poor phase-locking (coherence > 0.7 for the
former). The results may be seen in Figure 3.4 when considering the Panasonic and
Emkay signals. In the ﬁgure, different colors represent the different azimuthal angles
(relative to the top of the model) at which the Panasonic microphone was located. With
the exception off = 360 Hz, the phase-locking quality between the acoustic input and
output signal is quite good between 200 Hz and 700 Hz. This is true for all locations of
the Panasonic microphone. At frequencies higher than 850 Hz, considerable scatter is
found among the results obtained for different azimuthal locations. This indicates the
poor axisymmetric quality of the incident sound ﬁled at these high frequencies.

However, since this frequency range was not utilized in the actual experiments, this was

59

of no concern. Along with the power spectrum analysis done previously, the coherence
analysis demonstrates the high signal-to—noise ratio of the measurements within the
frequency range of interest, and gives evidence of the deterministic coupling between the

input and output acoustic signals.

 

1 i r i .
' 1 1 3 : : o

, x ,. . : 1+22.5
3 t : ' o

' ' :1 .1 5,. , +450
ae .. -. a - z , o
\_ I! . . . _. .‘ ’3‘. I.“ .- 3.. O

 

 

 

 

 

; . - 1 1 +1125
g g f )1 11 g 1350’
06"" .................. ........... ------------------- ----- I a" _' ' .1 +15750
. g g , . §+100.o°
a 5 E . t §+202.5:
0.4 _ -------------------------- +2250
é Threshold ; a 3 " “5 2415'
E g . a g , g+270.o°
0.2 292-5.
2 a a i 5 3 §+315.0°

 

 

 

  

 

I

 

+3375
900 300 400 500 600 700 800 900 1000
Frequency (Hz)

 

 

 

 

 

Figure 3.4. Coherence plot for the input and output acoustic signals of the resonator

A plot of the coherence between the acoustic input and ﬂuidic output signals (i.e.,
that measured by the Panasonic microphone and hotwire sensor) is shown in Figure 3.5
below. Unlike the coherence between the Emkay and Panasonic microphones, the
frequency range for good-quality phase-locking between the hotwire sensor and
Panasonic microphone signals is found to be much narrower. Speciﬁcally, the frequency
range with coherence value above 70% is seen to be conﬁned between 600 Hz and 700

Hz only. Hence, outside this frequency band, the Helmholtz resonator’s ﬂuidic response

60

cannot be determined reliably. This implies that ﬂuidic-response information will be

limited to the vicinity of the system’s resonance frequency.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 1+225°
Thres\hpld + 45.0.
§+e7.5°
W“ 1‘ +9011.
g+112.5
: g . ; 135.0”
_ 0.6 r............. 1 """"""""""" g """" '"“'"””‘é """"" _ I q I: : +157.50
3’ I i .‘f‘. i " ‘ ‘ +1800:
'— 5 i 5 . ‘ '. +2025
o.4~ -------------- ........... :11 342255
‘ 5 ' -' §+247.5'
-".1't s s 2 s 5 ~ 2 - .+270-0.
02-.....4. ...... .................... .............. 1‘ ~ 292.5’
-. . " V ~. - - - . - ‘ ' .(1' o
,4, $.17; are 3 a 2 s ., 1. . ,4—3150
'r *‘ ' g+337.5
.1 1 1 : : : z+m.o°

300 300 400 500 1000

Frequency (Hz)

Figure 3.5. Coherence plot for the hotwire and Panasonic signals

The resonator’s acoustic response determined from application of Equations 3.3
and 3.4 to the Panasonic and Emkay signals is shown in Figures 3.6 and 3.7 for the
magnitude and phase, respectively. As previously noted, the different-color lines
represent results obtained from different azimuthal locations of the Panasonic
microphone. Additionally, the solid black lines in the ﬁgures display the theoretical
response of a second-order, lumped—parameter, mechanical system for comparison

purposes. The equations for this response are given by:

61

 

 

2
/ j
’ :00 (3.6)

__J_ (3.7)

where 9' is the damping ratio and (00 is the natural frequency. Note that for the curves
shown in Figures 3.6 and 3.7, the values of g and too were taken as 0.11 and 4151 rad/s
(or 661 Hz), respectively. These values were not based on any curve ﬁtting but rather on
visual observation of the agreement between the measured and theoretical response. This
is sufﬁcient for the purposes of the discussion, which aims at examination of the
consistency of the measured response with the second-order-system behavior.

Overall, the magnitude results in Figure 3.6 are consistent for all angles except at
360 Hz, where there is large scatter. This is not too surprising since the coherence is low
at this frequency (refer to Figure 3.4). There is also some scatter at the resonance peak of
about 15% relative to the mean response over all azimuthal locations. This number is
reﬂective of the axisymmetric quality of the sound input to the resonator. For
convenience, the experimental curves at all azimuthal angles were averaged and
represented by solid red line. The behavior of this average seems consistent with that of
the second-order system. This is conﬁrmed further from consideration of the phase

results where the 180-degree phase change characteristic of the behavior of a second-

62

order system is observed as the frequency increases from low values to values beyond the

resonance frequency. The speciﬁc value of the resonance frequency is found to be 657

 

   

 

 

 

Hz.
3 5 i 1 1 +225
‘ l ' g. [:1 +450“
‘ 1 1'" +675“
3b I l "1'44 ' 35”" $90.0”
, 3g : +1125
_5 _ 11'.J H," ”.v. 1‘ . _ g
2 . it y}, 1' Theoretical second- + 1273(5)
-— _ M“ order curve ' .
3° 2 i ‘ [ff +1000
2 1 If" +2025
1.5~ all -' +2250
1. _ (1 +2475
1- a £35" +2700"
‘ s;- 5‘7} ""i" . 292.5°
. \ 7' + °
0'5” Average from ‘ ' +333:
1 1 different mns 1 1 1 + 360. 0.
900 300 400 500 600 700 800 900 1000
Frequency (Hz)

Figure 3.6. Magnitude of the acoustic system function of the Helmholtz resonator

63

 

 

0- 1 .
;. .( . “2" +22.5°
3?; J: :5, 11' :‘p‘azf _" ‘ Average from :450.
‘ ‘4" . different runs 67'5.
H900
+1125“
- ' 135.0“
; +1575"
+1800
+2025“
+2250"
+2475”
+2700
" 292.5”

.<'>
0'1
1

  

 

Theoretical second- 3;: ‘
‘2 ” order curve

.1, +3150"

. . ~~ .-'-*-337.5°

J l 1 l 1 l 1 J + ”0.0.

900 300 400 500 600 700 800 900 1000
Frequency (Hz)

 

 

 

 

 

Figure 3.7. Phase of the acoustic system function of the Helmholtz resonator

The consistency of the resonator’s acoustic response with that of a second-order
system provides a signiﬁcant veriﬁcation concerning the behavior of the device.
Speciﬁcally, unlike a simple Helmholtz resonator with small (relative to the wavelength
of sound) circular opening, the current resonator has an azimuthal slit and hence is
capable of sustaining inﬁnite azimuthal modes of oscillation of the air mass within the
resonator’s neck. When forcing the ﬂow, it is desired to control the oscillation mode to
be of the axisymmetric form; i.e., that corresponding to in-phase movement of the air
mass across the entire perimeter of the slit. This corresponds to the lowest order mode of
oscillation of the device, which is the Helmholtz resonance. Thus, the current results
conﬁrm the observance of Helmholtz-resonance behavior, leading to axisymmetric
forcing of the ﬂow.

The magnitude and phase plots of the ﬂuidic response of the resonator can be seen

in Figures 3.8 and 3.9 below. The results are consistent with the acoustic response in

64

depicting a resonance frequency of 657 Hz and a behavior consistent with second-order
system’s response. However, the reader is reminded here that conﬁdence in these results

is limited to the frequency band extending from 600 Hz to 700 Hz (see coherence results

 

 

 

above)
1110'4
7_ +225
+450“
+67.5°
6* 1 ““900
5 _ 1 Theoretical second- + 333.
\g order curve _._ 157.5.
>V 4-
_Q_
CT
1?.

 

 

 

 

Average from
different runs

 

 

 

£50 300 400 500 600 700
Frequency (Hz)

Figure 3.8. Magnitude of the ﬂuidic system function of the Helmholtz resonator

65

 

 

 

 

 

 

 

 

‘300 300 400 500 600
Frequency (Hz)

Figure 3.9. Phase of the ﬂuidic system function of the Helmholtz resonator

66

3.2 Characterization of the nature of the resonator’s ﬂuidic disturbance
3.2.1 Background

Prior to examination of the velocity ﬁeld produced by the excitation device, a
brief summary of some pertinent literature is provided here. The aim of this summary is
to use existing information to arrive at a “picture” of the expected nature of the ﬂuidic
disturbance. Subsequently, in sections 3.2.2 and 3.2.3, ﬂow-visualization and single-
hotwire results will be used to demonstrate the consistency of the resonator’s behavior
with that projected from consideration of the literature. The discussion should also help
clarify certain ﬂow physics about the behavior of driven resonators.

One of the early studies to examine the ﬂow physics in the vicinity of the oriﬁce
of a driven resonator is that by Ingard and Labate (1950). In their ﬂow visualization
experiments, they identiﬁed four distinct regions of ﬂow behavior depending on the
acoustic velocity, i.e., sound level, of the imposed acoustic ﬁeld. The brief descriptions

by Ingard and Labate (1950) of these four regions are as follows:

Region 1 A low intensity region with stationary circulation; the ﬂow is directed
out from the oriﬁce along the axis.

Region 2 A region of stationary circulation in which the direction of the ﬂow
along the axis is toward the oriﬁce, i.e., the reverse of that in region 1.

Region 3 A medium sound intensity region where pulsatory effects are
superposed upon circulation of the kind in region 2.

Region 4 A high sound intensity region in which pulsatory effects are
predominant, resulting in the formation of jets and vortex rings. The
jet consists of a strong airﬂow through the orifice, signiﬁed by a
sudden burst of air. This burst appears symmetrically on both sides of
the oriﬁce and is made up of pulses contributed by each cycle of the
sound wave.

