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Z V UBRARY
f CEC- Michigan State
, University

 

 

This is to certify that the
dissertation entitled

EFFECT OF FINITE CAVITY WIDTH ON THE SELF-
SUSTAINED OSCILLATION IN A LOW-MACH-NUMBER
CAVITY FLOW

presented by

Ke Zhang

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Mechanical Engineering

 

 

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Major Professor’s Signature/

lo /24/oq

Date

MS U is an Affirmative Action/Equal Opportunity Employer

 

 

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5/08 KilProleoc8iPrele|RCIDateDueJndd

 

EFFECT OF F INITE CAVITY WIDTH ON THE SELF-
SUSTATNED OSCILLATION IN A LOW-MACH-NUMBER
CAVITY FLOW

By

Ke Zhang

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
MECHANICAL ENGINEERING

2009

ABSTRACT

EFFECT OF F INITE CAVITY WIDTH ON THE SELF-SUSTAINED
OSCILLATION IN A LOW-MACH-NUMBER CAVITY FLOW

By
Ke Zhang

Cavity flow is known to be associated with high-level-noise generation, strong
vibration and substantial increase in the drag force on the object containing the cavity.
Most of the studies of the cavity flow are based on a two-dimensionality assumption. The
three-dimensional features of cavity unsteadiness/oscillation are rarely investigated. The
current study is focused on examining the effect of the cavity width and side walls on the
self-sustained oscillation in a low-Mach-number cavity flow with a turbulent boundary

layer at separation.

An axisymmetric cavity geometry is employed in the present research. The
axisymmetric configuration provides a distinct advantage in studying the side-wall effects
in that the configuration provides a reference condition that is free from any side-wall
influence; yet, the cavity could be partially filled to form a finite-width geometry.
Unsteady surface pressure, on the cavity floor along the streamwise direction and on the
downstream wall along the azimuthal direction, is acquired using microphone arrays. The
pressure data are recorded over a range of flow and geometrical parameters. In particular,
the Reynolds number based on cavity depth and free-stream velocity is changed from Re
= 4067 to 12200, and cavity length-to-depth ratio L/D from 2.6 to 4.1 for both the
axisymmetric geometry and finite-width cavities with width-to-depth ratio W/D in the

range 2.5 to 7.4. Based on the outcome of the analysis of the unsteady surface-pressure

 

field, velocity measurements using a two-component Laser Doppler Anemometer (LDA)
system are performed simultaneously with array measurements in different azimuthal
planes for a cavity with U0 = 3.3 and W/D = 7.4 at Re = 12200 to explore the effect of
the side wall on the mechanism driving the cavity oscillation. Evolution of coherent
structures generating the pressure oscillation on the downstream wall of the cavity is
evaluated using linear stochastic estimation (LSE) of the velocity field based on the wall-

pressure signature.

The results show that while no prominent oscillation is found in axisymmetric
cavities without side walls, strong harmonic pressure oscillation different from any of the
known modes in the literature is observed in finite-width cavities at an azimuthal location
of about one cavity depth away from the side wall. Analysis of the mean three-
dimensional flow inside the cavity and the stochastic estimation results lead to the
hypothesis that the flow structures in the symmetry plane of the finite-width cavities,
although have a weak pressure signature, they interact with the flow near the side wall,
providing the driving mechanism for the establishment of the oscillation near the side

wall.

Copyright by
Ke Zhang
2009

DEDICATION

To my parents and husband

for their love, support and encouragement

ACKNOWLEDGEMENTS

I would like to express my heartfelt gratitude to my advisor, Dr. Ahmed Naguib
for giving me the opportunity to work on this project and for his guidance and mentorship

during my four-years study at MSU. He is always there with patience.

I would also like to thank my committee members Dr. Changyi Wang, Dr. Mei
Zhuang and Dr. Manoochehr Koochesfahani for their helpful comments and discussion

throughout my Ph.D. study.

I really appreciate all the help from my colleagues and friends at MSU. I would
like to thank Barry and Tony for their help in getting me oriented around the Flow
Physics and Control Laboratory FPaCL at the beginning of my research work. I
appreciate all the help from Alan, Doug, Jason, Shahrarn, Rohit, Vibhav and Elliot
throughout my study at MSU. Special appreciations are given to Michael Mclean and

Roy Bailiff for their help with the experimental setups. Thanks for your fiiendship, J ing.

Finally, none of this would have been possible without the love and support from
my husband Min Tang. I could not imagine going through these years in the US without

his love and encouragement.

This study was supported by the National Science Foundation under grant number
CTS-O425374, monitored by Dr. William Schultz. Any opinions, findings and
conclusions or recommendations expressed in this material are those of the authors and

do not necessarily reflect the views of the National Science Foundation.

vi

TABLE OF CONTENTS

 

 

 

LIST OF TABLES ........................................................................ ix
LIST OF FIGURES ....................................................................... x
NOMENCLATURE .................................................................. xvii
1. INTRODUCTION ..................................................................... l
1.1 Background 1
1.2 Motivation 6
1.3 Objectives 9
1.4 References 11

 

2. EXPERIMENTAL SETUP AND MEASUREMENT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TECHNIQUES ............................................................................. 13
2.1 Wind Tunnel Facility and Cavity Model 13
2.2 Measurement Techniques - 17

2.2.1 Hot-wire Measurements 17
2.2.2 Static Pressure Taps 20
2.2.3 Microphone Arrays 21
2.2.4 Data Acquisition Hardware 28
2.2.5 Laser Doppler Anemometry 29
2.3 Experimental Procedure and Parameters 38
2.3.1 Model Alignment - 39
2.3.2 The Boundary Layer State at Separation - 40
2.4. References - 51

3. WALL-PRESSURE-MEASURMENTS RESULTS ............ 53

3.1 Unsteady Wall Pressure in the Cavity Symmetry Plane 53
3.1.1 The Unsteady Wall Pressure in the Axisymmetric Cavity Flow ............... 53
3.1.2 Effect of Cavity Width on the Unsteady Wall Pressure in the Symmetry

Plane 59

3.2 Azimuthal Distribution of the Unsteady Wall Pressure 70
3.2.1 Cavities with Finite Width 70
3.2.2 Axisymmetric Cavity 78

3.3 Summary 81

3.4 References - 83

4. SIMULTANEOUS VELOCITY AND PRESSURE

MEASURMENTS ........................................................................ 84
4.1 Velocity Results and Discussion 84

 

vii

4.1.1 Mean-Velocity Field - - - 84

 

 

 

 

 

 

 

 

 

4.1.2 F luctuating-Velocity Field 87
4.1.3 Reynolds Shear Stress 90
4.1.4 Mean Vorticity Field 92
4.1.5 Three-Dimensional Flow Field Close to Cavity Side Wall 96
4.2 Stochastic Estimation of the Coherent Structures Generating Wall-Pressure
Fluctuations - 112
4.2.1 Velocity-Pressure Correlation - 116
4.2.2 Evolution of the Coherent Structures in the Azimuthal-Symmetry Plane of
the Cavity 121

4. 2 .3 Evolution of the Coherent Structures Close to the Cavity Side Wall ..... 128

4 .3 The Oscillation Mechanism. a Hypothesis 136
4. 4 References 139

 

5. CONCLUSIONS AND RECOMMENDATIONS... 141

 

 

 

6.1 Wiring for Azimuthal Microphone Array 144
6.2 Schmitt Trigger Circuit 145
6.3 LDA system alignment - ..... 146

6.4 Azimuthal Traversing of the LDA Probe Volume 148

 

viii

LIST OF TABLES

Table 3.1 Cavity width-to-length ratio for oscillating (bold) and non-oscillating flow
in the symmetry plane _ 65

 

ix

LIST OF FIGURES

(Images in this dissertation are presented in color)

 

 

 

Figure 1.1 Illustration of the cavity geometry and coordinate system- 2
Figure 1.2 Cavity model: (a) axisymmetric cavity (b) finite-width cavity ................. 10
Figure 2.1 Schematic diagram of the wind tunnel 14
Figure 2.2 A schematic drawing of the axisymmetric test model 16

Figure 2.3 Three-dimensional drawing of the cavity model (note W is measured at
height of D/2 above the cavity bottom) 16

 

Figure 2.4 An image of the single hot wire above the axisymmetric model surface . 19

 

Figure 2.5 Example of a hot-wire calibration - -- 20
Figure 2.6 Configuration of the azimuthal Knowles Electronics microphone array 23

Figure 2.7 Photographic views of the PWT for calibration of the streamwise-army’s
microphones 24

 

Figure 2.8 Calibration of the PWT: amplitude ratio (top) and phase shift (bottom)25

Figure 2.9 Sensitivity of the streamwise-army’s microphones 26

 

Figure 2.10 Setup for calibration of the azimuthal-array’s microphones ................. 27

Figure 2.11 Sensitivity of the azimuthal-army’s microphones: (a) Knowles
microphones (see Figure 2.6 for microphone numbers) and (b) Panasonic
microphones (microphone numbers are based on viewing the cavity’s end wall from
inside the cavity. Number increases in the clockwise direction starting with
microphone 1 at the top) 28

 

Figure 2.12 Illustration of light scattering from a moving particle: (a) Single-beam
configuration (b) Dual-beam configuration 31

 

Figure 2.13 Image of the seeding system upstream of the wind tunnel inlet section 32

Figure 2.14 Picture of the FiberFlow optics system 34

 

Figure 2.15 Picture of the LDA probe mounted on the three-dimensional traversing
system 35

 

Figure 2.16 Block diagram demonstrating the synchronization of the LDA system
with the A/D boards used for acquiring pressure data 36
Figure 2.17 Streamwise distribution of the mean-pressure coefficient downstream of
the back step at different azimuthal locations around the model (left and right
indications in the legend are relative to viewing the model from upstream) ............. 40

 

Figure 2.18 Boundary layer mean-streamwise-velocity profile for (a) U 00 = 5 m/s

 

 

 

and (b) U a) = 20 m/s 42
Figure 2.19 Illustration of the definition of A¢ and k--- 46
Figure 2.20 LDA measurement grid - 46
Figure 2.21 Illustration of the pressure interpolation process 47

 

Figure 2.22 Convergence test for: (a) the mean streamwise velocity (b) cross-
correlation between the velocity and the pressure fluctuations normalized by the
velocity and pressure rms values 50

 

Figure 3.1 Streamwise distribution of the rms pressure acting on the bottom of the
axisymmetric cavity for [/0 = 3.3 and different Reynolds numbers 54

 

Figure 3.2 Wall-pressure frequency spectra on the bottom of the axisymmetric
cavity for W = 3.3 and Re =12200. Lines with different colors represent different
streamwise locations 55

 

Figure 3.3 Wall-pressure frequency spectra on the bottom of the axisymmetric
cavity for [/0 = 3.3 and Re =4067. Lines with different colors represent different
streamwise locations 57

 

Figure 3.4 Aspect ratio effect on the wall-pressure frequency spectra on the bottom
of the axisymmetric cavity at x/L = 0.95 and Re = 12200. Lines with different colors
represent cavities with different W 58

 

Figure 3.5 Cavity-width effect on wall-pressure spectra at x/L = 0.95 (Re = 12200).
Lines with different colors represent cavities with different width 60

 

Figure 3.6 Cavity-width effect on wall-pressure spectra at x/L = 0.95 for Re = 16267.
Lines with different colors represent cavities with different width 62

 

Figure 3.7 Cavity-width effect on wall-pressure spectra at x/L = 0.95 for Re = 8133.
Lines with different colors represent cavities with different width 63

 

Figure 3.8 Cavity-width effect on wall-pressure spectra at x/L = 0.95 for Re = 4067.
Lines with different colors represent cavities with different width 63

 

xi

Figure 3.9 Cavity-width effect on wall-pressure spectra at Re = 12200: (a) L/D =2.6
(b) L/D = 4.1. Lines with different colors represent cavities with different width.... 66
Figure 3.10 Coherence of pressure fluctuations across the cavity length relative to
pressure measured at x/L = 0.95 for W/D = 2.5 at Re = 12200. The color bar on the
right indicates the coherence value 68

 

Figure 3.11 Coherence of pressure fluctuations across the cavity length relative to
pressure measured at x/L = 0.95 for the axisymmetric cavity at Re = 12200. The
color bar on the right indicates the coherence value 69

 

Figure 3.12 Flood color-contour map showing the azimuthal variation of the
frequency spectra of the pressure acting on the end cavity wall for the cavity with
W/D = 2.5. The Color bar on the right indicates the magnitude of the spectra
normalized by the peak value 71

 

Figure 3.13 Frequency spectra at selected azimuthal locations for the cavity with
W/D = 2.5. Lines with different colors represent different azimuthal locations ....... 72

Figure 3.14 Flooded color-contour map showing the azimuthal variation of the
frequency spectra of the pressure acting on the end cavity wall for the cavity with
W/D = 3.7. The color bar on the right indicates the magnitude of the spectra
normalized by the peak value 73

 

Figure 3.15 Frequency spectra at selected azimuthal locations for the cavity with
W/D = 3.7. Lines with different colors represent different azimuthal locations ....... 73

Figure 3.16 Flooded color-contour map showing the azimuthal variation of the
frequency spectra of the pressure acting on the end cavity wall for the cavity with
W/D = 7.4. The color bar on the right indicates the magnitude of the spectra
normalized by the peak value 75

 

Figure 3.17 Frequency spectra at selected azimuthal locations for the cavity with
W/D = 7.4. Lines with different colors represent different azimuthal locations ....... 75

Figure 3.18 Frequency spectra at the azimuthal location of strongest harmonic
oscillation for cavities with different W/D values. Lines with different colors
represent cavities with different width -- 77

 

Figure 3.19 Coherence between the pressure fluctuations at different azimuthal
locations with those at z/D = 2.62 for W/D = 7.4 - -_ 78

 

Figure 3.20 Azimuthal distribution of the rms pressure for the axisymmetric cavity
with L/D = 3.3. Red line shows the average rms pressure and black lines represent
10% deviation from the mean value 79

 

xii

Figure 3.21 Frequency spectra at different azimuthal locations in the axisymmetric
cavity. Black solid line shows spectrum at d = 0°; Remaining lines depict the spectra
at angle increments of 22.5° 80
Figure 4.1 Mean-velocity vector field in the x - y planes at (a) A¢ = 0° and (b) A¢ = -
32°. The isolated vector near the top-:left corner of the plots represents the free-
stream velocity - 86

 

 

Figure 4.2 Mean streamlines in the x - y planes at (a) A¢ = 0° and (b) A¢ = -32° 87

Figure 4.3 Flooded color-contour map of the rms streamwise velocity u'mW co in

the x - y planes at (a) A¢= 0° and (b) A¢= -32°. The color bar at the bottom gives the
rms values normalized by the free-stream velocity 89

 

Figure 4.4 Flooded color-contour map of the rms wall-normal velocity V'rms/Um in

the x - y planes at (a) A¢= 0° and (b) A¢= -32°. The color bar at the bottom gives the
rms values normalized by the free-stream velocity 90

 

Figure 4.5 Flooded color-contour map of the Reynolds-shear-stress WI U 020 in the

x - y planes at (a) A¢ = 0° and (b) A¢ = -32°. The color bar shows the Reynolds shear
stress normalized by the square of the freestream velocity 92

 

Figure 4.6 Illustration of the layout of neighborhood points used in calculation of
velocity derivatives at point (i,j) 94

 

Figure 4.7 Flooded color-contour map of the mean vorticity EZD/UQo in the x-y

planes at (a) Ad = 0° and (b) A¢ = -32°. The color bar at the bottom indicates the
magnitude of vorticity normalized using the cavity depth and free-stream velocity 95

Figure 4.8 Mean streamlines in the x - y planes at (a) A¢ = -32°, (b) A¢ = -36° and
(c) A¢= -40° 98

 

Figure 4.9 Frequency spectra at selected azimuthal locations for the cavity with W/D
= 7.4. Lines with different colors represent different azimuthal locations ................ 99

Figure 4.10 Illustration of the cylindrical coordinate system with origin at the center
of the axsiymmetric model - 101

 

Figure 4.11 Wall-normal profiles of the mean azimuthal velocity it" ono at different

azimuthal locations and (a) x/D = 3.0, (b) x/D = 2.8, (c) x/D = 2.4, (d) x/D = 2.2, (e)
x/D = 2.0 and (f) x/D = 1.8 103

 

Figure 4.12 Mean streamlines in the y-¢ planes at (a) x/D = 3.0 and (b) x/D = 2.8.
The streamwise direction is into the paper 107

 

xiii

Figure 4.13 Mean streamlines in the y-¢ planes at (a) x/D = 2.6, (b) x/D = 2.4, (c) x/D
= 2.2 and (d) x/D = 2.0. The streamwise direction is into the paper 108

 

Figure 4.14 Mean three-dimensinal streamlines originating near the cavity’s side
wall: from x/D N 2.6 at A¢= -40° 109

 

Figure 4.15 Mean three-dimensional streamlines originating near the cavity’s side
wall and the center of recirculation: from x/D % 2.9 and y/D % 0.75 at A¢= -40° 110

Figure 4.16 Flooded color-contour maps of the velocity-pressure cross-correlation
(Rurpr) at A¢ = -32°. Pressure is measured at W = 3.3 and y/D = 0.5 (shown by the
blue circle on the end wall of the cavity). Correlation results are shown at zero time
delay (middle plot) as well as at time delays corresponding to :1: 1/4 the oscillation
period at [DU 00 as 0.21. The color bar at the bottom of the plot gives Rurpr valuesll7

Figure 4.17 Flooded color-contour maps of the velocity-pressure cross-correlation
(Rv'p') at A¢ = -32°. Pressure is measured at x/D = 3.3 and y/D = 0.5 (shown by the
blue circle on the end wall of the cavity). Correlation results are shown at zero time
delay (middle plot) as well as at time delays corresponding to :t: 1/4 the oscillation
period at leU co as 0.21. The color bar at the bottom of the plot gives Rv'p' values 118

Figure 4.18 Flooded color-contour maps of the velocity-pressure cross-correlation
(Ruvpr) at A¢ = 0°. Pressure is measured at x/D = 3.3 and y/D = 0.5 (shown by the
blue circle on the end wall of the cavity). Correlation results are shown at zero time
delay (middle plot) as well as at time delays corresponding to :t: 1/4 the oscillation
period at fl/Uoo z 0.21. The color bar at the bottom of the plot gives Ru'p' values

119

 

Figure 4.19 Flooded color-contour maps of the velocity-pressure cross-correlation
(Rv'p') at A¢ = 0°. Pressure is measured at x/D = 3.3 and y/D = 0.5 (shown by the

blue circle on the end wall of the cavity). Correlation results are shown at zero time
delay (middle plot) as well as at time delays corresponding to :1: 1/4 the oscillation

period at fl/U co as 0.21. The color bar at the bottom of the plot gives erpr values120

Figure 4.20 Streamlines and estimated fluctuating vorticity 5' z D/ UClo contours at
Ad = 0° for different time offsets (covering one period of the oscillation at fl/U 00 as
0.21 with a time interval of AtUoo/L = 0.365) preceding and following the occurrence
of a positive pressure (p’= Prms at At = 0 in (7)) at x/D = 3.3 and y/D = 0.5. The color
map on the bottom gives 07': D/ U a, values 125

 

xiv

Figure 4.21 Streamlines and vorticity sz/Uw contours at Ad = 0° for different
time offsets (covering one period of the oscillation at jL/Uoo as 0.21 with a time
interval of AtUoo/L = 0.365) preceding and following the occurrence of a positive
pressure (p’= Prms at At = 0 in (7)) at x/D = 3.3 and y/D = 0.5. The color map on the
bottom gives sz/U.Jo values - 126

 

Figure 4.22 Streamlines, estimated fluctuating vorticity 5' z D/Uw field and

concurrent surface pressure in the x-y plane of Ad = 0° corresponding to: (a) peak
positive pressure and (b) peak negative pressure on the cavity end wall at x/D = 3.3
and y/D = 0.5 (pressure value shown with red circle). Blue line shows the pressure
distribution on the cavity bottom, and color bar gives 5'2 D/Um values .............. 127

Figure 4.23 Streamlines and estimated fluctuating vorticity 5' z D/ Um contours at
Ad = -32° for different time offsets (covering one period of the oscillation at jL/U 00 as
0.21 with a time interval of AtU oo/L = 0.365) preceding and following the occurrence
of a positive pressure (p’= Prms at At = 0 in (7)) at x/D = 3.3 and y/D = 0.5. The color
map on the bottom gives 03'". D/ U a, values 130

 

Figure 4.24 Streamlines, estimated fluctuating vorticity 6' z D/Uao field and

concurrent surface pressure in the x-y plane of Ad = -32° corresponding to: (a) peak
positive pressure and (b) peak negative pressure on the cavity end wall at x/D = 3.3
and y/D = 0.5 (pressure value shown with red circle). Blue line shows the pressure
distribution on the cavity bottom, and color bar gives 6' z D/ U 00 values .............. 131

Figure 4.25 Streamlines and vorticity sz/ U00 contours at Ad = -32° for different
time offsets (covering one period of the oscillation at fl/Uoo as 0.21 with a time
interval of AtUoo/L = 0.365) preceding and following the occurrence of a positive
pressure (p’= Prms at At = 0 in (7)) at W = 3.3 and y/D = 0.5. The color map on the
bottom gives sz/Ua, values 132

 

Figure 4.26 Streamlines and estimated fluctuating vorticity 07". D/Uw contours at
Ad = -36° for different time offsets (covering one period of the oscillation at fL/U co m
0.21 with a time interval of AtUoo/L = 0.365) preceding and following the occurrence
of a positive pressure (p’= prmg at At = 0 in (7)) at x/D = 3.3 and y/D = 0.5. The color
map on the bottom gives 6'1 D/ U ,0 values - - 134

 

Figure 4.27 Streamlines and vorticity sz/ U Go contours at Ad = -36° for different
time offsets (covering one period of the oscillation at jL/Uoo 1: 0.21 with a time

XV

interval of AtUoo/L = 0.365) preceding and following the occurrence of a positive
pressure (p’= Prms at At = 0 in (7)) at W = 3.3 and y/D = 0.5. The color map on the

 

 

 

bottom gives sz/Uoo values 135
Figure 6.] Wiring for Knowles azimuthal microphone array _ - 144
Figure 6.2 Wiring for Panasonic azimuthal microphone array 145
Figure 6.3 Schematic of the Schmitt trigger circuit 146

 

Figure 6.4 Pin-out (left) and function diagram (right) of the Schmitt trigger (model
CD74HCT132E) Error! Bookmark not defined.

 

Figure 6.5 Frequency spectrum of the pressure acting on the cavity’s downstream
wall in the symmetry plane (at Ad= 0°) for [/0 = 3.3, W/D = 7.4 and Re = 12200.
Different colors correspond to different azimuthal locations of the symmetry plane
Error! Bookmark not defined.

