ABSTRACT
Title of Dissertation: RESPONSE OF HYPERSONIC
BOUNDARY-LAYER DISTURBANCES TO
COMPRESSION AND EXPANSION CORNERS
Cameron Butler
Doctor of Philosophy, 2021
Dissertation Directed by: Professor Stuart Laurence
Department of Aerospace Engineering
An experimental campaign was conducted at the University of Maryland - Col-
lege Park to examine the impact of abrupt changes in surface geometry on hypersonic
boundary-layer instability waves. A model consisting of a 5? conical forebody was
selected to encourage the dominance of second-mode wavepackets upstream of the
interaction region. Interchangeable afterbody attachments corresponding to flow
deflections of -5? to +15? in 5? increments were considered. The adverse pressure
gradient imposed by the +10? and +15? configurations caused the boundary layer
to separate upstream, creating a region of recirculating flow. High-speed schlieren
(440 ? 822 kHz) was employed as the primary means of flow interrogation, with
supplemental surface measurements provided by PCB132B38 pressure transducers.
A lens calibration was applied to the images to provide quantitative fluctuations in
density gradient.
The high frame rate made possible the use of spectral analysis techniques
throughout the entire field of view. This analysis reveals complex growth and decay
trends for incoming second-mode disturbances. Additional, low-frequency content is
generated by the deflected configurations. This is most pronounced for the separated
cases where distinct, shear-generated disturbances are observed. Spectral proper
orthogonal decomposition (SPOD) is demonstrated as a powerful tool for resolving
the flow structures tied to amplifying frequencies.
Nonlinear interactions are probed through bispectral analysis. Resonance of
low-frequency structures is found to play a large role in nonlinear energy transfer
downstream of the compression corners, particularly for the separated cases. Con-
cave streamline curvature appears to result in concentrated regions of increased
nonlinearity. These nonlinear interactions are shown to be spatially correlated with
coherent flow structures resolved through SPOD.
Finally, a limited computational study is carried out to demonstrate the ability
of linear stability theory and the parabolized stability equations to reproduce experi-
mental results obtained for the +10? extension. The development of the second-mode
and shear-generated disturbances resolved by the computational analysis shows ex-
cellent agreement with the experimental results.
RESPONSE OF HYPERSONIC BOUNDARY-LAYER
DISTURBANCES
TO COMPRESSION AND EXPANSION CORNERS
by
Cameron Butler
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2021
Advisory Committee:
Professor Stuart Laurence, Chair/Advisor
Professor James Baeder
Professor Kenneth Yu
Professor Christoph Brehm
Professor Johan Larsson
?c Copyright by
Cameron Butler
2021
Acknowledgments
It seems appropriate to begin by thanking my advisor, Stuart Laurence, with-
out whom this body of work would have been impossible. His guidance has proven
invaluable over the past six-and-a-half years and shaped me into the researcher I
am today. I would also like to extend my thanks to the members of my dissertation
committee: Professors Kenneth Yu, James Baeder, Christoph Brehm and Johan
Larsson. Their input led me to examine facets of this work which may otherwise
have gone unexplored and what I present here is undoubtedly improved as a result.
Equally important throughout this endeavor have been my loving wife, Emily,
and my beautiful dog, Korra. Having someone to celebrate in my triumphs has
meant more than I can say, and I could not ask for a better partner at my side
through the adventure that is life.
Thanks to all my colleagues in the High-speed Aerodynamics and Propulsion
Lab. Our discussions in the lab and office made my experience at Maryland far more
memorable and I count myself lucky for having worked alongside you. Special thanks
to Alvin Pee for being my partner in crime getting HyperTERP in a functional state,
and to Sam Maszkiewicz for his assistance in running the facility during many of
the tests presented here.
Others who deserve credit in this work include the talented machinists I have
relied on in designing and constructing HyperTERP and the test articles used
ii
throughout this work. Mike Perna, Bruce Rowley, Dave Cogsworth and Tommy
Tamarkin have always been there to talk me through any hardware design issues,
or to just shoot the breeze.
I would also like to acknowledge past mentors who have had a significant
impact on my trajectory towards a PhD. I first gained a sense of self-actualization
regarding research activities thanks to the guidance of Dr. Godfrey Sauti during
a summer internship at NASA Langley. Under the tutelage of Dr. Eric Loth at
the University of Virginia I became acquainted with CFD tools. I am a far more
well-rounded researcher today thanks to his expertise.
Finally, thanks to my parents, Gail and Wayne Butler, for their love and
support all my life. I know I would not be here today without them. And shouts
out to my brother and one of my best friends, Chris, who I can always count on for
a refreshing time.
I was supported financially by a National Defense Science and Engineering
Graduate Fellowship sponsored by the Air Force Office of Scientific Research, for
which I am exceedingly grateful.
iii
Table of Contents
Acknowledgements ii
Table of Contents iv
List of Tables vi
List of Figures vii
List of Abbreviations x
Chapter 1: Introduction 1
1.1 Second-mode Instability . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Shock-wave/Boundary-layer Interactions . . . . . . . . . . . . . . . . 5
1.3 Expansion Corner Interactions . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2: Experimental Methodology 14
2.1 HyperTERP Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Test Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Nozzle Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Test Article and Instrumentation . . . . . . . . . . . . . . . . . . . . 22
2.5 Calibrated Schlieren . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 3: Schlieren Observations 29
3.1 Straight (+0?) Extension . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 -5? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 +5? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 +10? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 +15? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Chapter 4: Low-order Spectral Analysis 56
4.1 Second-mode Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 N Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Low-frequency Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Bandpass-filtered Images . . . . . . . . . . . . . . . . . . . . . . . . . 76
Chapter 5: Spectral Proper Orthogonal Decomposition 82
iv
5.1 +0 ? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 -5 ? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3 +5 ? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4 +10 ? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.5 +15 ? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.6 Propagation Speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Chapter 6: Bispectral Analysis 114
6.1 +0 ? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 -5 ? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3 +5 ? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4 +10 ? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.5 +15 ? Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Chapter 7: LASTRAC 142
7.1 Mean Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2 LST Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 LPSE Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Chapter 8: Conclusions 150
Bibliography 158
v
List of Tables
2.1 Test condition matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 PCB sensor locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1 +10? configuration separation/reattachment locations. . . . . . . . . . 43
3.2 +15? configuration separation/reattachment locations. . . . . . . . . . 48
5.1 Parameters for SPOD computations. . . . . . . . . . . . . . . . . . . 83
5.2 Upstream and downstream propagation speeds computed with the
correlation method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.1 Inflow properties for STABL2D. . . . . . . . . . . . . . . . . . . . . . 144
vi
List of Figures
1.1 Mean flow topologies for compression and expansion corners. . . . . . 5
1.2 Computational results from Balakumar et al., 2002. . . . . . . . . . . 8
2.1 HyperTERP schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Schematic of double-diaphragm system. . . . . . . . . . . . . . . . . . 16
2.3 Sample reservoir pressure traces at each condition. . . . . . . . . . . . 20
2.4 Nozzle characterization results. . . . . . . . . . . . . . . . . . . . . . 23
2.5 Test article schematics. . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Alignment procedure for test article. . . . . . . . . . . . . . . . . . . 26
2.7 Sample alignment PCB spectra. . . . . . . . . . . . . . . . . . . . . . 26
2.8 Calibration image and intensity profile. . . . . . . . . . . . . . . . . . 28
3.1 +0? configuration image sequence at condition Re33. . . . . . . . . . 31
3.2 +0? configuration image sequence at condition Re45. . . . . . . . . . 32
3.3 +0? configuration image sequence at condition Re52. . . . . . . . . . 33
3.4 +0? configuration PCB spectra. . . . . . . . . . . . . . . . . . . . . . 33
3.5 -5? configuration image sequence at condition Re45. . . . . . . . . . . 35
3.6 -5? configuration image sequence at condition Re52. . . . . . . . . . . 36
3.7 -5? configuration PCB spectra. . . . . . . . . . . . . . . . . . . . . . . 37
3.8 +5? configuration image sequence at condition Re33. . . . . . . . . . 39
3.9 +5? configuration image sequence at condition Re45. . . . . . . . . . 40
3.10 +5? configuration image sequence at condition Re52. . . . . . . . . . 41
3.11 +5? configuration PCB spectra. . . . . . . . . . . . . . . . . . . . . . 42
3.12 +10? configuration image sequence at condition Re33. . . . . . . . . . 44
3.13 +10? configuration image sequence at condition Re45. . . . . . . . . . 45
3.14 +10? configuration image sequence at condition Re52. . . . . . . . . . 47
3.15 +10? configuration PCB spectra. . . . . . . . . . . . . . . . . . . . . . 47
3.16 +15? configuration shear-mode image sequence at condition Re33. . . 50
3.17 +15? configuration second-mode image sequence at condition Re33. . 51
3.18 +15? configuration shear-mode image sequence at condition Re45. . . 53
3.19 +15? configuration second-mode image sequence at condition Re45. . 54
3.20 +15? configuration image sequence at condition Re52. . . . . . . . . . 55
4.1 High-frequency spatial PSD contours for condition Re33. . . . . . . . 57
4.2 Pseudo-streamline spectra for condition Re33. . . . . . . . . . . . . . 58
4.3 High-frequency spatial PSD contours for condition Re45. . . . . . . . 60
vii
4.4 Pseudo-streamline spectra for condition Re45. . . . . . . . . . . . . . 61
4.5 High-frequency spatial PSD contours for condition Re52. . . . . . . . 64
4.6 Pseudo-streamline spectra for condition Re52. . . . . . . . . . . . . . 65
4.7 Pseudo-streamline N-factors for condition Re33. . . . . . . . . . . . . 67
4.8 Pseudo-streamline N-factors for condition Re45. . . . . . . . . . . . . 69
4.9 Pseudo-streamline N-factors for condition Re52. . . . . . . . . . . . . 71
4.10 Low-frequency spatial PSD contours for condition Re33. . . . . . . . 73
4.11 Low-frequency spatial PSD contours for condition Re45. . . . . . . . 74
4.12 Baroclinic instability production from Dwivedi et al., 2019. . . . . . . 74
4.13 Low-frequency spatial PSD contours for condition Re52. . . . . . . . 75
4.14 Example of second-mode bandpass filtering. . . . . . . . . . . . . . . 76
4.15 Example of shear-mode bandpass filtering. . . . . . . . . . . . . . . . 77
4.16 Shear-generated disturbance revealed for the +10? configuration. . . . 78
4.17 Low-frequency structures observed along +5? extension. . . . . . . . . 79
4.18 Second-mode harmonic revealed by bandpass-filtering. . . . . . . . . . 80
4.19 Harmonic/fundamental wavelength comparison. . . . . . . . . . . . . 81
5.1 SPOD mode energy for the +0? extension. . . . . . . . . . . . . . . . 84
5.2 SPOD mode shapes for the +0? configuration at condition Re33. . . . 85
5.3 SPOD mode shapes for the +0? configuration at condition Re45. . . . 86
5.4 SPOD mode shapes for the +0? configuration at condition Re52. . . . 87
5.5 SPOD mode energy for the -5? extension. . . . . . . . . . . . . . . . . 88
5.6 SPOD mode shapes for the -5? configuration at condition Re33. . . . 90
5.7 SPOD mode shapes for the -5? configuration at condition Re45. . . . 91
5.8 SPOD mode shapes for the -5? configuration at condition Re52. . . . 92
5.9 SPOD mode energy for the +5? extension. . . . . . . . . . . . . . . . 93
5.10 SPOD mode shapes for the +5? configuration at condition Re33. . . . 95
5.11 SPOD mode shapes for the +5? configuration at condition Re45. . . . 97
5.12 SPOD mode shapes for the +5? configuration at condition Re52. . . . 98
5.13 SPOD mode energy for the +10? extension. . . . . . . . . . . . . . . . 99
5.14 SPOD mode shapes for the +10? configuration at condition Re33. . . 100
5.15 SPOD mode shapes for the +10? configuration at condition Re45. . . 101
5.16 SPOD mode shapes for the +10? configuration at condition Re52. . . 102
5.17 SPOD mode energy for the +15? extension. . . . . . . . . . . . . . . . 103
5.18 SPOD mode shapes for the +15? configuration at condition Re33. . . 104
5.19 SPOD mode shapes for the +15? configuration at condition Re45. . . 106
5.20 SPOD mode shapes for the +15? configuration at condition Re52. . . 106
5.21 Propagation speeds computed at condition Re33 using the SPOD-
based methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.22 Second-mode propagation speeds for the -5? and +0? configurations. . 111
5.23 Second-mode propagation speeds for the +5? and +10? configurations. 113
6.1 Illustration of frequency domain for bicoherence spectra. . . . . . . . 116
6.2 Bicoherence results from Kimmel and Kendall, 1991, and Chokani,
1999. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
viii
6.3 Bicoherence spectra for the +0? configuration at condition Re33. . . . 120
6.4 Spatial bicoherence contours for the +0? configuration at condition
Re33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Bicoherence spectra for the +0? configuration at condition Re45. . . . 121
6.6 Spatial bicoherence contour for the +0? configuration at condition Re45.121
6.7 Bicoherence spectra for the -5? configuration at condition Re33. . . . 122
6.8 Spatial bicoherence contours for the -5? configuration at condition Re33.123
6.9 Bicoherence spectra for the -5? configuration at condition Re45. . . . 125
6.10 Spatial bicoherence contours for the -5? configuration at condition Re45.125
6.11 Bicoherence spectra for the +5? configuration at condition Re33. . . . 127
6.12 Spatial bicoherence contours for the +5? configuration at condition
Re33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.13 Bicoherence spectra for the +5? configuration at condition Re45. . . . 129
6.14 Spatial bicoherence contours for the +5? configuration at condition
Re45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.15 Bicoherence spectra for the +10? configuration at condition Re33. . . 132
6.16 Spatial bicoherence contours for the +10? configuration at condition
Re33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.17 Bicoherence spectra for the +10? configuration at condition Re45. . . 135
6.18 Spatial bicoherence contours for the +10? configuration at condition
Re45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.19 Bicoherence spectra for the +15? configuration at condition Re33. . . 138
6.20 Spatial bicoherence contours for the +15? configuration at condition
Re33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.21 Bicoherence spectra for the +15? configuration at condition Re45. . . 140
6.22 Spatial bicoherence contours for the +15? configuration at condition
Re45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.1 STABL2D computational grid. . . . . . . . . . . . . . . . . . . . . . . 145
7.2 Mean flow comparison to experiments. . . . . . . . . . . . . . . . . . 146
7.3 LST amplification rates and N-factors. . . . . . . . . . . . . . . . . . 147
7.4 LPSE eigenfunctions for second-mode disturbance. . . . . . . . . . . . 148
7.5 LPSE eigenfunctions for shear-generated disturbance. . . . . . . . . . 149
ix
List of Abbreviations
Acronyms
CFD Computational Fluid Dynamics
DNS Direct Numerical Simulation
FLDI Focused Laser Differential Interferometry
HyperTERP Hypersonic Tunnel for Educational and Research Purposes
IT Inner Triangle
LASTRAC Langley Stability and Transition Analysis Code
LST Linear Stability Theory
OT Outer Triangle
PSD Power Spectral Density
PSE Parabolized Stability Equations
QPC Quadratic Phase Coupling
RANS Reynolds Averaged Navier Stokes
SPOD Spectral Proper Orthogonal Decomposition
STABL Stability and Transition Analysis for Hypersonic Boundary Layers
SWBLI Shock-Wave/Boundary-Layer Interaction
Variables
b2 Squared Bicoherence
F Frequency
Fs Sampling Frequency
L Optical Integration Path Length
Lhann Hann Window Length
M Mach
n Index of Refraction
Nf Number of Frames
P Pressure
R Stability Reynolds Number
Rem Freestream Unit Reynolds Number
t Time
T Temperature
U Velocity
X Streamwise Surface Coordinate
x
Y Vertical Coordinate
? Complex Streamwise Wavenumber
? Complex Azimuthal Wavenumber
?N Change in N factor
? Boundary-Layer Thickness
 Deflection Angle
? Specific Heat Ratio
? Disturbance Wavelength
? Gladstone-Dale Constant
? Disturbance Shape Function
? Density
? Characteristic Vibrational Temperature
? Disturbance Mode Shape
xi
Chapter 1: Introduction
Throughout the history of hypersonic flight, interest in the field has waxed
and waned with global political dynamics. We currently find ourselves in a period
of renewed interest, with the United States pressing for the development of highly-
maneuverable hypersonic cruise vehicles. This is coupled with increasing interest
within the space sector for reusable space-launch systems. The design of practical
hypersonic vehicles is constrained by the extreme thermo-mechanical surface loads
which occur when travelling within the stratosphere at high Mach number (typi-
cally >5). This is further complicated by the dramatic increase in surface heat-flux
and skin friction which accompanies the laminar-turbulent transition of the vehicle
boundary layer.