67

As will be clariﬁed later, the velocity ﬁeld produced by the resonator here corresponds to
two distinct regions (i.e., regions 1 and 4). Within region 1, the sound level is large
enough to produce what is known as streaming, or steady, ﬂow in addition to the usual
oscillating ﬂow component in the vicinity of the oriﬁce. The streaming ﬂow is
characterized by a steady, outwards—directed jet along the centerline of the oriﬁce. Two
re-circulating-ﬂow regions just outside the edges of the oriﬁce surround this jet. Ingard
and Labate (1950) pointed out that no mass exchange takes place between the inside and
outside of the cavity in region 1. Therefore, the streaming motion consists entirely of the
ﬂuid from outside the resonator that is entrained into the jet.

On the other hand, a signiﬁcant regime for the operation of the resonator is that at
very high sound-intensity levels, or region 4. In this regime, formation of vortex
structures takes place at the oriﬁce during the blowing phase of the unsteady jet. These
vortices move away from the oriﬁce under the action of self-induced velocity, and hence
produce a jet ﬂow. Unlike region 1, however, this jet consists of the vortex structures (in
addition to the entrained ﬂuid), which originated from the ﬂuid within the resonator’s
cavity. Therefore, the ﬂow corresponding to region 4 involves mass exchange between
the inside and outside of the cavity according to Ingard and Labate (1950). Of course, the
mass outﬂow from the cavity must be balanced by the mass inﬂow to the cavity in a
given oscillation cycle. Otherwise, the resonator would become a material “source” or
“sink.” However, Glezer and Amitay (2002) contradicted that ﬁnding, suggesting that no
mass exchange took place with vortex formation. Glezer and Amitay (2002) pointed out

that the jet’s unique feature formed entirely from the working ﬂuid of the ﬂow system.

68

In recent history, utilization of jets with characteristics similar to those
corresponding to region 4 has become wide spread. These jets, which are known as
“synthetic jets”, are produced by devices, which are quite similar to the driven Helmholtz
resonator, except the driving mechanism is internal rather than external. The use of
internal driving mechanisms, such as a piezoelectric membrane, compression driver, etc.,
has been favored over an external sound ﬁeld due to the extremely loud sound required in
the latter case to produce a region 4 behavior.

A fairly comprehensive review article of synthetic (zero mass ﬂux) jets is that by
Glezer and Amitay (2002). These authors point out that two of the key non—dimensional
parameters characterizing the behavior of synthetic jets are the Reynolds number and
non-dimensional stroke length. The former parameter may be deﬁned utilizing the peak
or average blowing velocity, or the jet impulse during discharge. The non-dimensional
stroke length provides a measure of the distance traveled by a ﬂuid element during the
ejection phase relative to a characteristic dimension of the oriﬁce.

The Reynolds number inﬂuence primarily relates to the asymmetry of the ﬂow
ﬁeld during the ejection and suction phases. Speciﬁcally, at very low Reynolds numbers
(creeping ﬂow), the ejection ﬂow does not separate from the oriﬁce, and the ﬂow during
the ejection and suction phases will look like that of a “source” and “sink”, respectively.
As the Reynolds number increases, the “source” ﬂow separates from the edges of the
oriﬁce, forming a jet that becomes much narrower, and directed along the oriﬁce’s
centerline, than the “sink” ﬂow.

On the other hand, the non-dimensional stroke length seems to be the primary

parameter inﬂuencing whether or not periodic vortex-ring formation takes place.

69

Speciﬁcally, if the distance traveled by a ﬂuid particle during the ejection phase of the
cycle is too small in comparison to a characteristic oriﬁce scale, one would not expect the
ﬂuid mass to travel too far from the resonator. That is, the ejected mass is likely to be
drawn back again into the device during the suction phase. Under these conditions, no
mass exchange takes place between the cavity and outside, and region 1 or 2 behavior is
observed. However, if the non-dimensional stroke length is too large, ﬂuid mass will be
ejected from the cavity during the blowing phase, leading to region 4 behavior and
periodic formation of vortices.

More recently, Wu and Breuer (2003) pointed out that the non—dimensional stroke

length is essentially equivalent to the inverse of the Strouhal number, given by:

St .1 = (3.8)

 

Umax

where a) is the driving angular frequency, dis the jet-slit width, and Um is the maximum
jet velocity during the ejection phase. Using PIV measurements, they documented the
velocity ﬁeld of a rectangular synthetic jet driven by internally mounted miniature
speakers. Results were shown for two different cases corresponding to a Reynolds
number (Red) of 15 and 62 (based on d and Um) and Std values of 4 and 1, respectively.
The lower Reynolds number case exhibited practically similar ﬂow ﬁelds during the
ejection and suction phases (with the exception of reversal of the ﬂow direction). The
higher Reynolds number case (and lower Strouhal number/higher non-dimensional stroke
length) exhibited a high degree of asymmetry between the suction and ejection phases.

This was associated with a ﬂow velocity at and near the centerline of the jet that was

70

always directed away from the oriﬁce (i.e., in the ejection direction). However, the
Strouhal number was too large (or, the stroke length was too small) for vortex formation
to take place.

The velocity-ﬁeld measurements of Wu and Breuer (2003) at the higher Reynolds
number seems to correspond quite well with region 1 behavior described above.
Interestingly, the non-dimensional parameters of their jet are of the same order as found
in the current study. In particular, for the strongest sound used here during resonator
characterization under harmonic excitation, the Reynolds number (Red) and Std were
nominally 29.9 and 2.1 respectively. Based on this comparison, it is unlikely that
periodic vortex formation occurs in the present case under periodic driving conditions.
Under amplitude-modulated conditions, higher instantaneous sound amplitudes were
utilized, leading to peak instantaneous velocity values that are approximately twice that
of the strongest harmonic condition. The corresponding peak, instantaneous, Red ~ 60
and Std ~ 1, are almost identical to the strongest forcing case of Wu and Breuer (2003).
This, again, suggests that no unsteady vortex formation takes place here. This suggestion
should be treated with caution, though, since the jet is operating under unsteady,
amplitude-modulated conditions, which is different from the steady, harmonic excitation
of Wu and Breuer.

To illustrate further that no mass exchange between the cavity and outside is
expected to take place for the resonator used here, consider a slightly different form of

the Strouhal number:

71

l

51 =——— 3.9
’ (T/2)Um.g ( )

where T is the period of the forcing signal, I is the length of the resonator’s neck and Um.g
is the average velocity during the blowing stroke. The denominator in equation 3.9 gives
the distance traveled by a ﬂuid particle on the centerline of the jet during the blowing
stroke. Calculation of St, for the strongest jet velocity encountered here under harmonic
forcing conditions (order of 1 m/s) yields a value of nominally 8.1. This shows that
during the ejection stroke the particle travels a distance that is almost an order of
magnitude smaller than the length of the resonator’s neck. This provides additional
support to the idea that the current ﬂow-excitation device does not involve mass
exchange between the cavity and outside. The following section provides ﬂow-

visualization and hotwire-measurement results that are consistent with this statement.

3.2.2 Flow visualization of the ﬂow ﬁeld in the vicinity of resonator’s opening

Four sequences of ﬂow-visualization snapshots corresponding to four forcing
conﬁgurations under no-ﬂow condition are to be represented in the following order:
unforced-ﬂow case, sine-wave-forcing cases (at two different forcing amplitudes), and
arnplitude-modulation-forcing case. The aim of this ﬂow-visualization analysis is to
examine the consistency of the behavior of the resonator’s ﬂuidic disturbance with the
assessments made in section 3.2.1, and attempt to understand the nature of the ﬂow
behavior in the vicinity of the resonator’s oriﬁce.

The smoke-wire technique was employed for ﬂow-visualization. The set-up

involved the use of a couple of short-length, 0.1 mm-diameter, stainless steel wires

72

(“smoke wires”) that were mounted on two hotwire prongs, with wires lined-up in
parallel to each other. The use of two wires helped in the retention of an oil ﬁlm when
the wires were wetted with oil for model-train smoke. The wetted wires were then
positioned immediately above the resonators’ slit opening (see Figure 3.10a). A
homemade smoke-wire, controller box was used to initiate the smoke generation and
capturing of images using a CCD camera at standard video rate of 30 frame/s. This was
accomplished through ﬂipping of a toggle switch on the controller-box panel, which
immediately actuated a solenoid to feed electrical current through the smoke wire for
heating and subsequent evaporation of the smoke, for a period of time that is adjustable
through a timing knob in the controller box. Simultaneously, ﬂipping of the toggle
switch initiated a 0 — 5 V output-signal that was connected to the trigger-signal input of
the image grabber used for capturing the output of the CCD camera. Thus, the process of
image capturing and smoke generation commenced simultaneously. Given the initial
thermal transient associated with heating and evaporation of the oil, a typical test allowed
for capturing of enough number of images that covered the duration before, during and

aﬂer the generation of smoke.

3.2.2.1 Buoyancy effect (unforced condition)

Given that the heated smoke will have lower density than the surrounding air, it is
evident that, especially in the absence of ﬂow external to the resonator, the visualization
technique could introduce a signiﬁcant, buoyancy-induced, ﬂow pattern that is not related
to the actual ﬂuidic disturbance produced by the resonator. Therefore, it was desired to

initially capture images of the ﬂow pattern resulting from pure buoyancy effects; i.e.,

73

without driving the resonator. The resulting sequence of images is presented in Figures
3.10a-f, where the ﬁrst image (Figure 3.10a) was taken almost immediately after
triggering the smoke wires. These images are provided for reference in order to
demonstrate that subsequent observations, obtained when the resonator is actuated, are

not an artifact of buoyancy effects.

Smoke

/ wires

(:1)

 

Figure 3.10a-f. Smoke-wire images of the buoyancy-induced ﬂow under unforced
condition

3.2.2.2 Sine-wave forcing effect

For this case, a pure harmonic wave at frequency f z 630 Hz was used to drive the
speaker at two different forcing amplitudes: one representative of the forcing levels used
in this study, and the other at a level that is well above any employed here. The latter
was employed in order to examine if the resonator could be operated in the regime where
periodic vortex ejection takes place; i.e., similar to a synthetic jet (or region 4 of Ingard

and Labate, 1950).