 

Figure 6.6 Frequency spectrum of the pressure acting on the cavity’s downstream

wall at Ad= -32° for w = 3.3, W/D = 7.4 and Re = 12200. Different colors
correspond to different azimuthal locations of the symmetry planeError! Bookmark
not defined.

xvi

fb

fD

f,

NOMENCLATURE

Definition

mean-pressure coefficient = (ps-p,)/ (1/2p U002)

coherence between fluctuating pressures p l ' and p 2'

cavity depth

LDA measurement volume size in the streamwise direction
LDA measurement volume size in the wall-normal direction
LDA measurement volume size in the spanvvise direction

corrected output voltage of the hot wire
raw output voltage of the hot wire
frequency in Hz

frequency of the incident laser beam
Doppler frequency

frequency of the scattered laser light
shape factor of the boundary layer = 6'76
cavity length

Mach number

an integer Rossiter mode number

surface pressure fluctuation

xvii

I
p rms

pr

At

U-l-

root mean square value of p'
reference pressure taken as the static pressure in the free stream

mean pressure on the surface of the model

velocity magnitude in x-y plane = \l u2 + v2
Reynolds number based on cavity depth = UQOD/ v

normalized cross-correlation between the streamwise velocity and
pressure fluctuation

normalized cross-correlation between the wall-normal velocity and
pressure fluctuation

cross-correlation between the streamwise velocity and pressure fluctuation

cross-correlation between the wall-normal velocity and pressure
fluctuation

coordinate along the radius of the model
LDA measurement grid spacing in the radial direction

hot-wire temperature

flow temperature when the hot-wire calibration is performed
flow temperature during hot-wire data acquisition

transit time of particle crossing the LDA measuring volume
time offset

mean velocity normalized by wall units = U/u,

xviii

Cl

I
V rms

K1!

free-stream velocity
streamwise velocity
mean streamwise velocity

mean velocity along the azimuthal direction
mean velocity along the radial direction
mean velocity along the streamwise direction

estimated velocity fluctuation

rms of the streamwise velocity
friction velocity = (rh/pWZ

Reynolds shear stress
wall-normal velocity
mean wall-normal velocity

rms of the wall-normal velocity
seed particle velocity vector
cavity width

streamwise coordinate

grid spacing in the streamwise direction

wall-normal coordinate

xix

Ay

¢c

4¢

5¢

grid spacing in the wall-normal direction

wall-normal coordinate normalized by wall units = yu/v
spanwise coordinate

angle between the two incident LDA beams

boundary layer displacement thickness

frequency spectrum of p’

azimuthal angle relative to the top of the model

azimuthal location of the cavity's symmetry plane relative to the top of the

model

azimuthal location relative to the symmetry plane of the cavity = d -dc

LDA measurement grid spacing in the azimuthal location

wavelength of one-dimenaional acoustic planar waves propagating in the
plane wave tube

wavelength of the incident laser beam

kinematic viscosity of air

vorticity vector

out-of-plane vorticity in an x-y plane
estimated out-of-plane vorticity in an x-y plane

density of air

XX

Befiv

mean wall shear stress of the separating boundary layer

momentum thickness of the separating boundary layer

effective momentum thickness of the separating boundary layer

xxi

1. INTRODUCTION

1.1 Background

A cavity in a surface above which fluid flows (see Figure 1.1 for definition of
geometry and coordinate system) can cause large-amplitude unsteadiness in the flow field
that can result in the generation of high-level noise, strong vibration and, in the extreme
case, fatigue of the underlying surface. Moreover, the flow can also be associated with
substantial increase in the drag force on the object containing the cavity. Cavity flow has
drawn the attention of many researchers because of its significance to various engineering
applications. Examples include open automobile windows and sunroofs, dump
combustors, and airplane wheel wells. The importance of cavity flows is also reflected in
the number of recent studies aimed at attenuation of the flow unsteadiness through
various control methods. Recent reviews of these efforts may be found in Cattafesta et
al.1 and Rowley and Williamsz. However, in order to properly implement/optimize the
control techniques, understanding of the mechanisms responsible for the cavity

oscillations is necessary.

Uoo

Separation

'vy‘D/ Point yw
. /////

| L
I 'll

\\\\\.\\

s;

 

Figure 1.1 Illustration of the cavity geometry and coordinate system

A general description of cavity flow is given in the review by Rockwell and
Naudascher3. The behavior of the flow was found to be dramatically different depending
on the parameters of the cavity geometry (cavity length L and depth D: see Figure 1.1) in
addition to the condition of the boundary layer at separation. Sarohia4 defined a rough
boundary between deep and shallow cavities as L/D=1. Cavities with L/D<1 were
considered as deep cavities, while those with L/D>1 were taken as shallow. Deep cavities
act as Helmholtz resonators, and the shear layer above the cavity provides the forcing
mechanism. East5 experimentally studied deep cavities and validated that the resonance
frequencies corresponded to the acoustic depth modes of the cavity. Ma et 01.6 studied
flow-induced Helmholtz resonance experimentally and theoretically. They proposed a
model which successfully predicted the frequency and magnitude of the resonance based
on the momentum balance for a control volume surrounding the resonator’s opening. In
the numerical investigations by Tam7, he found that the acoustic depth modes were
strongly damped due to acoustic radiation for shallow cavities with L/D>l. Acoustic
depth modes could not be sustained in shallow cavities at low Mach number. The present

work is focused on studying shallow cavities only.

Shallow cavities are further classified into open or closed as defined by Charwat

et al.8. A cavity is said to closed if the shear layer reattaches on the bottom wall and
separates again ahead of the downstream edge; otherwise, the cavity is considered open.
In their study of a supersonic cavity flow with a turbulent separating boundary layer,
Charwat et al. found the transition from open to closed cavity flow takes place when the
aspect ratio L/D exceeded 11. On the other hand, in the investigation of avsubsonic cavity
flow with a laminar separation by Sarohia4, the transition occurred at L/D = 7 - 8. Only
open cavities are studied here.

Rossiter 9 identified the basic mechanism leading to strong, self-sustained
oscillations in open cavities at relatively high Mach number. Specifically, small
disturbances in the shear layer separating at the upstream edge of the cavity are amplified,
forming periodic vortex structures that travel downstream and interact with the aft edge
of the cavity, generating strong pressure fluctuations. These fluctuations propagate
(“feedback”) to the upstream edge acoustically and re-excite the shear layer at the
separation point. This forms strong, self-excited, self-sustained flow oscillations, or
“Rossiter” modes. Based on the idea that for self-sustained oscillation to occur, the time
duration for the travel of a disturbance from the upstream lip to the downstream edge of
the cavity and back must equal an integer multiple of the period of the oscillation,

Rossiter proposed the following formula for identifying the frequencies (f) of oscillation:

fl = m" 1.1
U00 M+1/k

 

where, L is the cavity length, U00 is the freestream velocity, M is the Mach number, m is

an integer mode number (1, 2, ...), k is the ratio of the convection speed of the vortex

structures (Uc) to the free-stream velocity, and y is a parameter representing the time

delay between the interaction of the vortex structure with the downstream lip and
subsequent generation of sound. 7 and k are treated as empirical parameters that are
adjusted to fit the observed frequencies of self-sustained oscillation in a particular
experiment. Rossiter employed values of 0.25 and 0.57, respectively, for these
parameters. The “Rossiter mechanism” and equation 1.1 have been validated in many of
the experimental and numerical studies of open cavity flow, especially at relatively high
Mach number (for instance, see Ahuja and Mendoza“), Rowley et a1.” and Murray and
Ukeileylz).

For open cavities, a notable deviation from the Rossiter mechanism that has been
reported at low Mach number, but generally received lesser attention, is the case where
the oscillations of the flow may not involve periodic vortex shedding from the upstream
edge, and the vortex-edge interaction is inefficient in producing significant acoustic
disturbances to excite the shear layer at separation. In this situation, self-sustained
oscillations can still exist, but they are driven by convective waves which cause large
lateral motion (or “flapping”) of the shear layer near the downstream lip of the cavity (e. g.
Sarohia4 and Chatellier eta1.13); and the observed frequencies of oscillation deviate from
those predicted from Rossiter’s equation. Unlike the Rossiter modes, it is believed that
the self-sustained oscillations in this case are driven by the fluctuation of the fluid volume
trapped inside the cavity by the shear layer. Martin et al.14 and Rockwell15 predicted the
frequency of the oscillation tones by invoking a 1: phase angle between the entrapped-

volume fluctuation and the lateral displacement of the shear layer at separation.

The flow dynamics of cavities is further complicated by the possible existence of
a completely different mode of oscillation. Gharib and Roshkol6 investigated the flow
over an axisymmetric cavity with a laminar boundary layer at separation and showed that
above a certain value of L/D, the flow field became unstable on a large scale, comparable
to the depth of the cavity. The resulting flow structures resembled those found in a bluff-
body wake. This behavior was referred to as ‘wake mode’. Gharib and Roshko observed
an abrupt, large increase in the drag force on the cavity associated with the wake mode
due to the large difference in pressure acting on the downstream and upstream cavity wall.
The wake mode was also found later in the numerical simulation by Najm and
Ghoniem17. Although not referred to as the wake mode, these authors observed that for
L/D > 2, the cavity flow switched from being driven by shear-layer instability to being
dominated by shedding of larger-scale, lower-frequency vortices from the recirculation
zone inside the cavity. More recently, Rowley et al.11 showed that the transition of the
flow from shear-layer (Rossiter) to wake mode of oscillation took place when the length
of the cavity became large compared to the upstream-boundary-layer thickness, or as the
Mach or Reynolds numbers were raised. Rowley et al. suggested that the wake mode was

sustained by an absolute instability of the flow.

1.2 Motivation

Even though there is a large body of literature on cavity flows, most of them are
based on the assumption that the flow is two-dimensional. However, in reality several
sources of three-dimensionality can be identified. Of these, the most prominent are: (1)
the effect of the side walls of the cavity; and (2) the instabilities arising from the
recirculation flow within the cavity. Very few studies exist of the three-dimensional
effects resulting from these influences and if/how these effects alter the self-sustained

cavity oscillations

In the experiments of Maull and East”, the flow in the cavity was found to have
cellular structures along the span. The number, configuration and distribution of the cells
were affected by the width-to-length ratio (W/L), depth-to-length ratio (D/L) and the
shape of the cavity. No quantitative measurements of the cavity unsteadiness were
reported. Rockwell and Knisely ‘9 utilized hydrogen bubble to visualize the three-
dimensional nature of the unsteady flow past a cavity. They found secondary, spanwise-
periodic, streamwise vortices to form and distort the primary spanwise vortex structures.
Although they could not elaborate on the mechanism that led to the formation of the
streamwise vortices, they suggested that the recirculation flow inside the cavity had an
influence on the formation process. More recently, three-dimensional-instability analysis
of a cavity flow that is two-dimensional in the mean was done by Bres and Coloniuszo.
They found that the most amplified three-dimensional mode had a spanwise wavelength
that scaled with the cavity depth, and atypical frequency that was an order of magnitude

smaller than the frequency of the self-sustained cavity oscillations. Bres and Colonius

related this mode to the centrifugal instability of the primary recirculation flow inside the

cavity.

None of the above studies reported how the cavity width and the side walls affect
the self-sustained oscillations. In experiments, the influence from the cavity side walls is
typically minimized by simply using a large cavity width-to-depth ratio and focusing only
on the symmetry plane along the span. Real cavities, however, have finite width; but
present understanding of the self-sustained oscillations in flows over cavities ignores this
fact. This implicitly assumes that the cavity width is immaterial for the establishment of

cavity oscillations.

Plumbee et al.2‘ observed that a factor of 2 change in the cavity width had no
effect on the frequency of the first two Rossiter modes at M = 2. The effect of the cavity
width on the amplitude of the oscillation was not reported. Studies on the radiated noise
from cavities by Block22 showed that the cavity width appeared to determine whether the
cavity acts as a narrowband or broadband source at a given velocity. She observed that
for cavities with L/D z 1 and 2, the radiated noise from the narrower cavity had a sharper-
shape spectrum and higher amplitude than that from the wider cavity even though the
spectra peaked at the same frequency. In contrast, data from Abuja and Mendozal0 on the
far-field acoustic radiation from the cavity flow indicated trends opposite to that reported
by Block. Ahuja and Mendoza noted that the flow was two-dimensional if W/L > 1, while
three-dimensional effects became notable for W/L < l. Far-field acoustic measurement
showed that the three-dimensional cavity flow (W/L < 1) produced lower levels of cavity
feedback tones as Well as broadband noise compared to the two-dimensional cavity flow

(W/L > 1). They attributed the reduction of sound pressure level to the loss of coherence

of the shear layer across the cavity width. However no evidence was provided regarding
the coherence of the shear layer along the span and how the coherence was related to the
mechanism driving the oscillation. Moreover, there were no measurements
demonstrating that the oscillations inside the cavity had the same characteristics as the
radiated noise in the far field. Both Block22 and Ahuja and Mendoza‘o’s studies also
focused only on the high-frequency far-field noise radiated from the cavity. The effect of
cavity width/side walls on the oscillations and three-dimensional flow structures within

the cavity were never studied in detail.

Larcheveque et 01.23 numerically studied a three-dimensional cavity flow with
L/D of 5 and W/L of 0.2 at M = 0.85. They reported that the turbulent kinetic energy
distribution in the mixing layer was less two-dimensional at low frequency, especially
near the afi wall. They claimed that the strong streamwise vortices ejected near the
comers induced strong low-fi'equency three-dimensionality. There was, however, no
detailed discussion of this low-frequency, the three-dimensionality and the vortices near
the comers since the pressure fluctuations on the floor of the cavity in their case were

dominated by Rossiter modes which were homogeneous in the spanwise direction.

1.3 Objectives

The current study is focused on examining the unsteady flow and surface-pressure
in a low-Mach-number cavity flow with a turbulent boundary layer at separation. The
primary concern is to investigate the effect of the cavity width and side walls on the flow.
In particular, answers to the following questions are sought: what is the effect of the
cavity width and side walls on the unsteady-flow behavior? If two-dimensional self-
sustained oscillations exist, does the cavity width/side wall alter the strength, frequency,
or mode shape of these oscillations? If so, what is the physical mechanism through which

this alteration takes place?

For the purpose of the investigation, an axisymmetric cavity geometry is
employed. The axisymmetric configuration has a distinct advantage in studying the
cavity-uddth/side-wall effects. In particular, the geometry is free from any side-wall
influence, as seen in Figure 1.2 (a), but could be partially filled to form finite-width
cavities, as depicted in Figure 1.2 (b). It is significant to contrast the present work to the
study of Block”, Ahuja and Mendoza10 and Larcheveque et al.23. These studies
employed rectangular cavity geometry that had to be terminated with end walls (even for
the widest cavity). One of the unique features of the present study is the “reference”
condition established by the axisymmetric cavity that is free of any side-wall influences
and the comparison of that condition with cavities with different widths. The handful
experimental studies involving axisymmetric cavity flows in the literature (Sarohia4 and
Gharib and Roshko”) did not include any examination of the effect of the cavity width.
Moreover these experimental studies only considered the laminar boundary layer case.

Another unique aspect of the present work is the extensive space-time unsteady wall

pressure information collected along the streamwise direction on the cavity floor and the
azimuthal direction on the downstream wall simultaneously with flow velocity

measurements.

(a) .
Upstream cavity wall /’ Downstream cavrty wall

1 /

    

4%
tr...»

0))

Upstream cavity wall / Downstream cavity wall

fit?

 

Figure 1.2 Cavity model: (a) axisymmetric cavity (b) finite-width cavity

1.4 References

 

I Cattafesta, L., Williams, D., Rowley, C., and Alvi, F., “Review of Active Control of
F low-Induced Cavity Resonance.” AIAA Paper 2003 ’ 3567 (2003)

2 Rowley, CW. and Williams, D.R., “Dynamics and Control of High-Reynolds-Number
Flow over Open Cavities,” Annual Review of Fluid Mechanics 38, pp. 251 7 276 (2006)

3 Rockwell, D. and Naudascher, E., “Review — self-sustaining oscillations of flow past
cavities,” Transactions of the ASME: Journal of Fluids Engineering 100, pp. 152 — 165
(1978)

4 Sarohia, V., “Experimental Investigation of Oscillations in Flows Over Shallow
Cavities,” AIAA Journal 15 (7), pp. 984' 991 (1977)

5 East, L.F., “Aerodynamic Induced Resonance in Rectangular Cavities,” Journal of
Sound and Vibrations 3, pp. 277‘ 287 (1966)

6 Ma, R., Slaboch, P. and Morris, S.C., “ Fluid Mechanics of the F low-Excited Helmholz
Resonator,” Journal of Fluid Mechanics 623, pp. 1-26 (2009)

7 Tam, C.K.W., “The Acoustic Modes of a Two-Dimensional Rectangular Cavity,”
Journal of Sound and Vibrations 49 (3), pp. 353 ' 364 (1976)

8 Charwat, A.F., Roos, J .N., Dewey, RC. and Hitz, J.A., “An Investigation of Separated
Flows. Part 1. The Pressure Field,” Journal of the Aerospace Sciences 28, pp. 457—470
(1961)

9 Rossiter, J.E., “Wind-tunnel Experiments on the Flow over Rectangular Cavities at
Subsonic and Transonic Speeds,” Technical Report 3438, Aeronautical Research Council
Reports and Memoranda (1964)

m Abuja, K.K and Mendoza, J. “Effects of Cavity Dimensions, Boundary Layer, and
Temperature on Cavity Noise with Emphasis on Benchmark Data to Validate
Computational Aeroacoustic Codes”, Contractor Report 4653, National Aeronautics and
Space Administration (1995)

H Rowley, C.W., Colonius, T. and Basu, A.J., “On Self-sustained Oscillations in Two-
dimensional Compressible Flow over Rectanguler Cavities”, Journal of Fluid Mechanics
455, pp. 315 — 346 (2002)

'2 Murray, N. and Ukeiley, L., “Estimationof the flowfield from surface pressure
measurements in an open cavity,” AIAA Journal, 41 (5), pp. 969-972 (2003)

ll

 

‘3 Chatellier, L., Laumonier, J., and Gervais, Y., “ Theoretical and Experimental
Investigations of Low Mach Number Turbulent Vavity Flows,” Experiments in Fluids
36, pp. 728 ’ 740 (2004)

M Martin, W.W., Naudascher, E.,and Padmanabhan, M., “F luid-dynamic Excitation
Involving Flow Instability,” Journal of the Hydraulics Division, ASCE 101 ( HY6), pp.
681 "698 (1975)

'5 Rockwell, D., “Prediction of Oscillation Frequencies for Unstable Flow Past Cavities,”
Journal of Fluids Engineering 99, pp. 294 ’ 300 (1977)

16 Gharib, M., Roshko, A.F., “The Effect of Flow Oscillations on Cavity Drag,” Journal
ofFluid Mechanics 177, pp. 501 ’ 530 (1987)

l7Najm, H.N., Ghoniem, A.F., “Numerical simulation of the convective instability in a
dump combustor,” AIAA Journal, 29 (6), pp. 911 ' 919 (1991)

'8 Maull, D.J., and East, L.F., “Three-dimensional Flow in Cavities,” Journal of Fluid
Mechanics 16, pp. 620' 632 (1963)

'9 Rockwell, D., and Knisely, C., “Observations of the Three-dimensional Nature of
Unstable F low Past a Cavity,” Physics of Fluids 23 (3), pp. 425 ’ 431 (1980)

20 Bres, G.A., and Colonius, T., “Three-dimensional Instability in Compressible Flow
over Open Cavities,” Journal of Fluid Mechanics 599, pp. 309-339, (2008)

2] Plumbee, H.E., Gibson, J .S. and Lassiter, L.W., “A Theoretical and Experimental
Investigation of the Acoustic Response of Cavities in an Aerodynamic Flow”, Technical
Report WADD—TR-61-75, US Air Force (1962)

22 Block, P.J.W., “Noise Response of Cavities of Varying Dimensions at Subsonic
Speeds”, Technical Notes D-835I, National Aeronautics and Space Administration (1976)

23 Larcheveque, L., Sagaut, P., Le, T. and Comte, P., “Large-eddy Simulation of a
Compressible Flow in a Three-dimensional Open Cavity at High Reynolds Number”,
Journal of Fluid Mechanics 516, pp. 265 ' 301 (2004)

12

2. EXPERIMENTAL SETUP AND MEASUREMENT

TECHNIQUES

2.1 Wind Tunnel Facility and Cavity Model

The present experiment is conducted in an open-circuit wind tunnel in the Flow
Physics and Control Laboratory (FPaCL) at MSU. A schematic drawing of the wind
tunnel is shown in Figure 2.1. The tunnel has a 1.829 m-long test section downstream of
a contraction with a 6.25:1 area ratio and a 1.549 m-wide by 1.549 m-high inlet cross
section. An aluminum honeycomb followed by three stainless steel mesh screens and a
non-woven fiberglass filter are mounted ahead of the contraction inlet to reduce the
turbulence intensity within the test section to less than 0.5% based on the streamwise
velocity component. A false ceiling with an adjustable angle is used to maintain zero
pressure gradient in the test section along the streamwise direction starting from a cross
section area of 0.610 m- by 0.610 m at the upstream end. Two slots with a width of
0.0127 m along the centerline of the test section’s false and main ceiling allow for the
insertion of probes. The air flow exiting the test section is slowed down through a
combination of a pre-diffuser and a diffuser. The bottom wall of the pre-diffuser has an
angle of 6 degrees with respect to the horizontal direction; while the top wall is hinged on
the downstream end and is set to a typically very small angle to accommodate the
position of the test-section’s movable ceiling at the exit of the latter. The diffuser has a
divergence angle of 5.9 degrees. A gap filled with rubber is left between the pre-diffuser

and the diffuser to minimize the transmission of vibration from the fan to the test section.

13

Acrylic windows in the side walls and bottom of the test section, as well as in the
pre-diffuser enable optical access to the tunnel. A Pitot tube with an outside diameter of 3
mm and an Omega (DP-25-TH) thermistor are inserted in the wind tunnel to monitor the
free-stream velocity and temperature respectively. The pressure differential produced by
the Pitot tube is monitored used 10 torr Baratron (model 223BD) pressure transducer with
0.0075 V/Pa sensitivity. On the other hand, the sensitivity of the temperature sensor is 0.1

V/°C.

Pitot
Tube

. . Test Adjustable .
Fan\ DUES Pre-Drffiser Section Ceiling /?tracthn

1.486 1.829 1.005 ‘ 1.829 2.096 7 ' i. . i
, >4 W >477 7>< 7 7 *7. w i ‘

Thermometer

   

Dimensions in m
Figure 2.1 Schematic diagram of the wind tunnel

The model used to establish the cavity flow is based on the axisymmetric
backward—facing-step model employed in different earlier studies at the FPaCL (Lil,
Hudyz, Trosin3 and Aditjandra4). The dimensions of the back-step model are shown in
Figure 2.2. A spherical nose is used at the leading edge of the model. Because it is
desired to establish a turbulent boundary layer at the separation point of the back step,
any disturbances caused by the curvature discontinuity between the nose and the
stationary section immediately downstream benefit the quicker development of the
boundary to a turbulent state. In addition, sandpaper is used at the upstream end of the

model to hasten the boundary—layer transition. Cavities with different lengths and widths

are formed on the model by assembling a movable ‘Downstream Cavity Wall’ and
‘Inserts for W/D Control’ on the model downstream of the back step, as seen in Figure
2.3. Both parts are made from two halves which could be assembled around the central
shaft inside the cavity. The downstream cavity wall is movable along the model’s axis in
order to enable variation of the aspect, or L/D, ratio of the cavity. The inserts for W/D
control are employed to fill a portion of the cavity along the azimuthal direction in order
to create finite-width cavities with different width-to-depth (W/D) ratios. By varying the
length and azimuthal extent of the inserts it is possible to vary L/D and W/D. Note that W
is defined here as the azimuthal arc length at a height of D/2 above the cavity floor (D =

12.2 mm).

The downstream cavity wall is composed of a 152 mm-long cylindrical shell
upstream of a 50.8 mm-long conical shell. The latter is employed to eliminate any
possible influence of disturbances from the flow past the downstream step of the
cylindrical shell on the flow structure in the cavity. Preliminary tests show that when the
cylindrical Shell’s length is larger than approximately 100 mm, the pressure signature on
the cavity floor remains independent of the Shell’s length. A 12.2 mm-long segment at the
upstream end of the cylindrical shell is detachable (labeled “Detachable Sensor Ring” in
Figure 2.3) and is employed to house an azimuthal microphone array for measuring the
unsteady pressure on the downstream cavity wall. Details of this array are described in

section 2.2.3.

15

R 0063 0.0122 m back

' step

< 00.126
9 cpo.102

Sand Paper

 

0.260

1.100 7 l 7 0.488 A ‘

1.219

¢ 77777 ”if , we >

2.383
Dimensions in m

Figure 2.2 A schematic drawing of the axisymmetric test model

Downstream Cavity Wall
Detachable Sensor Ring
Azimuthal Microphone Array
Cylindrical Shell

Inserts for W/D Control fl
\

    
   

Central Shaft \

Streamwise Microphone Array \ \

  
  

Upstream Cavity Wall

t - :Ffl\ (back step)

Figure 2.3 Three-dimensional drawing of the cavity model (note Wis measured at
height of D/2 above the cavity bottom)

2.2 Measurement Techniques

The measurement techniques employed in this study include hot wires, “static-
pressure taps”, microphones and Laser Doppler Anemometry (LDA). A single hot wire is
used to measure the boundary-layer streamwise-velocity profile at the separation point to
characterize the state of the boundary layer. 56 static pressure taps and 32 microphones
are embedded in the model downstream of the backward-facing step. The pressure taps
are used to align the model in the wind tunnel. The 32 microphones form a “streamwise
microphone array”, which is employed to measure the unsteady wall pressure on the
cavity bottom. In addition, arrays of microphones are integrated into the downstream wall
of the cavity to characterize the azimuthal distribution of the unsteady wall pressure. The
streamwise (u) and wall-normal (v) components (in the x and y directions respectively) of
the velocity field inside and above the cavity are measured using Laser Doppler

Anemometry. Further details of each of the employed methods are given below.
2.2.1 Hot-wire Measurements

A single hot wire is employed to measure the velocity profile of the boundary
layer at the separation point. The hot wire is made from tungsten with a diameter of 3.75
pm and a sensing length of about 0.8 mm. This corresponds to a wire length-to-diameter
ratio of more than 200 and a typical electrical resistance of 5 Q. A Constant Temperature
Anemometer (CTA), Dantec Mini CTA 54T30, is used to operate the hot wire, with the
overheat ratio set to 1.62. The hot wire prObe is mounted on a 15" Velmex A2515C-82.5
screw-drive Unislide traverse with 40 turns per inch of movement. The traverse is driven

by a TMG 56188-01 hybrid stepper motor which has an angular resolution of 800 steps

17

per revolution. The traverse system is employed to move the hot wire along the y

direction with a resolution of 0.823 um per step (based on calibration of the movement).