Failing to account for these factors in the design of a vehicle with a desired
mission duration of minutes can have catastrophic results, as evidenced by the X-
15A crash in 1967. Due to incomplete understanding of the hypersonic transition
process and inadequate modelling capabilities, these issues are presently avoided
by over-designed thermal protection systems; however, these come with significant
weight penalties, restricting vehicle and mission design. Safe and efficient vehicle
design thus requires advances in our understanding of hypersonic boundary-layer
1
transition and the underlying physics.
1.1 Second-mode Instability
In the low-disturbance environments typical of hypersonic flight (or most
ground-test facilities), the path to boundary-layer transition on slender, smooth
bodies at small incidence is characterized by linear growth of instabilities within the
boundary-layer leading up to nonlinear modal interactions and breakdown [5]. Early
stability theory was built upon the decomposition of the Navier Stokes equations
into mean and fluctuating terms and then linearizing based on the assumption of
small disturbances. Disturbances here take the form
q(x, y, z, t) = Q(y) exp[i(?x+ ?z ? ?t)], (1.1)
where ? and ? are the complex wavenumbers in the streamwise and lateral direc-
tions, respectively, and ? is the disturbance frequency. The dominant instability
for sharp, near-axisymmetric configurations at low incidence and high Mach num-
bers is the ?second mode,? defined by Mack [6]. The second mode is an acoustic
instability, largely consisting of density fluctuations. Furthermore, the most un-
stable second-mode disturbances are two-dimensional (? = 0). Early investigators
[7, 8, 9] were able to obtain single schlieren images of second-mode disturbances,
confirming their existence experimentally and revealing their ?rope-like? structure
with wavelengths of approximately twice the boundary-layer thickness. Extensive
studies performed by Stetson [10] and Stetson and Kimmel [11] on a 7? half-angle
2
cone at Mach 8 provided a characterization of second-mode behavior and variation
of stability with nose bluntness, wall-cooling and Reynolds number. A particularly
notable result from these tests was the appearance of the first harmonic of the sec-
ond mode, which was not predicted by linear analysis. These early studies relied
almost entirely on hot-wire anemometry. While this is a powerful technique capable
of high bandwidths, it is necessarily intrusive and requires long test durations for a
full traverse of the boundary layer.
In reviewing the effects of tunnel noise on transition, Schneider [12] noted that
measurements of transition location in conventional testing facilities may not be ex-
trapolated to flight conditions. Instead, measurements of disturbance development
are needed which elucidate the physics underlying the transition process. Advances
in optical technology have allowed researchers to obtain unobtrusive, time-resolved
measurements of hypersonic boundary-layer instability waves through high-speed
schlieren photography. Laurence et al. [13] first implemented high-speed schlieren
to track individual disturbances, employing image-processing techniques to com-
pute structural characteristics (such as disturbance wavelength) and propagation
speeds. Laurence et al. [14, 15] expanded upon this work by developing a pulsed-
laser schlieren apparatus, allowing image spacing as low as 2.8?s. This removed any
former ambiguity from the propagation speed calculations and providing a more de-
tailed view of instability growth. They were also able to probe the distribution of
second-mode energy as a function of wall-normal distance, observing two peaks of
roughly equal magnitude: one at the wall and one at y/? = 0.7 ? 0.75. Casper et
al. [16] performed similar tests on a 7? cone at Mach 5 and 8. Aside from deter-
3
mining propagation speeds, they developed a methodology to detect the presence
of second-mode wavepackets within individual images and computed intermittency
curves along the model surface.
More recently, Kennedy et al. [17] implemented these image processing tech-
niques with a calibrated schlieren setup to make global, quantitative measurements
of second-mode disturbances, leading to the compution of amplification rates. Good
agreement was obtained between the experimental amplification rates and computa-
tional predictions. They also employed a spatial reconstruction technique to increase
the effective framerate of the tests and were able to resolve higher harmonic content
generated from nonlinear resonance of the second-mode.
Piezoelectric surface pressure transducers have also proven useful for charac-
terizing second-mode disturbances. The previously mentioned schlieren observations
of Casper et al. [16] were accompanied by surface measurements from PCB132 pres-
sure transducers. These signals were sampled at 2.5 MHz but low-pass filtered at
1 MHz. Significant broadening of the dominant second-mode peak was observed
where the boundary layer was transitional. Grossir et al. [18] conducted tests on
a 7? cone at Mach 10 and performed a wavelet analysis on the observed surface
pressure fluctuations. They observed substantial energy content confined within a
narrow spectral range which showed excellent agreement with linear stablity theory
at the reported freestream conditions. They also identified harmonic content in the
PCB spectra.
All this is to say that the second-mode instability is today well-characterized
on simple model configurations. Unfortunately, the outer mold line of true flight
4
Reattachment
Shock
Leading 
Shock Separation 
Shock
Freestream
Expansion
Fan
Figure 1.1: Mean flow field over a supersonic (top) compression and (bottom) ex-
pansion corner.
vehicles does not always consist of smoothly varying surfaces, and may exhibit sud-
den changes in geometry, e.g., at a control surface or intake. It is thus of critical
importance to understand how the boundary layer and boundary-layer disturbances
may interact with such configurations.
1.2 Shock-wave/Boundary-layer Interactions
When supersonic flow encounters a compression corner (positive flow deflec-
tion), an oblique shock wave forms, creating a shock-wave/boundary-layer interac-
tion (SWBLI). This imposes an adverse pressure gradient on the incoming boundary
layer which, for sufficient turning angles, will cause the boundary layer upstream of
the corner to separate. This produces a complex mean flow field, illustrated by the
upper portion of figure 1.1, which incoming boundary-layer disturbances must then
traverse, potentially leading to complex unsteady interactions.
5
These SWBLIs can give rise to new instability mechanisms, particularly in the
separated case. For instance, the resulting streamline curvature has been seen to
induce a centrifugal instablity, leading to the formation of Go?rtler vortices. These
vortices were first observed for compression ramps in the surface oil flow visualization
of Miller et al. [19]. Researchers have since attributed surface heating streaks in
reattaching flows to these streamwise-oriented vortices. The impact of ramp angle on
the structure of Go?rtler vortices was examined by Roghelia et al. [20]. They found
the length scale required for breakdown of the vortices to decrease with increasing
ramp angle. A separate global instability was observed by Sidharth et al. [21] by
applying global stability analysis and direct numerical simulation to the study of a
12? -22? double-wedge configuration. This global instability was found to arise from a
local inviscid instability due to streamwise deceleration within the separation region.
Dwivedi et al.[2] applied input-output analysis to the study of Mach 8 flow over a
15? compression ramp to investigate the development of the aforementioned heat
streaks near reattachment. They observed substantial amplification of streamwise-
oriented vortical disturbances of small spanwise wavelength. Furthermore, the heat
streaks were found to be caused by a baroclinic instability as opposed to the oft-
attributed Go?rtler vortices.
Much of the prior work on 2-D SWBLIs has sought to characterize transitional
effects on flow topology (e.g. separation length and unsteadiness) and thermo-
mechanical surface loading. Heffner, Chpoun and Lengrand [22] conducted tests on
a hollow-cylinder/flare model at Mach 5 with interchangeable flares with 10? , 20? ,
and 30? half-angles. Critically, they confirmed the laminar state of the boundary-
6
layer upstream of separation, resulting in a transitional SWBLI. They employed
oil-flow and sublimation surface visualization techniques to obtain separation and
reattachment locations and revealed the development of Go?rtler vortices along the
flare. Surface heat-flux measurements rapidly attained turbulent levels in the vicin-
ity of reattachment for the larger flare angles; this occurred within several boundary-
layer thicknesses of reattachment for the 10? configuration. This work was expanded
upon by Benay et al. [23] who took a hybrid experimental/computational approach.
They focused largely on the formation of Go?rtler vortices, determining them to be
organized, three-dimensional flow structures and not purely a transition mechanism.
Bicone models with relatively large flare angles (?34?) have been tested in the
AFRL Mach 6 Ludwieg Tube by Running et al. [24, 25], where infrared thermogra-
phy was employed to study the thermal striations created by Go?rtler vortices. The
azimuthal extent of the thermal streaks was found to increase with increasing flare
angle. They also studied geometric effects, finding that increasing the nosetip blunt-
ess or compression angle caused an increase in the separation length while increasing
Re? caused the separation bubble to shrink.
Focused laser differential interferometry (FLDI) measurements obtained by
Benitez et al. [26] on a cone-cylinder-flare model in the Boeing/AFOSR Mach 6
Quiet Tunnel represent some of the only off-wall measurements of incoming boundary-
layer instabilities in an axisymmetric SWBLI. One of the primary results from this
campaign was the identification of low-frequency (50 ? 170 kHz) travelling waves
downstream of the compression corner. However, these waves were only observed
under quiet flow conditions and the researchers were unable to locate these distur-
7
Figure 1.2: Density fluctuations computed for a low-frequency mode within the
separation region taken from Balakumar et al. [1].
bances within the separation region. Furthermore, the point-like FLDI measure-
ments give no information regarding the structure of these instabilities. Second-
mode disturbances were found to amplify and increase in peak frequency along the
flare.
Only a limited number of studies have elucidated transition dynamics or
the impact of the SWBLI on pre-existing disturbances. Balakumar et al. [1]
used both linear stability theory (LST) and direct numerical simulation (DNS) to
study 2-D, fixed-frequency disturbances in a flat-plate boundary-layer encountering
a 5.5? compression corner at Mach 5.373. LST revealed the existence of multiple
unstable modes within narrow regions of the separation bubble, while DNS showed
the second-mode waves to be of neutral stability while traversing the separated
shear-layer but to grow exponentially upstream of separation and downstream of
reattachment. The unstable low-frequency mode within the separaton region was
found to have a frequency of 38% of the dominant second-mode frequency. This
low-frequency mode was also shown to emanate energy from the boundary layer at
a point just above the corner, as shown in figure 1.2.
8
Novikov et al. [27] carried out DNS of three-dimensional, broad-spectrum
wavepackets on this same configuration. Both oblique-wave- and second-mode-
dominated wavepackets were examined; the latter were found to be neutrally stable
within the upstream part of the separation region before amplifying downstream.
Strong forcing resulted in significant streamwise stretching of the wavepacket tail
and the formation of a turbulent spot downstream of reattachment.
Transition of a laminar boundary layer encountering an axisymmetric 15? compression
ramp at Mach 5 was computed by Lugrin et al. [28] using quasi-direct numerical
simulation. White noise forcing was used to excite a range of convective instabilities.
Spectral proper orthogonal decomposition was then applied to the unsteady results
to identify key flow structures. They observed a transition scenario dominated by
streamwise streaks resulting from nonlinear interaction of oblique first modes. The
shear layer and reattachment regions then triggered linear amplification of these
streaks, ultimately leading to breakdown.
1.3 Expansion Corner Interactions
The alternative configuration wherein the flow meets a convex corner and thus
passes through an expansion fan has received comparably little attention, though
it may also be encountered on high-speed vehicle configurations. In the case of a
laminar boundary layer, the expansion process is generally characterized by three
mechanisms: upstream influence of the corner through the subsonic portion of the
boundary layer; largely inviscid flow-turning in the vicinity of the corner; and down-
9
stream interaction between the boundary layer and the external flowfield to complete
the expansion process [29]. For hypersonic boundary layers with relatively mild turn-
ing angles, Sullivan theorized that the downstream interaction would dominate the
process.