74

3. 2. 2. 2.1 Forcing amplitude representative oﬂhose used in present study

A sequence of twelve images is displayed in Figures 3.11a-l for sinusoidal forcing
at RMS acoustic pressure value of 4.63 Pa at the resonator opening. The most signiﬁcant
point concerning the images in Figure 3.11 is that, consistent with the earlier projection
based on the analysis in section 3.2.1, there is no evidence of periodic formation of
vortices. Additionally, video observations of several recordings similar to that shown in
Figure 3.11 always revealed steady ﬂow of smoke in the upward direction. This provides
evidence for the establishment of streaming ﬂow. This ﬂow should not be an artifact of
the smoke-Wire’s buoyancy effects, given the big difference in the observed smoke
pattern between Figures 3.11 and 3.10. Of course, this does not imply that buoyancy
effects are negligible all together in the images shown in Figure 3.11. In fact, it is
believed that these effects are responsible for the meandering (ﬂame-like) appearance of
the smoke.

Another interesting point concerning the results in Figure 3.11 is the formation of
what appears to be two re-circulating ﬂow regions at the edges of resonator’s slit (most
evident in Figure 3.11j and 3.11k). The reader is cautioned, though, that these regions are
difficult to identify in the picture, since they are very small and are not resolved properly
in the given view. However, if correct, this would be consistent with the expected ﬂow

behavior in “region 1” of the Ingard and Labate (1950) study (see section 3.2.1).

75

 

Figure 3.11a-l. Smoke-wire images of the flow above the resonator’s slit under
sinusoidal forcing conditions at a level representative of that used to obtain
data in this study

3.2.2.2.2 Hi hintensi orcin am litude

The next sequence of images (Figures 3.12a-i) represents sinusoidal forcing at a
level higher than any employed for quantitative measurements in this study. It is clear
that under these forcing conditions the Strouhal number (or stroke length), see section

3.2.1, is sufﬁciently small (large) for periodic vortex formation to take place.

76

Additionally, observations of the temporal evolution of the image sequence shown in
Figure 3.12, and other similar sequences, reveal the existence of strong streaming ﬂow.
The behavior of the resonator disturbances is consistent with that of region 4 of the

Ingard and Labate (1950) study.

(d)

 

Figure 3.12a—i. Smoke-wire images of the flow above the resonator’s slit under
sinusoidal forcing conditions at a level stronger than that used to obtain data in
this study

77

3.2.2.3 Amplitude-modulation forcing effect

The last sequence of images (shown in Figures 3.13a-o) demonstrates ﬂow
visualization of the resonator’s ﬂow ﬁeld under the strongest amplitude-modulated
forcing used here. The forcing was achieved using a nominal carrier frequency, fc, of 630
Hz and a modulation frequency, fm, of 18 Hz.

The images in Figure 3.13 reveal periodic modulation in the width of the smoke
pattern. It is presumed that this modulation takes place at twice the modulation
frequency, which corresponds to the repetition of the cycle of build up of strong forcing
amplitude followed by decay to zero forcing. In the absence of existing literature
concerning the ﬂow structure of amplitude-modulated, synthetic jets and driven
resonators, it is difﬁcult to rely on the qualitative ﬂow visualization results shown in
Figure 3.13 to clarify the underlying ﬂow physics. A simple-minded possibility for the
observed modulation in the smoke width may relate to the spread rate of the jet
associated with the streaming ﬂow. During the modulation cycle, the jet is likely to
spread more widely at low forcing levels because of the lower Reynolds number of the jet.
At higher level, a narrower and faster jet will be produced. Periodic repetition of this
cycle would then cause the jet’s width to be modulated, as seen in the images. The actual
ﬂow behavior is likely to be substantially more complex than offered by this “quasi-

steady-ﬂow” description.

78

 

Figure 3.13a-o. Smoke-wire images of the ﬂow above the resonator’s slit under
amplitude-modulated forcing conditions at highest level of forcing

79

3.2.3 Hotwire measurements of the resonator’s velocity ﬁeld

To quantify the velocity disturbance produced by the Helmholtz resonator, a
single hotwire sensor was utilized. The adoption of the hotwire for the velocity-
measurement tool was motivated by the simplicity of its use, and its good temporal and
spatial resolution. However, the hotwire suffers from two important shortcomings that
limit the interpretation of the measurements. First, it is directionally blind, so it cannot
distinguish between the suction and ejection velocity. Second, it is equally sensitive to
the lateral (x) velocity component, as it is to wall-normal (y) velocity component (see
Figure 2.14 showing the x and y axes). Therefore, the hotwire sensor actually measures q,

or the total velocity component in the x-y plane, given by:

q=\/u2+v2 (3.10)

where u and v are the velocity components parallel to the x and y directions, respectively.
Note that the azimuthal velocity component is ignored here since: (1) it has an order of
magnitude lower cooling effect on the hotwire; (2) the resonator disturbance is
predominantly two-dimensional as demonstrated earlier from the velocity spectra (see
Figure 3.3). Finally, for the remainder of this work, q will be frequently decomposed into
the sum of a time average component Q, and an unsteady component q'.

Although the above discussion points out important limitations concerning the
information that can be extracted from the single-wire measurements, the data can be
used to reliably quantify the excitation level of the ﬂow. This can be done for instance by

calculating the relative magnitude of qmax to the freestream velocity (a more speciﬁc

80

deﬁnition will be introduced in Chapter 4). Additionally, q information may be
combined with physical reasoning to infer certain characteristics of the measured velocity
ﬁeld. These can then be examined against the nature of the ﬂuidic disturbance predicted
from analysis of the literature in section 3.2.1 and the ﬂow visualization results in section
3.2.2.

Figure 3.14 provides a plot of the phase-averaged q (<q>) measured at y = 0
above the slit exit for one cycle of forcing at a nominal frequency of 630 Hz. The phase
average is obtained relative to the phase of the forcing signal. For reference, the latter
signal is included in Figure 3.14 in a separate plot. The different-color lines in the ﬁgure
correspond to measurements at different x locations across the slit. Note that x = 0
corresponds to the “downstream” edge of the slit (see Figure 2.14). The driving acoustic
pressure for this particular data set was 43.1 Pa peak-to-peak.

The measurements indicated by the red line correspond to a location of the
hotwire sensor that is about 0.05 mm upstream of the slit (or x = -0.55 mm). Thus, the
wire is in the immediate vicinity of the test-model’s surface, which suggests that the
measurements primarily correspond to the u, or wall-parallel, velocity component. The
shape of the red line corresponds to a ‘rectiﬁed-like’ sinusoidal variation with two
velocity peaks per cycle. Presumably, these peaks correspond to the peak suction and
ejection phases; although it is not clear which is which. During the ejection phase, one
would expect that measurements at the location corresponding to the red line should
reﬂect the velocity of the entrainment ﬂow towards the jet. In addition, during the
suction phase the velocity will also be directed towards the centerline as the suction effect

draws ﬂuid mass towards the oriﬁce. Therefore, it is likely that the rectiﬁed waveform

81

seen at x = -0.55 mm is in fact reﬂective of ﬂow towards the jet for both the suction and
ejection phases.

Consideration of the hotwire traces close to the slit’s center helps to distinguish
the suction from the ejection phase. For instance, at x/d = -0.5, one can still identify two
local peaks per cycle in the hotwire signal. Since the jet ﬂow is substantially narrower
than the sink (suction) ﬂow, the velocity should be higher during the ejection phase.
Therefore, the peak of the ejection cycle is identiﬁed with the larger peaks in the hotwire
trace (see arrows shown in the ﬁgure). On the other hand, the peak of the suction cycle is
associated with the smaller velocity peak at x/d = -0.5. The velocity direction
corresponding to this peak cannot be determined without ambiguity. The high-Reynolds-
number synthetic jet of Wu and Breuer (2003) always sustained positive velocities, even
at the peak suction phase, on the centerline of the slit. In their case, however, the velocity
trace had a local minimum at the peak suction phase. This is unlike the results found here
where a local peak, albeit weak, is identiﬁed. Thus, in this study it is possible that the
ﬂow actually reverses direction for part of the suction cycle and/or the peak may be
produced by an instantaneous cross-stream (wall-parallel) component.

Another piece of information that is depicted from the results in Figure 3.14
relates to ﬂow symmetry relative to the center of the slit. Data measured at x locations
that are symmetric relative to the center of the slit are generally different. For example,
the velocity signal at x/d = -0.7 and -0.3 exhibit large differences during the suction phase.
At x/d = -1.1 and 0.1, substantial differences are seen for the ejection phase. This ﬂow
asymmetry should not be too surprising given that the geometry of the resonator is not

symmetric relative to the slit’s center.

82

 

 

[E

x
+__

 

 

 

 

 

 

 

 

+—0.9
+ -0.7
‘ +-o.5
——-O.3
- 01
Peak suct1on
+0.1
0 l l l l l l
0 45 90 135 180 225 270 315 360
phase (degree)
0.3 1‘ ,1»..- —
x", \\
.I' \
o 2 ~ / \ —
l; \\
0 1‘5 I \\ _
g / .
2 ,4
'5 o - f1" -
-5 '3. ,./
.§ -0.1 _ ‘\ // d
“\ /
-0 2 — \\ // -
x- l
03 ~ ‘\~«/ —
l I 1 l l 1 l
0 45 90 135 180 225 270 315 360
Phase (Wee)

Figure 3.14. Phase-average of hotwire output at different x locations and y = 0 (top);
forcing signal (bottom)

Some of the above observations may also be seen more easily in Figure 3.15,
which provides plots of the velocity proﬁles across the slit opening of the resonator taken

at the peaks of both the suction and ejection phases.