The velocity-profile measurement is conducted at. 5 mm upstream of the
separation point, starting from about 0.3 mm above the wall. The probe is traversed in the
y direction with an increment that varies from 0.08 mm closest to the wall to 5 mm in the
free stream. To determine the initial position of the hot wire above the wall, the probe is
imaged using a standard-video Sony CCD camera (XC-75), which is connected to a
National Instruments IMAQ PC-board PC1-1411 frame grabber. The imaging is achieved
using a Nikkor 500 mm lens and an illumination source consisting of a high-intensity
white light (a Stocker & Yale, Inc, Lite Mite Series Model 13 Plus). Figure 2.4 shows an
image exemplifying those used to determine the wire height above the wall. The height is
determined by the number of pixels in the image between the wire (i.e. end of the probe
prongs) and the surface of the model with known image scale (typically 0.03 mm/pixel)
that is calibrated by determining the average number of pixels by which the wire is

traversed in the image when it is moved 1 mm upwards in the y direction.
Because of the directional ambiguity of the hot wire, measurement in the

boundary layer yields the velocity magnitude q (q = \lu2 +v2 where u and v are the

streamwise and wall-normal velocity components respectively). Since the wall-normal
velocity component is much less than the streamwise velocity in the boundary layer, the

measurement could be interpreted as the streamwise velocity u.

18

Single hot-u ire probe

Flow
—.

 

Figure 2.4 An image of the single hot wire above the axisymmetric model surface

The hot wire is calibrated in the free stream in the wind tunnel against the Pitot
tube, shown in Figure 2.1, coupled with a pressure transducer (Baratron 223 BD-
000010ACU with measurement range 0-1333 Pa). The output of the hot wire, pressure
transducer and thermometer at different tunnel velocities, are sampled at a frequency of
1000 Hz for 10 seconds. The variation in the average voltage (E) of the hot wire with

free-stream velocity (U00) is curve-fitted to the “King’s-law” equation:

E2=A+BU;, 21

where n is set to 0.45. Temperature correction to the output voltage of the hot wire is

conducted using the equation

Tw _ Tcal
Tw - T,

5:5, 12

to eliminate the effect of any temperature variation between the calibration and data
acquisition; where E, is the raw output voltage of the hot wire, Tw is the hot-wire

temperature, Tea, is the flow temperature when the calibration is performed, and T, is the

flow temperature during acquisition. Figure 2.5 shows an example of the hot-wire
calibration data and King’s law curve fit. For all measurements reported here, the
maximum error between calibrations conducted before (pre) and after (post) acquisition

of data was less than 2%.

i” ' . I .
r —e-pre calibration i =

1.9 i —°T post calibrationj ‘

 

 

o 5 1o 15 2o 25

Uw(m/s)

Figure 2.5 Example of a hot-wire calibration
2.2.2 Static Pressure Taps

Downstream of the step (the separation point), there are four slots cut in the
model at 90° intervals around the perimeter for insertion of wall sensors. 32 pressure taps
are located next to the streamwise-array microphones in a plug fitted in the top slot. The
microphones, which are deployed along the centerline of the model, are offset from the
pressure taps by 3.2 mm center-to-center in the azimuthal direction. The taps have an

inside diameter of 1 mm and are spaced 4.75 mm apart in the streamwise direction,

20

starting from 5.3 mm downstream of the step. The remaining 24 pressure taps are
embedded in plugs inserted into two side and one bottom slots of the model; i.e. eight
taps per plug. These taps are spaced 9.5 mm apart in x starting at 19.55 mm downstream
of the step. All of the taps are connected using Urethane tubes to a Scanivalve pressure
scanner system. The output port of the Scanivalve is connected to the positive input of a
pressure transducer (Setra model 239 with measurement range 0-0.5 inch H20). The
negative input of the transducer is the reference pressure and is connected to the static-
pressure port of the free-stream Pitot tube. Stepping of the Scanivalve is controlled using
a digital-to-analog converter of a general purpose National Instruments data acquisition
PC board. The board is controlled using a LabView program running on a PC computer.
The static pressures are acquired for 10 seconds with a sampling frequency of 100 Hz.
One second time delay is used between measurements from consecutive ports to
eliminate transient effect associated with switching pressure ports. More details of the

pressure taps and the Scanivalve system can be found in section 2.3 of Hudyz.
2.2.3 Microphone Arrays

An array consisting of 32 Knowles Electronics FG-23629-P16 microphones are
mounted along the centerline of the top slot in the model; labeled ‘Streamwise
Microphone Array’ in Figure 2.3. The sensors, which have an outside diameter of 2.54
mm and sensing diameter of 0.75 mm, are spaced 4.75 mm apart in the streamwise
direction, starting from 5.3 mm downstream of the upstream cavity edge. Additional
information about the streamwise microphone array including the circuitry can be found

in section 8.4 of Hudyz.

21

Another wall-pressure-sensor array, labeled ‘Azimuthal Microphone Array’ in
Figure 2.3, is integrated into the downstream wall of the cavity at a height equal to half
the cavity depth (i.e. 6.1 mm) above the cavity floor. This array is embedded in the 12.2
mm-long Detachable Sensor Ring (shown in Figure 2.3), which could be replaced so that
arrays with different configurations could be employed along the azimuthal direction.
Two such arrays are used in this study. One consisting of 14 Knowles Electronics PC-
23629-Pl6 microphones, deployed over only a portion of the perimeter, is used for the
study of cavities with finite width; another array with 16 Panasonic WM-61-A
microphones covering the whole circumference of the model is used to characterize the

azimuthal variation of the unsteady pressure in the axisymmetric cavities.

The Knowles Electronics microphones have the same dimensions as the ones in
the streamwise array. The arrangement of these microphones in the azimuthal array is
shown in Figure 2.6. They extend over an azimuthal angle, d (see definition of d in
Figure 2.6), range of -40° to +20° with an inter-sensor spacing of 4° except near -40°
where a coarser spacing of 8° is used. This array is used for measurements in cavities
with azimuthal extent varying between 30° and 90°. In contrast, the Panasonic
microphones have an outside diameter of 6 mm and sensing diameter of 2 mm with a
uniform azimuthal spacing of 22.5°. 32-AWG wires are used to connect the microphones
with the operating circuits (see Appendix 6.] for detailed description of the construction
of the microphone arrays). The wires are threaded to the outside of the test section

through the gap between the cylindrical shell and the model’s surface.

22

Facing downstream Centerline aligned with

streamwise microphone array

  
  

Cavrty downstream Wall \\ 4'00

8.0°\ /- 1‘“

   

Microphone 14

 

Figure 2.6 Configuration of the azimuthal Knowles Electronics microphone array

The streamwise-army’s microphones are calibrated individually in-situ against a
B&K 4938-A-011 1/4" microphone with a manufacturer-supplied response. The
calibration is performed by subjecting the Knowles and B&K microphones to sound
waves in a custom made plane wave tube (PWT) that could be clamped on the model (see
Figure 2.7) in the wind tunnel. The tube is driven by a speaker located upstream of the
entrance of the wind tunnel. The PWT produces plane sound waves that travel parallel to

the axis of the tube. Such one-dimensional planar waves with wavelength 1,, are
achievable in a square duet with rigid walls and side length 2a when Xa>4a, or f<c/4a

(where f is the sound frequency and c is the speed of sound); see Kinsler et al5. The width
of the PWT in this study is 18.3 mm, which gives an upper frequency limit of 9.6 kHz for

a speed of sound of 350 m/s.

23

To conduct the calibration, the PWT is clamped on the model as depicted in
Figure 2.7 with the centerline of the tube aligned with the center of the microphone holes.
The B&K microphone is mounted on the centerline of the moveable upper wall of the
PWT. By sliding the upper wall it is possible to align the B&K microphone with each

Emkay microphone in the sensor array to perform the calibration.

B&K
\licroplmne

 

Figure 2.7 Photographic views of the PWT for calibration of the streamwise-army’s
microphones

To verify the planar-sound-wave nature of the sound traveling within the PWT,
measurements were conducted using two B&K microphones mounted at the same cross
section of the tube. This was done at three different cross sections, including those
corresponding to the beginning and end of the tube. The data were used to calculate the
ratio of sound amplitude measured by each of the B&K microphones and the phase shift
between the two measurements. Figure 2.8 provides the average of the results for the
three cross sections. The results in Figure 2.8 are shown for the frequency range
extending up to 1000 Hz only, which is well beyond the highest frequency of wall-
pressure fluctuations found in the cavity flow. As seen from the figure, the results

indicate uniform sound amplitude and phase across a given cross section of the PWT; i.e.,

24

corresponding to planar sound waves, up to the end of the frequency range of interest.
For reference, the individual sensitivity values for the upstream most 24 microphones
used in this experiment are plotted in Figure 2.9. The phase-angle calibration shows a
maximum 12 us time delay between any pair of Emkay microphones over all frequencies,

which is less than 0.4% of the shortest time scale of interest (300Hz) in the cavity flow.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.4
1.2-... ..................... é ...................... .
g
a) 08 ........................ .
“D g
g 0.6
5 0.4* ...................... ....................................................................... ...................
0.2 I
00 200 400 600 800 1000
f (HZ)
90
6() ..................................................
g’ 30 ................................................................................................................................
a?
.a 0 fl ,
m
g _30_ .................................................. é ........ .1
e
-60* ....................... . ..................................................... ., ....................... .
-q _ . ‘
) )0 200 400 _ 600 800 1000
1 (Hz)

Figure 2.8 Calibration of the PWT: amplitude ratio (top) and phase shift (bottom)

25

5 1’0 1‘5 20'" 25
microphone number

Figure 2.9 Sensitivity of the streamwise-array’s microphones

The azimuthal-army’s microphones were also calibrated individually, after
insertion into the cavity ’s end wall, against a B&K 4938-A-011 1/4" microphone in a
custom made PWT outside the wind tunnel, as shown in Figure 2.10. The PWT was
clamped on a metal plate which has a circular cut-out in the middle to accommodate the
ring into which the azimuthal-army’s microphones are fitted. The PWT is driven by a

speaker located at one of the tube’s ends.

26

‘ Azimuthal
Microphone

 

Figure 2.10 Setup for calibration of the azimuthal-army’s microphones

For reference, the individual sensitivity values for the azimuthal-army’s
microphones are given in Figure 2.11. Also, the calibration results show that the time
delay between any pair of microphones is less than 15.8 us and 10 us within the
frequency range of interest for the Knowles and Panasonic microphones respectively.

This is less than 0.5% of the shortest time scale of interest (300Hz) in the cavity flow.

27

 

 

 

 

i ,
28.- ; (a) _. 9.51 ~ (b)
26- i ‘ ‘- :
6‘3 24 g '
s s 8 5
E 22r . E .
20L ' . 8’ j
18» . i 7-5' i
160” s ' 1‘0 ’ 1'5 70 T 5 110 1'5 20
microphone number microphone number

Figure 2.11 Sensitivity of the azimuthal-array’s microphones: (a) Knowles
microphones (see Figure 2.6 for microphone numbers) and (b) Panasonic
microphones (microphone numbers are based on viewing the cavity’s end wall from
inside the cavity. Number increases in the clockwise direction starting with
microphone 1 at the top)

2.2.4 Data Acquisition Hardware

The microphone array data are acquired using two National Instruments A/D PC-
based boards (NI-6024E and AT-MIO-16E-10), each of which has 16 single-ended,
analog-input channels. The number of streamwise-army’s microphones used depends on
the length of cavities. Only the most upstream 12 microphones are used in most of this
study. In addition, data from a microphone under the downstream wall of the cavity are
acquired. These data are used to provide a reference signal for cancellation of any
electrical or acoustic fan noise using the optimal filtering technique described in Naguib
et al.6. The streamwise-army’s microphones, as well as signals from the pitot tube and
thermometer which are used to monitor the tunnel speed, are connected to the AT-MIO-
16E-10 board. The board has a 100 kHz maximum sampling rate and a 10 us inter-
channel time delay. On the other hand, the azimuthal-army’s sensors are connected to the

NI-6024E board which has a maximum sampling rate of 200 kHz and an inter-channel

28

time delay of 5 us. The inter-channel time delay between those wall-pressure sensors,
when accumulated across all acquired channels, produces a maximum time lag of 110 us ,
which is 3.3% of the period of the highest frequency of pressure fluctuations found for
the low-Mach-number cavity flow examined here (300 Hz).

The two A/D boards are synchronized by a trigger pulse (AI Scan Start Trigger)
that is sent from the AT-MIO-16E-10 board through the programmable firnction input
port 7 ( PFI 7) to PFI 0 of the NI—6024E board which is set to accept an external channel
scan clock. The AI Scan Start Trigger which reflects the actual start pulse that initiates a
scan of the AT-MIO-l6E-10 board acts as the channel scan clock for the NI-6024E board.
During operation, the NI-6024 board is started first, putting it in an idling state in which it
waits for the external clock trigger. Once, the AT-MIO-16E-10 board is run, it sends out
the AI Scan Start Trigger which initiates the data sampling of both boards at the same
exact time and at the same sampling rate. Synchronization tests are performed by sending
pure sine signals from a function generator with frequencies up to 1000Hz to different
channels on the two boards; no phase delay is observed between channels on the two
different boards (except that caused by the channel multiplexing; the inter-channel delay
discussed above).

2.2.5 Laser Doppler Anemometry

A two-component, dual-beam Laser Doppler Anemometer (LDA) is employed to
measure u and v at a number of points in the cavity. The LDA technique is based on the
Doppler frequency shift of the light scattered from particles that are seeded into the flow.

The frequency of the scattered light f, is given by equation 2.2 (see Adrian7), when

collecting the light using a stationary detector; where fb and 4b is the frequency and

29

wavelength of the incident light, 9", is the particle velocity. This process is illustrated in

Figure 2.12 (a).

T '(ér—é)
fr=fb+v—p—Ab—b‘a 2-2

where 5,, and 5, are unit vectors indicating the direction of the incident beam and

detected scattered light respectively. In the dual-beam configuration (Figure 2.12 b), the

frequencies of the scattered waves from each of the two incident beams can be written as:

17 '(é -él) ‘7 '(é —é2)
f1=fb+p—f—, f2=fb+ ” ’
b 4b

 

2.3

In practice, the light detector cannot respond fast enough to sense frequencies f1 and f2
which are of the order of the frequency of light, and hence the detector only measures the

light modulation at the frequency difference fD (normally called the Doppler frequency):

 

15p {52 —51) = 28in(a/2)v ,

, 2.4
4b 4b x

fD=f1—f2=

where vJ|C represents the particle velocity component along the x direction in Figure 2.12

(b); i.e. the direction perpendicular to the bisector of the angle, a; between the two beams.
Given that a and the wavelength of light are known parameters, the particle velocity can
be derived from measurement of f0 employing equation 2.4. Note that fD is independent

of the position of the detector in the dual-beam configuration.

30

(a)
Incident laser beam fb , lib

\eb\VP d€tector
'3 e, y
we

Moving particle

 

 

 

(b)

Incident laser beam fb , Ag,
61 Vp detector

 

 

 

Incident laser beam fl, , lib

 

Figure 2.12 Illustration of light scattering from a moving particle: (a) Single-beam
configuration (b) Dual-beam configuration

In the present study, a TSI six-jet atomizer (model 9306A) is used to generate the
seeding particles from Propylene Glycol solution (one part of Propylene Glycol mixed
with 5 parts of water by volume). The atomizer is driven using pressurized air from a
compressor line. The density of the generated particles can be controlled by the number
of jets and the input pressure. The seeding particles have a mean diameter of 0.35 pm and
a maximum diameter of 4 pm based on the specifications of the atomizer. This results in

a cut-off frequency of 3 kHz up to which a particle with the maximum diameter can

31

follow the flow (based on 3 dB cut-off: see Adrian7). Particles are injected into the flow
at the center of the wind tunnel inlet through a 25.4 mm-diameter tube as shown in Figure
2.13. This focuses the seeding only near the surface of the axisymmetric model and inside
the cavity. Measurements of the fluctuating wall-pressure measurements on the cavity
floor and the boundary-layer velocity profile at separation with and without the seeding
tube upstream of the tunnel entrance show that the presence of the tube upstream of the

tunnel inlet does not change the characteristics of the cavity flow.

/
/
/

lul‘c

Monti/er

 

Figure 2.13 Image of the seeding system upstream of the wind tunnel inlet section
The laser beam is generated from a 300 mW argon-ion laser (model 5500A, Ion
Laser Technology). This model has standard 300 mW multi-line output. The output beam
could be tuned to single line beams from 457 nm to 514 nm using a prism. Beams with

three different wavelengths of 514.5 nm (green), 488 nm (blue) and 476.5 nm (violet), at

32

single line energy of lOOmW, 100 mW and 30 mW respectively, are used here. A
FiberFlow transmitter from Dantec (Figure 2.14) couples the laser beam to an 85 mm-
diameter 2D Dantec probe (Figure 2.15) that steers and focuses the laser beams. Inside
the FiberFlow transmitter, the incident laser beam first passes through a Bragg cell, which
diffracts the light splitting its power, approximately equally, between the zero- and the
first-order diffracted beams and also shifts the frequency of the first-order beam by 40
MHz. The shifted and main beams are separated via a prism into three colors: green
(514.5 nm), blue (488 nm) and violet (476.5 nm) for measurements of the three velocity
components. Each of the resulting six beams is focused and coupled to a fiber optic cable
using micro manipulators to adjust the relative position and angle of the beam and fiber.
Only the green and blue beams are used 'in this study for measurement of u and v
respectively. The fibers transmitting these two beams are coupled with the 85 mm probe
which is mounted on a three-dimensional traversing system from Dantec to allow the
movement of the measurement point of the LDA system (see figure 2.15). The probe
contains provisions for steering each of the four beams before focusing them at the
measurement point using a lens with a focal length of 500 mm. The same lens is also
used to collect the light scattered from the moving particles and focus it on the receiving
fiber optic cable. In turn, the latter is coupled to two photo multipliers: one responding to

green and the other to blue light. The output of each of the photo multipliers (the

“Doppler burst signal ” ) is fed to and analyzed by a Dantec Burst Spectrum Analyzer

(BSA 57N1 l) to extract the Doppler frequency (fly) from each burst (resulting from light
scatter from a single particle passing thought the measurement volume) and convert it to

velocity.

33

Precise alignment had to be done before measurements to couple the laser beam to
the transmitter, couple the beams from the transmitter to the optical fibers and locate the
beams intersection at the focal point of the scattered-light receiving lens. The procedure
to accomplish these tasks is given in Appendix 6.3. For the measurements reported here,
the spacing of each of the green and blue beam pairs as they approach the 500 mm
focusing lens is 65mm. This gives a beam crossing angle of a = 7.44 degree. The
corresponding measurement volume size is dJr = 0.123 mm, dy = 0.123 mm and dz = 1.889
mm (in the x, y and azimuthal directions respectively) for the green beam and dx = 0.116

mm, dy = 0.116 mm and dz = 1.889 mm for the blue beam.

«1: Green beam fiber .
Argon-Ion laser

‘/Blue beam fiber
(il‘ccn ' Manipulator Transmitter f I
Photmnultiplier 1

(in con
Photomultiplier

 

Figure 2.14 Picture of the FiberFlow optics system

34

.l . -
11 ”E W --
85mm Test "

.-
pmbe section

 

Figure 2.15 Picture of the LDA probe mounted on the three-dimensional traversing
system

The burst signals of the scattered green and blue light are analyzed by individual
BSAs which are controlled by a host PC computer using an IEEE 488 interface. The
traversing system is also controlled by the same computer via a serial connection between
the computer and the controller of the traversing systems motors. The parameters and
execution of both data acquisition and probe traversing take place using a Dantec
application program: BAS Flow Software version 1.2. The two BSAs are synchronized
through a synchronization bus and they share the same clock, based on which the arrival
times of measurement samples are assigned. Furthermore, the LDA measurement is
synchronized with the A/D boards used to sample the unsteady wall pressure using two
signals (Sync Reset and CLK) available from the synchronization bus of the BSA. A
block diagram illustrating the synchronization with the A/D boards is shown in Figure
2.16. When the acquisition of the LDA system starts, the ‘Sync Reset’ trigger is sent to

the master A/D board (AT-MIO-16E-10). This ‘Sync Reset’ works as the acquisition-

35

start trigger of the master A/D board. The scan clock of the master A/D board is
generated by dividing down the 1 MHz CLK signal of the BSA, which guarantees the
arrival time assigned to the pressure data have the same exact time base used to determine
the arrival times of the LDA samples. Before it is sent to the master A/D board (AT-
MlO-16E-10), the 1 MHz CLK signal of the BSA is fed to a Schmitt trigger circuit so
that the CLK signal sent to the A/D board is regulated to a standard TTL signal. Detail of
the Schmitt trigger circuit is described in Appendix 6.2. The slave A/D board (N I-6024E)

is synchronized with the master board as described in 2.2.4.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PC PC
A33??? AI Scar; A/D Board
16E-10 Start Nl-6024E
11 l
Schmitt Sync Reset
Tngger
I
CLK
High
Traverse g“ ~— ...L. Green Voltage Multiplier
m1 0, ‘fiwW Green
a) 3
co in
‘1' o
m c
m > High Ph t
NJ (I) > O O
— __._u _ BSA Voltage Multiplier
PC -—~ Blue Blue
IEEE 488 Bus PM W

 

 

 

 

 

 

 

 

 

Figure 2.16 Block diagram demonstrating the synchronization of the LDA system
with the A/D boards used for acquiring pressure data

The BSA processors perform a validation of the burst signals. In analyzing the

spectrum of the burst signals, the BSA compares the two largest local maxima of the

36

spectrum. If the ratio between them is greater than 4, the spectrum is of good quality and
thus the burst is validated. Only the validated data is saved. Overall, the data rate of the
measurement is found to be higher in the opening of, than inside the cavity. The
maximum data rate in the region of the measurement is about 200 Hz and the lowest is
about 1 Hz. Because the data rate is much lower than the frequency of interest, no
frequency-spectrum analysis is performed on the velocity data. Instead, to obtain time-
dependent information concerning the velocity field, the cross-correlation between the
velocity and surface pressure signature is calculated. In turn the conditionally-averaged
velocity field associated with the dominated surface pressure signature could be extracted

using stochastic estimation (as described in Chapter 4).

Coincident data of the two velocity components in an interval window of 0.01 ms
are recorded. This interval window is about 0.3% of the shortest time scale of interest.
The measured instantaneous value of the two velocity components, the arriving time and
transient time of the particle in the measurement volume for each sample are saved in a

text file.

37

2.3 Experimental Procedure and Parameters

Axisymmetric cavities with variable aspect ratio (L/D) of 3.3, 5, 6.6, 8, and 9.7
were first studied using the streamwise surface-pressure-sensor array. For each aspect

ratio, data were acquired at four different free-stream velocities (5, 10, 15 and 20 m/s),

resulting in a corresponding Reynolds number based on cavity depth (Re = UOOD/ v) of

4067, 8133, 12200 and 16267, respectively. The mean and fluctuating pressure results
suggested that for cavities with L/D>5.0, the initial development of the flow structure
generating the wall pressure downstream of the. separation point is inherently similar to
the back-step flow of Hudyz. In the latter study, which employed the same test model
used here, it was demonstrated that the wall-pressure fluctuations were dominated by a
wake mode. In particular, unlike the traditionally held view of spatially growing vortex
structures that “impinge” on the wall at reattachment, Hudy et al. showed that in the
axisymmetric back step the vortex structures rolled up from the initially thin shear layer
and grew to a size comparable with the step height while remaining stationary at an x
location of 2 — 3 step heights (half way to reattachment). Subsequent to this formation
phase, the vortex accelerated to its terminal convection velocity as it traveled downstream.
In the present study, the longer cavity aspect ratios (L/D > 5) were found to be too large
to interfere with the development of the naturally-existing, back-step wake mode. The
inferred existence of the wake mode in longer cavities is also consistent with the
observations in the studies of Gharib and Roshkos, Najm and Ghoniem9 and Rowley et
al.10. Thus, given the similarity to the back-step flow for MB > 5, the scope of this thesis
is focused on cavities with length to depth ratio of less than five: L/D = 2.6, 3.3 and 4.1.