Sternberg performed ballistic tests on a 29? half-angle cone/cylinder model at
Mach 3.02 and 3.55 both with and without boundary-layer trips, the focus of this
campaign being surface recovery temperatures. In cases where the boundary layer
transitioned on the cone, the surface temperature fell to levels characteristic of a
laminar boundary layer for a significant region downstream of the corner before re-
verting to turbulent levels. Sternberg explained these results as ?relaminarization?
whereby the incoming turbulent boundary layer serves as the noisy ?free stream?
for a new laminar boundary layer which begins to grow at the corner. The signifi-
cantly elevated ?free stream? turbulence for this new laminar boundary layer means
it is highly susceptible to transition. Shadowgraph images also revealed a qualita-
tive reduction in the turbulence level of the incoming boundary layer due to the
acceleration around the corner. This idea of ?relaminarization? was strengthened
by the work of Zakkay, Toba and Kuo [30] who acheived good agreement between
between experimental heat-flux measurements on various cone-cylinder test arti-
cles and theoretical predications based on a laminar sub-layer model. It should be
noted, however, that the relatively large cone angle employed resulted in shock-
layer Mach numbers below 2 upstream of the corner for most cases. In the case
examined by Zakkay, Toba and Kuo [30] where the upstream Mach number was 3.8,
the downstream heat-flux more closely matched turbulent predictions. Similarly,
10
Bloy [31] observed heat transfer rates downstream of 5-15? expansions at Mach 16
which closely approximated turbulent predictions. It is thus ambiguous to what
extent this ?relaminarization? occurs at higher free-stream Mach numbers.
Chung and Lu [32] found that surface pressure fluctuations for a 2-D, Mach
8 turbulent flow were attenuated by at least 40% even for small (<4.25?) expan-
sions. A Mach 3, fully-developed, turbulent boundary layer interacting with 2-D
expansion configurations was studied experimentally by Dawson, Samimy and Ar-
nette [33]. They observed an immediate shift in surface pressure fluctuations to
lower frequencies across the expansion due to the dilation of small-scale structures.
The development of new 15 ? 30 kHz features with high spanwise coherence was
noted, though no physical structure could be assigned to these observations.
The prevailing theme in all these works is a bias towards turbulent boundary
layers and a focus on thermo-mechanical surface loading and global flow features
with little attention paid to the process of transition or incoming disturbances.
There is a particularly striking lack of off-surface measurements of individual dis-
turbances. Furthermore, the cone-cylinder models tested have historically been far
from slender, resulting in relatively low Mach numbers upstream of the corner.
1.4 Present Work
The preceding discussion should make plain that there exist sizable gaps in
the literature surrounding high-speed boundary-layer transition over geometries with
noncontinuous surface angles. The objective of the present work is thus to improve
11
our understanding of the interactions of hypersonic laminar boundary-layer distur-
bances - in particular second-mode waves - with the mean flow structures introduced
by a sudden change in surface inclination. To this end, an experimental campaign
was carried out in the Hypersonic Tunnel for Educational and Research Purposes
(HyperTERP) at the University of Maryland - College Park, a Mach 6 reflected-
shock tunnel. The forebody of the employed test article is a 5? half-angle cone,
meant to promote the growth of second-mode disturbances, with a nominally sharp
nosetip. We focus on compression and expansion-corner interactions with modest
(-5 to +15?) turning angles. The higher compression angles were sufficient to pro-
duce flow separation at the conditions examined but still within the range that one
might expect to encounter on a practical hypersonic vehicle. High-speed schlieren
is employed as the primary means of flow interrogation to provide a global view
on instability development. This is supplemented by high-speed surface pressure
fluctuations.
While the experimetal data set obtained throughout this study is of lower
quality than could be expected in a conventional (non-impulse) or quiet facility,
we demonstrate powerful image analysis techniques enabled by the extremely high
frame rates employed (440 ? 822 kHz). After providing details about the experi-
mental apparatus in chapter 2, the general, qualitative development of disturbances
observed for each configuration is described in chapter 3. The spatial development
of spectral content is then presented in chapter 4, revealing amplification and decay
trends for the boundary-layer disturbances. Spectral proper orthogonal decomposi-
tion is employed to resolve coherent disturbance structures in chapter 5. Nonlinear
12
interactions between these modal disturbances are investigated in chapter 6 through
bispectral analysis. Finally, in chapter 7, experimental results are compared to nu-
merical predictions for the +10? compression-corner.
13
Chapter 2: Experimental Methodology
2.1 HyperTERP Facility
All experiments were performed in HyperTERP, a small-scale tunnel oper-
ated by the University of Maryland. A schematic view of the facility, measuring
approximately 15 m in length, is presented in figure 2.1. HyperTERP is operated
as a reflected-shock-tunnel, consisting of a high pressure ?driver? section and a low
pressure ?driven? section that are separated by a diaphragm block. The driven tube
contains the test gas (typically air or nitrogen) and is isolated from the nozzle by
a secondary diaphragm. The desired driver gas composition, which is a mixture of
helium and air, is achieved according to Dalton?s Law of Partial Pressures. Fill-
ing typically takes around 15 minutes, allowing suffcient time for the driver gas to
become well-mixed.
Figure 2.1: Schematic of HyperTERP with components labelled: (A) driver tube;
(B) diaphragm block; (C) driven tube; (D) secondary diaphragm; (E) nozzle; (F)
test section; and (G) dump tank.
14
The tunnel employs a modular design, allowing the nozzle and test section
to be swapped out depending on experimental needs. For the present campaign,
HyperTERP was equipped with a contoured Mach-6 nozzle with a 22 cm exit diam-
eter exhausting into a 30.5 cm diameter free-jet test section. The test section was
designed with four 15.2 cm diameter windows for optical access.
We employ a double-diaphragm system using polyester film (thickness of 250?
500?m), illustrated by figure 2.2, to gain precise control over the timing and repeata-
bility of the tests. The filling procedure begins with valve A in the ?open? position
and valve B (an electronically-controlled solenoid) in the ?closed? position, with all
regions fully evacuated. The driven section is then filled with the desired test gas
before we begin filling the driver. Once the driver tube is approximately halfway
filled, valve A is closed; this isolates the gas in the diaphragm block and allows us
to continue increasing the driver pressure to the requisite level without exceeding
the burst pressure of the downstream diaphragm. To initiate a test, solenoid B is
opened, exhausting the small volume of gas within the diaphragm block into the
larger vacuum tank (which is separate from the facility?s dump tank). This creates
a pressure differential across the upstream diaphragm which causes the material to
burst.
Actuating the diaphragm block thus brings the driver and test gases into
contact, causing a shock wave to propagate downstream through the driven tube
and an expansion wave to travel in the opposite direction. This incident shock wave
then reflects off the driven-tube end wall, bursting the secondary diaphragm and
creating a stagnant, high-pressure, high-temperature reservoir which can then be
15
Vacuum
A B
Tank
Driver Driven
Section Section
Diaphragms
Figure 2.2: Schematic of double-diaphragm system.
16
expanded through the nozzle. The expansion wave likewise reflects from the driver
tube end wall and eventually arrives in the reservoir, bringing the steady-state test
to an end.
The propagation of the incident shock is measured by four PCB 113B26 pres-
sure transducers sampled at 100 kHz. The sensors are located at three different
stations along the driven-tube wall and the signals are conditioned using a PCB
482C05 signal conditioner. The sensor stations are positioned 186.7 cm, 58.4 cm
and 8.9 cm upstream of the nozzle. The most downstream station is instrumented
with two of the PCB sensors and is where the reservoir pressure is measured during
a test. This shock-speed measurement provides an indirect way to determine the
reservoir temperature. From the initial driven gas temperature (assumed to be room
temperature), the normal shock equations will give the temperature jump caused by
the incident shock. By then assuming the incident shock reflects so as to bring the
induced gas velocity back to zero, the initial reservoir conditions may be calculated.
Finally, the gas in this theoretical state is isentropically expanded or compressed to
the instantaneous pressure recorded by the PCB sensors, yielding the instantaneous
reservoir temperature. Because the air is heated substantially during this process, a
calorically imperfect gas model must be implemented where the specific heat ratio
varies with temperature according to
?p[? 1? = 1 + ] . (2.1)
?/T
1 + (?p + 1) (
? )2 e
T (e?/T?1)2
The subscript p here refers to the variable under a perfect-gas assumption and ? is
17
Condition Re33 Re45 Re52
Rem [10
6/m] 3.33 4.49 5.24
P0 [bar] 10.0 13.5 15.8
T0 [K] 890 890 890
U? [m/s] 1266 1266 1266
P? [Pa] 642 866 1006
T? [K] 111 111 111
? [kg/m3? ] 0.020 0.027 0.032
Table 2.1: HyperTERP test condition matrix.
the characteristic vibrational temperature of the test gas [34]. For air, a value for
? of 3056 K is a satisfactory approximation for the temperature range of interest.
Note that the energy equation across a normal shock for a thermally perfect gas
takes the form
( )
u22 ? u
2
1 ?p+ R(T2 ?
1 1
T1) +R? ? = 0. (2.2)
2 2 ? ? 1 e?/T2 ? 1 e?/T1p ? 1
2.2 Test Conditions
The total specific enthalpy was held approximately constant throughout the
campaign at 0.89 MJ/kg, with unit Reynolds numbers of 3.33, 4.49 and 5.24?106 m?1
achieved by varying the reservoir pressure. Reservoir and corresponding freestream
properties for each condition are detailed in table 2.1. Although HyperTERP is
capable of reaching reservoir temperatures as high as 1400 K, this lower enthalpy
configuration was selected to yield higher unit Reynolds numbers and dynamic pres-
sures, as well as longer test durations.
To optimize campaigns conducted in HyperTERP, some effort was expended
in tailoring of the facility. Tailoring refers to the proper selection of initial driver
18
gas composition and driver-to-driven-section pressure ratio to produce a station-
ary contact surface, thereby achieving steady conditions in the reservoir and test
section [35, 36]. If the initial conditions are improperly chosen, then additional dis-
turbances will be produced when the reflected shock intersects the contact surface,
perturbing the reservoir gas and resulting in temporal variations in the supply pres-
sure. Over-tailoring refers to the case where the incident shock is too weak and a
shock wave reflects off the contact surface, causing an increasing pressure with time.
Conversely, under-tailoring results in an expansion wave propagating back into the
reservoir and occurs when the incident shock is too strong. Although tailored con-
ditions are known theoretically, shock-wave attenuation and other imperfect-shock
effects mean that iteration over several facility runs is required to determine the
correct fill properties.
The result of these tailoring endeavors is demonstrated by the sample reser-
voir pressure traces in figure 2.3 for each condition. While there is some degree of
unsteadiness up until 5 ms, conditions are steady for the next 7 ms up until approx-
imately 12 ms when the expansion wave arrives. Across all tests and conditions, the
steady test time is ?6 ms, during which the reservoir-pressure unsteadiness is typi-
cally 2% (standard deviation). Shot-to-shot variation in the mean pressure is of the
order of 1.3%, while systematic uncertainty (combined calibration and nonlinearity)
is estimated as 1.6%. Shock-speed estimates are accurate to ?5 m/s, contributing
0.4% uncertainty in the reservoir temperature.
19
2
1.5
1
0.5 Re33
Re45
Re52
0
0 5 10 15
time [ms]
Figure 2.3: Sample reservoir pressure traces at each condition.
2.3 Nozzle Characterization
The contoured nozzle employed for this work was designed through a hybrid
approach using the method of charactistics (MoC) and computational fluid dynam-
ics (CFD). First, an inviscid nozzle contour was generated based on the rotational
method of characteristics including vibrational-relaxation effects. This contour was
then altered with a viscous boundary-layer correction estimated using the von Kar-
man momentum-integral equation. This boundary-layer correction was then cali-
brated iteratively by comparison to axisymmetric RANS computations. A full de-
scription of this iterative procedure and implementation is given by Starshak, Butler
and Laurence [37].
A limited pitot rake characterization study was performed with the rake po-
sitioned both at the nozzle exit plane and 160 mm downstream to ascertain flow
uniformity at condition Re33. The rake was instrumented with an array of Kulite
XCE-062 pressure transducers sampled at 100 kHz. Resulting values of P0/Ppitot
20
pressure [MPa]
as a function of radial distance from the nozzle centerline are given in figure 2.4.
Included on this plot are lines corresponding to freestream Mach numbers of 5.8
and 6.0. The high reservoir temperature here again means we cannot ignore the
effects vibrational excitation, leading to two theoretical scenarios for the expansion
through the nozzle. The first of these assumes the gas is in vibrational equilibrium
throughout the nozzle, meaning the specific heat ratio varies with temperature ac-
cording to equation 2.1. The integrated form of the energy equation for a thermally
perfect gas,
[ ( ) ( )]
2 2T0 ?p T ? 1 1M = 1? + ? , (2.3)
?T ? ? 1 T T e?/T0 ? 1 e?/Tp 0 ? 1
may be used to determine the temperature at the nozzle exit for a given freestream
Mach number. The pressure may then be determined from the isentropic relation-
ship
( ) ?p [( ) ( ) ]
P e?/T0 ? ( )1 T ? ?/T ?/Tp?1 ? e ? ? e 0= exp . (2.4)
P e?/T ? 1 T T e?/T ? 1 T e?/T00 0 0 ? 1
At these low-density conditions, however, there is a high likelihood of vibra-
tional freezing within the nozzle. For the frozen scenario, we consider the gas to
be in vibrational equilibrium up until the throat, at which point we revert to a
calorically perfect gas assumption with a ? of 1.4 until the nozzle exit. The primary
effect of noneqilibrium here is to increase the freestream pressure for a given Mach
number, reducing the expected P0/Ppitot.
21
Regardless of the flow model employed for the nozzle expansion, the normal
shock upstream of the probe and the subsequent isentropic expansion are treated
with a calorically perfect gas assumption (the calculated vibrational-relaxation time
behind the shock being orders of manitude larger than the shock stand-off distance),
meaning the pressure ratio is given by
?p
P02 (? + 1)
2M2 ?p?1p 1? ? 2p + 2?pM
= , (2.5)
P? 4?pM2 ? 2(? ? 1) ?p + 1
where P02 refers to the pressure measured by the pitot rake.
The experimental data presented in figure 2.4 most closely aligns with the
frozen, Mach 6 pressure ratio; moreover, visualization of the shock generated by a
sharp-edged planar ramp at various angles of incidence indicated a Mach number of
5.99? 0.09. In light of this and the fact that the frozen scenario is more physically
realistic, it is for this condition that the freestream properties of table 2.1 are com-
puted. This figure also shows little variation as a function of radial or streamwise
location, suggesting that the core flow is fairly uniform with minimal flow divergence.