83

 

0.9r _

<q> (m/S)

.0
O)
I

l

 

 

   

 

 

 

 

 

-0.8 -0.6 -0.4 41.2
x/d
1 I I I I I I
o.9~ -
0.8— -
15 0.7 — 4
E,
A
6’ 0.6— 3
0.5» _
l l l 1 La
-O.8 -0.6 -0.4 41.2 0 0.2
x/d

Figure 3.15. Velocity proﬁles across the resonator’s slit opening during peak
ejection phase (t0p) and peak suction phase (bottom)
The next analysis is concerned with the inﬂuence of the driving-sound level on
the generated hydrodynamic disturbance. To this end, velocity data were sampled with
the hotwire sensor located at the center of, and just above the resonator’s slit (x/d = -0.5

and y/d = 0). The forcing amplitude was varied to produce ﬁve different levels of RMS

84

acoustic-pressure ﬂuctuations at the resonator’s opening: 4.65, 2.71, 4.63, 5.83 and 7.68
Pa. The corresponding hotwire signals are displayed in Figure 3.16 for two forcing
cycles. Note the displayed signals have been phase—averaged, and therefore the peaks

and valleys in the results for the different forcing levels occur at the same time.

0.35 1 1 r 1 1 1

 

0.3L

 

0.25~

 

 

0.2~

 

<q> (m/S)

 

 

 

l l l

l
0 0.5 1 1.5 2 2.5 3 3.5
time (milliseconds)

 

Figure 3.16. Conditional average of the hotwire output for different forcing levels
(data obtained at x/d = -0.5 and y/d = 0)

Two primary effects of forcing level can be depicted from Figure 3.16. Not
surprisingly, as the forcing amplitude is increased, the ﬂuctuating velocity component
associated with the jet ensuing from the resonator becomes more signiﬁcant. More
interestingly, however, is the vertical shift in the mean (“DC” component) of the
measurements that becomes progressively stronger with increasing forcing level. This
provides evidence that the purely sinusoidal acoustic forcing imposed by the speaker

produces not only a sinusoidal ﬂuidic disturbance of the ﬂow, but also a steady

85

component as well. These observations, although seem counter intuitive, they are
consistent with the ﬁndings of Ingard and Labate (1950) discussed in section 3.2.1 and
the ﬂow visualization results presented in section 3.2.2 earlier.

The formation of the mean motion is essentially reﬂective of the non-linearity of
the resonator at large levels of driving acoustic pressures. As explained in section 3.2.1,
this mean motion is formed entirely from entrained ﬂuid from outside the cavity at high
Strouhal numbers, but its motion does contain ﬂuid from the cavity at low Strouhal
numbers. In the latter case, the ﬂuid ejected from the cavity must be replaced during the
suction stroke since the net average mass ﬂow rate through the resonator must be zero.
Given the qualitative similarity of the hotwire signal shapes in Figure 3.16 for all forcing
levels that extend down to the weakest level, it is apparent that no change in the nature of
the resonator’s ﬂuidic disturbance takes place as the forcing level is increased. Moreover,
it is also most likely that the resonator’s operating regime corresponds to the weakest-
forced regime of Ingard and Labate (1950) since the lowest forcing level used here
corresponds to a quite low sound pressure level of 98.1 dB. In fact, St1(see Equation 3.9)
for the lowest forcing level is 39.9, which provides further support that no mass exchange
takes place between the inside and outside of the resonator’s cavity in this study, and that
the streaming motion is formed from ﬂuid entrained from outside the cavity. This is also
consistent with the ﬂow visualization results in Figure 3.11, where no evidence of
periodic vortex formation was found.

Finally, the dependence of the ﬂuidic disturbance on the distance from the
resonator’s exit was investigated. This was accomplished by locating the hotwire at the

center (x/d = -O.5) of the resonator’s slit and different heights in the range of y/d = -O.ll

86

to y/d = 1.89. For all y locations, the driving acoustic pressure was kept at a level of 16.7
Pa. The resulting phase—averaged velocity traces can be seen in Figure 3.17 for the
duration of two forcing cycles. As expected, the strength of the streaming and unsteady
disturbance velocity decreases as the distance between the hotwire sensor and the slit

opening increases.

 

       

 

 

 

0‘3 1 T T I l I
0.25~ A
0.2 r _
E
E.
A 0.15 _
O- 1
v \ ,
0.1 K \kﬁ—V —
0.05 ,r- *1 «1.-.,» _
,1/ ”53:7.— .\__ “‘ *—g “ ‘__ -..
zit-tiﬁinﬁm “33:3:- "— g
e l l | | l
0 0.5 1 2 5 3 3 5

1.5 2
lime (millisecond)
Figure 3.17. Conditional average of hotwire output at 11 different y—positions (y/d =

-0.11, 0.09, 0.29, 0.49, 0.69, 0.89, 1.09, 1.29, 1.49, 1,69, and 1.89 from top to
bottom)

87

3.3 Resonator’s disturbance characteristics under modulated forcing conditions

As discussed in the introduction (section 1.2), periodic excitation of separated
ﬂows near the separation point is most effective if the excitation frequency is selected

such that the reduced frequency F, deﬁned as follows, is of order 1:

 

F“ = {19L (3.11)

where, ﬂ, is the excitation frequency of the ﬂow, L is a length scale characteristic of the
separation bubble size (typically taken as the reattachment length, x,, for
separating/reattaching ﬂows), and U... is the freestream velocity. For the present ﬂow
ﬁeld, an F+ = 1 corresponds to a frequency of 67 Hz. Clearly, this is an order of
magnitude lower than the resonant frequency of the Helmholtz resonator, and hence
simple driving of the resonator at its resonance frequency is not anticipated to lead to
effective control of the separation bubble. On the other hand, if one operates the
resonator directly at the desired low frequencies, the resulting ﬂuidic disturbance is
extremely weak and incapable of exciting the ﬂow (e.g., see Figure 3.3, where the
disturbance velocity is indistinguishable from the hotwire anemometer’s background
noise for frequencies below 400 Hz).

To remedy the mismatch between the resonator’s operating frequency and the
effective ﬂow forcing frequency, it was decided to modulate the amplitude of the main
resonator’s driving, or carrier, acoustic signal (with frequency fc = 630 Hz) at the desired
low frequency (fm). In order for this approach to be successful, a non-linear coupling

mechanism is required between the input acoustic disturbance and the ﬂow ﬁeld.

88

Speciﬁcally, modulation alone will result in exciting the ﬂow at two different frequencies
simultaneously: (fc—fm) and (fc + f,,,) as described in section 2.1.3.5. Since the modulated
frequency is much smaller than the carrier frequency (fm << fc), these two new
frequencies do not deviate substantially from the high operating frequency and would not
produce the desired result. However, if a non-linear coupling mechanism exists between
the modulated acoustic signal and the ﬂow, then frequencies corresponding to various
sums and differences of (fc — fm) and (ﬂ. + fm) will be generated. In particular the
difference between these two frequencies produces a sinusoidal signal at a frequency of
2f,,,, which is much lower than fc.

For this study, the Helmholtz resonator, when driven at high sound intensities,
provides the necessary non-linear coupling mechanism. As demonstrated earlier, the
non-linearity of the resonator results in the formation of a streaming disturbance
component when driven using a pure harmonic signal. Since the magnitude of the
streaming disturbance increases with the amplitude of the harmonic signal, then
amplitude modulation of the latter will result in a time-varying magnitude of this
streaming disturbance at a frequency of 2f,,,. It is this sinusoidal disturbance that should
couple to the separated ﬂow effectively. Consequently, it is likely that the ﬂow response
to this type of amplitude-modulated forcing is similar to direct harmonic excitation at
frequency of 2f,,, (although this statement requires veriﬁcation via measurements of the
ﬂow ﬁeld under forced conditions). This is to be distinguished from other types of
amplitude-modulated frequency, such as that by Wiltse and Glezer (1993), where the

non-linear response of the ﬂow to high-amplitude forcing by piezoelectric actuators, is

89

argued to provide the coupling mechanism between the amplitude-modulated excitation
and the ﬂow.

The range of modulation frequencies employed here is fm = 10 - 40 Hz. As will
become clear in the following chapter, an optimal (largest ﬂow change for a given
forcing level) modulation frequency of 181Hz (note that actual ﬂow excitation frequency
is 12 = 2f,,, = 36 Hz) was found. Therefore, the main focus of the discussion in this section,
and the remainder of the thesis, will be on cases corresponding to 10 Hz and 18 Hz
modulation frequencies. The ﬁrst case provides an example of forcing that produces
hardly any change to the ﬂow, whereas the latter is where substantial ﬂow changes are
observed.

Figure 3.18 provides plots of the imposed acoustic pressure at the resonator’s slit
for the two different forcing levels employed with 10 Hz modulation frequency. It can be
seen that good modulation quality (amplitude change ranging from full amplitude down
to zero) is attained. This was possible because of the use of two independent function

generators to synthesize the amplitude-modulated signal as described in section 2.1.3.5.

90

 

 

 

 

 

l I | l l l l

l l l
20 30 4O 50 60 70 80 90 100 1 10 120
time (milliseconds)

 

Figure 3.18. Sample of the driving acoustic signal at 10 Hz modulation frequency
and two forcing levels

The velocity measurements at the exit of the slit (x/d = -0.5 and y/d = 0)
corresponding to the excitation sound signals shown in Figure 3.18 are shown in Figure
3.19. In order to distinguish the lower-frequency modulation from the higher-frequency
signal, a low-pass ﬁlter was used in the data processing. The cut-off frequency of the
ﬁlter was set to 200 Hz. The resulting ﬁltered signal is shown using a black line in
Figure 3.19. It is clear that the black line “traces” the low-frequency component (i.e.,
modulated streaming ﬂow) that is employed to affect the ﬂow. Also, note that, as
discussed above, two cycles of the low frequency forcing are realized during a single
modulation cycle (100 milliseconds for 10 Hz). This means that the actual frequency

affecting the ﬂow is in fact twice the modulating frequency, i.e., 20 Hz.

91

 

 

 

 

 

d
I

0 (m/s)

.JHIHHH M“

0.5— (will ”MM” W11?”