In addition, cases corresponding to width to depth ratio W/D of 2.5, 3.7, 7.4 and "NW"

38

(go-sidewall, or axisymmetric) and four different free-stream velocities (5, 10, 15 and 20
m/s) are examined. The given W/D values correspond to cavities with an angular

azimuthal extent of 30°, 45°, and 90°. Also, the Re (Re = UOOD/ v) values corresponding to

the given free-stream velocities are 4067, 8133, 12200 and 16267, respectively.

2.3.1 Model Alignment

The static pressure taps downstream of the separation point (described in 2.2.2)
are used to align the axisymmetric model inside the wind tunnel. This is done prior to
mounting of the downstream cavity wall in order to avoid the possibility of establishment
of any three-dimensional (azimuthally-varying) mean flow inside the cavity similar to
that visualized by Maull and EastI 1. The alignment procedure involves utilization of
traversing provisions in the model stand to adjust the yaw and pitch angles of the model
and its lateral location in the test section. Starting with the model centered in the wind
tunnel and its axis parallel to the test-section walls, small adjustments to the yaw and
pitch angles are made until the streamwise mean pressure distributions downstream of the
step measured from the top, bottom and sides of the model agree to the best possible
extent.

The mean static pressure is acquired at 100 Hz for 10 seconds. Figure 2.17 shows
the mean-pressure profile on the four sides of the model for Re = 12200. The mean-

pressure coefficient Cp= (ps-p,)/ (l/2p U002), is plotted versus the streamwise coordinate
normalized by the height of the step. p, is the mean pressure on the surface of the model
and p, is a reference pressure taken as the static pressure measured by the Pitot tube in the

free stream. The results in Figure 2.17 show that the pressure distributions from the

39

different sides collapse well with the biggest deviation of 4.3% of the overall pressure

variation (found at around x/D= 4).

 

 

0.2 1 _ -._-__, a __ I A I

ll 0 top I ‘ i i ‘3
0.15, <1 bottom 0L -

. 1 o .

i x mm ; 0‘, E000 i
“i- I=__1~:L -- it ----- * ------- :-~-°°o «
0.05i —————————— . o ........

it

0 _ _ _,

.. a l
-035. i ,,,,,,,,,, . _______________

Q o T
-0.1 ...........................................

! if i
-o.15i 3 eeeeeeeeeeee

- ' i
-0.2r O 00‘ ;,-,o, .......

i 9 I , ; I
025' ______.___, 1 *e . . _

0 2 4 6 8 10

x/D

Figure 2.17 Streamwise distribution of the mean-pressure coefficient downstream of
the back step at different azimuthal locations around the model (left and right
indications in the legend are relative to viewing the model from upstream)

2.3.2 The Boundary Layer State at Separation

The state of the boundary layer at separation highly affects the subsequent
development of the shear layer instability. For example, Morris and Foss12 demonstrated
that only the near-wall, or inner-layer, vorticity participated in the initial shear-layer
instability in the case of turbulent-flow separation. They showed that “an effective

momentum thickness” of 69]] = 0.0526 is the appropriate length scale characterizing

turbulent separation rather than 6, which is appropriate for the laminar case but has
traditionally been used for both laminar and turbulent separations. To characterize the

state of the boundary layer at separation, mean streamwise velocity profiles are measured

40

using a single hot wire. Figure 2.18 shows the velocity profile represented by the blue
circles on a semi-log plot at a free-stream velocity of 5 m/s (Figure 2.18 a) and 20 m/s
(Figure 2.18 b); the corresponding Reynolds number based on momentum thickness is
1830 and 4387 respectively. The velocity and wall-normal coordinate are normalized by
wall units, i.e. U+ = U/u, and y+ = yut/v, in which u, = (r,,/,0)”2 , ‘l'w is the mean wall
shear stress, and p and v are the density and the kinematic viscosity, respectively, of air.

The Motion velocity u, is estimated by fitting the first ten points of the measured profile
to the equation of Spalding”:

_ (km)2 _ (km)3

+
y+ =U+ +e-kB[ekU _1_kU+ 2 6

 

] 2.5

The constants used here are k = 0.41 and B = 5.0. This method of estimating the wall

shear stress was examined by Kendall and Koochesfahani 14. They showed that the
method has a mean error less than 2% in estimating the friction velocity using Spalding

profile.

41

h v—A! v v—vfi- 7— .. 77 . 'f"v v v‘~—roy 1, . Viriv—fiQV

     

 

 

 

 

 

 

 

 

 

252' To FFesSthdEE '7
I l—Spalding profile- (a) 3
. ; Log-law ; l
20. _1.--, *_* ----’ — 1‘
157
+ .
D
10;
5t
01 0 1 + Wm”:- 3 4
10 1O 10 10 10
25f. ' F T 'T—_ T
. 0 Present data l
| ——Spa|ding profile ' (b)
l
20- : . .'-°_9_'?_VL___ __
+
3
10° 101 1o2 103 104
+
Y

Figure 2.18 Boundary layer mean-streamwise-velocity profile for (a) U00 = 5 m/s and
(b) UGo = 20 m/s

The friction velocity was determined to be 0.23 and 0.80 m/s for the free-stream

velocity of 5 m/s and 20 m/s respectively. The red line in Figure 2.18 represents the

42

Spalding profile, and the black solid line shows the well known “log law”. At the lower
free-stream velocity the data undershoot the log law. This could indicate that although the
boundary layer is turbulent, the Reynolds number may not be sufficiently high for the
boundary layer to develop a proper log region. At the higher free-stream velocity,
coincidence between the log law and the measured data is seen between 50<y+<300.
Above y+ z 300, the measured velocity profile overshoots the log law, consistent with the
known behavior of “the wake region” in a turbulent boundary layer. Overall, the results
indicate that the boundary layer at separation is turbulent at 5 m/s and 20 m/s. Similar
boundary layer characteristics are also found at the other free-stream velocities (10 and
15 m/s). The corresponding momentum thickness is 5.49 mm, 4.03 mm 3.54 mm and
3.29 mm in order of increasing velocity. The shape factor of the boundary layer defined
as the ratio between the displacement and momentum thickness, H = 670 is about 1.35.
For reference, the shape factor is 2.6 for a laminar zero-pressure-gradient boundary layer
(Blasius profile) and is 1.3 for a zero-pressure-gradient turbulent boundary profile with
momentum-thickness Reynolds number of 8000 (see Pope15).
2.3.3 Microphone Array Measurements

Two main explorations are performed using the microphone arrays: one to study
the effect of cavity width on the self-sustained oscillation and the other to examine the
characteristics of the oscillation along the azimuthal direction of the cavity. In the former,
the unsteady-wall-pressure measurement beneath the central plane of the cavity is
conducted for L/D = 2.6, 3.3 and 4.1, and W/D = 2.5, 3.7, 7.4 and "NW" (no-side-wall, or
axisymmetric) using the streamwise microphone array at four different free-stream

velocities (5, 10, 15 and 20 m/s), resulting in a corresponding Re = 4067, 8133, 12200

43

and 16267, respectively. For the second study, the unsteady wall pressure on the
downstream cavity wall (at y/D = 0.5) is acquired for cavities with W/D of 2.5, 3.7, 7.4
and NW and L/D = 3.3 at a free-stream velocity of 15 m/s. All data are acquired at a
sampling frequency of 2000 Hz, resulting in a Nyquist frequency of 1000 Hz, which is
sufficiently higher than the highest frequency of interest (approximately 300 Hz). The
absence of frequency aliasing is checked by comparing the resulting pressure spectra to
those produced from acquisition at a higher sampling frequency. 262144 numbers of
samples are acquired for about 131 seconds. More than 6000 cycles of the main
oscillation frequency (between 50 Hz to 100 Hz) are contained in such time series.

When the unsteady wall pressure is acquired simultaneously with the LDA, longer
acquisition time is required because the LDA has a substantially lower data rate. In this
case, the wall-pressure sampling frequency is lowered to 1000 Hz to reduce the size of
the resulting data files. The Nyquist frequency in this situation is 500 Hz, which remains
higher than the highest frequency of interest.

2.3.4 LDA Measurements

Two-component velocity data are acquired over x-y planes at different azimuthal
locations using the LDA system simultaneously with the unsteady wall pressure. For
reasons that will become clear in Chapter 3, these measurements are conducted only for
one aspect ratio of L/D = 3.3 and W/D = 7.4 over x-y planes at different azimuthal
locations. To accomplish this, the LDA probe volume must be movable along the cavity
perimeter. There are two drawbacks of doing so: 1) attaining a probe volume movement
that is concentric with the axisymmetric model axis, though doable, is not easy; 2)

measurements in azimuthal planes other than at ¢ = 0° (see definition of in ¢ Figure 2.6)

44

don’t benefit from the existence of the streamwise array beneath the measurements.
Therefore, during the velocity measurements the focal point of the LDA is maintained
within the plane on top of the model (¢ = 0°) while the insert for W/D control is rotated
around the central shaft (see Figure 2.3). Thus the azimuthal location of the LDA

measurement plane relative to the symmetry plane of the cavity (18¢) can be varied.

To accomplish the above, one must ensure that the characteristics of the pressure
fluctuations in a given azimuthal plane depend only on the offset angle relative to the

cavity symmetry plane (13¢) regardless of the azimuthal location of the symmetry plane

relative to the top of the model ((156): see Figure 2.19 for definition of M5 and ¢c. That is,

the wall-pressure fluctuation, and hence its generating flow structures, in a plane with a
given azimuthal offset relative to the symmetry plane are not changed when the cavity is
rotated around the central shaft. This is verified using pressure measurements as
discussed in Appendix 6.4.

LDA measurements are performed in x—y planes at A¢ = 0° (the azimuthal
symmetry plane), -24°, -28°, -32°, -36° and -40°. The latter five planes are selected in the
vicinity of the azimuthal location where strong harmonic oscillations are found. Figure

2.20 shows the LDA measurement grid using red “+” symbols. The grid extends from

1.2 mm to 16.2 mm above the cavity bottom in increment of 1mm in the y direction, and
from 2.9 mm downstream of separation to 4 mm ahead of the downstream cavity wall in
increment of 2.4 mm in the streamwise direction. The blue circles in Figure 2.20 mark the
locations of the microphones, the signals of which are sampled simultaneously with the

LDA.

45

Facing downstream

¢c/ A¢/

Top plane aligned with '1
streamwise microphone array

/

     

 

Azimuthal Symmetry plane

    
  

/ Azimuthal microphone array

Cavity side wall

 

Cavity side wall

Figure 2.19 Illustration of the definition of A¢ and ¢c

 

 

  

' + +' + + '+ + +' + + l + +j + + 4
+ + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + +
1 + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + +
Q + + + + + + + + + + + + + + +
> + + + + + + + + + + + + + + +
0.5 + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + +
U + + + + + + + + + + + + + + +
I] 1.5 2 2.5 3 3.5

     

XID
Figure 2.20 LDA measurement grid

The time of acquisition of LDA samples depends on the arrival time of seeding
particles which is random. For the present measurements, the mean data rate varies
between 1 Hz to 200 Hz depending on the location of the measurement. It is significant to
note that in general the LDA data are not coincident in time with the pressure samples.
Hence, in the data processing, the wall-pressure data, which have a substantially higher
sampling frequency, are linearly interpolated using samples immediately preceding and
following the arrival time of the LDA samples. This process is illustrated in Figure 2.21,

in which the black squares represent the pressure samples interpolated from the raw data

46

(the blue circles) at the same arrival time as the velocity samples, shown by the red
diamonds. The resulting simultaneous wall-pressure and velocity data allow computation
of pressure-velocity correlations and stochastic estimation of the velocity field based on

the unsteady wall pressure (see Chapter 4 for details).

25‘ ~ - ,rs - 7 7
—0— u (m/s)
20 -°— p (Pa)
El”, interpolated p

 

  

8‘0 7 '"100

t(ms)
Figure 2.21 Illustration of the pressure interpolation process

LDA measurements tend to have more samples of high than of low velocity.
Specifically, a larger number of particles will be swept through the measuring volume
during a high velocity period. As a result, the computation of statistics based on
arithmetic averaging has a bias towards higher velocity. To correct this bias, a weighting
factor (see Adrian7) based on the transit time, ti, of the particle in crossing the
measurement volume is introduced in the calculation of the statistics. More specifically,

the average of any variable, g, subjected to the weighting factor is:

47

Zgi’i
— I
g = , 2.6
Zn-
I

 

where g; is, for example, a velocity sample if calculating the mean velocity or product of

two variables if computing second-order statistics.

In addition to accounting for the LDA bias, convergence tests were done for
determining the number of samples and associated random uncertainty in calculating the
mean and rms velocity and cross-correlation between the velocity and the pressure
fluctuation. Figure 2.22 shows an example of the convergence test for the mean velocity
(27 : plot a) and pressure-velocity correlation (Rurpv: plot b). The abscissa displays the
number of samples involved in averaging and the ordinate shows the corresponding value
of the statistical quantity. Different colors represent data records acquired at different

times. The black dashed lines in Figure 2.22(a) represent the region of i5% deviation

from the converged value and those in Figure 2.22(b) represent the region of i10%

deviation. Figure 2.22(a) shows that more than 400 data points are required to attain
better than 5% convergence uncertainty for the mean velocity. On the other hand, Figure
2.22(b) indicates that at least 10000 points are needed for the velocity-pressure cross
correlation to have an uncertainty of better than 10%. Hence, 10,000 velocity samples are
acquired in all five azimuthal planes (A¢ = 0°, -24°, -28°, -32° and -36°). This results in
uncertainty for the mean of less than 0.3% of the free stream velocity, for the rrns
velocity of less than 1% of the maximum value, and for the velocity-pressure correlation
of less than 7% of the maximum value. In the azimuthal plane close to the cavity side

wall (A¢= -40°), because of the low data rate (around 1-5 Hz), only 2000 samples are

48

collected to guarantee the convergence of the mean velocity. No velocity-pressure

correlation is obtained for this plane.

49

 

-0.16 - _
-0.18 - -
-02 - -
-022 - -

a -024 — -

 

ulU

 

 

-0.3 - ~
-0.32 - -

 

 

0 0.5 1 1.5 2
number of samples x 104

0 l l l

 

 

 

f

2 -05 — —
Er
-0.6- _
-0.r- -
-0.8 - -

-09 - _

 

 

‘10 0.5 1 1.5 2

number of samples x 104
Figure 2.22 Convergence test for: (a) the mean streamwise velocity (b) cross-
correlation between the velocity and the pressure fluctuations normalized by the
velocity and pressure rms values

 

SO

2.4. References

 

I Li, Y.X., “Investigation of the Wall-Shear-Stress Signature in a Backward-Facing-Step
Flow Using Oscillation Hot-Wire Sensors,” Ph.D. thesis, pp 20-32 (2004)

2 Hudy, L.M., “Simultaneous Wall-Pressure and Velocity Measurements in the Flow
Field Downstream of an Axisymmetric Backward-Facing Step,” Ph.D. thesis, (2005)

3 Trosin, B.J., “Velocity-Field Measurements of an Axisymmetric Separated Flow
Subjected to Amplitude-Modulated Excitation,” Master thesis, pp 12-15 (2006)

4 Aditijandra, A.K., “Amplotude-Modulated Excitation of a Separated Flow Using an
Externally Driven Helmholtz Resonator,” Master thesis, pp 19-22 (2006)

5 Kinsler, L.E., Frey, A.R., Coppen, A.R. AND Sanders, J.V., “Fundamentals of
Acoustics,” 3rd edition, John Wiley and Sons, New York (1982)

6 Naguib, A.M., Gravante, SR, and Wark, C.E., “Extraction of Turbulent Wall-
pressure Time-series Using an Optimal Filtering Scheme,” Experiments in Fluids 22 (I),
pp. 14-22 (1996)

7 Adrian, J.R., “Fluid Mechanics Measurements, Chapter 5: Laser Velocimetry,” Taylor
& Francis, Bristol, pp. 157-159 (1983)

8 Gharib, M., Roshko, A.F., “The Effect of Flow Oscillations on Cavity Drag,” Journal
of Fluid Mechanics 177, pp. 501-530 (1987)

9 Najm, H.N., Ghoniem, A.F., “Numerical Simulation of the Convective Instability in a
Dump Combustor,” AIAA Journal 29(6), pp. 911-919 (1991)

'0 Rowley, C.W., Colonius, T. and Basu, A.J., “On Self-sustained Oscillations in Two-

dimensional Compressible Flow over Rectanguler Cavities”, Journal of Fluid Mechanisc
455, pp. 315-346 (2002)

“ Maull, D.J., and East, L.F., “Three-dimensional Flow in Cavities,” Journal of Fluid
Mechanics 16, pp. 620‘ 632 (1963)

'2 Morris, SC, and Foss, J .F., “Turbulent Boundary Layer to Single-stream Shear Layer:
the Transition Region,” Journal of Fluid Mechanics 494, pp. 187-221, (2003)

'3 Spalding, D.B., “A single Formula for the Law of the Wall,” Journal of Applied
Mechanics 28, pp. 455-457 (1961)

'4 Kendall, A. and Koochesfahani, M., “A Method for Estimating Wall Friction in
Turbulent Wall-Bounded Flows,” Experiments in Fluids 44, pp. 773-780 (2008)

51

 

‘5 Pope, S.B., “Turbulent Flows,” Cambridge University Press, Cambridge, pp. 303 (2000)

52

3. WALL-PRESSURE-MEASURMENTS RESULTS

In this chapter, the unsteady pressure on the bottom and downstream wall of
cavities with different aspect ratios will be discussed with an emphasis on the effect of
the cavity width on the pressure signature. The discussion will focus on cavities with L/D

= 2.6, 3.3 and 4.1 and W/D = 2.5, 3.7, 7.4 and "NW" (go-side-yyall, or axisymmetric) at

four different freestream velocities (5, 10, 15 and 20 m/s), corresponding to Re = UwD/ v

= 4067, 8133, 12200 and 16267, respectively.
3.1 Unsteady Wall Pressure in the Cavity Symmetry Plane

Most of the literature on cavity flow is based on the assumption that the flow is
two dimensional. Experimental studies usually use cavities covering the whole span of
the tunnel and focus only on the symmetry plane in the spanwise direction. Therefore, as
a starting point, the exploration of the unsteady wall pressure here will be focused on
comparing the wall-pressure unsteadiness in the axisymmetric cavity with that in the
symmetry plane of cavities with side walls. The aim of the comparison is to explore the
effect of cavity width and side walls on the unsteady wall pressure using the streamwise

microphone array.
3.1.1 The Unsteady Wall Pressure in the Axisymmetric Cavity Flow

Figure 3.1 shows the streamwise distribution of the root-mean-square (rms) of the
fluctuating pressure on the bottom of the axisymmetric cavity with U0 of 3.3 at different

Reynolds numbers. The normalized rms value (p'ms/ 1/2pU002) is plotted as function of

the streamwise distance from the cavity upstream wall normalized by the cavity length,

53

x/L. The rms represents the strength of the pressure fluctuations. The plot indicates that
the wall-pressure unsteadiness increases along the streamwise direction of the cavity. The
largest rrns value is found nearest to the downstream corner and this value appears to
decrease as the Reynolds number increases. Hudy1 found a similar trend in the wall-
pressure fluctuation downstream of a backward facing step. The reason for the decrease

in the rrns level with increasing Reynolds number is not clear.

 

 

 

 

 

 

0.05 l,’____fi __’_; _ 771**- 1
; —e—Re=4067
l —e—Re=8133
0.041 —6—Re=12200 .
l -9-Re=162671
Q 0.03; -
Q .
E:
E 0.02 ~ -
E
0.01 ,
0* 4 -* - 4 .1 1 41. .1 __1l_1 1. J
0 0.2 0.4 0.6 0.8 1

x/L

Figure 3.1 Streamwise distribution of the rms pressure acting on the bottom of the
axisymmetric cavity for [/D = 3.3 and different Reynolds numbers

The frequency spectra ¢ p' p'( f) = <P( f )* - P( f )> provide information

concerning the distribution of the energy of the pressure fluctuations over different

frequencies f; where< > denotes the average over different records, P(f) is the Fourier

transform of the pressure fluctuation signal p’(t), and P(f)* is the conjugate of P(/). 1024

54

records of data with 256 samples in each record are used in calculating the spectra
resulting in a frequency resolution of 7.8 Hz and random uncertainty of 3.1%. Figure 3.2
provides a plot of the frequency spectra of the unsteady wall pressure on the bottom of
the axisymmetric cavity for L/D = 3.3 and Re =12200. The magnitude of the spectra
normalized by the dynamic pressure of the free stream is plotted as function of the

normalized frequency fl/UOO. Lines with different colors represent different streamwise

  
 
 
   

locations.
-5

12X10» . . . . .M--, ,7, ,.

. —x/L=0.25 '2

1. ——-x/L=O.36 ,_

l 1 —-x/L=0.48 5

‘ —x/L=0.60

g8 0'8 —x/L=0.71
“a ; l—x/L=0.83i;
5:. 0 6 ——x/L=0.95

en.

E 0.41 y
l

0.2 ,;
10'2 10'1 10° 101

fL/U00

Figure 3.2 Wall-pressure frequency spectra on the bottom of the axisymmetric
cavity for L/D = 3.3 and Re =12200. Lines with different colors represent different
streamwise locations

Inspection of Figure 3.2 shows that no strong harmonic peaks are found in the
spectra at all measurement locations. A small “bump” (pointed-to by an arrow in the

figure) at fI/U‘,0 z 0.19 is seen at x/L = 0.95 where the fluctuations are strongest. The lack

of strong harmonic spectral peaks suggests that no strong self-sustained oscillations exist

55

in the present axisymmetric cavity flow. Overall, the spectra are broadband with their
largest magnitude found at low frequencies. Even though disturbances at low frequencies
seem dominant at x/L = 0.95, 80% of the fluctuation energy is contained in the frequency

region offl/U00 > 0.1.

The spectra for the same cavity geometry as above but at the lowest Reynolds
number examined here (Re = 4067) are shown in Figure 3.3. As seen from the figure, the

spectra remain broadband, although they exhibit a preferred peak at a frequency of fL/UOo

z 0.25 at x/L = 0.95. Another interesting observation regarding the lower Reynolds
number case, is the absence of the strong low-frequency unsteadiness found at Re =
12200. The dependence of the low-frequency fluctuations on the Reynolds number
suggests that this unsteadiness might be related to some form of instability. This will be

discussed later.

56

fL/UOo

Figure 3.3 Wall-pressure frequency spectra on the bottom of the axisymmetric
cavity for L/D = 3.3 and Re =4067. Lines with different colors represent different
streamwise locations

 

-.1-;i2 Lew“... V

101

Wall-pressure measurements in axisymmetric cavities with L/D of 2.6 and 4.1

also do not show any evidence of existence of strong self-sustained oscillations. The

spectra of the unsteady wall pressure at x/L = 0.95 are shown in Figure 3.4 for all aspect

ratios investigated. No harmonic peak is found in the spectra; although curiously all cases

exhibit the non-prominent peak at fL/Uco z 0.19 (within i 0.02, which is the spectrum

resolution). Please note here, 80% of the fluctuation energy is contained in the frequency

region of jL/UGo > 0.1 even though disturbances at the low frequency end are strong. In

addition, the spectra for different L/D values are similar, which suggests that the pressure

fluctuation is dominated by similar flow structures for all aspect ratios investigated. Also,

spectra results for other Reynolds numbers show similar characteristics as found in

57

Figure 3.4; i.e. showing no prominent self-sustained oscillations in the axisymmetric

cavities studied here.