2.4 Test Article and Instrumentation
Two configurations of the primary test article are pictured in figure 2.5; both
consist of a 410 mm, 5? half-angle frustum and a 76.2 mm extension. The top
schematic depicts the -5? configuration, which creates a 5? expansion by employing
a cylindrical afterbody. The afterbody of the lower configuration has a 15? half-
angle, creating a +10? compression. Compression angles of +5? and +15?were also
22
40
0mm
160mm
Equilibrium
M = 6.00
35 Frozen
M = 5.80
30
25
-50 0 50
Radial Distance [mm]
Figure 2.4: Pitot pressure ratio as a function of radial distance from the centerline.
studied, though these configurations are not pictured. A straight-cone configuration
(5? half-angle extension) was tested to provide comparisons against an undisturbed
boundary layer. Thus, this study includes five distinct model configurations (-5? ,
+0? , +5? , +10? , and +15?). The same, nominally-sharp nosetip was used in all
tests; the true bluntness radius was measured to be 0.10 mm using a SmartScope
optical gage.
A single streamwise row of PCB 132B38 pressure transducers was installed
along the upper surface of the cone as shown in figure 2.5 (U1-D5), held in place
using clear nail polish following the methodology of Ort and Dosch [38]. The surface
coordinate of each station relative to the nosetip is given in table 2.2. These sensors
were sampled at 2 MHz with a 600 kHz lowpass filter to remove aliasing effects,
though it should be noted that [38] observed resonances at frequencies above 300 kHz
23
Figure 2.5: Model schematic for the (top) -5? and (bottom) +10? configurations
showing sensor layout with dimensions in millimeters.
Station U1 U2 U3 U4 D1 D2 D3 D4 D5
s [mm] 326 353 381 399 415 422 434 447 460
Table 2.2: Surface coordinate of PCB stations.
for this model. Indeed, a 300 kHz peak was observed in many of the spectra in the
present experiments, and results near this frequency should be treated with caution.
The model was initially installed at approximately zero incidence by sight with
the aid of a laser level, as illustrated in figure 2.6, by aligning the model seam and
PCB array with the horizontal and vertical centerlines of the nozzle. Obviously, this
method suffers from significant inaccuracy as it relies entirely on the researcher?s
eyesight. For this reason, tests were performed late in the campaign where an
additional PCB sensor was instrumented on the underside of the model opposite
station U1. The results from these tests were extrapolated to gain confidence in
tests conducted early in the campaign. The dominant fundamental frequency of
24
second-mode disturbances is highly sensitive to angle of attack, meaning that zero
incidence is achieved when there is a frequency match between the top and bottom
surfaces. Resulting spectra from two tests at condition Re45 are shown in figure 2.7
for which the angle of attack was altered by 0.3? between tests using shims. The
peak frequency on the upper surface is seen to increase from 235 kHz in the left plot
to 275 kHz in the right plot. The match in frequencies in the right plot suggests
nearly zero incidence. The dominant second-mode frequencies observed at station
U1 for all tests analyzed throughout this work (at condition Re45) were between
250?275 kHz. Assuming the second-mode frequency varies linearly for small changes
in incidence, this suggests the error in angle of attack for this study is less than 0.2?.
Furthermore, because the dominant frequency is consistently below the matching
frequency of 275 kHz, we can conclude that the upper surface of the model is also
the leeward side throughout this work. Although no comparable corrections were
made regarding model yaw, small yaw variations are expected to have a negligible
effect on the flow behavior in the regions of interest.
2.5 Calibrated Schlieren
High-speed flowfield visualization was obtained from a standard Z-type schlieren
with a horizontal knife-edge and serves as the primary means of flow interrogation
throughout this work. Light pulses of 20 ns duration were provided by a Cavilux HF
laser and a Phantom v2512 camera was used to capture the images at 440?822 kHz,
allowing resolution of spectral content up to 411 kHz. The frame rates were chosen
25
Figure 2.6: Images of the alignment procedure when installing the model.
2 2
10 10
U1 - top U1 - top
U1 - bot U1 - bot
0 0
10 10
-2 -2
10 10
-4 -4
10 10
100 200 300 400 500 600 100 200 300 400 500 600
Figure 2.7: PCB spectra for the top and bottom model surfaces with the angle of
attack increased by 0.3? between runs.
26
to be higher than the Nyquist sampling rate of the dominant boundary-layer distur-
bances (typically second-mode waves). The field of view ranged from 512?32 pixels
at 822 kHz to 640? 64 pixels at 440 kHz, with the camera rotated to maximize the
region of flow visualized. The magnification of the optical setup resulted in a length
scale of 0.112 mm/pixel for the -5? and +15? configurations and 0.139 mm/pixel for
all others (+0? , +5? , and +10?). The theoretical, undisturbed boundary-layer thick-
ness at the corner junction ranged from 1.44?1.81 mm depending on the condition.
This gives a boundary layer resolution of at least 10 pixels in the lesser magnifica-
tion runs, though it should be noted that this number will be lower far upstream
of the corner and along the compression flares due to the reduced boundary-layer
thickness.
By using a calibration procedure described by Hargather and Settles [39] and
Kennedy [40], the pixel intensities of each image were converted to density gradients.
Following their implementation, a spherical lens with a known deflection angle profile
was placed into the field of view and a reference image captured, as in the left image
of figure 2.8. A plano-convex lens was employed for this study with a focal length, f ,
of 10 m and a diameter of 25.4 mm. The intensity profile along the lens face was then
mapped to the known deflection profile by assuming that zero deflection corresponds
to 92% of the background pixel intensity (to account for the 8% absorption specific
to the calibration lens). A sample intensity profile is shown in the right image of
figure 2.8. For a weak lens where the focal length is much greater than the diameter,
the deflection angle, , is given as a function of radial distance from the lens center,
27
4
10
7
6
5
4
3
2
1 Raw Values
Fit
0
-4 -2 0 2 4
Radial distance [mm]
Figure 2.8: (Left) Sample calibration lens image and (right) pixel intensity profile
over lens face.
r, according to
r
= tan  ? , (2.6)
f
The deflection profile is then mapped to the density gradient of the test gas by
?L ??
 = , (2.7)
n? ?y
where L is the integration path-length of the light, ? is the Gladstone-Dale constant,
? refers to the gas density and n? is the index of refraction of air at laboratory
conditions.
28
Pixel Intensity
Chapter 3: Schlieren Observations
3.1 Straight (+0?) Extension
We begin by examining the behavior of the undisturbed boundary layer at each
condition in the +0? configuration through the time-resolved schlieren sequences
presented in figures 3.1, 3.2, and 3.3 with inter-image spacings of 9.7?s, 9.4?s, and
10.2?s, respectively. The images have been rotated to align the abscissa with the
frustum surface and X refers to the distance from the nosetip. At the top of each
sequence is shown an average flow-on image to highlight the mean flow features.
Subsequent images are reference-subtracted using a 40-image sliding average to em-
phasize transient features. Red triangles denote the locations of PCB sensors, which
are labelled in the mean flow image, while red bars are used to bracket regions of
interest and approximate the propagation of the disturbances. Power spectral den-
sities (PSDs) for select PCB sensors are presented in figure 3.4 corresponding to
each image sequence.
Distinct, rope-like waves are seen upstream of U3 in the first image of fig-
ure 3.1 (condition Re33), corresponding to a second-mode wavepacket, with addi-
tional second-mode content visible within the vicinity of D3. The PCB spectra show
that the peak second-mode frequency shifts from approximately 210 kHz to 190 kHz
29
as the waves propagate from U1 to U3. Note that these frequencies are somewhat
lower than the peak frequencies observed at this condition in other tests, likely
due to the previously discussed angle of attack errors. The disturbances amplify
over this region, increasing in N-factor (integrated amplification rate) by 0.59. The
wavepacket appears to intensify as it propagates along the cone surface, correlating
with the continued amplification observed in the PCB spectra from U3 to D3 where
the N-factor increases by an additional 0.61. The last three images of the sequence
show the packet in a transitional state with its periodic structure deteriorating. This
behavior aligns with the spectral broadening observed in the spectra for sensors D3
and D5. Note that the 300 kHz peak at station D5 is likely an example of anomalous
sensor resonance and not attributable to flow features. While this schlieren sequence
is intended to be representative, the transition process is stochastic by nature and
the freestream noise environment of impulse facilities like HyperTERP is somewhat
more variable than in conventional facilities. This resulted in occasionally signifi-
cant variation in wavepacket behavior, with some breaking down over the frustum
and others traversing the entire cone surface without obviously transitioning. This
variation in behavior is a consistent theme throughout this work.
A wavepacket exhibiting transitional behavior at condition Re45 is highlighted
in figure 3.2. The wavepacket does not obviously grow in intensity from the first
image and begins to lost its structure at around X=405 mm in the fourth and fifth
images of the sequence. The PCB spectra indicate saturation occurs between U1
and U3, as the fundamental peak (now at 260 kHz) experiences no growth while
frequencies above and below the second mode amplify by nearly an order of mag-
30
U3 D1 D3 D5
350 360 371 381 391 402 412 423 433 444 454
X [mm]
Figure 3.1: Reference-subtracted image sequence captured for the +0? configuration
at condition Re33 with inter-image spacing of 9.7?s.
nitude. This spectral broadening continues along the length of the model, though
the second-mode peak is still strongly apparent at D5, indicating that (on average)
the boundary-layer is not fully turbulent. The final image of the sequence shows
transitional content entering the field of view, once again demonstrating the range
of boundary-layer states which occur within individual experiments. There appears
to be harmonic second-mode content in the U1 spectrum at 520 kHz which may
be linked to the 480 kHz peak at D1 and the 440 kHz peak at D3, though these
observations must be treated with skepticism due to sensor resonance.
The upstream movement of the transition front continues at condition Re52,
as exemplified by figure 3.3. The highlighted wavepacket in this sequence breaks
down as it passes over X=400 mm, resulting in a young turbulent spot over much of
the extension. The PCB spectra in figure 3.4 reveal essentially the same trends re-
garding saturation and spectral broadening as those for condition Re45. Resonance
31
U3 D1 D3
367 378 388 399 409 419 430 440
X [mm]
Figure 3.2: Reference-subtracted image sequence captured for the +0? configuration
at condition Re45 with inter-image spacing of 9.4?s.
in sensors D3 and D5 now overlaps with the second-mode peak making it difficult
to discern the true amplitude of the disturbances.
32
U3 D1 D3
359 369 380 390 401 411 422 432 442 453
X [mm]
Figure 3.3: Reference-subtracted image sequence captured for the +0? configuration
at condition Re52 with inter-image spacing of 10.2?s.
U1
U3
0
10 D1
D3
D5
-2
10
-4
10
200 400 600 200 400 600 200 400 600
Figure 3.4: Power spectral densities at select PCB stations for the +0? configuration
at (left) condition Re33, (middle) condition Re45, and (right) condition Re52.
33
3.2 -5?Extension
The image sequences of figures 3.5 and 3.6 demonstrate the interaction of
second-mode wavepackets with the 5? expansion at conditions Re45 and Re52, re-
spectively; figure 3.5 is also representative of the wavepacket behavior at condition
Re33. At conditions Re33 and Re45, the second-mode disturbances are seen to
dampen significantly as they propagate through the expansion and along the ex-
tension. The wavepacket in figure 3.5 appears to stretch out and gradually vanish
as it approaches the edge of the field of view. This is accompanied by a marked
decrease in the PCB spectra (see figure 3.7) for frequencies at and above the second-
mode fundamental in sensors D3 and D5; the second-mode N-factor drops by 0.55
between U3 and D3. The wavepackets also propagate at a greater off-wall distance
downstream of the corner, though this may simply be due to the increased momen-
tum thickness of the boundary layer. At condition Re33, a significant 420 kHz peak
persists just downstream of the corner at D1 which may correspond to the first
harmonic of the second mode; a similar peak is observed around 480 kHz at con-
dition Re45. For both conditions, D5 demonstrates an increase in spectral energy
in lower frequencies. This makes sense, as the increased boundary layer thickness
downstream of the corner will tend to promote the amplification of lower-frequency
disturbances. A notable phenomenon observed in the image sequence of figure 3.5
is emanation of high-frequency energy along the head of the expansion fan. This
behavior, highlighted by red arrows in the third and fourth images of the sequence,
is more easily identified when viewing the full videos.
34
Figure 3.5: Reference-subtracted image sequence for the -5? extension at condition
Re45 with inter-image spacing of 10.9?s.
35
Figure 3.6: Reference-subtracted image sequence for the -5? extension at condition
Re52 with inter-image spacing of 10.7?s.
Transitional behavior prior to the corner becomes commonplace at condition
Re52, with one such occurence illustrated in figure 3.6; in this case the incoming
wavepacket seemingly breaks down as it passes over the corner. The expansion
still has a damping effect on the turbulent content, however, with the structures
becoming largely indistinct downstream. Though there is an obvious reduction in
second-mode power in the PCB spectra at stations D3 and D5, sensor resonance
makes it difficult to quantify the decay due to overlap with the fundamental fre-
quency of the disturbances. These sensors also demonstrate greater persistance of
high frequencies (> 400 kHz) at the model surface than at lower Rem.
36
U1
U3
0
10 D1
D3
D5
-2
10
-4
10
200 400 600 200 400 600 200 400 600
Figure 3.7: Power spectral densities at select PCB stations for the -5? configuration
at (left) condition Re33, (middle) condition Re45, and (right) condition Re52.
3.3 +5?Extension
Contrasting starkly with the expansion case, the compression-flare configura-
tions and the resulting corner SWBLIs serve largely to promote disturbance growth
and transition. Figure 3.8 presents an image sequence captured at condition Re33
with the +5? configuration, again focusing on an incoming second-mode wavepacket.
The field of view for this test is focused primarily on the flare to better capture the
downstream development of the boundary-layer disturbances. When the wavepacket
encounters the SWBLI, energy is radiated away along the shock (indicated by the
red arrow in the fifth and sixth images of the sequence). This radiation is appar-
ent for nearly all wavepackets, has a periodicity similar to the second-mode, and
typically appears to emanate from the tail of a wavepacket. A similar phenomenon
appeared in computations performed by Sawaya et al. [41] of second-mode waves
37
interacting with two-dimensional wall deformations. Downstream of the corner the
wavepackets retain their ?rope-like? appearance, but the structure angle of the dis-
turbances decreases such that they appear flatter relative to the model surface. This
continued growth is confirmed by the PCB spectra of figure 3.11, as the wavepack-
ets do not appear to saturate until downstream of D3. The tail of the wavepacket
then appears to transition around X=450 mm as the head of the packet leaves the
field of view, seemingly without breaking down. Note that the first three images of
figure 3.8 depict the tail of a wavepacket exiting the field of view intact, providing
a counter-example to this transitional behavior. The PCB spectra along the flare,
particularly D5, demonstrate a notable shift in the second-mode peak to higher fre-
quencies brought about by the reduced boundary-layer thickness. The compression
may also promote the development of harmonic content, as the D3 and D5 spectra
display strong peaks at 435 kHz and 470 kHz, respectively. The spectra also display
substantial low-frequency growth downstream of the corner.