“HHHIHIH‘ ‘

A l l l l l l l 1 I
”)0 30 4o 50 60 7o 80 90 100 110 120
time (milliseconds)

1,4

 

lI

 

 

Figure 3.19. Velocity measurements at x/d = -0.5 and y/d = 0 corresponding to the
amplitude-modulated acoustic forcing shown in Figure 3.18

Similar results to those shown in Figures 3.18 and 3.19, but for 18 Hz modulation
frequency are shown in Figures 3.20 and 3.21. In this case, ﬁve different forcing levels
were employed (since this was found to be the optimal modulation frequency, as will be
discussed in Chapter 4). The corresponding driving acoustic pressure signals can be seen
in Figure 3.20. The peak-to-peak values of these signals, from the weakest to the
strongest are 10.6, 24.9, 33.9, 49.2, and 69.8 Pa (shown using cyan, magenta, green, blue,
and red colored lines, respectively). The resulting velocity traces of the ﬂuidic
disturbance at x/d = -O.5 and y/d = O are shown in Figure 3.21, along with the low-
frequency component highlighted using a black line. The details of these velocity traces
may be seen more clearly from Figure 3.22, where “magniﬁed” plots of the traces for the

two, largest, forcing amplitudes are given.

92

 

 

 

 

 

 

 

 

 

  

13 20- 20» «
“P-zo- -20 ‘
10 20 3b 4b 5b 6b 10 ’ 2b 3b 4b 50 6b
a 20 f ‘ 20 '
a _ n n l . v i l
v” 0 41!le \AANV WAN) JVVVV‘V mum WMAM 0
°--20» < -20- 1
2o ab 4o 56 so 76 1b 2b 3b 46 so
. , . time (milliseconds)
A 20 ' . ‘
(g 0 ,va 1. ([1511 ll 1 ‘1" ii. (1 1‘ it 1" V.‘ MM ,1 1,1,1”) Jq l 1 if) 1. fl NM
a.“ _ ”ilsllll'll“‘ “WNW“ _

 

 

 

1b 26 3o 40 so
time (milliseconds)

Figure 3.20. Sample of the driving acoustic signal at 18 Hz modulation frequency
and five forcing levels

 

 

 

 

 

 

 

 

 

ll
(1 j'l‘i
“ll ‘.

 

30 40 so
time (milliseconds)

 

Figure 3.2]. Velocity measurements at x/d = -0.5 and y/d = 0 corresponding to the
amplitude-modulated acoustic forcing shown in Figure 3.20

93

 

0 (m/s)

 

 

 

2” I lllll
50.. llllllll‘ll. I‘lllll

1025 30 35 40
time (milliseconds)

 

5345;. ‘

 

 

Figure 3.22. Magnified velocity measurements at x/d = -0.5 and y/d = 0
corresponding to the two largest amplitude-modulated acoustic forcing shown
in Figure 3.20
A quantitatively better way of representing the input pressure and resulting
excitation velocity relationship is by plotting the RMS of the low-pass-ﬁltered velocity
against the input, peak-to-peak pressure. Such plots are done for the two forcing levels

employed at 10 Hz modulation frequency, and the ﬁve forcing levels used at 18 Hz

modulation frequency. The plots can be seen in Figure 3.23.

94

 

 

 

 

 

 

 

 

I I I l I T
0‘3” O 10 Hz i A
I 18 Hz
0.25— s
O
0.2~ s
’6? I
E:
E 0.15" ‘i
O
O I
0_1~ s
0.05» I —
0 - l I l l l l
10 20 3O 40 50 60 70
pa“, (Pa)

Figure 3.23. RMS of the low-frequency-disturbance velocity versus the input peak-
to-peak of the driving acoustic pressure

It is also important to examine the harmonic content of the excitation (low-
frequency) signal. To this end, the power spectrum of the low-pass ﬁltered velocity
component in the case of 18 Hz modulation frequency is obtained. A sample of this
spectrum for peak-to-peak acoustic pressure of 49.2 Pa is provided in Figure 3.24. Two
different plots are included in the ﬁgure: the top plot (Figure 3.24a) displays the spectrum
using logarithmic scale, and the bottom plot applying linear ordinate scale. The former
scale indicates certain amount of harmonic distortion, where a second harmonic of the
excitation frequency is evident at 72 Hz. However, this distortion is practically
negligible, as evident from the dominance of the peak at 36 Hz. In fact, the second

harmonic peak cannot be detected altogether in the semi-log plot.

95

 

 

 

 

 

 

 

 

    

10‘ 102 10
Frequency (Hz)

Figure 3.24. A sample power spectrum of the low-frequency ﬂuidic disturbance
associated with 18 Hz modulation frequency: (a) logarithmic scale, (b) linear
ordinate scale

Finally, as indicated in the motivation section, one of the signiﬁcant aspects of the
ﬂow-excitation device developed here is that it is acoustically quite at the excitation
frequency. More speciﬁcally, the above discussion indicates that the acoustic input used
to drive the speaker will occur at frequencies of (fc + fm) and (ﬂ — fm), which are
nominally at 612 and 648 Hz for a modulation frequency of 18 Hz. Under these
conditions, the actual excitation to the ﬂow takes place via a ﬂuidic disturbance at a
frequency of 36 Hz, or 2f,,,. Thus, it is evident that the major sound contamination from
the driven-resonator is at a much higher frequency than the excitation frequency, and
hence this contamination could be removed easily via low-pass ﬁltering. This point is
demonstrated using the power spectrum results in Figure 3.25 of the acoustic pressure
measured at the resonator’s lip, under no-ﬂow conditions, for the two lowest forcing

levels and modulation frequency of 18 Hz. Note that only the two lowest forcing levels

96

 

are examined here since no wall-pressure measurements were conducted for higher
forcing levels (because of a dynamic-range limitation of the surface-pressure sensors, as

discussed in Chapter 4).

 

pa pl
8.
l

 

 

 

 

 

 

 

JLJLJ l l 1 111.111 1 1 1 llllll

 

1 102
Frequency (Hz)

10

Figure 3.25. Sample power spectra of the acoustic pressure associated with 18 Hz
modulation frequency and two lowest forcing levels: (a) 10.6 Pa peak-to-peak,
(b) 24.9 Pa peak-to-peak

The spectra in Figure 3.25 show that no pressure disturbances are detected at 2fm,
or 36 Hz, at the lowest forcing level. A very weak disturbance is seen at 36 Hz for the
higher forcing level. This is likely related to the hydrodynamic pressure ﬂuctuations of
the oscillating jet, given that the measurements were conducted in the immediate vicinity
of the resonator’s slit. Moreover, the pressure ﬂuctuations energy contained in this peak
is equal to only 1.6% of the peak, hydrodynamic surface-pressure ﬂuctuations measured

here. It is noted, though, that we believe that the driven resonator is acoustically quite at

forcing levels typical of those used here. At substantially higher forcing levels, the

97

oscillating jet velocity could become sufﬁciently high such that the unsteady ﬂuidic

disturbance itself may become an efﬁcient noise generator.

98

4 EFFECT OF FORCING ON THE SEPARATED FLOW

In the following sections, an analysis focused on assessing the inﬂuence of
forcing on the axisymmetric back-step ﬂow will be presented. In the analysis, two issues
are of speciﬁc interest. First, it is desired to assess whether amplitude-modulated forcing
using the externally driven Helmholtz” resonator is capable of altering the global
characteristics of the separation bubble. To this end, the mean wall-pressure data beneath
the bubble will be examined under different levels and frequencies of forcing. It is also
desired to learn if alteration to the mean wall—pressure ﬁeld characteristics caused by the
amplitude-modulated forcing is similar to (or different from) those obtained under the
more conventional, periodic way of ﬂow control. The second issue to be investigated
here is the effect of forcing on the unsteady wall-pressure measurements. This study is
preliminary and is primarily focused on the effect of forcing amplitude and modulation
frequency on the RMS pressure ﬂuctuations, the corresponding Power Spectral Density
(PSD), and the frequency-wavenumber (f-kx) representation of the spatio-temporal

characteristics of the unsteady wall-pressure ﬁeld.

99

4.1 The effect of forcing on the mean-pressure distribution

As indicated in Chapter 2, mean-wall-pressure measurements were acquired

downstream of the Helmholtz resonator at a Reynolds number (Rey) of 2525 based on the

step height. These measurements were done for the unforced as well as forced ﬂow

conditions at different levels and frequencies of forcing. For each of the different forcing

conditions, the measurements were done ﬁve times to ensure repeatability of the data.

The average of these ﬁve data sets is then used to calculate the mean-pressure coefﬁcient

(Cp), given by Equation 2.4, which is plotted along the ordinate versus the normalized

streamwise coordinate (x/H) in Figure 4.1 for the unforced case. The different colors

represent the ﬁve different runs and the average of these ﬁve runs is shown in orange.

0.09

 

0.06 -
0.03 -
0.00

 

 

 

 

 

a -o.03 .2 ,,
-0.06 - 2
-009 4.

-0.12 1

 

-0.15

 

 

 

 

 

 

 

xIH

Figure 4.1. Mean-pressure distribution under no forcing condition

The mean-pressure distribution in Figure 4.1 is consistent with the classical

backward-facing-step, wall-pressure proﬁle, exhibiting an initial decrease to a minimum,

followed by recovery through reattachment to a peak before gradually decreasing to

100

approach the freestream pressure. In the study of Hudy (2005), the peak location was
found to be nominally 1.5 step heights downstream of x,. For the unforced case in this
study, the peak value is found at 5.5H. Thus, the corresponding x, value is 4H. It is
important to note that this is an approximate estimate and that a precise value can only be
found from wall-shear-stress measurements. Nevertheless, the estimate compares well
with data from the study of Li (2004) who did conduct wall-shear measurements on the
same test mode] employed here. This comparison is shown in Figure 4.2, which contains

a plot of the reattachment length normalized by the step height versus Re”.

14000

 

e Li(2004)h Q
12000 0 Present data

10000

8000

6000

4000 O

 

 

2000

 

3.8 4.0 4.2 4.4 4.6 4.8 5.0
xrlH

Figure 4.2. Reattachment length estimate from the present study and Li (2004) data

As evident from the plot, the estimated x,/H value reasonably follows the
Reynolds number trend of Li’s data. This estimate is used in subsequent analysis for the

calculation of the non-dimensional forcing frequency:

101

F" zgfifi (4_1)
U00

Prior to discussing additional results, it is also signiﬁcant to note that the level of
forcing is expressed here using a non-dimensional momentum coefﬁcient (C y). The
deﬁnition of C y, which is the most commonly accepted form for expressing the forcing
level for periodically excited ﬂows (e.g., see Greenblatt and Wygnanski, 2000), is given

by:

2du- 2
Cﬂ = ___!;""2_5 (4.2)
HUoo

where d is the resonator’s slit opening width, H is the step height, ujm, is the RMS jet
velocity at a frequency of 2fm (i.e., that produced by the low-frequency modulated
streaming ﬂow), and U22. is the mean ﬂow velocity.