—L/D=2.6
.—L/D=3.3
‘ —. LID-747.1

   

A

2 V “1 0 7 ‘9'1

10‘ 10' 10 10
fL/Uso

Figure 3.4 Aspect ratio effect on the wall-pressure frequency spectra on the bottom
of the axisymmetric cavity atx/L = 0.95 and Re = 12200. Lines with different colors
represent cavities with different L/D

It is interesting to note that the frequency of the non-prominent peak at fL/Uco z
0.19 i 0.02 when normalized by the cavity depth (ID/U00 = 0.07, 0.06 and 0.05 for L/D =
2.6, 3.3 and 4.1 respectively) is close to the normalized frequency of the wake mode in
Rowley et al.2 (fD/U,C = 0.06 and 0.05 for L/D = 4 and 5 respectively). In their simulation
of cavity flow with a laminar boundary layer at separation, the wake mode is found to
exist for cases with [/6 > 75. U6 in this study varies between 5.6 and 15.2. However, the
boundary layer at separation is turbulent. Morris and Foss3 showed that only the near-

wall, or inner-layer, vorticity participated in the initial shear layer instability in the case

58

of turbulent-flow separation. An “effective momentum thickness” of 9e)?“ U /ut) z 10 is
found (based on their results: Be = 0.5mm, U0 = 7.1m/s, Cf= 2(uT/U0)2 = 0.00295) to be

the appropriate length scale characterizing turbulent separation rather than 6’. Based on

this result, 128 S L/Hefirs 210 for Re = 12200, which falls in the region of wake-mode
oscillation found by Rowley et al. Therefore, it is suspected that the broad peak at fl/U00

1: 0.19 i 0.01 in the axisymmetric cavity corresponds to a wake mode. The wall-pressure
signature of this mode produces a spectral peak that is easily seen in the low Reynolds
number case and is embedded in broadband turbulence in the high Reynolds number case.

This finding is supported by the stochastic estimation results in Chapter 4.
3.1.2 Effect of Cavity Width on the Unsteady Wall Pressure in the Symmetry Plane

As discussed above, except for the possibility of a wake mode that is non-
prominent at high Reynolds number, no strong self-sustained oscillations are found in the
axisymmetric cavity with L/D = 2.6, 3.3 and 4.1 at different Reynolds numbers. To
investigate if this observation depends on the cavity width, measurements are first
performed for the cavity with L/D of 3.3 and different widths. The specific W/D values
are 2.5, 3.7, and 7.4. For each of these cases, the unsteady pressure on the floor in the

symmetry plane is investigated at Re = 4067 and 12200.
3.1.2.1 Frequency Spectra

Figure 3.5 shows the frequency spectra of the unsteady pressure measured in the
symmetry plane near the downstream wall (x/L = 0.95) on the cavity floor at Re = 12200
for different values of W/D. There is a striking change in the nature of the cavity

oscillation as the cavity becomes narrower. For the narrower cavities (W/D = 2.5 and 3.7),

59

a strong harmonic peak is observed at fL/Uw z 0.21, suggesting the possible

establishment of self-sustained oscillations. On the other hand, there is no evidence of
such oscillations in the spectra for the wider cavities (W/D = 7.4 and NW) and instead the

strongest unsteadiness is found at low frequency.

14 . . . 1

i 1 ‘ . —W/D=2.5 E
Vu—W/D=3.7
—W/D=7.4
—NW

     

_L

-2 i -1 ‘h'io '1

10 10 1O 10
fL/U00

Figure 3.5 Cavity-width effect on wall-pressure spectra at x/L = 0.95 (Re = 12200).
Lines with different colors represent cavities with different width

The frequency of the harmonic oscillation (fl/U,o 1: 0.21) is substantially lower
than the typical value for the lowest Rossiter mode (fl/U00 z 0.4 to 0.5). This suggests

that the self-sustained oscillation observed here is not of the Rossiter type. Interestingly,
however, the harmonic peak occurs at practically the same frequency where it is believed
that a non-prominent wake mode exists in the axisymmetric cavity. This could imply that
the strong harmonic oscillation found in the narrow cavities maybe caused by

“intensification” of the naturally existing wake mode in the wider cavities. This idea will

60

be explored in the following chapter, where the mechanism leading to the establishment
of the harmonic oscillation is investigated using the simultaneous wall-pressure and

velocity measurements.

The suppression of the self-sustained oscillation in the wider cavities is also
observed for other Reynolds numbers, but the effect of the cavity width seems to
decrease as the Reynolds number is reduced. This can be seen by comparing the results
shown in Figures 3.6 through 3.8 for Re = 16267, 8133 and 4067 respectively. The
spectra in Figure 3.6 for Re = 16267 show a distinguished difference between the narrow
and wide cavities, similar to that observed in Figure 3.5 for Re = 12200. That is, a

harmonic peak at fl/Uco z 0.21 for W/D = 2.5 and 3.7 and the attenuation of this peak

accompanied by the strong low-frequency unsteadiness for W/D = 7.4 and NW. For Re =

4067, the spectra shown in Figure 3.8, the harmonic peak at fL/Uoo z 0.26 is not as

pronounced as in the higher-Reynolds-number case, but it remains discemable for the
narrower cavities. The peak gets weaker with increasing cavity width, but the attenuation
is not as strong as in the higher-Reynolds-number case, and the strongest unsteadiness

never switches to low frequencies.

Also noteworthy is the effect of the Reynolds number on the wake mode
prominence for the axisymmetric cavity. The results in Figures 3.5 through 3.8 clearly
show how the spectral peak associated with this mode is prominent at the lowest
Reynolds number, but gradually becomes less observable with increasing Reynolds
number, accompanied by concurrent increase in the low-frequency broadband

fluctuations.

61

 

10‘ 10‘1 100 101
fL/Um

Figure 3.6 Cavity-width effect on wall-pressure spectra at x/L = 0.95 for Re = 16267.
Lines with different colors represent cavities with different width

62

 

fL/U
(D
Figure 3.7 Cavity-width effect on wall-pressure spectra at x/L = 0.95 for Re = 8133.
Lines with different colors represent cavities with different width

5x10-5

 

10'2 10'1 100 101
fL/Uw

Figure 3.8 Cavity-width effect on wall-pressure spectra at x/L = 0.95 for Re = 4067.
Lines with different colors represent cavities with different width

63

The increase in the strength of the low-frequency unsteadiness with Reynolds
number leads us to believe that this unsteadiness could be related to some form of
instability. As discussed in the introduction, Bres and Colonius4 discovered that a three-
dimensional centrifugal instability of the two-dimensional re-circulating flow inside a
cavity “kicks in” above a critical Reynolds number. Furthermore, the frequency of this
instability was found to be much smaller than that of the self-sustained cavity oscillations.
These characteristics agree with the present observations, where the rise in the spectra at
the low-frequency end for the wide cavities is only found at the higher Reynolds number
and at frequencies that are substantially lower than that of the harmonic oscillations (see
Figure 3.5, 3.6 and 3.7). Thus, it is possible that the low-frequency disturbances observed

here are caused by the three-dimensional instability of the flow inside the cavity.

Frequency spectra similar to those shown in Figure 3.5 but for L/D values other
than 3.3 in the range L/D < 5 show that the observed cavity-width effect is not specific to
L/D = 3.3. Figure 3.9 gives the frequency spectra of the unsteady wall pressure near the
downstream comer at Re = 12200 for cavities with L/D= 2.6 and 4.1, and W/D = 2.4, 3.5
and 7.1 and NW. Similar to Figure 3.5, the spectra in Figure 3.9 for narrow cavities are
characterized by a clear harmonic peak which suggests the establishment of cavity
oscillation. The spectral peak, however, is highly damped as the cavity becomes wider.
An interesting observation is that for the cavity with L/D = 2.6, the harmonic oscillation
is weakened for W/D = 3.7 and wider cavities (Figure 3.11a). In contrast, for cavities with
L/D = 3.3 and 4.1, the oscillation can be sustained for W/D = 3.7. Table 3.1 compares the
cavity aspect ratio L/D and W/D for oscillating (bold) and non-oscillating flow conditions

in the symmetry plane. Noting that in all cases strong oscillations are observed in the

64

symmetry plane when W/L is approximately less than or equal to one, it is apparent that
the ratio between the cavity width and length, W/L, is a significant parameter to include
in setting criteria for the establishment of the observed harmonic oscillations in the
symmetry plane of the cavity. This conclusion will be re-examined and refined in light of

section 3.2 results.

Table 3.1 Cavity width-to-Iength ratio for oscillating (bold) and non-oscillating flow
in the symmetry plane

 

 

 

 

 

 

 

 

 

W/D=2.5 W/D=3 .7 W/D=7.4 NW
I/D=2.6 W/L=0.96 W/L=l .42 W/L=2.85 NA
L/D=3.3 W/L=0.76 W/L=l.12 W/L=2.24 NA
L/D=4. 1 W/L=0.61 W/L=0.90 W/L=l .80 NA

 

65

 

 

1-5 1 .1. 1 “M11
- 1 1—W/D=2.51*‘
‘ —W/D=3.7153

    
  

1-—-W/D=7.4 3

._ 1 1. . .NWv1..
v 8 1 1 11
3 1 1.
N3 1 1:
>— 1 ‘ ‘l
e- ‘ 11;:
E 0.5 «l
‘ 1 i

l 1 l:

0 3 ._. ,: 111.1...1, 11;:

10’2 10'1 100 101

fL/Uw

Figure 3.9 Cavity-width effect on wall-pressure spectra at Re = 12200: (a) L/D =2.6
(b) W = 4.1. Lines with different colors represent cavities with different width

66

Since cavities with different L/D exhibit the same width influences, subsequent
results will be focused only on cavities with L/D = 3.3 in order to further explore the
characteristics of the harmonic oscillation and the effect of the cavity width on the

mechanism driving the oscillation.
3.1.2.2 Coherence

The harmonic oscillations found in the narrower cavities exhibit a large degree of
phase locking across the entire cavity length. This can be seen from the contour plot of
the coherence between the wall-pressure fluctuations measured near the cavity
downstream edge and those at different locations on the cavity floor; shown in Figure
3.10 for a cavity with L/D = 3.3 and W/D = 2.5 at Re = 12200. The coherence, as defined
by equation 3.1 below, gives a measure of the degree of phase locking between two

signals at a given frequency:

K1101)" -P2<f)>|

Cp'lp'2(f)= l/2 l/2
¢P'IP'1(f).¢P'2P'2(f)

3.1

 

where< > denotes the average over different records, P,( f ) and P2( f ) are the Fourier

transform of the signals p'1(t) and p'2(t) respectively. P1( f )* is the conjugate of P1( f ).

1024 records of data with 256 samples in each record are used in calculation resulting in

a frequency resolution of 7.8 Hz and random uncertainty of 3. l %.

67

  
 
 
  

100

Oscillation
Frequency

 

0.2 0.4 0.6 0.8
x/L
Figure 3.10 Coherence of pressure fluctuations across the cavity length relative to
pressure measured at x/L= 0. 95 for W/D= 2. 5 at Re— - 12200. The color bar on the
right indicates the coherence value

The coherence value in Figure 3.10 is represented by the color bar displayed next
to the coherence plot. As seen from the figure, substantial coherence magnitude is
sustained across the length of the cavity at fL/U,0 z 0.21. This confirms the global
organization of the flow unsteadiness at this frequency and the establishment of self-
sustained oscillation in the narrow cavity. On the other hand, no substantial coherence is

found across the whole cavity length for the axsiymmetric case; shown in Figure 3.11.

68

fL/Uoo

 

0 .2 0 .4 0.6 0 .8
X/L
Figure 3.11 Coherence of pressure fluctuations across the cavity length relative to
pressure measured at x/L = 0.95 for the axisymmetric cavity at Re = 12200. The
color bar on the right indicates the coherence value

In summary, it is found here that the occurrence of strong harmonic oscillation in
the symmetry plane depends on the cavity width, this width effect‘becomes more
pronounced with increasing Reynolds number. At high Re, the cessation of the oscillation
is accompanied by increased unsteadiness at very low frequencies. Also the observed
oscillation is not of Rossiter type and it occurs at the same frequency where it is believed
that a wake mode with non-prominent pressure signature exists in the axisymmetric

cavity.

69

3.2 Azimuthal Distribution of the Unsteady Wall Pressure

Results from above show that the characteristics of the pressure signature in the
symmetry plane of the wide cavity (W/L > 1; approximately) are the same as for the
axisymmetric cavity, i.e. exhibiting strong attenuation of the self-sustained, harmonic
oscillation accompanied by increased unsteadiness at very low frequencies at high
Reynolds number. In contrast, narrow cavities (W/L < 1; approximately) exhibit
dramatically different behavior in the symmetry plane, as reflected in the presence of
self-sustained harmonic oscillation. To better understand the effect of cavity width on the
cavity unsteadiness and its three-dimensional characteristics, the fluctuation wall pressure
is measured using the azimuthal microphone array on the downstream cavity wall. All

studies are done for a cavity with [/0 of 3.3 and Re = 12200.
3.2.1 Cavities with Finite Width

The azimuthal microphone array, with 14 microphones covering 60° along the
azimuthal direction, is used to explore the three-dimensional features of the oscillations
in the finite width cavity. Figure 3.12 displays a flooded color-contour plot for the
frequency spectra of the unsteady wall pressure on the downstream cavity wall at
different azimuthal locations for the narrowest cavity (W/D = 2.5). The magnitude of the
spectra normalized by the peak value is represented by the color bar on the right side of
the figure. The spectra are plotted as a function of the frequency and the azimuthal
location, 2, which is defined as the arch length between the measurement location and the
symmetry plane at a height of D/Z from the cavity bottom. Positive values of 2 show
positions to the right of the symmetry plane when facing the cavity’s end wall from

upstream. The azimuthal location is normalized by the cavity depth.

70

The color contours in Figure 3.12 show variations along the azimuthal direction at
the oscillation frequency, indicating the strength of the oscillation is not uniform along
the span even for W/D = 2.5 (the narrowest cavity studied here). The oscillation is strong
in the symmetry plane, z/D = 0 (indicated by the broken red line), and close to the cavity
side walls (indicated by the dashed blue line for z/D = 0.98 near the right side wall); In
between, at z/D = 0.66 (indicated by the dashed black line), the oscillation is suppressed.
This can be seen more clearly in the line plots in Figure 3.13, which provides a
comparison between the frequency spectra at these three different azimuthal locations.
The results show that the observed oscillation in the narrowest cavity is three-
dimensional: a finding that could not be made from examination of the streamwise array

data only.

 

z/D

Figure 3.12 Flood color-contour map showing the azimuthal variation of the
frequency spectra of the pressure acting on the end cavity wall for the cavity with
W/D = 2.5. The Color bar on the right indicates the magnitude of the spectra
normalized by the peak value

71

 

fL/UC10

Figure 3.13 Frequency spectra at selected azimuthal locations for the cavity with
W/D = 2.5. Lines with different colors represent different azimuthal locations

The spectral characteristics of the oscillation along the cavity span change when
the cavity width is varied. This can be seen by comparing the spectra contour plot for
W/D = 2.5 in Figure 3.12 with a similar plot for W/D = 3.7 in Figure 3.14. For this cavity
width, the oscillation at fL/U00 z 0.21, detected earlier in the symmetry plane, shows a
peak at z/D = 0.66 instead of at z/D = 0 as found in Figure 3.12 for W/D = 2.5. Moreover,
disturbances at a lower frequency of fL/ U00 2 0.06 seem to be strongest close to the cavity
side walls, which could be a sub-harmonic of the oscillation at fL/U,o z 0.21 (to within
the spectral resolution of AflJUgo = 0.02). Further details of the spectra may be seen using
line plots in Figure 3.15 for the symmetry plane, and the azimuthal locations of the
highest fluctuations at fL/ U00 z 0.21 and 0.06. These locations are shown by broken lines

inFigure 3.14.

72

10

Oscillation
Frequency

 

Z/D
Figure 3.14 Flooded color-contour map showing the azimuthal variation of the
frequency spectra of the pressure acting on the end cavity wall for the cavity with
W/D = 3.7. The color bar on the right indicates the magnitude of the spectra
normalized by the peak value

-5
1.2510 -

 

fL/U00

Figure 3.15 Frequency spectra at selected azimuthal locations for the cavity with
W/D = 3.7. Lines with different colors represent different azimuthal locations

73

Because the contour plots in Figure 3.12 and 3.14 show that the spectra of the
disturbances on opposite sides of the symmetry plane have similar features, the azimuthal
microphone measurements for the wide cavity with W/D = 7.4 are conducted in only one
half of the cavity. The spectra contour plot is shown in Figure 3.16. Given that

examination of the streamwise-array results showed no harmonic oscillation at fL/Uc,O z

0.21 (Figure 3.6) in the symmetry plane of this wide cavity, it is somewhat surprising that
such a harmonic peak is now found in Figure 3.16. That such a peak was not observed in
the measurements using the streamwise array is caused by the fact that the peak occurs at
an azimuthal location of z/D = 2.62, which is away from the symmetry plane. Also worth
noting is that similar to what is observed for the cavity with W/D = 3.7, the unsteadiness

at the lower frequency of fL/ U00 as 0.06 is strong close to the side wall at z/D = 3.28.
Figure 3.17 provides the frequency spectra of the wall-pressure fluctuations at z/D = 0
(symmetry plane), 2.62 (where the peak at fL/Uoo z 0.21 is found) and 3.28 (where
disturbances at fl/ U00 z 0.06 are strong). The peak in the spectrum at fL/ U00 "z 0.21 at z/D

= 2.62 suggests the existence of self-sustained oscillation. Note that most of the
disturbances (80% of the total fluctuation energy) are contained in the frequency band

jL/U,O >0.1 even at the azimuthal locations where the magnitude of the spectra at the low-

frequency end is highest.

74

fL/Uoo

 

0 0.5 1 1 .5 2 2.5 3
z/D
Figure 3.16 Flooded color-contour map showing the azimuthal variation of the
frequency spectra of the pressure acting on the end cavity wall for the cavity with
W/D = 7.4. The color bar on the right indicates the magnitude of the spectra
normalized by the peak value

 

fL/U00

Figure 3.17 Frequency spectra at selected azimuthal locations for the cavity with
W/D = 7.4. Lines with different colors represent different azimuthal locations

75

The above results indicate that there are strong harmonic pressure oscillations in
all cavities having side walls; including the widest cavity with W/D = 7.4. It appears that
these oscillations are generally located on the two sides of the cavity central plane.
However, when the cavity becomes narrower, these two locations “merge” in the
symmetry plane for the narrowest cavity with W/D =2.5. Figure 3.18 yields the wall-
pressure frequency spectra at the azimuthal location where the peak in the oscillation is
detected for cavities with different widths. It can be seen that the strength of the
oscillations is similar for cavities with different widths. The primary difference is the
azimuthal location where the oscillation happens. It is interesting to note that as the
cavity width increases, the azimuthal distance between the peak-oscillation location and
the cavity side wall, AW remains approximately invariant relative to the cavity depth:
AW/D z 1.25, 1.1 and 1.1, for cavities with W/D = 2.5, 3.7 and 7.4 respectively. This,
coupled with the fact that similar harmonic oscillations are absent at all azimuthal
locations for the axisymmetric cavity (see the following section), leads us to suspect that

the oscillation found here is related to the existence of the cavity side walls.

76

     
    

8x10 1 . _ ‘ ,,-.
q—W/D=7.4 z/D=27.672 1
71 ;—W/D=3.7 le=0.66 i
6} l 1—W/D=2.5 z/D=0
‘ ‘ I "’7’ 1
g8 51 777777777777777777
s“ 3
2
1 1
10‘2 10'1 100 101

fL/U00

Figure 3.18 Frequency spectra at the azimuthal location of strongest harmonic
oscillation for cavities with different W/D values. Lines with different colors
represent cavities with different width

Another noteworthy feature of the harmonic oscillation can be seen from
examination of the coherence between the pressure fluctuation at fL/U00 z 0.21 at
different azimuthal locations and that at the location of peak oscillation. Figure 3.19
depicts the coherence value for the widest cavity (where the peak oscillation location is at
z/D = 2.62). The figure indicates that even though high coherence is maintained over a
relatively narrow zone around z/D = 2.62, non-zero coherence (approximately 20%) is
found with disturbances at fL/Uw z 0.21 in the symmetry plane (corresponding to the
small “bump” in the spectra in Figure 3.17). This suggests the flow structures near the

side wall and the flow in the middle of the cavity are correlated.

77

Finally, it is noted here that even though the harmonic oscillation shows three-
dimensional characteristics, it doesn’t posses the azimuthally-periodic behavior of the
centrifugal instability observed by Bres and Colonius4. This provides additional evidence
that the observed oscillation is not caused by the 3-D instability of the re-circulating flow

in the cavity.

coherence

 

 

Figure 3.19 Coherence between the pressure fluctuations at different azimuthal
locations with those at z/D = 2.62 for W/D = 7.4

3.2.2 Axisymmetric Cavity

The variation of the wall-pressure unsteadiness along the azimuthal direction for
the axisymmetric cavity (NW) is investigated using the Panasonic azimuthal microphone
array which covers the whole perimeter of the cavity. Figure 3.20 displays the
distribution of the rms pressure along the azimuthal direction. The horizontal axis of the

plot shows the azimuthal angle¢ , as defined in Figure 2.6, between the measurement

78

location and the top plane where the streamwise microphone array is embedded. The rms
pressure is seen to exhibit some variation along the azimuthal direction with a deviation
of 7% (based on the standard deviation) from the mean value. There are two possible
reasons for the observed variation: 1) the existence of three-dimensional features inside
the cavity (as found, for example in the visualization study of Maull and East5, or the
instability analysis of Bres and Colonius4; 2) noting that the highest rrns values are found
around ¢ =l80° (on the bottom side of the model), it is possible that there is some
influence of the support of the model, which is located 0.61 m(more then 500 cavity

depths) downstream of the separation point.

0.1 I I I I I I I
0.09

 

I
l

 

0.08 - o o o O O _
U 0
0.071, 0 ° 0 -

0.06 - -

 

 

I
L

0.05
0.04
0.03 r -
0.02
0.01

I
l

owl/[112.0301

I
L

I

‘

 

 

00 50 100 150 200 250 300 350
¢(degrees)
Figure 3.20 Azimuthal distribution of the rms pressure for the axisymmetric cavity

with W = 3.3. Red line shows the average rms pressure and black lines represent
10% deviation from the mean value

 

79

Regardless of the cause of the azimuthal variation in the strength of the pressure
fluctuations, the spectral shapes remain invariant. This can be seen from frequency
spectra depicted in Figure 3.21 for different azimuthal locations (represented by lines
with different colors). All spectra exhibit similar features with the strongest pressure
fluctuations taking place at the low-frequency end of the spectrum. Most significant is the
absence of any harmonic peak in the spectra for the axisymmetric cavity at all azimuthal
locations. This reinforces the statements in the previous section that the harmonic
oscillation observed in the finite—width cavity is related to the existence of the side walls.
It is, however, significant to note that a local, broad peak is found in all spectra in the
axisymmetric case (pointed to by an arrow in Figure 3.21) at the same frequency (fl/U,o

z 0.21) as the self-sustained oscillation.

 

fL/Uno

Figure 3.21 Frequency spectra at different azimuthal locations in the axisymmetric
cavity. Black solid line shows spectrum at ¢ = 0°; Remaining lines depict the spectra
at angle increments of 22.5°

80

3.3 Summary

In conclusion, strong harmonic pressure oscillations are detected in cavities
having side walls. These oscillations are generally located at a distance of the order of the
cavity depth from the side walls. However, when the cavity becomes sufficiently
narrower, such as the case for the narrowest cavity examined here (W/D =2.5), the
oscillations on opposite sides of the centerline “merge” on the symmetry plane. The
frequency of the harmonic oscillation is much lower than the lowest Rossiter mode, and
close to the frequency of the wake mode found by Rowley et afl. No prominent
oscillation exists in the axisymmetric cavity without side walls. This shows that the
existence of a side-wall is essential for the establishment of the strong harmonic

oscillation found here.

Interestingly, a local, non-prominent peak is found in the wall-pressure frequency
spectra for the axisymmetric cavity at the same frequency as the self-sustained oscillation
in finite-width cavities. At high Re, the cessation of strong oscillation in the axisymmetric
cavity is accompanied by increased unsteadiness at very low frequencies. The
characteristics of the pressure signature in the symmetry plane of the wide cavity are the
same as for the axisymmetric cavity, i.e. exhibiting strong attenuation of the self-
sustained oscillation accompanied by increased unsteadiness at very low frequencies at

high Reynolds number.