The PCB spectra in the middle plot of figure 3.11, corresponding to condition
Re45, show reduced growth in the second-mode peak along the frustum relative to
condition Re33. In other respects, the behavior observed at condition Re45 mimics
that at lower Rem, including the apparent saturation and decay of wavepackets
in the vicinity of D3. Importantly, the broadband amplification observed between
sensors D1 and D3 exceeds that of the +0? configuration, implying acceleration of
the transition process relative to the undisturbed boundary-layer. The sequence
in figure 3.9 again demonstrates the tendency for the radiation to propagate from
the wavepacket tail and for this to be the portion of the disturbance which first
38
D5
D3
D1
394 405 415 426 436 447 457 468 478 488
X [mm]
Figure 3.8: Reference-subtracted image sequence captured for the +5? configuration
at condition Re33 with inter-image spacing of 8.8?s.
39
D5
D3
D1
412 423 433 444 454 464
X [mm]
Figure 3.9: Reference-subtracted image sequence captured for the +5? configuration
at condition Re45 with inter-image spacing of 7.3?s.
transitions.
At condition Re52, the PCB spectra along the flare (D1, D3 and D5) in the
rightmost plot of figure 3.11 show significant amplification in frequencies above and
below the second-mode peak relative to the undisturbed spectra obtained with the
+0? configuration. It is unclear to what extent the spike in second-mode power at
D3 and D5 is due to sensor resonance, but it is clear the signal becomes largely
broadband along the flare, indicative of the onset of a turbulent boundary layer.
40
D3
U3 D1
356 366 377 387 398 408 419 429 440 450
X [mm]
Figure 3.10: Reference-subtracted image sequence captured for the
+5? configuration at condition Re52 with inter-image spacing of 8.8?s.
Indeed, the full test video shows this is typically the case, with many packets already
transitioning upstream of the flare. The selected image sequence in figure 3.10 shows
a packet which retains some periodicity through the compression, though the tail
of the packet transitions immediately upon reaching the flare. Red arrows in the
fifth and sixth images highlight radiation away from the packet even as it enters a
transitional state. When contrasted with the undisturbed case, the results for this
configuration suggest that the non-separated SWBLI here accelerates boundary-
layer transition for sufficiently high Rem.
41
U1
U3
0
10 D1
D3
D5
-2
10
-4
10
200 400 600 200 400 600 200 400 600
Figure 3.11: Power spectral densities for select PCB stations at the +5? configuration
at (left) condition Re33, (middle) condition Re45, and (right) condition Re52.
3.4 +10?Extension
The visualization sequences presented in figures 3.12, 3.13, and 3.14 show the
development of second-mode wavepackets at each condition for the +10? configuration.
The adverse pressure gradient imposed by this compression is sufficient to separate
the boundary layer upstream of the flare. Approximate separation and reattach-
ment locations are represented in each image sequence by vertical, dashed lines
and are given in table 3.1. These locations were identified as points at which the
pseudo-streamline profiles reported in section 4.1 exhibited a sudden change in slope;
reasonable agreement was obtained between these results and preliminary numerical
simulations presented in chapter 7. The uncertainty in each location is estimated
as ?3 mm.
The images in figure 3.12 depict a single wavepacket traversing the resulting
42
Re33 Re45 Re52
Xseparation [mm] 391 396 403
Xreattachment [mm] 432 428 424
Table 3.1: Approximate separation and reattachment locations for the
+10? configuration.
separation region at condition Re33. In the mean image, the boundary layer sep-
arates roughly halfway between sensor U3 and the corner junction, and reattaches
slightly upstream of D3. In the PCB spectra for this condition (figure 3.15), sub-
stantial growth in the fundamental second-mode range (175? 275 kHz) is observed
between U1 and U3, with the disturbance peak increasing in N-factor by 0.70 (sim-
ilar to the other cases) and decreasing in frequency from 247 kHz to 220 kHz. These
sensors also reveal a potential harmonic developing at 440 kHz. At the separation
point, the instability waves lift off the cone surface and propagate largely within
the separated shear layer. This behavior is reflected in the PSD for sensor D1,
which shows a marked decrease in second-mode power; note, however, the increase
in low-frequency content peaking at 73 kHz. We will later demonstrate that this
low-frequency peak corresponds to new, shear-generated disturbances. The waves
appear to stretch and flatten out as they propagate downstream, but retain their
periodic structure through reattachment, with sensor D3 displaying a recovery in
second-mode power. The reduced boundary-layer thickness along the flare results
in the amplification of higher-frequency disturbances, as evidenced by sensor D5.
Figure 3.13 depicts the typical transitional behavior of wavepackets which en-
counter the separated region at condition Re45. The incoming wavepacket can be
seen to morph substantially downstream of separation, with some features propa-
43
D3
U3 D1
382 393 403 414 424 435
X [mm]
Figure 3.12: Reference-subtracted image sequence captured for the +10? flare at
condition Re33 with inter-image spacing of 9.7?s.
44
D3
U3 D1
386 396 406 417 427 438
X [mm]
Figure 3.13: Reference-subtracted image sequence captured for the +10? flare at
condition Re45 with inter-image spacing of 9.7?s.
45
gating near the wall within the recirculation zone. These images show the apparent
radiation of disturbance energy along the separation shock, highlighted by red arrows
in the second, third and fourth images of the sequence, matching observations seen
with the +5? configuration. As at condition Re33, there is evidence of a harmonic
developing in the U3 spectra at 500 kHz while sensor D1 reveals substantial growth
of low-frequencies within the separated region, now peaking at 85 kHz. Upon reat-
tachment the packet?s structure has distorted to include a larger range of scales and
lost its ?rope-like? appearance, explaining the broadband spectra of sensors D3 and
D5. However, such behavior is not fully representative of condition Re45; the state
of the incoming wavepacket may substantially alter its development through the
SWBLI. In the first image of the sequence in figure 3.13, for example, a wavepacket
is visible downstream of D3 that has largely retained its structure.
For condition Re52, the separation region has shrunk significantly and the
second-mode wave packets are, on average, close to saturation when incident upon it.
Figure 3.14 depicts a wavepacket approaching the SWBLI with persistent turbulence
enveloping the flare. Such behavior mimicks that observed with the +5? configuration
and is reflected in the PCB spectra, where even sensor D1 displays primarily broad-
band content. Radiation events still occur at this condition and are indicated by
red arrows in the fourth and fifth reference-subtracted images.
46
D3
U3 D1
380 391 401 412 422 433
X [mm]
Figure 3.14: Reference-subtracted image sequence captured for the +10? flare at
condition Re52 with inter-image spacing of 9.7?s.
U1
U3
0
10 D1
D3
D5
-2
10
-4
10
200 400 600 200 400 600 200 400 600
Figure 3.15: Representative power spectral densities at select PCB stations for the
+10? configuration at (left) condition Re33, (middle) condition Re45, and (right)
condition Re52.
47
Re33 Re45 Re52
Xseparation [mm] N/A N/A 394
Xreattachment [mm] 440 433 427
Table 3.2: Approximate separation and reattachment locations for the
+15? configuration.
3.5 +15?Extension
As evident from the mean flow-on image in figure 3.16, the +15? compression
flare produces a separation bubble which (at condition Re33) extends far enough
upstream to fully envelop sensor U3. Approximate separation and reattachment
locations for this configuration are given in table 3.2. These were determined in the
same manner as before, though the separation location could not be determined at
conditions Re33 and Re45 as the boundary layer appeared to separate upstream of
the field of view in both cases.
Second-mode wavepackets appear only intermittently within the field of view
with this configuration (at Re33 and Re45) and the dominant boundary-layer feature
instead appears to be a shear-generated instability which forms over the separation
bubble. This instability takes the form of long, braided structures aligned with the
direction of propagation which may be seen in the instantaneous image sequence
of figure 3.16. The disturbances undergo substantial amplification once the shear
layer begins to recompress from X=412 mm. As the waves encounter the flare near
reattachment, they emanate energy into the shock-layer as annotated by red arrows
in the third and fourth images of the sequence. This energy has a wavy, ribbon-
like structure. Around reattachment, the features take on a thin, layered structure
48
but the streamwise extent of the field of view is insufficient to fully characterize
their downstream behavior. Excessive broadband sensor resonance contaminated the
PCB data for most experimental runs with this configuration and thus no spectra
are presented. It is also worth noting that the method used for determining the
separation and reattachment locations is particularly fraught at these conditions
due to the lack of second-mode wavepackets to define the pseudo-streamline.
Though rare at this condition, the propagation of one second-mode wavepacket
at condition Re33 is presented in figure 3.17. As the wavepacket traverses the
separation bubble, it loses its distinctive structure and appears to merge with the
shear-waves. That said, the boundary layer over the flare in the final image of
the sequence appears perturbed by the arrival of the remnants of this disturbance
relative to its uniform, layered structure in the first three images of the sequence.
At condition Re45 the boundary layer is once again dominated by the shear-
generated disturbances. The image sequence of figure 3.18 shows the structure of
the disturbances is almost identical to that observed at condition Re33, but the
emanation of disturbance energy is more distinct (see the first three images of the
sequence). The reduced separation bubble size relative to condition Re33 allows us
to observe the behavior of these disturbances further downstream of reattachment
(in the final three images) where they retain a thin, periodic structure. The final
image of figure 3.18 shows a second ribbon of energy detaching from the separated
shear layer as another shear disturbance approaches the flare.
Although second-mode wavepackets remain rare at this condition, those that
do appear retain their structure through the separation bubble to a greater extent
49
D3
D1
U3
380 388 397 405 413 422 430 439
X [mm]
Figure 3.16: Reference-subtracted image sequence captured for the
+15? configuration at condition Re33. Successive images are spaced by 6.8?s.
50
D3
D1
U3
380 388 397 405 413 422 430 439
X [mm]
Figure 3.17: Reference-subtracted image sequence captured for the
+15? configuration at condition Re33 showing second-mode behavior. Succes-
sive images are spaced by 9.1?s.
51
than at condition Re33. This is exemplified by figure 3.19 where the incoming
wavepacket is seen to flatten out significantly over the separated region but imposes
a periodic, wavy structure on the separated shear layer near reattachment at around
X=430 mm in the final image of the sequence.
At condition Re52, second-mode waves re-emerge as the primary disturbance
with shear waves appearing only intermittently. Several particularly intense bursts
of turbulence were observed in this experiment which caused the separation bub-
ble to collapse and reform in a transient process. The image sequence of fig-
ure 3.20 shows a wavepacket passing through the SWBLI during one such recovery
event. The behavior of the packet in many ways mimics that observed with the
+10? configuration. The third image of the sequence shows radiation emanating
from the head of the wavepacket as it reaches the flare (annotated with a green
arrow). The wavepacket immediately enters a transitional state along the flare, as
evidenced by the turbulent structures seen around X=427 mm in the final image.
This behavior is not, however, universal: the first image of the sequence shows a
wavepacket on the flare (again indicated by a green arrow) which does not appear
obviously transitional. The full video shows that the boundary layer over the flare
is often fully turbulent.
52
D3
D1
U3
380 388 396 405 413 422 430 438
X [mm]
Figure 3.18: Reference-subtracted image sequence captured for the
+15? configuration at condition Re45. Successive images are spaced by 5.5?s.
53
D3
D1
U3
380 388 396 405 413 422 430 438
X [mm]
Figure 3.19: Reference-subtracted image sequence captured for the
+15? configuration at condition Re45 showing second-mode behavior. Succes-
sive images are spaced by 9.1?s.
54
D3
D1
U3
387 396 404 412 421 429
X [mm]
Figure 3.20: Reference-subtracted image sequence captured for the
+15? configuration at condition Re52. Successive images are spaced by 8.2?s.
55
Chapter 4: Low-order Spectral Analysis
4.1 Second-mode Behavior
The ultra-high frame rate employed throughout this work allows us to perform
spectral analysis on the schlieren data without assuming a propagation speed for the
disturbances (as in Kennedy et al. [17]). To this end, PSD curves were computed
for each pixel in the field of view for each test over the steady test duration using
Welch?s method with Hann windows of length 64. This process then allows us to
visualize the spatial distribution of frequency content anywhere within the field of
view. The number of frames included in these calculation varied from 2400-5000
depending on the framerate of the test. The one exception to this is the condition
Re45 test of the +15? configuration. The latter half of this test was contaminated
by anomalous turbulence and thus only 1300 images were kept. The exact number
of frames used for each test was the same as that used for the SPOD analysis, as
reported in table 5.1.
In figure 4.1 we present spatial distributions of the integrated disturbance
power for each configuration at condition Re33 over the frequency band 170?270 kHz
(corresponding the dominant second-mode frequencies determined from the PCB
measurements). Note that the data for the +15? case was obtained at only 440 kHz,
56
4 -9.5
0
-10
-4
-9.5
5
-10
0
-9.5
5
-10
0
-9.5
5
0 -10
-9.5
5
0 -10
365 375 385 395 405 415 425 435 445 455
X [mm]
Figure 4.1: Spatial contours for each flare configuration of average PSD from 170?
270 kHz at condition Re33.
meaning that frequency content between 220 ? 270 kHz would be aliased down to
170 ? 220 kHz. The dashed line in each image traces the location of maximum
PSD strength within this second-mode frequency range and can be interpreted as a
pseudo-streamline along which the disturbances tend to propagate. The full-range
spectra corresponding to each pixel along these pseudo-streamlines are presented in
figure 4.2.
The second contour in figure 4.1 illustrates the behavior of the undisturbed
second-mode content, which gradually diminishes downstream of X=435 mm, likely
due to saturation. The streamline spectra support this interpretation, as spectral
broadening is observed even upstream of the corner. This behavior contrasts sharply
with the -5? configuration (top contour), in which the second-mode power rapidly
decays downstream of the expansion corner; this matches the qualitative behavior
observed in the schlieren image sequences. This decay is experienced across all
57
Y [mm]
Figure 4.2: Power spectra computed along pseudo-streamlines at condition Re33
for the (top-left) -5? , (top-right) +0? , (middle-left) +5? , (middle-right) +10? , and
(bottom) +15? configurations.
58
frequencies along the pseudo-streamline, though there appears to be some growth
downstream of X = 435 mm for content at frequencies below 200 kHz.