Figures 4.3 and 4.4 show the mean-wall-pressure proﬁles for different forcing
frequencies and levels, respectively. Provided in these ﬁgures is the benchmark mean-
pressure distribution corresponding to the unforced case (blue line). In Figure 4.3, the
different forced cases correspond to ﬁve different F” values (i.e., fm of 10 Hz, 15 Hz, 18

Hz, 20 Hz, and 25 Hz) at the same level of forcing of C y = 0.0690%.

102

 

Ft 2 l+011forced +0300 —¢—0.450 —o—0.541 +0601 40.751 '

 

0.08

 

0.04 -

 

 

 

0.00

 

 

 

6' -0.04 -

 

 

-0.08 -

 

 

-0.12 -~

 

 

 

 

 

 

 

-0.16

 

 

Figure 4.3. Forcing-frequency effect on the mean-pressure distribution beneath the
separation bubble at C” = 0.0690%

A couple of interesting observations may be made from the data in Figure 4.3.
First, is the apparent strengthening of the negative peak at a nominal value of 1.5H with
increasing frequency up to F+ of 0.541. If the frequency is increased beyond this value,
the negative peak becomes less pronounced and gradually approaches the unforced value.
Moreover, the start of the pressure recovery for the forced cases is evidently sooner
(farther upstream) than that of the unforced case: the quicker the start of the recovery, the
more pronounced is the negative Cp peak. This is especially seen for the black line (i.e.,
F” 2' 0.541), where the pressure begins to recover at x = 2H, in comparison to 2.5H for
the unforced case (blue line).

The above demonstrates that the forcing device developed here has the capability
of altering the entire separation bubble (as reﬂected in its mean-wall-pressure imprint).
Moreover, it is also found that there is an optimum F” value of 0.541 (f,,. = 18 Hz), which
is evident from the strongest negative Cp peak along with the fact that the pressure begins

to recover soonest in comparison to all other cases. The quicker recovery causes a shiﬁ

103

in the mean-pressure distribution towards the upstream direction, which suggests
shrinking of the separation bubble and shortening of the reattachment length.

The observations concerning the forcing frequency effect above is consistent with
that found in the literature by Chun and Sung (1996, 1998). They showed that increasing
the forcing frequency beyond a critical value (SI/1 2 0.8) caused the reattachment length
to become independent of the forcing frequency. In fact, Chun and Sung (1996) found
that forcing frequencies that were higher than that of the critical value had an adverse
effect on the reattachment length; hence, x, became longer than the unforced case. They
also found an optimal, non-dimensional, forcing frequency of 0.275 based on the step
height and freestream velocity (this compares to 0.541 in the present study).

Greenblatt et al. (2005) associated the improved pressure recovery near
reattachment with an intensifying, near-wall, reverse ﬂow region that reduced
reattachment length in their investigation of low control frequency (17+ < 1). However,
they found out that although the pressure drop immediately downstream of the control
slot was smaller, the pressure recovery near reattachment was adversely affected when P”
was increased beyond its optimum value (F 2 1.35). In another study, Glezer et a1.
(2005) observed the same effect of increasing actuation frequency. They observed a
trend in which the actuation became less effective with increasing actuation frequency
beyond the optimal value. Interestingly, though, in the Glezer et al. (2005) study, when
the frequency was increased substantially beyond the optimal value (i.e., more than ten
times), it was possible to effectively control the ﬂow. However, this type of control

differs substantially from oscillatory control that capitalizes on ﬂow instabilities.

104

The effect of increasing C), at the optimum F” of 0.541 may be examined with the
aid of Figure 4.4. Five different forcing levels (i.e., C p of 0.0003%, 0.0095%, 0.0280%,
0.0690%, and 0.1953%) were exercised. The mean-wall-pressure data for those cases, in
addition to the unforced case, are displayed in Figure 4.4. It is seen that the effect of
increasing the strength of the sound ﬁeld driving the Helmholtz resonator produces a
gradual, overall, shift of the mean-pressure distribution curves towards the upstream
direction. The reader is reminded that the unforced case (blue line) serves as the
benchmark pressure proﬁle in this analysis. Moreover, it is evident that as the strength of

the forcing is increased, the stronger is the negative Cp peak.

C y = l—o—Unforced —a-0.0003% —a—0009?%;0—0-0280% +0.0690‘16 —+-—0.1—953%
0.08 if j l _ ﬂ __ _-_ i
l
0.04 - - . 22W 2

0.00 T

004.1 _2 l , 1 2 l“ 42 722222222 .2
6-008- — 2222 2 2 2. 2 2‘22 l

20.12. 22 ' 22 22 A2 22 l . e222 2_2
0164222222222 2 22 2 2 2’2 2 2 222— . 22222—
1 1 . l
3

 

 

 

 

 

 

 

 

 

 

 

 

__._

 

 

 

 

-0.20 L

 

xIH

Figure 4.4. Forcing-amplitude effect on mean-pressure distributions at F+ z 0.541
The above ﬁnding is consistent with that of the literature (e.g., Chun and Sung,
1996, Greenblatt and Wygnanski, 2000, and Greenblatt et al., 2005). However, unlike

these studies, where stronger forcing levels were exercised, it appears that the forcing

device developed here produces disturbances that are not sufﬁciently strong to reach

105

saturation levels, beyond which increased forcing does not cause a proportional change in
ﬂow response.

An example of forcing saturation may be found in the study of Greenblatt et al.
(2005). They found that the shortening of the separation bubble, seen from the pressure
recovery, continued with increasing C), until Cy z 0.11%. For higher values of C”, the

forcing started to weaken the strength of the bubble.

106

4.2 The unsteady-wall-pressm signature beneath the forced bubble

4.2.1 RMS pressure distributions

The microphone array embedded in the surface of the model was used to measure
the spatial distribution of pressure ﬂuctuations resulting from ﬂow structures within the
separating/reattaching ﬂow. These pressure ﬂuctuations correspond to deviation of the
instantaneous pressure from the mean-pressure value. To examine the strength of the
wall-pressure ﬂuctuations at different locations on the wall, one may analyze the root-
mean-square (RMS) of the pressure ﬂuctuations captured by each of the microphones
embedded in the surface of the model.

The normalized RMS pressure ﬂuctuation plots are shown in Figures 4.5 and 4.6.
The normalizing factor for the RMS pressure ﬂuctuations is the freestream dynamic
pressure, given by ’/2p(,,-,2U.n2. Three-point averaging was used to remove some data
scatter associated with microphone-calibration uncertainty. The abscissa represents the
streamwise location of each of the microphones downstream of the separation point,
normalized by the step height (H). The differently colored curves represent data for the
unforced case and different forced cases.

In Figure 4.5, the forced cases correspond to the same C y of 0.0095% but two
different excitation frequencies that include the optimal frequency: F” 2 0.300 and 0.541.
It is important to note that for the ﬂuctuating pressure results, it was not possible to show
results for C,, values larger than 0.0095% because of dynamic-range limitation of the
measurement microphones. Speciﬁcally, although the sound ﬁeld from the speaker did
not contaminate the measurements at the forcing frequency (2f,,,), it did so at (fc —f,,,) and

(fc + fm). Because of the high intensity of the sound, the overall pressure ﬂuctuations

107

exceeded the full-scale threshold of the wall microphones for cases corresponding to C y >
0.0095%.

Examination of the data in Figure 4.5 shows that directly downstream of the step,
at the point of separation, the microphones sensed low RMS pressure ﬂuctuations. Hudy
(2001, 2005) and Hudy et al. (2003) found that the shear layer at the point of separation
is too far away from the wall-pressure sensors to inﬂuence the measured ﬂuctuations near
the step. The small RMS values near separation have been generally linked to the
unsteadiness or ‘ﬂapping’ of the shear layer. Beyond x/H = 2, there is a signiﬁcant
increase in the wall-pressure ﬂuctuations, which is consistent with that of the literature
(e.g., see Lee and Sung, 2001). It is generally believed that the increase of the RMS
pressure ﬂuctuations is associated with the grth of the shear layer vortical structures.
These vortical structures grow in size and strength, and eventually convect downstream
while moving closer to the wall. This ﬂow behavior causes the increase in the RMS
pressure signature up to a maximum level where the ﬂow “impinges” on the wall as
described by Farabee and Casarella (1986). This scenario of a downwards-curved shear
layer containing spatially evolving vortex structures, was contradicted recently by the
ﬁndings of Hudy (2005). In her study, Hudy found the vortex structures to develop to
full size (of the order of the step size) in place near the middle of the reattachment zone,
before convecting downstream. Albeit different, the scenario was found to be consistent
with the shape of the RMS distributions shown in Figure 4.5. In fact, Hudy’s
measurements were conducted on the same test model as that used here.

The unforced case, shown using the blue line in Figure 4.5, has the lowest RMS

pressure. Stronger level of wall-pressure ﬂuctuations, reﬂective of the inﬂuence of ﬂow

108

control, is found for the forced cases. It is notable that the increase in ﬂuctuation level is
not associated with a change in the streamwise location of the peak RMS value (within
the uncertainty of the measurements). This peak is known to occur at or slightly
upstream of the reattachment location. Therefore, the results suggest that for the forcing
conditions of the cases examined in Figure 4.5, there is no substantial shrinkage in the
separation bubble. This is consistent with the results given in Figure 4.4, where it is seen
that the change in the mean-pressure distribution for C y = 0.0095% reﬂects very small

bubble shrinkage. Such a small change is difﬁcult to detect in the RMS pressure

 

 

 

 

 

 

 

 

measurements.
0.035 1 1 1 1 1 1
cp = 0.0095% ‘
0.03» , , - .
' ' 7‘;\
0.025 - A , ~- J -
N 8
:21.
\> 0.02 — f —
E
‘3.
Q 0015— 1
0 012 +No Forcing _
+F“ ~ 0300
+F‘ .1 0.541
1 I I I l T
0‘0050 1 2 3 4 5 6 7

x/H

Figure 4.5. Effect of forcing frequency (F‘) on RMS wall-pressure measurements

To examine the effect of the forcing amplitude, RMS pressure results obtained at
the most effective F” but two different C x4 values are shown in Figure 4.6. For reference,

the ﬁgure also contains the RMS pressure ﬂuctuations for the unforced case. It is evident

109

that the lower forcing amplitude yielded pressure ﬂuctuation level that is hardly
distinguishable from the unforced case. On the other hand, the increased level of

pressure ﬂuctuations at the forcing level of 0.0095% is clearly visible.