Because of the three-dimensional nature of the harmonic oscillation found here
and its relation to the cavity side wall, this type of cavity unsteadiness could not be
identified in existing numerical simulations of cavity flows, which are based on infinitely

wide geometry. Moreover, because the oscillation is confined close to the cavity side

81

walls, experimental studies focusing only on the symmetry plane and employing cavities
with large width could not observe this oscillation either. However, since in practice all
cavities are finite in width, the oscillation could be pronounced producing high level of
noise and vibration of the geometry containing the cavity. In the following chapter, the
mechanism leading to the establishment of this new mode of oscillation will be

investigated using simultaneous velocity and wall-pressure measurements.

82

3.4 References

 

l Hudy, L.M., “Simultaneous Wall-Pressure and Velocity Measurements in the Flow
Field Downstream of an Axisymmetric Backward-Facing Step,” Ph.D. thesis, pp 64-67
(2005)

2 Rowley, C.W., Colonius, T. and Basu, A.J., “On Self-sustained Oscillations in Two-
dimensional Compressible Flow over Rectanguler Cavities”, Journal of Fluid Mechanics
455, pp. 315 -346 (2002)

3 Morris, SC, and Foss, J .F., “Turbulent Boundary Layer to Single-stream Shear Layer:
the Transition Region,” Journal of Fluid Mechanics, Vol. 494, Nov. 2003, pp. 187-221.

4 Bres, G.A., and Colonius, T., “Three-dimensional Instability in Compressible Flow over
Open Cavities,” Journal of Fluid Mechanics 599, pp. 309-339, (2008)

5 Maull, D.J., and East, L.F., “Three-dimensional Flow in Cavities,” Journal of Fluid
Mechanics 16, pp. 620-632 (1963)

83

4. SIMULTANEOUS VELOCITY AND PRESSURE

MEASURMENTS

To explore the mechanism of the oscillation in the non-axisymmetic cavity and
how this mechanism is affected by the side walls, two-component velocity data are
acquired using LDA simultaneously with the unsteady wall-pressure signals. This is done
for the cavity with L/D = 3.3 and W/D = 7.4 over x-y planes at the azimuthal symmetry
plane (A¢ = 0°, see Figure 2.19 for the definition of 11¢) and five other planes (A¢ = -24°,
-28°, -32°, -36° and -40°) in the vicinity of the azimuthal location where the harmonic
oscillation at fL/ U00 z 0.21 is strongest (Ail = -32° or z/D = -2.62). All measurements are
performed for a free-stream velocity of 15 m/s. In this chapter, the velocity-field

characteristics will be discussed first, followed by stochastic estimation of the flow field

to extract the coherent structures generating the surface-pressure fluctuation.

4.1 Velocity Results and Discussion

4.1.1 Mean-Velocity Field

The mean velocity fields in the x-y planes at A¢ = 0° and A¢ = -32° are compared
in Figure 4.1, which displays the mean-velocity vector field for (a) A¢ = 0° and (b) A¢ =
-32°. The reference vector at x/D = 0.24 and y/D = 1.89 represents the free-stream
velocity and is provided to define the plotting scale of the vector field. The uncertainty of
the mean velocity is less than 0.3% of the free stream velocity. Figure 4.2 shows the

corresponding streamlines. It can be seen from Figure 4.1 and Figure 4.2 that the mean

84

flow field in the central plane of the cavity is substantially different from that in the plane

close to the cavity side wall where the surface-pressure oscillation is strong.

Figure 4.1 indicates that at Art = 0°, the main re-circulating flow in the cavity
produces back flow that occupies most of the bottom part of the cavity and can reach all
the way to the. upstream cavity lip. However, at MS = -32°, the back flow is constrained in
the downstream half of the cavity and close to the bottom. The streamlines in Figure 4.2
show that the main recirculation at Art = 0° covers the whole cavity length with its center
at x/D z 2.6 and y/D z 0.7. At A¢ = -32°, the flow is dominated by a recirculation flow
that is confined in the downstream half of the cavity with the recirculation center at x/D z
2.4 and y/D z 0.4. Upstream of the recirculation bubble, the flow is directed towards the

downstream direction.

The features of the mean flow field in the central plane, i.e. the presence of a
recirculation covering the whole cavity length, is consistent with what is found in open
cavity flow (e.g. see Grace et a1 1, Ozsoy et al 2, Ukeiley and Murray3 and Ashcroft and
Zhang4). No similar mean flow structure as that shown in Figures 4.1 (b) and 4.2 (b), that
is, the confined, low-centered recirculation flow in the downstream half of the cavity,
could be found in any of the literature, which generally show measurements only in the
central plane of the cavity. It is suspected that this constrained recirculation is somehow

related to the presence of side wall.

85

—\
'01

 

 

 

 

 

 

 

' if ' ' , ‘1" - 77 :. — -— —_- ,. , _ . , . ,v— 17, , "v” w—k , j.»
_. _. . . . :p ,-* r, : ifp._..:i_;:>._ .7372-.. a. Q 3.. __'7. 2b __ 157:. ;- 1 ,p ‘ .*.___9-: “_._... —__._~ .__ V ‘3.
, ‘7»... . , 71., , 4,1,1” ---i':’> jg. _j, g j» —-5;':->—~. —- -~ —-_ —"~‘¥ -—"-1:r —" “:1.- — —‘”—s>“,
\0 1 -—- .. 5.. 71:7:- HfiH; 4:.» ,. — far- —- - i)- -—. V,_a-——' -.> - --- w—i' - g 1 3.3 — _,_“~
> ‘ ‘ 4- —-a __+. _._,u. . ~~1~— _. 17' -— ——'>- 131->111 , ":1... 3,1, . 1, .
.1 1“; -»_~-
1 - > .. _. ..- _.I- .._ . .‘b ._—~" _._ _(zv- _ , w. ‘:.. __ .1 1:5 _ "—4 i ,7”
‘ - 72’- ~;\
* > 1 I . 1 .- .J A) ”_.- .z—
1 ‘ -‘ ~‘ * .—. — _. r~w m2 .
~ ‘5 ‘*A -.
m
. . . 1 . . . I ,~7 I
. - ..—> 14. -_ , 1, .
,_ ,
__ g
. l g \
O o g 1 . ‘ ——‘ v 1‘
a
‘ | - . s
n r O ' Q . l l .' _ ‘
‘~
‘
I l ‘. \ ‘
' - o o . . ~ . .
l
, . . . i
I V O . - U ‘.
\ ‘ i if
‘
.
l 1 O . . O > "‘- ¢_ 9- .
’1
\, ‘1 w ’
- 1 o . . . o - - . ~' w». ‘_ - _ ... _
. 4'-
_ , I
l . o . _ , . o -.. »._,_ <5- .-1 q... _ .. . __ ‘
. ,1 1 _ . _-
‘~ A": :-
. l
0 1,1 _1, . _ __ ..__ .1 l___ _ 11L“_.___4__

0 0.5 1 1.5 2 2.5 3

Figure 4.1 Mean-velocity vector field in the x - y planes at (a) A¢ = 0° and (b) A¢= -
32°. The isolated vector near the top-left corner of the plots represents the free-
stream velocity

86

 

 

 

 

 

  
 

1
D '
3 1 I“ «
0 O 5 1 1 5 2 2 5 3 3 5
1.5
l fLTvfl‘ Naif??? ”3 (a)
1“ i: “ 5 5 - - ; w"
o - ~
3. 1 , _ - ‘ I: 3
0 0 5 1 1.5 2 2.5 3 3.5
x/D

Figure 4.2 Mean streamlines in the x - y planes at (a) A¢ = 0° and (b) A¢ = -32°

4.1.2 F luctuating—Velocity Field

Figure 4.3 shows flooded color—contour maps of the rms streamwise velocity

u’nm/U00 at (a) A¢ = 0° and (b) A¢ = -32°; and Figure 4.4 displays similar contour plots
for the rms wall-normal velocity v’mu/Uw. The magnitude of the rrns is represented by
the color bar provided beneath the plots. The figures indicate that the level of the
streamwise-velocity and wall-norrnal-velocity fluctuation at the cavity opening is higher
in the central plane than in the plane at Art = -32°. More specifically, the maximum
u’ms/U,O in the central plane is larger than that at Art = -32° by 21% (note that the

uncertainty of the rrns velocity is less than 1% of the maximum u’ms value in the central

87

plane). The shape of the contours also exhibits some differences between the two cases.
Noteworthy is the observation that if one characterizes the width of the separating shear
layer and its lateral spread using the rrns of streamwise velocity, u'mS/UOO, it is seen that
(for example, by following the yellow contour in Figure 4.3) the shear layer in the central

plane spreads downwards reaching to half the cavity depth at approximately x/D aV2.0. In

contrast, at A¢ = -32°, the yellow contour stays confined near the cavity opening for most
of the length of the cavity before it starts spreading near the downstream cavity wall. As
will be seen in section 4.2.2, the substantial spreading of the shear layer in the
downstream half of the cavity at Art = 0° is the result of the penetration of the shear layer
vortices into the cavity, and their subsequent grth to a scale comparable to the cavity

depth. Similar behavior is absent in the plane where the harmonic oscillation are strongest.

For the nns wall-normal velocity, in the central plane, the region with high rrns
gradually grows from the upstream lip to the downstream edge (Figure 4.4). At A¢ = -32°,
there is a somewhat sudden lateral expansion of the region with high rrns at x/D ~15
after the initial gradual growth. Noting that this is the location of upstream boundary of
the confined recirculation (Figure 4.2 (b)), it is believed that this sudden expansion in the
contours is related to flow structures in the confined recirculation flow at Art = -32°. This
suggests that the behavior of the v',,,,, contours is affected by both the shear layer spread

and the unsteady recirculation flow.

88

 

    

 

0.06 0.08 01 0.12 0.14

Figure 4.3 Flooded color-contour map of the rms streamwise velocity u'rm/U,O in

thex - y planes at (a) A¢ = 0° and (b) A¢= -32°. The color bar at the bottom gives the
rms values normalized by the free-stream velocity

89

 

       

 

0.03 0.04 0.055006 0.07 0.08

Figure 4.4 Flooded color-contour map of the rms wall-normal velocity V'rms/Uoo in
the x - y planes at (a) A¢= 0° and (b) A¢= -32°. The color bar at the bottom gives the
rms values normalized by the free-stream velocity

4.1.3 Reynolds Shear Stress

Figure 4.5 displays the color contours of the Reynolds-shear-stress u'—v'/U°2° in

the x-y planes at (a) A¢ = 0° and (b) A¢ = -32°. The figure shows that Tau/3, has its
maximum value at the cavity opening for both cases. However, the magnitude of the peak

u'—v'/ U 3, at the cavity opening in the central plane is 29% higher than that at A¢ = -32°

90

(note that the uncertainty of the Reynolds shear stress is less than 2% of the peak value in
the central plane). In addition, the large spread of the shear layer into the cavity in the
downstream half of the cavity at Art = 0° discussed above is also reflected in the spread of
the Reynolds- shear-stress contours in Figure 4.5 (a). At A¢ = 32°, most of the contours
in Figure 4.5 (b) remain confined to the cavity opening; again suggesting that in this
plane the shear layer does not penetrate into the cavity for most of the cavity length. The
“local” penetration of the contours deep into the cavity at x/D z 2.5 is likely associated

with the activity of the recirculation flow structure as will be seen in section 4.2.3.

91

 

 

x10-3

Figure 4.5 Flooded color-contour map of the Reynolds-shear-stress u—'17/U 3, in the
x - y planes at (a) A¢= 0° and (b) A¢= -32°. The color bar shows the Reynolds shear
stress normalized by the square of the freestream velocity

4.1.4 Mean Vorticity Field

Vorticity (a‘) ), the curl of the velocity vector field (I7 ), is used to characterize the

rotation rate of a fluid particle:
" 4.1

92

Since only two components of the velocity vector are measured in x-y planes, only the

out-of-plane vorticity (0),) will be discussed. The out-of-plane vorticity is calculated

using

wZ=———, 4.2

in which 3 is the partial derivative of the wall-normal velocity with respect to the

. . 6a . . . . . . .
streamwrse coordinate, and — rs the partial derivative of the streamwrse velocrty With

respect to the wall-normal coordinate.

The spatial derivative of velocity at each point of the measurement grid, except
those at the boundary, is calculated using the central finite-difference approximation

(expressed below in equations 4.3 and 4.4) employing the neighborhood points shown in

 

 

 

 

Figure4.6.
v. .—v._ .
fl = r+l,_/ I 1,] , 4.3
{9ij 2Ax
u. . —u. .—
211 = l,j+l 1,} l , 44
6x i.j 2Ay

in which Ax and Ay is the grid spacing in the streamwise and wall-normal direction

respectively, and i and j are integer indices indicating the grid node number in the x and y

directions respectively. 3:— on the upstream and 2u_ on the bottom boundary of the

measurement grid are calculated using the forward difference given by equation 4.5, and

their counterparts on the downstream and the top boundary of the measurement grid are

93

computed employing the backward difference (equation 4.6). Three-point Gaussian

smoothing is applied to the mean velocity field before the finite difference is calculated.

 

 

 

 

 

 

v. . _V' . u. . —u. .
fl = 1H,} 1,] & .61 = l,j+l 1,} 4.5
v. . —v._ . u. . —u. .—
21: = 1,] I 1,} & _afi : 1,] 1,] l 4.6
3x 1.1 M ay 1.1 my
i,j+1

O
y . i-1,j 1,1,1 . i+1,j

. i,j-1

 

 

$X

Figure 4.6 Illustration of the layout of neighborhood points used in calculation of
velocity derivatives at point (i, j)

Figure 4.7 displays the mean normalized out-of-plane vorticity 0720/6/00 in the
x-y planes at (a) A¢ = 0° and (b) A¢ = -32°. The uncertainty of the mean out-of-plane
vorticity is better than 0.4% of the peak value in the central plane. The distribution of
vorticity in these two planes looks similar, exhibiting high vorticity concentration close to
the separation point. The magnitude of vorticity decreases from the upstream to the
downstream lip as the vorticity is diffused laterally into and out of the cavity. As will
become clear in section 4.2.2, at M) = 0°, the spread of vorticity into the cavity is
associated with the growth of the shear layer vortices and their gradual movement into

the cavity as they travel downstream. In contrast, at A¢ = -32°, the shear-layer vorticity is

94

“pulled” into the cavity near the downstream wall of the cavity. This vorticity is then
directed downwards along the downstream wall, then upstream along the bottom wall.
This results in higher mean vorticity magnitude at x/D : 2.5 ~ 3.0 and y/D z 0.1 ~ 0.3 at
A¢ = -32° (bottom right corner of the plot in Figure 4.7 (b)) than that at A¢ = 0°. This
again indicates that the features of the flow close to the downstream comer at Art = -32°

are different from that in the central plane of the cavity.

 

Aisha-.1 . r '4‘ I} ’f;
a' v

    

-2.5 -2 -1.5’”’-1 '05 0

Figure 4.7 Flooded color-contour map of the mean vorticity a_)zD/Uno in the x-y

planes at (a) A¢= 0° and (b) A¢= -32°. The color bar at the bottom indicates the
magnitude of vorticity normalized using the cavity depth and free-stream velocity

95

4.1.5 Three-Dimensional Flow Field Close to Cavity Side Wall

It is shown from the above results that the mean flow and normal/shear Reynolds
stresses in the x-y plane at A46 = -32°, where the harmonic pressure oscillation is largest,
exhibit pronounced differences from that in the central plane of the cavity. To further
investigate how the side wall affects the mean flow locally, the mean-velocity field in x-y
planes near the side wall at A¢ = -45° will be examined. Figure 4.8 shows the streamlines
in the x-y planes at (a) A¢ = -32°, (b) A¢ = -36° and (c) A¢ = -40°. The streamlines pattern

at A¢ = -24° and A¢ = -28° are similar to those at A¢ = -32°, so they are not shown here.

The streamlines pattern at A¢ = -36° and A¢ = -40° is evidently different from
that at A¢ = -32°. Instead of being confined in the downstream half of the cavity as that at
A¢ = -32°, the recirculation flow in the x-y planes at A¢ = -36° and Ari = -40° covers the
whole length of the cavity, similar to the one in the central plane of the cavity. It is worth
noting that the center of the recirculation shifts downstream and away from the cavity
bottom as the side wall is approached: from Ag) = -32° to A¢ = -40°. Specifically, the
center of the recirculation is at x/D z 2.4 and y/D z 0.4 for A¢ = -32°, at x/D z 2.8 and y/D

z 0.5 for Ad = -36° and at x/D z 3.0 and y/D 7: 0.75 for A¢ = -40°.
In addition to sharing cavity-scale recirculation pattern, it is interesting to note
that the strength of the pressure fluctuations at fL/U,O z 0.21 on the downstream wall of

the cavity at A¢ = -36° and A90 = -40° is also similar to that at A¢ = 0°. This can be seen
from the frequency spectra plot in Figure 4.9. The figure demonstrates the weakness of

the pressure fluctuations at fl/ U00 z 0.21 relative to the broadband disturbances at A¢ = -

36°, A¢ = -40° and Ari = 0° compared to that at A¢ = -32°, even though a small peak at

96

fL/UO0 z 0.21 remains visible in the spectra for the former three azimuthal locations.

Together, the mean flow and unsteady pressure characteristics indicate that the strong

pressure oscillation at fL/UC,O z 0.21 at azimuthal location of A¢ = -32° is related to flow

structures and/or a mechanism that is causing the confined recirculation in the

downstream half of the cavity.

 

97

 

 

 

Figure 4.8 Mean streamlines in the x - y planes at (a) A¢= -32°, (b) A¢ = -36° and
(c) A¢= -40°

98

  

 

1X10 ,
0.8 _ ,
‘58 0.6
”a
e“ 0.4
E 1v , 1 .‘ , ‘
— A¢=-40°(z/D=-3.28) 1
0.2 g — A¢=-36°(yD=-2.95)
— A¢=-32°(z/D=-2.62) 1
‘1 A¢f0°(z/D=0) _‘ . , g :1 s,
10‘2 10'1 100
fL/U00

Figure 4.9 Frequency spectra at selected azimuthal locations for the cavity with W/D
= 7.4. Lines with different colors represent different azimuthal locations

99

To further understand the three-dimensional flow field close to the cavity side
wall, the third mean-velocity component normal to the x-y plane (i.e. in the azimuthal

direction), 17¢ is calculated within the cavity (0 < y/D < l) in five x-y planes at A¢ = -40°,

A¢ = -36°, A¢ = -32°, A¢ = -28° and A¢ = -24°. This is accomplished by solving the
continuity equation for the mean flow with known streamwise and wall-normal velocity
and the no-slip boundary condition at the cavity side wall (A¢ = -45°). The continuity

equation is given by:

l 0
;5(ru,)+——(u¢)+—(ux)= 0, 4.7

51¢

where r, ¢ and x are cylindrical coordinates and 17,, 17¢ and 17x are the corresponding

mean-velocity components. Figure 4.10 illustrates the cylindrical coordinate system, in
which x points in the streamwise direction along the axis of the model and is
perpendicular to the plane of the figure, and r is the radial coordinate measured from the

center of the model. Note that r and y, as defined in Figure 1.1, are related by r = y + r0,

where r0 is the model radius (50.8 mm). Thus, 17,, is the streamwise velocity 17 and L7, is

the wall-normal velocity 17 .

100

  
   
  

Azimuthal Symmetry plane “

Cavity side wall ' 1
Cavity bottom wall ‘

 

Figure 4.10 Illustration of the cylindrical coordinate system with origin at the center
of the axsiymmetric model

317,

Employing backward-difference approximation for—a; and central difference for

fl
ax

and y , equation 4.7 can be written as
r

17 l —17 l 1 — - —
¢ i,j,k ¢ i,j—l,k __ ui,j,k+1_ui,j,k—l + (rur)i+l,j,k —(rur)i—l,j,k

. . , 4.8
6g) [ IBM 2Ax 2Ar ]

 

in which Ar, §¢ and Ax are the grid spacing in the radial, azimuthal and streamwise
direction respectively, and i, j and k are integer indices indicating the grid node number in

the same directions respectively. On the upstream and the bottom boundary of the

x

 

measurement grid in an x-y plane, the finite-difference approximation of and

at?) in equation 4.8 employs forward difference; and their counterparts on the

downstream and the top boundary of the measurement grid are computed employing the

backward difference . Thus, 17¢ in five x-y planes at A¢ = -40°, A¢ = -36°, A¢ = -32°,

101

A¢ = -28° and A¢ = -24° within the cavity (0 < y/D < 1) is calculated by solving equation

4.8 with known streamwise (fix) and wall-normal (17,) velocity and the no-slip boundary

condition (l7r = 0) at the cavity side wall (71¢ = -

lA¢=—45° : l7¢lA¢=_450 = ExlA¢=—45°
45°),
Figure 4.11 shows profiles of the mean azimuthal velocity 17¢/Uw along the

wall-normal direction y at different azimuthal and streamwise locations. Near the
downstream cavity wall, at x/D = 3.0 and 2.8, shown in Figure 4.11 (a) and (b)
respectively, the azimuthal velocity is negative everywhere, which means the mean flow
is directed towards the side wall. Farther upstream, at x/D = 2.4, the mean azimuthal
velocity changes to positive in the top part of the cavity, which indicates the mean flow
moves away from the side wall. At x/D = 2.2 the region of positive azimuthal velocity
along the y direction increases and the magnitude of azimuthal velocity also increases. At
x/D = 2.0, the azimuthal velocity reaches its maximum positive value at y/D = 0.92 (the

upper boundary of the calculation) and the region with positive 5,, is also the largest
found within the measurement domain. The peak ii¢ magnitude is about the same as the
peak 27,, or 17, magnitude (about 10% of the free-stream velocity). At x/D = 1.8 the
extent of the region with positive 17,, (i.e with net flow towards the center of the cavity)

starts to decrease, accompanied with a reduction in the magnitude of the azimuthal
velocity. Overall, the data in Figure 4.11 show that there is a net mean flow towards the
center of the cavity in the top half of the cavity in the approximate range 1.8 < x/D < 2.5:
a point that will have significance when discussing the oscillation mechanism in section

4.2.

102

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

(a) A¢=-40° A¢=-36° A¢=-32° A¢=-28° A¢:-24°
1 ‘ ' 1 ' 1 1 . 1 . . 1 . .
0.9- -0.9- - 0.9- - 0.9- -0.9- .
0.81' «08 l- 108 _ 70.8" J08- .
07* 1 0.7- . 0.7- - 0.7- -0.7- .
0.6- -0.6- - 0.6- « 0.6- 105. .
gs 055 .0-5' “0.5' “0.5” -0_51- -1
04" '04- 104- -04- ~041 .
0.3r "0.3 ' -0_3 . _03_ _0.3_ ‘
0-2' "0.2 - ~02 . .02. ,0.2_ 4
0.1” '0.1 1' -01- .‘01 _, ‘01 _
0 . - 0 1 . 0 - 1 0 1 . 0 . .
(b) 1 ' ' 1 v t 1 r 1 1 T 1 1 . r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.9- 0 -0.91 -0.9- -0.9- 03 -

0.8- ' -0.8- ' -O.8- «08- -0.8- -

0.7- - 0.7» -0.7- 1 0.7- -0.7- .

0.6- - 0.31 -0.6- - 03- 03 ~
g 0.5- -0.5- 10.5» «0.5- «0.5-
04- -04- -04- 104- -041
0.3- -0.3- - 0.3- « 0.3- 03-
02- «0.21 .02. -0.2- 1 ~02-
0.1+ -0.1- 10.1- -0.1- 011

0-0110011 0.0110011 0.0110011 0.010011 0.010011

ago”

Figure 4.11 Wall-normal profiles of the mean azimuthal velocity 5, lU00 at different

azimuthal locations and (a) x/D = 3.0, (b) x/D = 2.8, (e) W = 2.4, (d) x/D = 2.2, (e)
x/D = 2.0 and (Ox/D = 1.8

103

Figure 4.11 continued:

 

 

 

 

 

 

(C) A¢=-40° A¢=-36° A¢=-32° A¢=-28° A¢=-24°
1 - . 1 . . 1 . , 1 . . 1 f T
0.9» ~0.9- ‘ -0.9- 0 «0.9- ' «0.9- ‘
0.8~ « 0.8- - 0.8- ‘ « 0.8- -0.8- -
0.7» 4 0.7? - 0.7 - - 0.7- ~0.7~
0.6» o -0.6- - 0.6r . ~ 0.6- 40.6-
g05~ -0.5- * -o.5b «0.5- -0.5~ ' -
04- -04- «04- ~04» ~04~ +
0.3- - 0.3- - 0.3h - 0.3- -0.3#
02- ~02» ‘ 0.2- - 0.2» ~02»-

0.1~ b -o.1L B ~0.1- \o-m- \o -0.1~ \9-

0: 10L 101 101 101 1
((1)1. 11. r1f r1. .1fi,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.9 - + 0.9 - -0.9- a 0.9 b -0.9» 4
0.8- +0.8- . 0.8- - 0.8- «0.8»
0.7» 40.7- d 0.7- 4 0.7% -0.7- -
0.6~ ~ 0.6- - 0.6» a 0.6- q0.6-
g 0.5L 0 -0.5- ' «0.5- * ~o.5- ' «0.5- -
04~ -04- " -04- " -04- .04, -
<: c.
0.3- ~o.3~ ~ 0.3» - 03- ~03»
a; o
02- «0.2L 4;, «0.2_ K-oz ~02-
0.1» «0.1- E -o.1- ‘01- -0.1-
0-0.1004 0.0.1001 0.0.1001 0.0.1004 0.0.1001
WU.»