The growth/decay characteristics of the second mode become more interest-
ing when the boundary-layer interacts with a compression corner. In the case of
a +5? compression (middle contour of figure 4.1) the wavepackets amplify substan-
tially on the flare, peaking around X=450 mm. The pseudo-streamline spectrum
downstream of this point broadens significantly, correlating well with the transi-
tional features observed in the instantaneous flow images and the spectrum of PCB
D5.
In the +10? case, the fundamental second-mode energy is seen to amplify along
the frustum until the separation point at approximately X=391 mm, where it un-
dergoes a rapid decay; this is perhaps linked to the radiation phenomenon noted
earlier, though other effects may be at play. The amplitude of the second-mode
fluctuations appears to freeze within the seperated shear-layer before growing sub-
stantially downstream of reattachment alongside low-frequency content near 50 kHz
(seen in figure 4.2). The pronounced growth in frequencies below 100 kHz which oc-
curs from X = 415? 429 mm is completely absent on the straight cone, suggesting
a new instability generated by the separated shear layer.
The bottom contour of figure 4.1 (+15? flare) shows consistent growth of high-
frequency content within the separation bubble, in contrast to the +10? results. It is
worth noting, however, that the streamline spectra in the bottom plot of figure 4.2
show no distinct peak in the second-mode range, meaning that the behavior of the
second-mode is likely not captured by this spatial contour and the apparent growth
59
4
-9.5
0
-4 -10
-9.2
5 -9.4
-9.6
0 -9.8
-9
5
-9.5
0
-9
5
-9.5
0
-9.5
5
-10
0
365 375 385 395 405 415 425 435 445 455
X [mm]
Figure 4.3: Spatial contours for each flare configuration of average PSD from 200?
300 kHz at condition Re45.
is instead caused by broadband amplification. This makes sense given the relative
lack of distinct wavepackets observed instantaneously. Instead, low-frequency dis-
turbances can be seen developing as far upstream as X=390 mm in the streamlie
spectra, dropping from 65 kHz at onset to 40 kHz at the corner. Amplification of
higher frequencies does not begin until downstream of X=432 mm.
Many of the previous observations regarding the second-mode hold at condi-
tion Re45, where the second-mode range depicted by figure 4.3 is now 200?300 kHz.
Pseudo-streamline spectra are given in figure 4.4. The undisturbed boundary layer
again shows consistent amplification of second-mode content leading up to satu-
ration, though the magnitude of the fluctuations appears to have increased. The
only notable difference from condition Re33 for the -5? case is that the radiation
highlighted in figure 3.5 is visible as a line of elevated PSD emanating from the
corner.
60
Y [mm]
Figure 4.4: Power spectra computed along pseudo-streamlines at condition Re45
for the (top-left) -5? , (top-right) +0? , (middle-left) +5? , (middle-right) +10? , and
(bottom) +15? configurations.
61
The +5?Re45 configuration shows second-mode content reaching off-wall dis-
tances greater than the upstream boundary-layer thickness downstream ofX=440 mm;
this may be attributed to intermittent turbulent behavior of the flare boundary layer.
This behavior is tied to dramatic broadband excitation which is seen along the
pseudo-streamline in figure 4.4 and suggests that the SWBLI promotes transition at
this condition. All this is consistent with the transitional wavepacket development
discussed previously in section 3.3.
As at condition Re33, the second-mode disturbances freeze in amplitude within
the upstream portion of the separation bubble in the +10? case, but begin to amplify
within the downstream portion of the bubble on the approach to reattachment, as in
the computations of Novikov et al. [27]. Dramatic spectral broadening is observed
in the vicinity of reattachment near X=430 mm. The pseudo-streamline spectra do
not demonstrate a distinct shear-generated instability at this condition, though we
will see that it is indeed present.
This second-mode-dominated behavior is once again contrasted slightly by the
+15? results: these show amplification through the entire separation bubble in the
spatial contour, though the most significant growth again occurs near reattachment.
There also appears to be elevated content along the reattachment shock, similar
to previously discussed radiation phenomena. However, as in the Re33 case, the
pseudo-streamline spectrum pictured in figure 4.4 indicates no second-mode peak.
The lack of mature second-mode waves for the +15? configuration is perhaps unsur-
prising given the upstream extent of the separation bubble and the freezing effect
this is seen to have in the +10? configuration. Instead, the doinant feature of the
62
+15? pseudo-streamline spectrum is the shear-generated instability which is seen
from X = 387 mm and reduces in frequency from approximately 85 kHz to 50 kHz
at the corner.
Spatial PSD contours (now corresponding to 230 ? 330 kHz) for the condi-
tion Re52 tests are shown in figure 4.5, where the intermittently turbulent nature
of the boundary layer now has a significant effect in all configurations. For the
+0? case, this manifests itself as high-frequency content further lying from the wall
downstream of X=403 mm, where the disturbance strength now peaks. This also
corresponds to a point beyond which significant spectral broadening is seen in the
streamline spectra of figure 4.6. As with lower Rem, the high-frequency content
drops appreciably just downstream of the -5? corner but then appears to re-amplify
along the flare until X=445 mm. There is also now substantial spectral broadening
along the flare, likely due to wavepackets which reach the corner in a transitional
state.
Just downstream of the +5? compression, the high frequency content am-
plifies rapidly and apparently saturates. This is accompanied by nearly instan-
taneous spectral broadening, with the power of lower frequencies rising to meet
or exceed the second-mode power. The high-frequency content in the +10? case
grows steadily along the frustum and decays slightly downstream of separation
from X=403 ? 413 mm. As the flow reattaches, this content amplifies and spreads
out rapidly to cover an off-wall distance significantly greater than the upstream
boundary-layer thickness, again implying transition. This behavior is mirrored by
the +15? configuration, where the turbulent state of the flare boundary layer is
63
4
0 -9.5
-4 -10
-8.8
5 -9
-9.2
0 -9.4
-8.8
5 -9
-9.2
0 -9.4
-8.8
5 -9
-9.2
0 -9.4
-8.5
5
-9
0
365 375 385 395 405 415 425 435 445 455
X [mm]
Figure 4.5: Spatial contours for each flare configuration of average PSD from 230?
330 kHz at condition Re52.
even more obvious. Both separated configurations display the same instantaneous
spectral broadening immediately at the corner in figure 4.6. The low-frequency sig-
nature of the shear-mode has largely vanished from the streamline spectra for the
+15? extension even upstream of this point.
4.2 N Factors
A common metric for quantifying disturbance growth is the N factor, which
is a spatial integration of the amplification rate. The spectra along the pseudo-
streamlines may also be used to compute changes in N factors for each frequency
according to ( )
1 PSD(f, xi)
?N(f, xi) = ln , (4.1)
2 PSD(f, x0)
64
Y [mm]
Figure 4.6: Power spectra computed along pseudo-streamlines at condition Re52
for the (top-left) -5? , (top-right) +0? , (middle-left) +5? , (middle-right) +10? , and
(bottom) +15? configurations.
65
where PSD(f, xi) refers to the power of frequency f at streamwise location xi, and
x0 refers to the upstream edge of the field of view. The change in N factor calculated
in this manner applies to the density-gradient fluctuations recorded by the schlieren
apparatus; if we are to extend these results to the density fluctuations that would
typically be of more interest, we must invoke the parallel flow assumption [17].
This assumption will become questionable across and immediately downstream of
the corner and within regions of flow separation, so caution should be exercised in
interpreting these results (quantitatively).
The N factors for each configuration at condition Re33 are given in figure 4.7
?
plotted against the stability Reynolds number (R = Res) which is based on
freestream conditions and the surface coordinate. Nondimensional frequencies are
included based on the theoretical boundary-layer thickness at the corner, ?c, and
the theoretical edge velocity along the frustum, Ue. These contours demonstrate
relative amplification of disturbances and thus may reveal key spectral development
at frequencies that are not apparent in the raw PSD.
The undisturbed boundary layer only demonstrates strong amplification within
the second-mode band. The expansion causes a rapid drop in N factor for the dom-
inant second-mode frequency of approximately 0.4; the magnitude of this drop is
consistent across all conditions. Disturbances from 175?200 kHz then amplify down-
stream of the corner, likely corresponding to lower-frequency second-mode waves,
and bands of growth appear at 60, 90, and 140 kHz. Flow structures which may ac-
count for these and other peaks throughout this section are discussed in chapter 5.
The amplification of content below 50 kHz caused by the +5? compression,
66
Figure 4.7: N-factor increase computed along pseudo-streamlines at condition Re33
for the (top-left) -5? , (top-right) +0? , (middle-left) +5? , (middle-right) +10? , and
(bottom) +15? configurations.
67
which was noted in figure 4.2, is more distinct here, and there is evidence of an
additional disturbance developing at 75 kHz. Figure 4.7 also shows that the second-
mode amplification extends to higher frequencies along the flare, matching the PCB
observations. The N-factor spectra also help us better identify key low-frequency
content for the separated cases. Within the +10? separation bubble, two bands of
growth are observed: one at 70?80 kHz which peaks within the separation region and
decays slightly as the boundary layer reattaches, and another at 30? 40 kHz which
continues to amplify downstream. We will demonstrate that the upper-band cor-
rresponds to a shear-generated instability similar to that observed instantaneously
with the +15? extension. The N-factor contour for the +15? compression is similarly
dominated by two bands of growth, now at approximately 15 kHz and 45 kHz. It
is worth noting that the N factors for the +15? case reach much greater levels than
any other configuration, which would indicate that the amplification rates for the
shear-layer instabilities are higher than for any second-mode.
At condition Re45 (figure 4.8), second-mode disturbances in the +0? configuration
do not attain the same levels of amplification as at condition Re33 due to the earlier
onset of saturation. Strong bands of relative growth are observed around 155 kHz
and 190 kHz in the expansion case, though it should be noted that these frequencies
are particularly weak at the upstream edge of the field of view (where the refer-
ence power is considered). The +5? configuration causes the growth of content at
approximately 150 kHz and 80 kHz, both of which appear as distinct bands and
may correspond to new flow structures. Amplification for the +10? configuration is
largely concentrated downstream of reattachment, though there is evidence of weakly
68
Figure 4.8: N-factor increase computed along pseudo-streamlines at condition Re45
for the (top-left) -5? , (top-right) +0? , (middle-left) +5? , (middle-right) +10? , and
(bottom) +15? configurations.
69
amplifying disturbances at 80 kHz from R = 1300? 1330 which may correspond to
intermittent shear waves. Strong growth concentrated around 50 kHz is observed for
the +15? configuration, again corresponding to a distinct, shear-generated instabil-
ity. Substantial low-frequency amplification occurs downstream of the corner near
the reattachment zone.
Switching focus to condition Re52 in figure 4.9, the N factors for the
-5? configuration demonstrate significant amplification of frequencies both higher
and lower than the upstream second-mode fundamental. The low-frequency growth
is considerably less concentrated than at lower Rem, though it appears to peak
around 120 kHz. All three compression cases are characterized by rapid, broadband
amplification downstream of the corner, with no clearly dominant frequencies. Note
that the contour limits are increased for higher flare angles as the amplification
becomes much stronger.
4.3 Low-frequency Behavior
The consistent generation of low-frequency content, particularly in the sepa-
rated cases, justifies a spatial analysis with the frequency range of interest instead
chosen to bracket the potential shear-generated instability. Figure 4.10 shows energy
generated from 40?80 kHz for each configuration at condition Re33. Downstream of
the expansion corner in the first contour, content at these frequencies undergoes an
initial decay similar to that observed for the second-mode disturbances. This is then
followed by another period of amplification near X=435 mm, which is also captured
70
Figure 4.9: N-factor increase computed along pseudo-streamlines at condition Re52
for the (top-left) -5? , (top-right) +0? , (middle-left) +5? , (middle-right) +10? , and
(bottom) +15? configurations.
71
by the N factors in figure 4.7. Growth also occurs downstream of the +5? corner in
the vicinity of the pseudo-streamline.
This is dwarfed by the content generation observed in the separated regions of
the +10? and +15? configurations. For the +10? extension, much of the amplification
occurs from X=410?430 mm as the separation bubble is compressing towards reat-
tachment. As mentioned previously, both pseudo-streamline contours (figures 4.2
and 4.7) show growth of a distinct 75 kHz band within this region, whereas the am-
plification seen along the flare is likely due to the 40 kHz peak seen in the N-factor
plot. These observations largely hold for the +15? case, but the disturbance ampli-
fication can be discerned as far upstream as X=390 mm and reaches a much greater
magnitude. There is also evidence of energy emanating away from the separation
bubble at approximately X=415 mm, which correlates well with the behavior noted
in the reference-subtracted sequences. The low-frequency content along the flare for
this configuration is more likely due to the persistence of the shear waves given the
lack of multiple N-factor peaks within this frequency range.
Figure 4.11 shows little change regarding low-frequency content generation
at condition Re45 (now 50 ? 90 kHz) for the -5? and +0? cases. As noted in the
streamline discussion, however, the +5? compression causes greater low-frequency
amplification than at condition Re33. The spatial amplification seen along the flare
for this configuration is likely most attributable to the N-factor peak around 80 kHz,
as both peaks begin to dissipate at approximately X = 455 mm.
The +10? compression shows evidence of the shear-generated instability de-
veloping within the separation bubble; this was obscured in the pseudo-streamline
72
4
-9.5
0
-4 -10
-9
5
-9.5
0
-9
5
-9.5
0
-9
5
-9.5
0
-8.5
5
0 -9
365 375 385 395 405 415 425 435 445 455
X [mm]
Figure 4.10: Spatial contours for each flare configuration of average PSD from 40?
80 kHz at condition Re33.
spectra, though the growth is concentrated more within the downstream portion
from X=415 ? 425 mm. This content amplifies further downstream of reattach-
ment, but the pseudo-streamline spectra of figures 4.4 and 4.8 suggest this is due
more to spectral broadening and transition.
The shear-generated instability for the +15? configuration again develops far
upstream of the corner. There is a brief but notable drop in PSD directly above
the corner at X=412 mm where significant energy is seen emanating from the shear
layer. This drop is not captured in figure 4.4 because the maximum low-frequency
fluctuations occur further from the wall than the high-frequency fluctuations and
therefore do not lie along the pseudo-streamline. The growth and slight decay of
the shear disturbances illustrated here correlate well with the production terms for
the baroclinic instability identified by Dwivedi et al. [2] for a 2D compression ramp,
depicted in figure 4.12, suggesting a similar mechanism may be at work. Another
73
Y [mm]
4
0 -9.5
-4 -10
-9
5
0 -9.5
-8.8
5 -9
-9.2
0 -9.4
-8.6
-8.8
5 -9
-9.2
0
-8.6
-8.8
5 -9
-9.2
0
365 375 385 395 405 415 425 435 445 455
X [mm]
Figure 4.11: Spatial contours for each flare configuration of average PSD from 50?