0.035 I I I I I I
F+ z 0541

 

0.03e

0.025~

1/2pU

0.02~

p'w.rm3/

0.015~

 

 

+No Forcing
00‘” 2 42¢“ = 0.000394

—o—Cll = 0.0095%

 

 

 

 

 

y.-
p—
.—

 

l l I
00050 1 2 3 4 5 6 7

x/H

Figure 4.6. Effect of forcing level (C,,) on RMS wall-pressure measurements

4.2.2 Power Spectrum Density (PSD)

Figure 4.7 show seven plots of the wall-pressure power spectral density at the
streamwise location of all microphones. The speciﬁc x locations are: x/H = 1.35, 2.14,
2.93, 3.72, 4.50, 5.29, and 6.08. Generally speaking, PSD plots provide frequency-
content information of a time signal (e.g., a power spectrum plot can identify at which
frequencies the largest ﬂuctuation in a signal occur). In the present analysis, this
information can be useful in characterizing the frequency content of the pressure
signature of the ﬂow structures. The equation used for calculating the discrete, one-sided,

PSD was given earlier (Equation 3.1).

110

In the seven PSD plots, the abscissa represents frequency plotted up to 150 Hz,
since no signiﬁcant wall-pressure ﬂuctuation exists at higher frequencies, while the
ordinate shows the power spectrum magnitude. Each of the plots contains spectra for
four different test conﬁgurations. Theseare the unforced-ﬂow case along with three
forced-ﬂow cases corresponding to two. different F values (0.300 and 0.541) at C), =
0.0095%, and two different C ,, values (0.0003% and 0.0095%) at F” 2 0.541.

The ﬁrst plot shown in Figure 4.7 shows the power spectra at the streamwise
location closest to the point of separation. At this location (x/H = 1.35), all four data sets
show very low spectrum values in accordance with the low level of pressure ﬂuctuations
found near separation (see Figure 4.5 and 4.6). However, the effect of the forcing
imposed at the higher forcing amplitude (C y = 0.0095%) is clearly visible in the small

harmonic peaks found at 20 Hz and 36 Hz (F = 0.300 and 0.541, respectively).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

X 10.3 ' ' x10”1 . .
9 2 9 .
6 x/H=1.35 . 6 x/H=2.14 .
3 4 3 1
x10” 50 100 150 x10” 50 100 150
9 ' ' 2 9 ' F .
61’" 6 x/H = 2.93 . 6 x/H = 3.72 .
m 3 2 3 .
g
. 3 x104 50 100 150 “04 50 100 150
CL 2 1 2 .
. 3 9 2 9
a“ 6 x/H = 4.50 . 6 x/H = 5.29
3 - 3
x10" 50 100 150 50 100 150
9 ' ' 1 +No Forcing
6 x/H = 6.08 2 + F" 4 0.300 2 on = 0.0095%.
3 ‘ + F” .1 0.541 2 C1- = 0.0003911
50 100 150 + + - 2
Frequency (Hz) F 4 0.541 on 0.0095%.

 

 

 

Figure 4.7. One-sided power spectra of the surface pressure at different streamwise
locations downstream of the step

111

The harmonic peak at the forcing frequency becomes stronger with increasing
downstream distance up to a certain x location before it starts to decay again. This is
consistent with that found in the recent study by Liu et al. (2005), who also employed a
wall microphone array in a study of a harmonically-excited back-step ﬂow. Although,
this is true for both forcing frequencies, the x location at which the peak is highest is
more upstream for the lower frequency. Speciﬁcally, for F” = 0.300, the highest peak is
found at x/H = 3.72, while for F” = 0.541, the corresponding location is x/H = 2.93. This
may be seen more clearly in Figure 4.8, where the spectrum magnitude at the frequency
of the forcing is plotted (after subtraction of the spectrum value corresponding to the
unforced case) for all x locations (note that, similar to the RMS results, three-point
averaging was employed to remove some data scatter associated with microphone-
calibration uncertainty).

Further examination of Figure 4.8 shows that although the spectrum peak at the
higher, forcing frequency is attained farther downstream, the associated pressure
ﬂuctuations actually start to amplify quicker (i.e., farther upstream) relative to the lower
forcing frequency. Initially, the ampliﬁcation rate is strong, followed by slow growth, or
possibly a ﬂat proﬁle within the uncertainty of the data, for a good stretch of the
measurement domain. In contrast, the data at the lower, off-optimal, frequency show the
associated pressure ﬂuctuations to start to grow at a farther downstream location relative
to the higher forcing frequency. Subsequently, these ﬂuctuations amplify quickly, but the
data do not exhibit a streamwise stretch over which the ﬂuctuations strength is sustained.
Instead, grth of the pressure ﬂuctuations is immediately followed by rapid decay.

These observations are consistent with the ﬁndings of Greenblatt et al. (2005), who

112

measured the pressure ﬂuctuations beneath a separation bubble downstream of a wall-

mounted “hump” under different forcing conditions.

 

 

 

 

 

 

 

 

0-012 7 r T ! 1 ‘
, A F+ in 0.300 - Cu = 0.009596
0.01 _ .......................... .......................... .......... o F+ a 0,541 2 cll = 0,0095% ..-
NA 0 008~- ............................................... 1 ......................... .......................... , .................... 1 ........................ , ....................... -
m ' E :
Q_ 3 e 0 i0
V o. i
a % 0 A ;
a"; 00m_ .......................... .......................... ,A... ............... .0g ................................................. _
~ : o
O. : . 2
3 : :
‘9‘? 0.004.... ............................. .0. Q ........................................................ A .................................................. ..
A o A? 0
0m2_ ......................................... .A... ..... AMHMA. ................................... ......................... inuA ..... Ammg ...... o .......... _.
A
0 l l l l l l
0 1 2 3 4 5 6 7
x/H

Figure 4.8. Streamwise distribution of the pressure-spectrum magnitude at the
forcing frequency for forcing level of C” = 0.0095%

4.2.3 Frequency-wavenumber spectra

The frequency-wavenumber (f-kx) spectrum corresponds to the two-dimensional
Fourier transformation of the spatio-temporal, wall-pressure signal’s [p'w(x,t)] auto-
correlation. The resulting spectra breakdown the energy content of the wall-pressure
ﬁeld into contributions at different frequency and wavenumber combinations. Here, data
from ﬁfteen microphones were used in obtaining the f-kJr spectra. The sampling rate used
for each of the microphones during the measurements was 8 kHz for 33 seconds. A total
of 262,144 (2‘8) number of samples for each of the microphones were used to create a

262,144 x 15 (time x space) two-dimensional array that made up the entire data set. This

113

data set was then divided into sub-arrays of 1024 rows x 32 columns, which allowed
averaging of the spectrum over 256 records and produced a random uncertainty of 6.25%.
It should be noted that the space array was padded with zeros to increase its size from 15
to 32 in order to obtain the resulting spectrum at smaller wavenumber increments via
interpolation. To calculate the fit, spectrum, Fast Fourier Transform (FF T) of each of the
1024 x 32 records was ﬁrst taken along the time dimension. A second FFT was then
calculated along the space dimension to change the spatial domain representation into the
wavenumber domain. Finally, the resulting 2D FFT was multiplied by its conjugate to
arrive at the frequency-wavenumber spectrum.

Frequency-wavenumber (f-kx) spectrum plots representing the four different cases
considered for the RMS and PSD analyses above, are shown in Figures 4.9 through 4.12.
For all the f—kJr plots, the ordinate gives the frequency normalized by the step height (H)
and freestream velocity (U00). The abscissa represents the wavenumber normalized with
the step height. The color bar on the right side represents the magnitude of the spectrum
normalized by its maximum peak value, resulting in a “zero—to-one” scale.

For all the plots, there are two identiﬁable regions where the pressure-ﬂuctuation
energy is concentrated. The bulk of the pressure ﬂuctuation is found to be associated
with a “ridge” that is inclined at an angle and located in the right-hand plane (kJr > O).
Noting that the slope of a line drawn from the origin of the f-kx spectrum to any point in
the f-kx plane gives the convection velocity of the corresponding disturbance, it is evident
that this ridge represents the signature of downstream-traveling (positive convection
velocity) disturbances. In the unforced case, Hudy (2005) showed that this ridge is the

result of the pressure ﬂuctuations produced by large-scale (order of H) vortices that form

114

via the roll-up of the separated shear layer, near the center of the re-circulation zone, and
convect downstream. In addition to the “convective ridge”, some weak pressure
ﬂuctuations are found in the k, < 0 half plane. This signiﬁes that some of the wall-
pressure ﬂuctuations beneath the separation bubble are associated with upstream
traveling disturbances. These presumably are related to the back-ﬂow within the
separation bubble.

Focusing on the convective ridge in the unforced-ﬂow case shown in Figure 4.9,
it is seen that there is a preferred frequency/wavenumber where the ridge attains its
highest value. The peak associated with this preferred mode is identiﬁed at fH/U.no 2

0.103 and ka of 0.289.

 

spectrum for the unforced-ﬂow

115

Figure 4.10 shows the f-k, spectrum when forcing the ﬂow at P : 0.300 (10 Hz
modulation frequency, or ﬁli/U.o = 0.075) and C” of 0.0095%. Two main observations
are found here. First, it is evident that the frequency of the peak of the convective ridge
shifts from the unforced value identiﬁed earlier to a value equal to that of the forcing
frequency. The corresponding wavenumber is found to be ka = 0.268. The second
observation that can be made from Figure 4.10 is that the convective ridge is found to be
less broad in comparison with the spectrum of the unforced case (Figure 4.9). This
suggests that the forcing leads to better organization of the wall-pressure-generating ﬂow

structures: consistent with expectation.