104

Figure 4.11 continued:

(6) ¢'=-40° ¢=-36°
1 . 1 1 1 .
0.9- -0.9- -
0.8- -0.8- -
0% -0.7- .
0.6- ¢ 00- < .
gos- 0 ‘05- 0 ~
04_ ‘9 -04- ‘1’ -
(D 4b
0.3L -0.3- -
II 0
0.21 -0.2- c, -
0.1- ‘01- o -
0 4 0 . 1
(f) 1 - 1 .
0.9- ‘09- -
0.8~ «0.8» -
0.7- ¢ 07- : -
0.6- m -0.6- 1, -
O . . q
3.05 0 J05 0
0.4- 0 10.4» " ~
4L 0
0.31 10.3» f -
0.2» -0.2- -
0.1? -0.1~ .
0 1 1 1 1
0100.1 0.0.1001

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

¢=-32° ¢=—28° 0:240
1 . . 1 . . 1 1 .
0.91 - 0.9- -0.9~ .
0.8 L " 08L ~08 h- ..
0.7% C ‘ 0.7 ’ $07 - .
0.6- p 40.6- 0 -0.6- (J .
0.5- J> 05+ 0 .05. 0 .

04- ‘° 04- " .041 '>

‘D t) d)
0.3- - 0.3? -0.3. .

<9 (1 n

0.2 _ ‘i ~ 0.2 - (1: “0.2 ' 4:

0.1_ -0.1- 0 .04. A.
0 ' 0 ‘ . 0 . .
1 1 . . 1 T .
0.9 - . 0.9- -0.9- 1
0.8- ~ 0.8- -08. .
0.7- « 0.7» .07. ‘ i
0.6 ‘06- 406- 0 .
05- 05? -0,5_ «3 .
04- «04- 404- '5 .

‘5

0.3- - 0.3» 10.3-
0.2- - 0.2L 10.2-
0.1- ~0.1+ -0_1. -
0.0.1001 0.0.1 00.1 0.0.1 00.1
0.1000

105

 

To provide a more complete view of the cross-flow found above, the mean
streamlines in y-¢ planes at different streamwise locations are shown in Figure 4.12 and
Figure 4.13. At x/D = 3.0, downstream of the recirculation core in all measured x-y planes
(the most downstream recirculation center is at x/D = 3.0 at A¢ = -40°, as shown in Figure
4.8), the flow is directed towards the side wall from above and from the center part of the
cavity. At x/D = 2.8, upstream of the circulation center in the x-y plane at A¢ = -40°, the
flow remains directed towards the side wall but it now appears to “spill over” the top side
corner of the cavity. Between x/D z 2.6 - 2.4, in the vicinity of the main recirculation
center in the x-y plane at A05 z -36° - -24°, the flow is directed upwards after interaction
with the side wall and a circulatory flow pattern forms in the y-¢ planes. At x/D = 2.2
and 2, upstream of the recirculation center in all x—y planes, the flow has strong directivity
towards the center of the cavity in the upper half of the cavity. Overall, the results in
Figures 4.12 and 4.13, highlight the highly complex three-dimensional ‘flow character

inside the cavity.

106

a:

 

(a) y/D = 0.92 —

‘S,

L

 

y/D= 0.10 .
M): «092., y
V“ L,
y/D= 0.10 %
A¢= 40° A¢=-24°

Figure 4.12 Mean streamlines in the y-¢ planes at (a) x/D = 3.0 and (b) x/D = 2.8.
The streamwise direction is into the paper

107

(a)

(b)

(C)

(d)

 

y/D=0.10

y/D = 0.92 if '

 

 

y/D= 0.10

A05 = —40° A05 = -24°
Figure 4.13 Mean streamlines in the y- 90 planes at (a) x/D = 2.6, (b) x/D = 2.4, (c) x/D
= 2.2 and (d) x/D = 2.0. The streamwise direction is into the paper

108

Additional insight into the three-dimensional character of the mean flow can be
gained through inspection of Figures 4.14 and 4.15, which display selected three-
dimensional streamlines inside the cavity within the region between A¢ = -40° to A¢ = -
24°. Figure 4.14 demonstrates the streamlines originating from x/D 2 2.6 at A¢ = -40°.
The streamlines indicate that the mean flow field originating close to the side wall swirls
around the main recirculation flow as it is driven towards the middle potion of the cavity.
The cross-flow towards the cavity center can be highlighted further by considering the

streamlines near the center of the recirculation flow (displayed in Figure 4.15).

 

Figure 4.14 Mean three-dimensinal streamlines originating near the cavity’s side
wall: from x/D z 2.6 at A¢ = -40°

109

 

 

 

Figure 4.15 Mean three-dimensional streamlines originating near the cavity’s side
wall and the center of recirculation: from x/D z 2.9 and y/D z 0.75 at A¢ = -40°

It is useful to compare the complex three-dimensional flow found here inside the
cavity to the flow inside a lid-driven cavity (in which the shear layer is replaced by a
solid wall moving in the streamwise direction). The comparison is motivated by two
factors: 1) no numerical or experimental studies could be found of shear-driven cavity
flows, in which the cavity width is finite and details regarding the 3D flow inside the
cavity are provided; 2) it is believed that the flow features inside a finite-width cavity,
whether it is driven by a moving wall or a shear layer, should bear some similarity. Of
particular relevance to the present findings is the comparison with the detailed numerical

study of a finite-width rectangular, lid-driven cavity at relatively low Reynolds number

110

by Chiang et al. 5 . These authors found particles to circuit around the primary
recirculation flow inside the cavity towards the side wall, and then (similar to the present
flow) spiral to the symmetry plane along the core of the primary recirculation. Chiang et
al. found the cross-flow towards the center to be driven by a spanwise pressure gradient.
Although caused by a different mechanism, the cross-flow found here is also believed to
be driven by a spanwise pressure gradient. How this pressure gradient is established, and
the significance of the resulting cross flow to the observed harmonic oscillations near the

side wall will be discussed in the following section.

111

4.2 Stochastic Estimation of the Coherent Structures Generating Wall-

Pressure Fluctuations

Stochastic Estimation (SE) as an approach to extract coherent structures in
turbulent flows was first proposed by Adrian6. It is a conditional-averaging technique that
uses unconditional statistics (two-point correlations) to estimate a stochastic flow variable
at a particular point in space or time based on the information of a known variable at the
same or other point. Adrian7 and Guezennec8 showed the capability of SE in estimating
the turbulent velocity field based on velocity measurements at different locations. Naguib
et al 9 was the first study to estimate the flow sources related to surface pressure events in
a turbulent boundary layer. They estimated the flow field using both linear stochastic
estimation (LSE) and quadratic stochastic estimation (QSE), and found it was necessary
to include the quadratic term in the estimation. This was related to the generation of the
wall pressure by the nonlinear sources of the conditional velocity field. Murray and
Ukeileylo applied stochastic estimation to predict the temporal evolution of the velocity
field in a 2D open cavity using surface-pressure information. They compared the
estimation with direct numerical simulation and showed that adding more pressure
“measurement” locations in LSE only improves the estimation in regions close to where
the pressure was acquired. They found that the linear estimation was able to predict most
of the time-resolved flow evolution, but the quadratic term was necessary in capturing the
turbulent energy and finer details of the vorticity field. Hudy1| successfully predicted the
evolution of the flow structures generating the surface-pressure signature on the same

axisymmetric model employed here (but for a backward-facing-step geometry) using

112

LSE estimation based on the wall-pressure array measurements beneath the

separating/reattaching flow.

Since the wall-pressure signature inside the cavity is dominated by unsteadiness
near the cavity’s downstream wall (see Figures 3.1, 3.2 and 3.3 for the axisymmetric
case), in this section the evolution of the flow structures in the cavity is done based on the
surface-pressure signal on the downstream wall. In particular, the velocity fluctuation at a

certain location and time, u'(x0 + Ax, yo + Ay,zo + A2,! + At) is estimated using the wall-
pressure fluctuation (also known as “condition”, or “event”) p'(x0, y0,zo,t). Within the

SE framework, the estimated velocity fluctuation, 17' , is expressed in a Taylor series

expansion in terms of the pressure condition:

l7'(xo + Ax, yo + Ay. 20 + Az,t + At) = AMAy,&,A0xo;yo;zo)fl(xo,yo,zo,t)
+B(Ar,Ay.Az,Ar;xo;yo;zo)p'2(xo,yo.zo.0+0(p3) 4,9
where x0 , yo and 20 are the streamwise, wall-normal and spanwise (or azimuthal)
location of the reference pressure, and t is the instance when the pressure event occurs.
Ax , Ay , Azand At are the streamwise, wall-normal, spanwise distance and time offset

between the estimated velocity and reference pressure. A is the linear estimation
coefficient and B is the quadratic estimation coefficient. The estimation coefficients are
determined by minimizing the expectation, E , of the squared error between the estimated

and measured velocity:

E(e2) =< {17'(x0 + Ax,y0 + Ay,zO +Az,t + At)—u'(x0 +Ax,y0 +Ay,zo +Az,t +At)}2 >

4.10

113

 

 

 

2 2
The minimization problem leads to: 6H8 ) = 0 and 6E“: ) = 0.
6A BB
In LSE,
A _ < u'(x0 + Ax,y0 + Ay,zO + Az,t+ At)p’(x0,y0,zo,t) > _ ru'p'
<p'(x0,yo,zo,t)p'(x0,y0,zo,t)> [73,13 ,
and B = 0, 4.11

where ru'pv is the correlation between the velocity and pressure. Equations 4.9 and 4.11
yield single-point LSE of the streamwise velocity. Similar equations with u’ replaced
with v' give LSE of the wall-normal velocity. In most cases, it is desired to estimate the
instantaneous flow field from concurrent pressure information, and hence the estimation

coefficient is only a function of space (Ax , Ay and A2) while the time offset At = 0. As

a result, the temporal evolution of the velocity field is estimated based on the evolution of
the pressure event. The method works well at locations where there is good velocity-
pressure correlation at time offset of zero. To ensure that such correlation is maintained

with all points within the flow, multiple pressure events measured at distributed points on

the wall are typically used; this leads to the so-called, multi-point LSE (e.g. see Hudyl I).

Although in the present study, multi-point LSE could be used based on
information from the streamwise and spanwise microphone arrays, no good correlation
could be found at a time offset of zero between any of the microphone’s signals and the
velocity signature of the shear layer in the upstream half of the cavity. Hence, the

approach didn’t capture the shear-layer evolution well.

Generally speaking, the pressure signal measured at a particular location

correlates well with the velocity field within its vicinity at zero time offset. If this signal

114

is produced by a coherent structure, such as the shear-layer vortices, the pressure signal is
also well correlated with the velocity at remote spatial locations along which this
structure evolves; although in this case, the correlation would be found at positive or
negative time offsets. To take advantage of this temporal relationship in LSE, equations
4.9 and 4.11 are employed here in the following sense: a pressure event with a given
magnitude is specified at location x0, yo and 20, which is taken on the downstream wall of
the cavity. The temporal evolution of the velocity field is then captured by considering
the estimated velocity field at different time offsets At before, at the same time, and after
the event has occurred. Note that in this estimation, which can be considered as single-
point multi-time-offs‘et estimation, the event is fixed but the estimation coefficient (which
expresses the pressure-velocity correlation at different time offsets) changes with time

relative to the occurrence of the event.

More generally, the estimation used here can be viewed as a special case of
single-point, multi-time estimation, in which, the velocity at a given point in the flow is
estimated from past, current, and future values of the pressure measured at a point on the
wall (see Durgesh‘z). The approach leads to a linear algebraic set of equations of order
equal to the number of temporal pressure points used in the estimation. The equation set
is solved to obtain the estimation coefficients. In this thesis, the simpler approach
described above is used since it is found to capture the shear-layer evolution satisfactorily.
Finally, it is noted that quadratic estimation was also conducted in this work but found
not to add information about the propagation/evolution of the shear layer structures but

only regarding the re-circulating flow inside the cavity.

115

4.2.1 Velocity-Pressure Correlation

As seen above, linear stochastic estimation requires information about the
velocity-pressure correlation in order to calculate the estimation coefficient (see equation
4.11). Example of such correlation is given in Figure 4.16 at different time offsets. The
figure contains flooded color contour maps of the correlation between the fluctuating
streamwise velocity in the x-y plane at A¢ = -32° and the wall-pressure fluctuations on the
downstream wall of the cavity at x/D = 3.3 and y/D = 0.5 (indicated by the blue open

circle). The correlation value is normalized by the freestream velocity U ,0 and dynamic

pressure 1/2pU020, i.e. Rurpr = rurp/( 1/ ZpUgo), to emphasize the global importance of
regions with high correlation. If normalized by local rms values, the correlation falls

between -O.3 and 0.3. The reference surface-pressure signature are band-pass filtered

between 50 Hz to 150 Hz (IL/U00 = 0.137- 0.410) in order to focus the analysis on
disturbances generating the pressure oscillation at fZ/U00 z 0.21. Figure 4.17 shows
similar contours for the wall-normal velocity component: vapr. Figure 4.18 and Figure
4.19 display correlation contours of Rurp: and erpr respectively in the x-y plane at A05 = 0°.

The region with high correlation indicates locations within the flow where the
disturbances are correlated with the reference pressure signature. The evolution of the
correlation with time offset in Figures 4.16 through 4.19 tracks the propagation of
velocity disturbances that are related to the surface-pressure oscillation. It is interesting to
note that the region with high correlation in the symmetry plane extends more upstream,
close to the separation point, compared to that in the x—y plane at A¢ = -32°. The flow

structure associated with these disturbances will become evident through the use of these

116

 

 

correlation results to arrive at linear stochastic estimation of the flow field based on the

unsteady pressure on the end wall.

AtUw/L= -1.1O

1.5r 7. , .7. a ,,,, 477‘ , -

 

x104

Figure 4.16 Flooded color-contour maps of the velocity-pressure cross-correlation
(Ruvpr) at Ad= -32°. Pressure is measured at x/D = 3.3 and y/D = 0.5 (shown by the
blue circle on the end wall of the cavity). Correlation results are shown at zero time
delay (middle plot) as well as at time delays corresponding to :1: 1/4 the oscillation
period at fL/Uw a: 0.21. The color bar at the bottom of the plot gives Rurprvalues

117

AtUm/L= -1.10

 

0 AA A. AA A 7 L A , in

AtUwIL= 1.10 7

 

    

-5-4-3-2-1012 34 5
x 10“

Figure 4.17 Flooded color-contour maps of the velocity-pressure cross-correlation
(Rv'p') at A¢= -32°. Pressure is measured at x/D = 3.3 and y/D = 0.5 (shown by the
blue circle on the end wall of the cavity). Correlation results are shown at zero time
delay (middle plot) as well as at time delays corresponding to :1: 1/4 the oscillation
period at leU,o as 0.21. The color bar at the bottom of the plot gives Rv'p' values

118

 

AtU /L=-1.1O
(X)
1.5.. W. ,1. .7~ f 7. 7 ~

 

x104
Figure 4.18 Flooded color-contour maps of the velocity-pressure cross-correlation
(Rurpr) at A¢= 0°. Pressure is measured at x/D = 3.3 and y/D = 0.5 (shown by the
blue circle on the end wall of the cavity). Correlation results are shown at zero time
delay (middle plot) as well as at time delays corresponding to :1: 1/4 the oscillation
period at jL/Uw z 0.21. The color bar at the bottom of the plot gives Rurpr values

119

AtUm/L= -1.10

 

y/D

 

x 10“

Figure 4.19 Flooded color-contour maps of the velocity-pressure cross-correlation
(erp') at A¢= 0°. Pressure is measured at x/D = 3.3 and y/D = 0.5 (shown by the
blue circle on the end wall of the cavity). Correlation results are shown at zero time
delay (middle plot) as well as at time delays corresponding to :1: 1/4 the oscillation
period at jL/Uoo z 0.21. The color bar at the bottom of the plot gives Rv'p' values

120

4.2.2 Evolution of the Coherent Structures in the Azimuthal-Symmetry Plane of the

Cavity

The fluctuating velocity in the azimuthal symmetry plane (A¢ = 0°) is estimated at
different time lags relative to a reference positive pressure on the downstream wall in the
same plane (at x/D=3.3 and y/D=0.5). For convenience, the reference pressure is taken as

' = prms. Because of the linearity of the estimation, stronger or weaker pressure

magnitude will only increase or decrease the estimated velocity magnitude but not its
direction; thus, the features of the estimated flow structures are independent of the
specific choice of the strength of the pressure event. Figure 4.20 (l) - (12) show the
evolution of the vorticity contours of the estimated flow field (without the addition of the
mean flow) and the streamlines of the total velocity (with the addition of the mean flow;
details of the superposition of the estimated and mean velocity field is explained in the
following paragraph) at different time offsets preceding and following the occurrence of
the positive pressure condition (for At = 0: shown in Figure 4.20 (7)) with a time interval
of AtUw/L = 0.375 covering one period of the oscillation at fL/ U00 z 0.21. The blue circle
on the cavity’s downstream wall marks the location of the reference wall pressure. Note
that showing the vorticity contours based on the estimated fluctuating velocity in Figure
4.20 is done to emphasize the unsteady flow structure. Also, three-point Gaussian

smoothing is applied to the estimated velocity field before the vorticity is calculated.

Figure 4.20 depicts a small concentration of vorticity apparently originating from

the shear layer, near the upstream lip (pointed to by an arrow in Figure 4.20 (1) at AtUw/L

= -2.20). This vorticity concentration seems to correspond to a vortex which grows as it

propagates downstream and into the cavity (see Figure 4.20 (2) through (7) from AtUoo/L

121

 

 

= -1.83 to 0). Subsequently, the vortex is shed out of the cavity and decays (see Figure

4.20 (7) through (12) from AtUoo/L = 0 to 1.83). The process of vortex growing while

penetrating into the cavity and shedding resembles that for the cavity wake mode (see
Najm and Ghoniem13 and Rowley etal.14). In fact, as discussed previously in section 3.1,
the frequency of the weak peak in the wall-pressure spectra at A¢ = 0° shown in Figure
4.9 is 0.06 if normalized by the cavity depth, which is the same as the frequency of the
wake mode found by Rowley et al. ‘4. A significant point is that when the vortex structure
penetrates into the cavity, it overlaps the x-y domain over which mean cross-flow towards
the cavity center is found in the analysis of the mean three-dimensional flow. The extent
of this domain, as found from the results of section 4.1.5, is outlined in Figure 4.20 using
a broken-line rectangle. Outside the highlighted domain, the vortex is practically outside
of the cavity. These observations suggest that the cross-flow towards the center of the
cavity may in fact be related to the penetration of the shear-layer’s vortex structure into
the cavity and its subsequent growth/strengthening. It is important to note that such
penetration is a feature of the wake mode, and it does not take place in the shear-layer

(Rossiter) type cavity unsteadiness.

Figure 4.21 displays the contours of vorticity, sz/Uoo and streamlines,

calculated from the total velocity in the x-y plane of A¢= 0° at different time offsets
relative to a positive pressure condition on the downstream wall at x/D = 3.3 and y/D =
0.5. The plot is obtained by multiplying the estimated velocity field by a factor then
superposing the outcome on the mean velocity field. The multiplication by a factor is
done to strengthen the estimated flow structures relative to the mean velocity. This is

necessary since the conditional averaging associated with stochastic estimation inevitably

122

weakens the strength of the estimated structures. Hence, when superposing the estimated
and mean fields, the latter could dominate the outcome. A factor of 15 is chosen so that
the peak vorticity of the vortex structure during its evolution is of similar magnitude to
the peak mean vorticity at separation. This accounts for the fact that the vortex structure
forms from the roll-up of the shear layer, while ignoring the effects of viscous diffusion
and vorticity stretching/tilting. It is important to note, however, that the features of the
unsteady structures is not affected by the multiplication factor but only the strength of
them relative to the mean flow. The purpose of superposing the estimated and mean

fields is to help visualize the unsteady structures in relation to the separating shear layer.

The region with large negative vorticity value in Figure 4.21 highlights the
location of the shear layer. The figure illustrates that the shear layer dips into the cavity

during the time period of Ath/L = -2.20 ~ 0 (from Figure 4.21 (1) to (7)) when vortex

structures grow and propagate (shown in Figure 4.20 (1) through (7))), and it moves out

of the cavity during the period of AtUOO/L = 0 ~ 1.83 (see Figure 4.21 (7) through (12))

when vortex structures are shed out of the cavity and decay.

Figure 4.22 shows the fluctuating wall-pressure on the cavity bottom (blue line in
Figure 4.22) and on the downstream wall (red circle in Figure 4.22) beneath flooded
color-contour maps of the streamlines and estimated vorticity at two instances: when the
pressure on the downstream wall is peak positive and peak negative. The fluctuating
wall-pressure is estimated using the same method as that for the velocity. The figure
shows that the pressure on the downstream wall reaches a positive peak when the vortex
structure inside the cavity grows to a size compatible to the cavity depth, and penetrates

into the cavity, directing the high-speed flow from above towards the downstream wall;

123

the pressure reaches a negative peak when the vortex structure is shed out of the cavity.
This is associated with pumping of fluid out of the cavity’s downstream corner, as shown

by the streamlines.

In summary, the stochastic estimation results suggest that the wall-pressure
fluctuations in the symmetry plane are generated by the growth and shedding of vortex
structures accompanied by the flapping of the shear layer into and out of the cavity. The
coherent structures represent a cavity wake mode even though the pressure oscillation is
not pronounced in the symmetry plane. However, it is believed that the wake mode
vortex structures are responsible for “pumping” fluid from the side-wall vicinity towards
the cavity center, ultimately leading to the strong harmonic oscillations found near the

wall (as will be discussed below).

124

 

 

 

 

 

 

 

-0.05 *0. 0.05

Figure 4.20 Streamlines and estimated fluctuating vorticity 5", D/ U a, contours at
A¢= 0° for different time offsets (covering one period of the oscillation at jL/Uw =1
0.21 with a time interval of AtUoc/L = 0.365) preceding and following the occurrence
of a positive pressure (p’= Prms at At = 0 in (7)) at x/D = 3.3 and y/D = 0.5. The color
map on the bottom gives 07'z D/ U00 values

125

 

 

 

 

 

 

 

 

 

 

Figure 4.21 Streamlines and vorticity sz / U a, contours at A¢= 0° for different
time offsets (covering one period of the oscillation at [Ll U,O a 0.21 with a time
interval of AtUa/L = 0.365) preceding and following the occurrence of a positive
pressure (p’= pm” at At = 0 in (7)) at x/D = 3.3 and y/D = 0.5. The color map on the
bottom gives sz/Um values

126

 

 

 

L

0' '1’ ’2" 3

x/D
Figure 4.22 Streamlines, estimated fluctuating vorticity 07' z D/ U a, field and
concurrent surface pressure in the x-y plane of A¢= 0° corresponding to: (a) peak
positive pressure and (b) peak negative pressure on the cavity end wall at x/D = 3.3
and y/D = 0.5 (pressure value shown with red circle). Blue line shows the pressure
distribution on the cavity bottom, and color bar gives 5'2 D/ U 00 values

127

4.2.3 Evolution of the Coherent Structures Close to the Cavity Side Wall

Figure 4.23 shows the streamlines and estimated fluctuating vorticity (5'2 D/ U 00)
field in the x-y plane at A¢ = -32° for different time offsets (extending over one period of
the oscillation cycle at fL/Uc,O z 0.21) relative to a positive pressure (p'= prms) on the
downstream wall at x/D = 3.3 and y/D = 0.5. Recall that the wall-pressure signature in

this plane is dominated by the pronounced harmonic oscillation at fL/Uoo z 0.21 (shown

in Figure 3.21 and Figure 4.9). No obvious vortex growing and shedding as found at A05 =

 

0° is observed in Figure 4.23. Instead, the flow seems to be dominated by the “splitting”
and pulling of negative vorticity from the shear layer (i.e. at the top of the cavity) into the
cavity near the downstream lip: see the top-right comer of the measurement domain in

Figure 4.23 (1) through (3) for time offsets AtUw/L = -2.20 ~ -l.46. Subsequently, this

vorticity is intensified and grows inside the cavity, leading to the establishment of a
recirculation flow in the downstream half of the cavity. As this happens, the vorticity
progressively moves towards the cavity bottom, then upstream before it weakens again
and disappears from the field of view. The strengthening and growth of the negative
vorticity pulled into the cavity near the lower downstream comer of the cavity induces
movement of high-speed fluid from above the cavity towards the comer. This can be seen
best by focusing on the time offset at which the peak positive pressure on the downstream
wall takes place, seen in Figure 4.24 (a). Particularly interesting is that the concentration
of negative vorticity in this case is substantially lower and farther downstream than that
leading to the peak positive pressure at A0) = 0° (Figure 4.22 (a)). Thus, in the former case,
the penetration of the high-speed fluid into the cavity is deeper, causing a stronger peak

positive pressure. On the other hand, the peak negative pressure on the downstream wall

128

 

is seen to be associated with the movement of fluid away from the wall, as seen from the

streamlines pattern in the vicinity of the pressure-measurement location in Figure 4.24 (b).