90 kHz at condition Re45.
Figure 4.12: Distribution of the production term P for steady perturbations. Taken
from Dwivedi et al. [2].
particularly strong ray of energy is observed originating from the reattachment zone
in the spatial contour. The pseudo-streamline spectrum shows significant spectral
broadening around this same point at approximately X=430 mm.
At condition Re52, the low-frequency content experiences transition-related
amplification in all configurations (figure 4.13). Note that the large artifact for
the straight case at around X=387 mm is due to debris which passes through the
field of view, contaminating the spectra. This also appears as a broadband spike in
figure 4.6. Contrasting with the results at lower Re , the -5?m case now shows the
74
Y [mm]
4
-9.2
0 -9.4
-9.6
-4 -9.8
5 -9
0 -9.5
-8.5
5
-9
0
-8.5
5
0 -9
-8.2
5 -8.4
-8.6
-8.8
0
365 375 385 395 405 415 425 435 445 455
X [mm]
Figure 4.13: Spatial contours for each flare configuration of average PSD from 60?
100 kHz at condition Re52.
low-frequency band reaches levels towards the end of the flare which exceed those
seen upstream of the corner and are not confined to the pseudo-streamline. This is
unsurprising given the previously discussed spectral broadening along the extension
which was attributed to turbulent structures. The +0? and +5? configurations show
the same qualitative trends as at condition Re45 but with greater amplification. The
separated cases no longer show significant evidence of the shear-generated instability,
though low-frequency content does amplify within the separation bubbles. Both
configurations then show abrupt amplification upon reattachment, as was seen in
the streamline spectra. This is to be expected given the turbulent nature of the
flare boundary layer observed in the schlieren images.
75
Y [mm]
412 423 433 444 454 464
X [mm]
Figure 4.14: (Top) Reference-subtracted image and (bottom) bandpass-filtered
(200? 300 kHz) image obtained at condition Re45 with the +5? extension.
4.4 Bandpass-filtered Images
The evolution of modal content identified through the preceding analysis can
be observed in a time-resolved sense by applying a temporal bandpass filter to each
pixel time series, then stitching these filtered series back into image sequences. An
example is given in figure 4.18 where a reference-subtracted image obtained with the
+5? configuration at condition Re45 is displayed above the corresponding image after
applying a 200?300 kHz bandpass filter to the raw time series. Both images clearly
depict a second-mode wavepacket traversing the corner, but the filter enhances the
visibility of the second-mode flow structures as well as the radiation phenomenon
observed at the corner (though this is still more obvious when viewing the videos).
In similar fashion, this technique may be employed to highlight the low-
frequency, shear-generated disturbances present in the +15? configuration. An im-
age sequence created with a passband of 30 ? 80 kHz is depicted in figure 4.15 for
condition Re45. We can now distinguish the disturbance development even at the
76
380 388 396 405 413 422 430 438
X [mm]
Figure 4.15: Bandpass-filtered (30? 80 kHz) sequence for the +15? configuration at
condition Re45; images are spaced by 10.9?s.
upstream edge of the field of view. This image sequence emphasizes the ray of en-
ergy emanating away from the shear layer at X = 412 mm, just above the corner.
It is beyond this point at which the disturbances develop their distinctly braided
appearance.
One particularly useful application of this technique is to probe for distur-
bances that are less easily identified in the reference-subtracted images. Figure 4.16
depicts a sequence obtained by filtering +10? configuration, condition Re33 images
with a passband of 70?86 kHz, intended to isolate the shear-generated disturbances
which should clearly exist based on the N-factor peak in figure 4.8. A braided struc-
ture is clearly visible developing within the separation region that correlates spatially
with the amplitude peak in figure 4.10, and has a similar structure to the distur-
77
382 393 403 414 424 435
X [mm]
Figure 4.16: Bandpass-filtered (70? 86 kHz) sequence for the +10? configuration at
condition Re33; images are spaced by 3.65?s.
bances observed within the separation bubble of the +15? configuration. A distinct
ray of energy is once again seen coming from the shear layer just above the cor-
ner. These low-frequency braided structures are found propagating when there is
no evidence of second-mode content, indicating that they correspond to independent
instability mechanisms.
The +5? configuration image sequence from condition Re33 was bandpass fil-
tered from 10? 50 kHz to probe the low-frequency amplification observed along the
pseudo-streamline in the N-factor plot of figure 4.7. A sample sequence is shown
in figure 4.17 with an inter-image spacing of 14.6?s. Periodic features are seen
propagating along the flare with a braided structure very similar to that of the
shear-generated disturbances which appear in the separated cases. These distur-
bances are found coexisting with second-mode wavepackets, but also appear when
there is no evidence of high-frequency content. Thus, it appears that the streamline
78
412 423 433 444 454 464
X [mm]
Figure 4.17: Bandpass-filtered (10 ? 50 kHz) sequence for the +5? configuration at
condition Re33; images are spaced by 14.6?s.
curvature at the corner may provoke the development of new disturbances even in
the absence of a separation bubble.
This strategy of bandpass-filtering the images also allows us to observe har-
monic second-mode content despite the insufficient framerate by taking advan-
tage of aliasing. We will demonstrate this for a test at condition Re45 on the
-5? configuration. The dominant second-mode frequency for this test was approxi-
mately 250? 260 kHz (based on the PCB spectra), which would imply a harmonic
frequency of 500?520 kHz. Based on the camera framerate of 550 kHz, this harmonic
would be aliased down to 30 ? 50 kHz. Shown in figure 4.18 is a single reference-
subtracted image (top) and the corresponding image after applying a 30 ? 80 kHz
bandpass filter (bottom). A second-mode wavepacket is clearly seen in the reference-
subtracted image upstream of the corner. The filtered image displays a rope-like
structure propagating alongside this wavepacket, but with a much shorter wave-
79
397 405 414 422 430 439 447 456
X [mm]
Figure 4.18: (Top) Reference-subtracted image and (bottom) bandpass-filtered (30?
80 kHz) image for the -5? configuration at condition Re45.
length. We can confirm this to be the second-mode harmonic by extracting the
pixel signal along the pseudo-streamline for these images and computing the spatial
PSD to determine the dominant wavelength in each. Figure 4.19 shows a compar-
ison of the PSD for the two images computed with pixels only up to X = 422 mm
(beyond which the wavepacket becomes indistinct). The reference-subtracted im-
age peak wavelength corresponds to the second-mode disturbance and lies between
4.1 mm and 4.4 mm. The peak wavelength of the bandpass-filtered image is 2.2 mm,
approximately half that of the fundamental.
80
4
10
3
10
2
10
1
10
Reference-subtracted
Bandpass-filtered
0
10
0 2 4 6 8
Wavelength [mm]
Figure 4.19: Comparison of dominant wavelengths within the reference-subtracted
and bandpass-filtered images.
81
PSD [Arb.]
Chapter 5: Spectral Proper Orthogonal Decomposition
The structure and evolution of disturbances may also be analyzed through
the Spectral Proper Orthogonal Decomposition (SPOD) methodology of Towne et
al. [42]. This technique provides a set of orthogonal modes which oscillate at distinct
frequencies and describe the coherent evolution of flow structures in both time and
space. By resolving these modes, we may be able to identify the flow structures
responsible for the amplification peaks observed in chapter 4. Also, as will be shown
in chapter 6, this technique is particularly useful when coupled with bispectral
analysis, as it provides a method to probe the physical structure of frequencies
which dominate nonlinear growth mechanisms. In the present implementation, Hann
windows of length 64, 128, or 256 were used (depending on the framerate of the test)
with 50% overlap to group the images of each sequence, resulting in a minimum of 34
flow realizations for an individual test. The specific parameters for each computation
are given in table 5.1.
5.1 +0 ?Extension
As has been established, the boundary layer on the straight-cone configuration
is largely dominated by the second-mode instability; this is confirmed by the SPOD
82
Extension Condition Nf [frames] Lhann [frames] Realizations
Re33 3350 128 51
-5? Re45 3305 128 50
Re52 4490 128 69
Re33 3720 128 56
+0? Re45 3800 128 58
Re52 3690 128 56
Re33 4900 256 37
+5? Re45 6000 256 45
Re52 4500 256 34
Re33 5000 256 38
+10? Re45 5920 256 45
Re52 5000 256 38
Re33 2400 128 36
+15? Re45 1300 64 39
Re52 3000 128 45
Table 5.1: Parameters for SPOD computations.
analysis. The energy of the SPOD modes computed for each condition are given
in figure 5.1, where we see a distinct peak in the highest-ranked mode around the
fundamental second-mode frequency of each test. Throughout this chapter, only the
highest-ranked mode will be considered for analysis, as it respresents the greatest
energy content and appears to capture the dominant instability mechanisms. Note
that the second-mode peak appears reduced at Re52; this is due to turbulent content
which posseses significant flow energy but is less structurally coherent.
Eigenvalue contours for select SPOD modes for condition Re33 on the straight
cone are given in figure 5.2. The 160, 184, and 228 kHz contours show the typical
rope-like structure of the second-mode disturbances at three different frequencies.
The 160 kHz mode does not develop a coherent structure until X=400 mm, which is
also the point at which the streamline spectrum of figure 4.2 demonstrated significant
spectral broadening. At 184 and 228 kHz, the disturbances amplify from the up-
83
0 0
10 10
-1 -1
10 10
-2 -2
10 10
-3 -3
10 10
-4 -4
10 10
0 50 100 150 200 250 300 0 100 200 300
Frequency [kHz] Frequency [kHz]
0
10
-1
10
-2
10
-3
10
-4
10
0 50 100 150 200 250 300
Frequency [kHz]
Figure 5.1: SPOD mode energy for the +0? extension at (top-left) condition Re33,
(top-right) condition Re45 and (bottom) condition Re52.
84
Energy [Arb.]
Energy [Arb.]
Energy [Arb.]
2.5
0
2.5
0
2.5
0
2.5
0
2.5
0
340 360 380 400 420 440 460
X [mm]
Figure 5.2: SPOD mode shapes for the +0? configuration at condition Re33 for
frequencies of (top to bottom) 34 kHz, 160 kHz, 184 kHz, 228 kHz, and 280 kHz.
stream end of the FOV until approximately X=440 mm where both modes begin to
dissipate; this behavior also matches observations made along the pseudo-streamline.
While these second-mode disturbances are dominant, the aforementioned spectral
broadening extends to low-frequency content. The top contour of figure 5.2 shows
coherent structures at 34 kHz amplifying as far upstream as X=380 mm. These
structures appear similar to the braided structures observed along the +5? flare in
the 10 ? 50 kHz bandpass-filtered sequence of figure 4.17. The presence of these
structures in the undisturbed case is somewhat surprising; however, the significance
of SPOD modes must be determined within the context of other results and the
lack of an amplification peak for this frequency reduces its relevance. Furthermore,
the freestream noise environment of HyperTERP is likely to excite low-frequency
disturbances which may not be present in the flight environment. The large spike in
SPOD energy from 250? 300 kHz is exemplified by the 280 kHz mode, which corre-
sponds to uniform flickering; this is thus a noise mode from the schlieren apparatus
and can be safely ignored.
The enhanced growth of low-frequency content at condition Re45 is demon-
85
Y [mm]
2.5
0
2.5
0
2.5
0
2.5
0
2.5
0
2.5
0
2.5
0
360 370 380 390 400 410 420 430 440
X [mm]
Figure 5.3: SPOD mode shapes for the +0? configuration at condition Re45 for
frequencies of (top to bottom) 35 kHz, 70 kHz, 111 kHz, 204 kHz, 257 kHz, 274 kHz,
and 350 kHz.
strated by the first three contours of figure 5.3, with the 35 kHz mode now visible
at X=360 mm. The 70 and 111 kHz modes develop alongside the spectral broad-
ening observed in figure 4.4 and may thus correspond to transitional structures.
The second-mode SPOD contours show the expected behavior in that the dominant
(257 kHz) and lower-sideband (204 kHz) frequencies amplify over much of the field of
view while the upper-sideband (274 kHz) decays. A high-frequency (350 kHz) mode
is illustrated in the final contour of figure 5.3. While much of the content at this
frequency is likely caused by intermittent turbulence, particularly towards the edge
of the field of view, a rope-like structure can be seen from X=390 ? 400 mm. It
is possible that this corresponds to harmonic content at 400 kHz which would be
aliased by the 747 kHz framerate of this test.
Low-frequency disturbances generated at condition Re52 (figure 5.4) largely
resemble those observed at condition Re45. Second-mode structures are now ob-
86
Y [mm]
2.5
0
2.5
0
2.5
0
2.5
0
2.5
0
2.5
0
350 360 370 380 390 400 410 420 430 440 450
X [mm]
Figure 5.4: SPOD mode shapes for the +0? configuration at condition Re52 for fre-
quencies of (top to bottom) 32 kHz, 75 kHz, 144 kHz, 235 kHz, 257 kHz, and 294 kHz.
served at higher frequencies and overpower the flickering noise. The 294 kHz mode
demonstrates strong disturbances even at the upstream edge of the field of view. It
is worth noting that all modal structures begin to dissipate or lose coherence down-
stream of X=430 mm. This is attributed to the transitional/turbulent nature of the
boundary layer at such high Rem, which degrades the coherent flow structures.
5.2 -5 ?Extension
The SPOD energy curves (see figure 5.5) again demonstrate the dominance of
the second mode for this configuration. Flickering noise now manifests itself as spikes
around 130 kHz which may be ignored. An interesting phenomenon is observed at
condition Re45, where the second-ranked SPOD mode shows elevated energy at the
upper end of the frequency range. This is likely an effect of aliasing, as second-mode
structures will exist beyond the Nyquist limit of 275 kHz for this test.
Select mode shapes computed for condition Re33 are given in figure 5.6. As
should be expected based on the preceding discussion, at 241 kHz a distinctly rope-
87
Y [mm]
0 0
10 10
-1 -1
10 10
-2 -2
10 10
-3 -3
10 10
-4 -4
10 10
0 50 100 150 200 250 0 50 100 150 200 250
Frequency [kHz] Frequency [kHz]
0
10
-1
10
-2
10
-3
10
-4
10
0 100 200 300
Frequency [kHz]
Figure 5.5: SPOD mode energy for the -5? extension at (top-left) condition Re33,
(top-right) condition Re45 and (bottom) condition Re52.