 

Figure 4.10. Wall-pressure”.-
(F+~ 0. 300 [feH/Uw- — 0. 075] and C,,— "0. 0095%)

spectrum for the forced ﬂow

1

Figure 4.11 represents the f-kx spectrum obtained when forcing the ﬂow at the

optimal forcing frequency F 3 0.541 (18 Hz modulation frequency, or feH/U.,D = 0.135)

116

forcing condition, but very low forcing amplitude (C), = 0.0003%). In concert with the
RMS and frequency spectrum results, very little difference is seen in the f-kx spectrum
results between the forced and unforced cases because of the minute level of forcing. At
the higher forcing level of C ,1 = 0.0095% (Figure 4.12), while forced at the optimum
frequency, the f-k, spectrum shows the strongest deviation from the spectrum of the
natural ﬂow. Similar to the results shown in Figure 4.10 for F = 0.300, the global
spectrum peak is found at the forcing frequency, and the overall spectrum becomes
substantially narrower than for the natural case, indicating better organization of the ﬂow
features generating the surface-pressure ﬂuctuations. The wavenumber corresponding to

the global spectrum peak is found at ka = 0.350.

 

L

spectrum for the forced flow
(F+~ o. 541 [ﬂH/Um- — 0.135] and c,,= 0. 0003%)

Figure 4.11. Wall-pressuref. 1

117

 

Figure 4.12. Wall-pressure‘u ' spectrum for the forced flow
(F’~ 0. 541 [f,H/U.,, = 0.135] and C = 0. 0095%)

An interesting outcome of the frequency-wavenumber analysis is that it shows
that the wavenumber of the forced disturbance could be predicted a priori (at least within
the forcing and ﬂow parameters employed here). To demonstrate this, consider Figure
4.13 where the f-k,r of the unforced-ﬂow case is re-examined with focus on positive
wavenumbers only. The peaks of the convective ridge at different frequencies are
identiﬁed and shown in yellow circles in the ﬁgure. A linear ﬁt to the circles (red dashed
line) is employed to provide an analytic equation describing the top of the ridge (or most
energetic disturbances at different frequencies). As seen from Figure 4.13, the straight
line provides good representation of the ridge top (the data scatter around the line is
related to the resolution of the spectrum). Using the obtained linear relation in
conjunction with the forcing-frequency values, it was then possible to predict the

wavenumbers of the most energetic disturbances in the natural ﬂow at the same

118

frequency as the forced disturbances. This is illustrated with the broken white lines in
Figure 4.13 forﬁH/Uw = 0.075 (orfm = 10 Hz) andﬁ.H/U0° = 0.135 (orﬁn = 18 Hz). The
corresponding wavenumber estimates are ka = 0.256 and 0.354 respectively, which
when compared to the wavenumbers identiﬁed earlier from the spectrum peak for the
forced ﬂow cases at C” = 0.0095% (ka = 0.268 and 0.350) yield an error percentage of
less than 5%: a value that is well within the wavenumber resolution of the spectrum.
Thus, it appears that the spatial scale of the forced disturbance is pre-set by the natural-
ﬂow’s frequency-wavenumber characteristics. This ﬁnding is interesting in the sense that
it provides means for predicting the wavenumber for a given forcing frequency. This
may be used, for instance, to shiﬁ the wavenumber of a dominant existing disturbance to
be far removed from resonant wavenumbers of an underlying surface in order to reduce

or eliminate ﬂow-induced noise and vibration.

\

linear ﬁt

peaks

 

 

Figure 4.13. Wall-pressure‘u 1 ' spectrum for the unforced flow,
showing straight-line fit to the peak of the convective ridge

119

5 CONCLUSIONS AND RECOMMENDATIONS
The results from the present investigation are recapitulated below, followed by

recommendations for future work.

5.1 Conclusions

An externally driven Helmholtz resonator was designed and implemented for
forcing the separated ﬂow over an axisymmetric, backward-facing step at a Reynolds
number based on step height of Key = 2525. The new device, which allowed the
introduction of disturbances to the ﬂow at the point of separation, was used to investigate
the effect of the forcing on the wall-pressure signature beneath the separating/reattaching
ﬂow downstream of the step.

The acoustic and ﬂuidic disturbance produced by the Helmholtz resonator was
characterized utilizing microphone and single-hotwire measurements. The results
showed that the frequency at which the strongest ﬂuidic disturbance was produced did
not coincide with the resonance frequency of the device. The strongest ﬂuidic
disturbance was found at 630 Hz, while the Helmholtz resonator had a resonance
frequency of 657 Hz. The difference was due to resonance of the wind tunnel that
ampliﬁed the device’s disturbance at 630 Hz. Nevertheless, the resonator’s frequency
response (both amplitude and phase) agreed well with the response of a second-order,
lumped-parameter system. This conﬁrmed that the oscillations produced by the resonator
corresponded to the lowest-order, or Helmholtz, resonance.

Because the frequency at which the ﬂow was most receptive to disturbances was

more than an order of magnitude lower than the operating frequency of the device of 630

120

Hz, a low-frequency amplitude-modulation of the operating (carrier) frequency was
employed to effectively couple the perturbation to the ﬂow. The success of this coupling
was linked to modulation of the streaming (steady) ﬂow component produced by the
driven resonator. In particular, it was found that driving the resonator using a pure
harmonic signal produced both an oscillating and streaming ﬂuidic disturbance. The
latter is a result of non-linearity of Helmholtz resonators when driven at high intensities
of sound (e.g., see Ingard and Labate, 1950). Thus, by modulating the driving acoustic
signal of the resonator, the streaming ﬂow component produced a low-frequency jet that
excited the ﬂow at the appropriate frequencies.

An additional advantage of the forcing scheme adopted here, which is particularly
important to surface-pressure studies, is that the ﬂow disturbance was ﬂuidic rather than
acoustic in nature. The acoustic signature of the forcing scheme (primarily from the
driving speaker) was conﬁned to high frequencies (near the carrier frequency) and hence
it could be removed from the surface-pressure measurements via simple low-pass
ﬁltering. It was noted, though, that at forcing levels substantially higher than those
employed here, the ﬂuidic disturbance itself might become an effective noise source.
Under these conditions the device could not be considered acoustically quite.

The optimum modulation frequency at which coupling of the resonator’s
disturbance to the ﬂow was most effective was found to be P 2 0.541. At this frequency,
a pronounced shift in the mean-pressure distribution towards the upstream direction
suggested shrinking of the separation bubble and shortening of the mean reattachment
length, x,. The optimum F value found in the present study is twice the value found by

Greenblatt and Wygnanski (2000), Chun and Sung (1996), and Chun et al. (1999). The

121

difference might be due to the geometry. Unlike the axisymmetric model, the planar
geometry might be affected by end-walls effect.

The effect of excitation level, C,,, on the mean-pressure distribution was also
examined at the optimum F . The results showed monotonic reduction in the separation
bubble size with increasing forcing level. 1 However, the range of excitation levels
investigated here did not reach the saturation level suggested in the literature at very large
forcing levels. Chun and Sung (1996), Greenblatt and Wygnanski (2000), and Greenblatt
et al. (2005) suggested that ﬂow excitation at saturation levels does not cause
proportional change in ﬂow response. Overall, the above demonstrates that the forcing
device developed here has the capability of altering the behavior of the entire separation
bubble, and not just the local characteristics of the separated shear layer.

The unsteady-wall-pressure imprint beneath the separating/reattaching ﬂow
region was measured using an array of 15 wall-pressure microphones. Fluctuating-
pressure RMS, power spectrum density (PSD), and frequency-wavenumber (f-kx) spectra
were analyzed. However, the C ,, values employed for this analysis were limited to within
0.0095% due to the upper limit of the dynamic-range of the measurement microphones.
Overall, it was found that the wall-pressure ﬂuctuations became more energetic and
organized as a result of the forcing. Additionally, PSD results showed that the harmonic
peak at the forcing frequency became stronger with increasing downstream distance up to
a certain x location before it started to decay again. The streamwise range over which
sustained grth of the forced disturbance was found was substantially larger when

forcing at the optimal, in comparison to other, frequencies. When forcing at an off-

122

optimal frequency, the pressure oscillations at the forcing frequency exhibited rapid
decay following the initial growth.

An interesting outcome of the frequency-wavenumber—spectrum analysis was the
ﬁnding that the spatial scale, i.e., streamwise wavenumber, for the wall-pressure
disturbance at a given forcing frequency could be predicted using the peak-locus of the
convective ridge in the frequency-wavenumber spectrum of the unforced ﬂow. This
ﬁnding may be used, for instance, as basis for selecting the forcing frequency in order to
produce a particular wavenumber associated with the dominant wall-pressure disturbance.
This could be useful in reducing or eliminating ﬂow-induced noise and vibration by
selecting a wavenumber that is far removed from the resonant wavenumbers of an

underlying surface.

123

5.2 Recommendations for future work

There are still several interesting aspects to explore in attempt to further
investigate the effect of amplitude-modulated forcing on the axisymmetric, backward-
facing-step ﬂow. Increasing the excitation level beyond that attained here would be
useﬁrl to conﬁrm the saturation behavior suggested by Chun and Sung (1996) and
Greenblatt and Wygnanski (2000). Additionally, unsteady wall-pressure measurements
using microphones with wider dynamic range can facilitate higher levels of excitation
than was possible here.

Aside from wall-pressure information, it is also signiﬁcant to examine if the
amplitude-modulated excitation alters the ﬂow ﬁeld in a manner that is similar to the
more common method of harmonic excitation. To accomplish this, a velocity-ﬁeld study
is of interest. Measurements using a single-hotwire sensor have been recently completed
in the Flow Physics and Control Laboratory (FPaCL) to address this point (Trosin, 2006).
Finally, a nice complement to the present data would be oil-ﬁlm interferometry (OFI) to

directly determine the reattachment length under different forcing conditions.

124

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