Finally, for completeness, the total vorticity field at A¢ = -32° can be seen in
Figure 4.25. No strong vertical displacement of the shear-layer in/out of the cavity is seen

here. Recall that such flapping motion was found to be associated with the formation of

the wake mode at A40 = 0°.

129

 

 

 

 

 

 

 

   

005 ”‘0 i 0.05

Figure 4.23 Streamlines and estimated fluctuating vorticity 5'z D/Uw contours at

A¢= -32° for different time offsets (covering one period of the oscillation at fl./ U00 N

0.21 with a time interval of AtUw/L = 0.365) preceding and following the occurrence

of a positive pressure (p’= Prms at At = 0 in (7)) at x/D = 3.3 and y/D = 0.5. The color
map on the bottom gives 5'z D/ U a, values

130

 

Pa
O

(b) 1.5. 7

y/D

 

 

Pa
O

0 1 ' 2 3
x/D
Figure 4.24 Streamlines, estimated fluctuating vorticity 5'1 D/ U 00 field and
concurrent surface pressure in the x-y plane of A¢ = -32° corresponding to: (a) peak
positive pressure and (b) peak negative pressure on the cavity end wall at x/D = 3.3
and y/D = 0.5 (pressure value shown with red circle). Blue line shows the pressure
distribution on the cavity bottom, and color bar gives 5'2 D/ Um values

131

 

 

 

 

 

 

Figure 4.25 Streamlines and vorticity atzD / U a, contours at A¢= -32° for different
time offsets (covering one period of the oscillation at [Ll U00 as 0.21 with a time
interval of AtUw/L = 0.365) preceding and following the occurrence of a positive
pressure (p ’= pm” at At = 0 in (7)) at x/D = 3.3 and y/D = 0.5. The color map on the
bottom gives szI U a, values

132

Figure 4.26 displays the streamlines and estimated vorticity 5'2 D/ U ,0 field in x-
y plane at A¢ = -36° for different time offsets (over one period of the oscillation at fL/ U00

z 0.21) relative to a positive pressure (p ’= prms) at x/D = 3.3, y/D = 0.5 and A¢ = -32°.

The observed flow structures combine the features of the structures found at A¢ = 0° and -
32°. Similar to what happens in the central plane, vortex structures from the shear layer
grow and propagate downstream in the x-y plane at A¢ = -36°. In addition, negative
vorticity that is trapped near the downstream wall and pulled into the downstream half of
the cavity (as at A¢ = -32°) is found. The downstream convecting vortex structures do not
penetrate much into the cavity or grow to a size comparable to the cavity depth as in the

central plane before they interact with the trapped vorticity from the downstream comer

(see Figure 4.26 (6) to (8) at AtUoo/L = -0.37 to 0.37).

Figure 4.27 shows the evolution of the vorticity field in the x-y plane at A¢ = -36°.
It suggests that there may be some lateral displacement of the shear layer into the cavity

between AtUoo/L = -1.83 to -0.37 (seen Figure Figure 4.27 (2) to (6)). The displacement is

not as pronounced as in the central plane.

133

 

 

 

 

 

 

 

 

Figure 4.26 Streamlines and estimciigid fluctiiating vciig'ficity 'ai'z D/Uw contours at

A¢= -36° for different time offsets (covering one period of the oscillation at jL/U00 z

0.21 with a time interval of AtUw/L = 0.365) preceding and following the occurrence

of a positive pressure (p’= Prms at At = 0 in (7)) at x/D = 3.3 and y/D = 0.5. The color
map on the bottom gives (3' z D/ U 0., values

134

 

 

 

 

 

 

 

 

Figure 4.27 Streamlines and vo-rgticity-1wZD IOU“, cimtoulz's at A¢= -36° for different
time offsets (covering one period of the oscillation at jL/Uw as 0.21 with a time
interval of AtUa/L = 0.365) preceding and following the occurrence of a positive
pressure (p’= pm” at At = 0 in (7)) at x/D = 3.3 and y/D = 0.5. The color. map on the
bottom gives sz/ U00 values

135

4.3 The Oscillation Mechanism: a Hypothesis

The results presented so far clearly show that strong, harmonic pressure
oscillation are established close to the side wall (about one cavity depth away from the
side wall) at the downstream end of the cavity. Although this oscillation is absent from
the symmetry plane of the cavity (for sufficiently wide cavities), evidence leads to the
hypothesis that the flow structures in the symmetry plane (which are similar to those
found in the axi-symmetric cavity, where no strong harmonic oscillations are found at all)
might be responsible (i.e. provide the driving mechanism) for the establishment of the

oscillation near the side wall. This evidence consists of two components:

1. The frequency of the pressure oscillation caused by the harmonic oscillation is the
same as that of the structure found in the symmetry plane and axi-symmetric cavities

(see Figure 3.21 and 4.9).

2. At the frequency of the harmonic oscillation, coherence of about 20% is maintained
between the pressure fluctuations in the symmetry plane and the harmonic oscillation
near the side wall (see Figure 3.19). Although the coherence value is low, it is not
zero and shows that correlated structure exists at the harmonic oscillation frequency
in the symmetry and near-side-wall planes. The low coherence value should not
undermine the significance of this correlation as this low value is likely the result of
the weak pressure signature of the structure in the symmetry plane relative to the

signature of the broadband turbulence.

The flow structure in the symmetry plane is found to be wake-like, consisting of

vortex structures that form from the roll-up of the separating shear layer, and penetrate

136

 

into the cavity down to approximately half the cavity depth. Over the x-y domain inside
the cavity into which these vortex structures penetrate, a cross-flow towards the
symmetry plane is found near the side wall. This is believed to be the “communication
mechanism” between the symmetry and oscillation planes. In particular, it is well known
that the core of vortex structures corresponds to a low pressure zone. For example,
Bradshaw and KohIS manipulated Poisson’s equation, which governs the pressure field in
incompressible flow, to show that negative pressures are generated wherever vorticity
exists. Thus, it is hypothesized here that when the wake mode vortex structures penetrate
into the cavity they create low-pressure regions in the central zone of the cavity. This
leads to the establishment of a spanwise pressure gradient that drives the flow from near

the side wall towards the symmetry plane.

The cross flow, which will occur at the same frequency as the shedding frequency
of the central-plane vortex structures, is believed to cause two effects in the zone of
strong pressure oscillation. First, it pumps fluid away from the side/end-wall zone,
creating strong suction locally (corresponding to plot (1) in Figure 4.23). Second, to
replace the fluid that is pumped from the side wall, vortical fluid is pulled from the shear
layer into the cavity near the end wall (as seen from the LSE results in Figure 4.23 in
plots (1) through (7)) towards the cavity downstream corner. The deep penetration of this
vorticity and its subsequent intensification induces the flow of high-speed fluid from
above the cavity towards the cavity downstream comer creating large positive pressure
(plot (7) in Figure 4.23). Because of the “down pulling” of vorticity, the induced flow
towards the cavity corner penetrates deeper than in the symmetry plane, causing larger

pressure swings than found in the latter.

137

 

 

In summary, the above implies that the pressure fluctuations generated by the
flow structure in the axi-symmetric cavity, which are not too strong and are comparable
with those produced by background turbulence, can be intensified in the presence of
,cavity side walls. The intensification mechanism appears to be related to the
establishment of complex, unsteady, three-dimensional flow resulting from the
interaction of the flow structure near the cavity center with the flow near the side wall.
The hypothesized mechanism linking the two flow zones requires fitrther validation
through SE, or whole-field measurements of the flow in y-¢ planes that span half the
cavity width and extend over the downstream half of the cavity. This is left for a future

follow-up study.

138

4.4 References

 

1 Grace, S.M., Dewar, G.W. and Wroblewski, D.E., “Experimental Investigation of the
Flow Characteristics within a Shallow Wall Cavity for both Laminar and Turbulent
Upstream Boundary Layers,” Experiments in Fluids, 36, 2004, pp. 791—804.

2 Ozsoy, E., Rambaud, P. and Riethmuller, M.L., “Vortex Characteristics in Laminar
Cavity Flow at Very Low Mach Number,” Experiments in Fluid, 38, 2005, pp. 133-145.

3 Ukeiley, L. and Murray, N., “Velocity and Surface Pressure Measurements in an Open
Cavity,” Experiments in Fluids, 38, 2005, pp. 656-671.

4 Ashcroft, G. and Zhang, X., “Vortical Structures over Rectangular Cavities at Low
Speed,” Physics of Fluids, 17 (015104), 2005, pp. 1-8.

5 Chiang, T.P., Hwang, RR. and Sheu, W.H., “Finite Volume Analysis of Spiral Motion
in a Rectangular Lid-Driven Cavity,” International Journal for Numerical Methods in
Fluids, 23, 1996, pp. 325-346.

6 Adrian, R.J., “On the Role of Conditional Averages in Turbulence Theory,”
Proceedings of the Fourth Biennial Symposium on Turbulence in Liquids, Rolla, Mo.,
September, 1975. (A77-40426 18-34) Princeton, N.J., Science Presss, 1977, pp. 323-332.

7 Adrian, R.J., “Conditional Eddys in Isotropic Turbulence,” Physics of Fluids, 22, 1979,
pp. 2065-2070.

8 Guezennec, Y.G., “Stochastic Estimationof Coherent Structures in Turbulent Boundary
Layer,” Physics of Fluids A, l, 1989, pp. 1054-1060.

9 Naguib, A.M., Wark, CE, and Juchenhofel, 0., “Stochastic Estimation and Flow
Sources Associated with Surface Pressure Events in a Turbulent Boundary Layer,”
Physics of Fluids, 13(9), 2001 , pp. 22611-2626.

'0 Murray, N. and Ukeiley, L., “Estimationof the flowfield from surface pressure
measurements in an open cavity,” AIAA Journal, 41 (5), pp. 969-972 (2003)

H Hudy, L.M., “Simultaneous Wall-Pressure and Velocity Measurements in the Flow
Field Downstream of an Axisymmetric Backward-Facing Step,” Ph.D. thesis, pp 13 -16
(2005)

'2 Durgesh, V., “Experimental Study of Near Wake Dynamics Associated with Bluff
Body Based Drag Reduction,” Ph.D. thesis, (2008)

'3 Najm, H.N. and Ghoniem, A.F., “Numerical simulation of the convective instability
in a dump combustor,” AIAA Journal, 29 (6), pp. 911 - 919 (1991)

139

 

'4 Rowley, C.W., Colonius, T. and Basu, A.J., “On Self-sustained Oscillations in Two-
dimensional Compressible Flow over Rectanguler Cavities”, Journal of Fluid Mechanics

455, pp. 315 - 346 (2002)

'5 Bradshaw, P. and Kog, Y.M., “A note on Poisson’s Equation for Pressure in a
Turbulent Flow,” Physics of Fluids, 24 (4), pp. 777, (1981)

140

5. CONCLUSIONS AND RECOMMENDATIONS

Examined in the current study is the effect of cavity width and side walls on the
self-sustained oscillations in a low-Mach-number cavity flow with a turbulent boundary
layer at separation. An axisymmetric cavity geometry which is free from any side-wall
influence but could be partially filled to form finite-width cavities is employed. Unsteady
surface pressure is measured on the cavity bottom along the streamwise direction and on
the downstream wall along the azimuthal direction to explore the three—dimensional
features of cavity oscillation. Two-component velocity is measured using an LDA system,
simultaneously with the surface pressure in x-y (i.e. streamwise-wall normal) planes at

different azimuthal locations to understand the mechanism driving the oscillation.

Unlike the axisymmetric geometry, strong harmonic pressure oscillations are
detected in cavities having side walls. The oscillations are generally located at a distance
of approximately one cavity depth away from the side walls. However, when the cavity
becomes sufficiently narrow, the two locations where the oscillation is found (near each
of the two side walls) “merge” in the symmetry plane. This is the case for the narrowest
cavity examined (W/D = 2.5). No prominent oscillation is observed in the symmetry
plane in the widest finite-width cavity investigated (W/D = 7.4). In this case the flow
structure in the symmetry plane is found to share the same features as in the
axisymmetric cavity. The distinction between “narrow” and “wide” cavities (i.e.
corresponding to whether strong harmonic oscillations are observed in the symmetry
plane or not) is found to take place at a cavity width that scales with the cavity length:

W/Lzl

141

Mean-velocity results for the wide cavity show that there is a net mean flow from
the region near the side walls towards the center of the cavity in the approximate range
1.8 < x/D < 2.5. Stochastic estimation of the velocity field suggests that the flow structure
in the symmetry plane is wake-like. Vortex structures form as a result of the roll-up of the
separating shear layer, and penetrate into the cavity down to approximately half the
cavity depth. Interestingly, the mean cross-flow from the side-wall region towards the
symmetry plane is found to occur over the same x-y domain inside the cavity into which
the vortex structures penetrate. It is hypothesized that when the vortex structures
penetrate into the cavity they create low-pressure regions in the central zone of the cavity.
This leads to the establishment of a spanwise pressure gradient that drives the flow from
near the side wall towards the symmetry plane. This cross flow is ultimately linked to the
intensification of pressure oscillation near the side walls. In other words, although the
flow structure in the axisymmetic cavity (and symmetry plane of wide finite-width
cavities) does not have a prominent pressure signature, it can interact with the side walls
in finite-width cavities, leading to the establishment of strong harmonic pressure

oscillation near the side wall.

The hypothesized oscillation mechanism is based on a complex, unsteady, three-
dimensional flow field that results from the interaction of the structures near the cavity
center with the flow near the side walls. Rigorous validation of this mechanism linking
the two flow zones is recommended for a firture follow-up study. This could be done
through whole-field measurements of the flow in y-¢ planes that span half the cavity
width and extend over the downstream half of the cavity. To “enhance the visibility” of

the oscillation mechanism (relative to background broadband turbulence), these

142

measurements could be done while forcing the wake mode near the middle of the cavity.
This will cause the wake mode to be strong and more organized than under natural
conditions, and hence all measurements can be phase-locked to the forcing signal and

data sampled in all azimuthal planes can be correlated across the cavity.

Another way to check the proposed mechanism is to change the boundary—layer
condition at separation so that a shear-layer (with or without Rossiter oscillation) rather
than a wake mode exist in the axisymmetric cavity (and hence, also in the symmetry
plane of wide finite-width cavity). Since the vortex structures remain above the cavity
opening in this case, no cross-flow should be established at the passage frequency of the
vortex structures of the shear layer (based on the above hypothesis). Consequently, the
intensification of oscillations near the side wall would not happen in cavities with shear-

layer mode.
Finally, this work addresses the mechanism driving the harmonic wall-pressure
oscillation at jL/U00 z 0.21 near the side wall in cavities with finite width (because this

oscillation occurs at the same frequency as that of the weak peak found in the
axisymmetric cavity). Understanding of the nature of disturbances found at the low-

frequency end near the side walls is left for future studies.

143

6. APPENDEX

6.1 Wiring for Azimuthal Microphone Array

Figure 6.1 shows a picture of the wiring for the Knowles azimuthal microphone
array. Each microphone has three 25.4 mm-long leads for power, ground and output
signals. Leads carrying power and ground signals were soldered to copper tape (forming
power and ground bus respectively) on the back side of the Detachable Sensor Ring (see
Figure 2.3). 32-AWG wires connected the power bus (red wire) and ground bus (black
wire) to the power and read-out circuits. 1.5 volts voltage is required to operate the
microphones. Yellow leads connected with 32-AWG wires carry the output signal. 32-
AWG wires are directed out of the model through the gap between the cylindrical shell

and the model’s surface.

 

Figure 6.1 Wiring for Knowles azimuthal microphone array

Figure 6.2 shows a picture of the wiring for the Panisonic azimuthal microphone

array. Red wire connected to the out put terminal and black wire connected to the ground

144

terminal of each microphone are directed out of the model through the gap between the

cylindrical shell and the model’s surface.

 

Figure 6.2 Wiring for Panasonic azimuthal microphone array

6.2 Schmitt Trigger Circuit

The A/D boards used to acquire the microphone-array signals share the same 1
MHz clock signal with the LDA system. A Schmitt trigger (CD74HCT132E) is used to
regulate the 1 MHz CLK signal from the BSA to a TTL compatible signal before it is
sent to the master A/D board (AT-MIO-l6E-10). Figure 6.3 shows a schematic of the
Schmitt trigger circuit. The pin-out and function diagram of the Schmitt trigger IC are
displayed in Figure 6.4. 5 V DC power supply is required to drive the Schmitt trigger
circuit. The lMHz CLK signal is fed to the Schmitt trigger as the input (V I)- The output
signal (V0) is zero if V1 is over a high-voltage threshold around 2 V and V0 switch to
VCC if V] is below a low voltage threshold around 1 V. Thus the output signal of the
Schmitt trigger circuit used as the clock signal of the A/D boards is a square wave having

the same frequency as the clock signal of the LDA system.

145

¢ Vl

 

VCC = 5 V . .
Schmitt Trigger V0

 

Figure 6.3 Schematic of the Schmitt trigger circuit

 

 

 

 

1 14
1A — -— Vec
2 13
ME m... w— —«
3 12
13 E [E] 48 1v __ — 4A
4 11
"E [E 4A 2A a
211E m 4v 28 s 10 38
28 E m 38 2v 6 9 3A
21! E 5 3A 8
GND E a 3V GND —4 — 3v

 

 

Figure 6.4 Pin-out (left) and function diagram (right) of the Schmitt trigger (model
CD74HCT132E)

6.3 LDA system alignment

The following is a description of the procedure used to align the LDA system.
First, align the transmitter with the argon-ion laser by adjusting the support—feet thumb
screws on the side of the transmitter until the energy of the beams coming out of each
manipulator is maximized. For this step, the optical fibers leading to the probe head

should be disconnected from the manipulator and the beam shutter should be open. This

146

will cause the beams coming out of the manipulator to be directed vertically up into the
open air. The strength of the beams can be gauged visually from their projection on the
ceiling of the lab. The transmitter should be aligned to attain two conditions: (1) the
overall beam intensity is maximized; (2) the intensity of the shifted and un-shifted beams

for a given wavelength is balanced.

Second, connect the optical fibers of the probe head to the manipulators. Direct
the beams coming out from the probe at a photo detector that accommodates light
intensity of 100 mW or higher (or run the laser at low power during alignment). For each
of the four beams (two green and two blue), close the shutter on the other three beams
and turn the adjustment thumb screws on the manipulator (two for focusing, one for
displacement and one for angle of the fiber optic end) to maximize the power of the beam
(i.e. the output of the photo detector). After the above adjustment, the power of the two
beams with the same color coming out of the probe should be equal. If for a beam pair of
a given wavelength, the un-shified beam is stronger than the shifted beam, or vice versa,

the equality of beam intensity can be achieved by weakening the stronger beam.

Third, to precisely place the beams intersection point at the receiving lens focal
point, mount the end of receiver fiber in the output aperture of the spare color (violet here)
in the manipulator. Close the shutters of the manipulator for the green and blue beams
and open the one for the violet beam. This will cause the violet beam to emerge from the
center of the probe head and is directed towards the measurement plane. Place a
microscope objective in the focal point of the violet beam. To identify this point, adjust
the position of the objective until the projection of the violet beam on a screen placed on

the opposite side of the objective is minimum. Open the shutter of the manipulator for

147

each of the green and blue beams; one at a time. Adjust the ‘BEAM ADJ’ screws on the
side of the probe head to move the projection (from the microscope objective) of each
beam to coincide with that of the violet beam. This guarantees the intersection of the

beams to coincide with the focus of the receiver fiber. The beam spacing (lateral distance
between beam pair of a given wavelength) can be varied by adjusting the ‘BEAM
SEP ’ screw on the side of the probe. Finally, close the beam shutter in the violet beam

manipulator. Disconnect the end of the receiving fiber from the manipulator and connect

it to the appropriate adaptor attached to the photomultiplier tubes.
6.4 Azimuthal Traversing of the LDA Probe Volume

Wall pressure measurements are performed to ensure that the characteristics of
the pressure fluctuations in a given azimuthal plane depend only on the offset angle

relative to the cavity symmetry plane (13¢) regardless of the azimuthal location of the

symmetry plane relative to the top of the model ((156): see Figure 2.19 for definition of A¢

and ¢c.

Figure 6.5 displays frequency spectra of the unsteady wall pressure acting on the
cavity's downstream wall at y/D = 0.5 in the symmetry plane (A¢ = 0°) of the cavity with
L/D = 3.3 and W/D = 7.4, at Re = 12200. Lines with different colors represent different
azimuthal locations of the symmetry plane, (be, obtained by rotating the cavity relative to
the top plane of the model. The legend shows the azimuthal angle of the symmetry plane,
¢c, relative to the top plane of the model. Note that positive angle values correspond to

the symmetry plane being on the right of the top plane of the model when viewing the

model from the upstream direction. Also, the magnitude of the spectra is normalized by

148

the energy of the fluctuations, i.e. the square of the rms pressure, to compare the

spectrum shape for the different cases.

 

 

 

 

I I I I I I I I
O l__ ._ .14 ,4 ,_L wk L_L ..LJ ‘ _ _g a;
-2

1o 10'1
* fL/Uoo

Figure 6.5 Frequency spectrum of the pressure acting on the cavity’s downstream
wall in the symmetry plane (at A¢= 0°) for W = 3.3, W/D = 7.4 and Re = 12200.
Different colors correspond to different azimuthal locations of the symmetry plane

Figure 6.5 shows that the spectra for different cases collapse except those
fluctuations at very low frequency. The spectral characteristics of the pressure
fluctuations (particularly within the frequency band of interest, which surrounds the

frequency fL/ U 00 z 0.21) remain the same for different azimuthal locations of the cavity.

Results similar to those in Figure 6.5 but in the azimuthal plane where the strength of the
harmonic oscillation reaches a peak value (z/D = -2.62 or A¢= -32°) are displayed in
Figure 6.6 for the cavity with W/D = 7.4. Lines with different colors represent the

different azimuthal angles by which the cavity is rotated. The plot shows that prominent

149

 

harmonic oscillation is established within the azimuthal plane corresponding to A¢ = -32°
regardless of the azimuthal location of the symmetry plane relative to the top of the
model (¢c). The above results confirm that the wall-pressure characteristics and its

generating flow structure remain unaltered when the cavity is rotated to different

azimuthal locations around the axisymmetric model.

0.08
0.07*
0.06I

r—
U)

005

,2
rm

0.04 ‘7

[4¢p'p']/[p

0.03f ,

0.02I

l
0.01l

 

0 . , A . , I . . , ,, , I , , , l .
10‘2 10" 10° 1 o1
fL/U
00
Figure 6.6 Frequency spectrum of the pressure acting on the cavity’s downstream
wall at A¢ = -32° for L/D = 3.3, W/D = 7.4 and Re = 12200. Different colors
correspond to different azimuthal locations of the symmetry plane

150

   
 

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