88
Energy [Arb.]
Energy [Arb.]
Energy [Arb.]
like, second-mode structure is observed concentrated about the pseudo-streamline
which decays downstream of the corner. The mode shape for 176 kHz possesses this
same second-mode structure but instead amplifies along the cylindrical afterbody,
demonstrating that the band of growth observed in figure 4.7 indeed corresponds
to second-mode structures which develop at lower frequencies downstream of the
corner. This is also consistent with the low-frequency shift seen at sensor D5 in fig-
ure 3.7. The 60 kHz and 90 kHz contours also correspond to bands of growth identi-
fied in the N-factor plot and seemingly are associated with coherent flow structures
in the downstream part of the field of view; the origin of these features is uncertain.
There also appear to be very low-frequency structures even upstream of the corner
at 13 kHz.
At condition Re45, SPOD again resolves the decaying second mode, now at
258 kHz, shown in the lowermost contour of figure 5.7. This mode shape also cap-
tures the radiation of high-frequency content along the expansion wave. In contrast,
the lower side-band of the second mode at 236 kHz maintains its amplitude through
the expansion, decaying only weakly towards the end of the field of view. Frequency-
shifted second-mode waves are seen amplifying in the 193 kHz contour, and similar
behavior is noted for the 155 kHz mode shape, representing the dramatic N-factor
peak in figure 4.8. In this latter case, however, the structures remain relatively in-
distinct, consistent with growth beginning from a weak initial state. Amplification
of braided, 34 kHz structures is again observed with this configuration in the top
contour of figure 5.7.
Figure 5.8 shows similar SPOD mode shapes for condition Re52. Along the
89
2.5
0
-2.5
2.5
0
-2.5
390 400 410 420 430 440 450 460
2.5 X [mm]
0
-2.5
390 400 410 420 430 440 450 460
2.5 X [mm]
0
-2.5
390 400 410 420 430 440 450 460
2.5 X [mm]
0
-2.5
390 400 410 420 430 440 450 460
2.5 X [mm]
0
-2.5
390 400 410 420 430 440 450 460
X [mm]
Figure 5.6: SPOD mode shapes at condition Re33 for frequencies of (top to bottom)
13 kHz, 60 kHz, 95 kHz, 163 kHz, 176 kHz, and 241 kHz.
90
Y [mm] Y [mm] Y [mm] Y [mm] Y [mm] Y [mm]
2.5
0
-2.5
390 400 410 420 430 440 450 460
X [mm]
2.5
0
-2.5
390 400 410 420 430 440 450 460
X [mm]
2.5
0
-2.5
390 400 410 420 430 440 450 460
X [mm]
2.5
0
-2.5
390 400 410 420 430 440 450 460
X [mm]
2.5
0
-2.5
390 400 410 420 430 440 450 460
X [mm]
Figure 5.7: SPOD mode shapes at condition Re45 for frequencies of (top to bottom)
34 kHz, 155 kHz, 193 kHz, 236 kHz, and 258 kHz.
afterbody, decaying and amplifying rope-like structures are observed at 274 kHz and
204 kHz, respectively. Additional structures with a shallower inclination angle are
observed extending out from near the wall at 117 kHz, corresponding to the N-
factor peak along the afterbody, though it is unclear if these are truly new modal
disturbances or merely turbulent artifacts. The high-frequency content at 344 kHz is
almost certainly due to intermittent turbulence. As with condition Re45, significant
content was seen to develop at very low frequencies along the pseudo-streamline (see
figure 4.6). The structure of this content is exemplified by the 47 kHz mode shape
in figure 5.8, though turbulence appears to be contaminating and blurring these
features.
91
Y [mm] Y [mm] Y [mm] Y [mm] Y [mm]
2.5
0
-2.5
400 410 420 430 440 450 460
2.5 X [mm]
0
-2.5
400 410 420 430 440 450 460
2.5 X [mm]
0
-2.5
400 410 420 430 440 450 460
2.5 X [mm]
0
-2.5
400 410 420 430 440 450 460
2.5 X [mm]
0
-2.5
400 410 420 430 440 450 460
X [mm]
Figure 5.8: SPOD mode shapes at condition Re52 for frequencies of (top to bottom)
47 kHz, 117 kHz, 204 kHz, 274 kHz, and 344 kHz.
5.3 +5 ?Extension
The SPOD energy spectra for the +5? compression (5.9) still demonstrate dis-
tinct peaks surrounding the second-mode fundamental, but with additional regions
of elevation. At condition Re33, for instance, the highest-ranked mode contains
substantially more energy than the second-highest around 70? 80 kHz; this is also
seen near 150 kHz at condition Re45. The 300 kHz peaks in these spectra again
correspond to flickering noise in the schlieren system.
SPOD mode shapes computed for the +5? case at condition Re33 reveal com-
plex growth of disturbances along the flare, as seen in the contours of figure 5.10.
Second-mode content at 199 kHz can be seen in figure 5.10(g) passing through the
92
Y [mm] Y [mm] Y [mm] Y [mm] Y [mm]
0 0
10 10
-1 -1
10 10
-2 -2
10 10
-3 -3
10 10
-4 -4
10 10
0 100 200 300 400 0 100 200 300 400
Frequency [kHz] Frequency [kHz]
0
10
-1
10
-2
10
-3
10
-4
10
0 50 100 150 200 250 300
Frequency [kHz]
Figure 5.9: SPOD mode energy for the +5? extension at (top-left) condition Re33,
(top-right) condition Re45 and (bottom) condition Re52.
93
Energy [Arb.]
Energy [Arb.]
Energy [Arb.]
corner and amplifying until approximately X=450 mm, matching the saturation
point seen in the pseudo-streamline spectrum of figure 4.2. Rope-like structures
can also be seen developing just downstream of the corner at 180 kHz and closer
to X=440 mm at 238 kHz. The growth of low-frequency second-mode disturbances
is somewhat surprising given the reduced boundary-layer thickness on the flare.
The low-frequency growth identified downstream of the compression corner in chap-
ter 4 is exemplified by the coherent structures appearing at 32 and 77 kHz around
X=420 mm, both of which seemingly correspond to significant N-factor peaks along
the pseudo-streamline. The 32 kHz mode has the same appearance as the braided
structures observed in the bandpass-filtered sequence of figure 4.17. An additional
structure appears in this region at 125 kHz (figure 5.10d), though it possesses an ir-
regular structure. As will be discussed in chapter 6, this growth region correlates well
with the appearance of nonlinear interactions, particularly for the 77 and 125 kHz
structures. Additional SPOD mode shapes with nonlinear relevance are show for
frequencies of 141 and 161 kHz which amplify downstream of the corner but then
lose coherence towards the end of the flare. High-frequency content at 343 kHz is
illustrated in the final contour of figure 5.10 with some coherent structure amplifying
around X=440 mm, but may again be attributed to turbulence.
Low-frequency content is again seen to develop downstream of the corner at
condition Re45 in figure 5.11. Braided structures are observed in the 35 and 61 kHz
mode shapes. At 100, 151, and 177 kHz, flow features are seen developing halfway
along the flare which are structurally similar to those observed in the transitional
region of the undisturbed case (the second and third contours of figure 5.3). Par-
94
5
2.5
a) 0
5
2.5
b) 0
5
2.5
c) 0
5
2.5
d) 0
5
2.5
e) 0
5
2.5
f) 0
5
2.5
g) 0
5
2.5
h) 0
5
2.5
0
400 410 420 430 440 450 460 470
i) X [mm]
Figure 5.10: SPOD mode shapes for the +5? configuration at condition Re33 for
frequencies of (a) 32 kHz, (b) 77 kHz, (c) 125 kHz, (d) 141 kHz, (e) 161 kHz, (f)
180 kHz, (g) 199 kHz, (h) 238 kHz, and (i) 343 kHz.
95
Y [mm]
ticular note should be taken of the 151 kHz structure, as this corresponds to a
significant N-factor peak for the test, but this appears to be largely due to a com-
plete lack of structure upstream. The SPOD mode shapes within the second-mode
band at 225 and 244 kHz show evidence of dissipation due to transition downstream
of X = 450 mm, and energy can be seen radiating away from the boundary layer
along the corner shock. At this condition, coherent structures are observed at higher
frequencies of 292 and 382 kHz from around X=420 mm, which appear related to
turbulent content.
The rapid transition induced by the compression corner at condition Re52 is
evident in the structure of the SPOD modes shown in figure 5.12. Low frequency con-
tent appears almost out of nowhere at the corner in the 27, 61, and 136 kHz modes.
These structures have the ?fuzzy? appearance indicative of turbulent contamination.
The lack of coherent structures along the frustum at 136 kHz is somewhat surpris-
ing given the spectral peak exhibited by the U3 PCB sensor in figure 3.11. The
higher-frequency mode shapes associated with the second mode show an immediate
breakdown in structure at the flare due to the turbulent nature of the boundary
layer. All this is to be expected based on the immediate spectral broadening and
dissapearance of the second-mode peak observed along the pseudo-streamline. The
270 kHz mode, which corresponds closely to the U3 peak second-mode frequency at
this condition, does persist for a short distance downstream of the corner thanks
to several wavepackets which manage to traverse the SWBLI without immediately
breaking down.
96
5
2.5
a) 0
5
2.5
b) 0
5
2.5
c) 0
5
2.5
d) 0
5
2.5
e) 0
5
2.5
f) 0
5
2.5
g) 0
5
2.5
h) 0
5
2.5
0
410 420 430 440 450 460 470
i) X [mm]
Figure 5.11: SPOD mode shapes for the +5? configuration at condition Re45 for
frequencies of (a) 35 kHz, (b) 61 kHz, (c) 100 kHz, (d) 151 kHz, (e) 177 kHz, (f)
225 kHz, (g) 244 kHz, (h) 292 kHz, and (i) 382 kHz.
97
Y [mm]
2.5
a) 0
2.5
b) 0
2.5
c) 0
2.5
d) 0
2.5
e) 0
2.5
0
350 360 370 380 390 400 410 420 430 440 450
f) X [mm]
Figure 5.12: SPOD mode shapes for the +5? configuration at condition Re52 for
frequencies of (a) 27 kHz, (b) 61 kHz, (c) 136 kHz, (d) 238 kHz, (e) 270 kHz, and (f)
294 kHz.
5.4 +10 ?Extension
Aside from the expected second-mode peak, the SPOD energy spectrum for
condition Re33 (figure 5.13) demonstrates a distinct peak at 71 kHz, matching the
expected frequency of the previously identified shear-generated instability. Note
that the turbulence on the flare has swamped the energy spectrum for condition
Re52 (and to a lesser extent condition Re45), causing the second-mode peak to
vanish. This is a good example of why the significance of SPOD modes should
not be determined entirely from the energy content, which is averaged over the
entire field of view, as the presence of second-mode waves was shown in the time-
resolved images and the pseudo-streamline spectra to characterize the boundary
layer upstream of the corner.
Eigenvalue contours computed for 42, 71, and 215 kHz at condition Re33 are
presented in figure 5.14. The second-mode waves at 215 kHz exhibit the expected
98
Y [mm]
0 0
10 10
-1 -1
10 10
-2 -2
10 10
-3 -3
10 10
-4 -4
10 10
0 100 200 300 400 0 100 200 300 400
Frequency [kHz] Frequency [kHz]
0
10
-1
10
-2
10
-3
10
-4
10
0 100 200 300 400
Frequency [kHz]
Figure 5.13: SPOD mode energy for the +10? extension at (top-left) condition Re33,
(top-right) condition Re45 and (bottom) condition Re52.
99
Energy [Arb.]
Energy [Arb.]
Energy [Arb.]
5
2.5
0
5
2.5
0
5
2.5
0
380 390 400 410 420 430 440
X [mm]
Figure 5.14: SPOD mode shapes for the +10? configuration at condition Re33 at
(top to bottom) 42 kHz, 71 kHz, and 218 kHz.
rope-like structure, weakening only slightly as they traverse the corner separation
with some evidence of second-mode energy passing into the separation bubble. The
71 kHz contour reveals the same braided structure forming in the separated shear
layer as in the bandpass-filtered series of figure 4.16. These flow features appear
to weaken slightly downstream of reattachment, as was seen along the pseudo-
streamline spectra. The 42 kHz mode shape consists of similar, braided features,
but they amplify significantly immediately downstream of separation and main-
tain amplitude through reattachment. This behavior is consistent with that of
the 40 kHz N-factor peak which develops along the pseudo-streamline in figure 4.7.
Although it is unclear if the separation bubble is responsible for producing these
42 kHz disturbances (which seems unlikely given the presence of similar features in
the +0? configuration), it clearly serves to amplify them.
Figure 5.15 displays mode shapes for various frequencies at condition Re45,
with the second-mode sidebands presented at 257 kHz and 225 kHz. Although these
frequencies both lie within the second-mode range, the waves at 257 kHz begin to
100
Y [mm]
5
2.5
0
5
2.5
0
5
2.5
0
5
2.5
0
5
2.5
0
5
2.5
0
380 390 400 410 420 430 440
X [mm]
Figure 5.15: SPOD mode shapes for the +10? configuration at condition Re45 at
(a) 61 kHz, (b) 83 kHz, (c) 119 kHz, (d) 164 kHz, (e) 225 kHz, and (f) 257 kHz.
decay downstream of separation while the 225 kHz content maintains its amplitude
over the same region. Both frequencies then take on a modified structure down-
stream of reattachment. A shear-generated disturbance is visualized in the 83 kHz
contour and is seen to lose its braided structure upon reattachment. This loss of
structure is likely due more to transition than the natural evolution of the dis-
turbance. Additional modes are presented at 61, 119, and 164 kHz which amplify
alongside the shear-generated disturbance within the downstream portion of the
separation bubble. The eigenvalue contours at 119 kHz and 164 kHz appear signifi-
cantly augmented by turbulence downstream of reattachment and are included here
primarily to facilitate the discussion of nonlinear interactions in the next chapter.
101
Y [mm]
5
2.5
0
5
2.5
0
5
2.5
0
380 390 400 410 420 430 440
X [mm]
Figure 5.16: SPOD mode shapes for the +10? configuration at condition Re52 at
(top to bottom) 90 kHz, 289 kHz and 318 kHz.
The middle mode shape in figure 5.16 (289 kHz) demonstrates the second-mode
content is still well-resolved upstream of the corner at condition Re52, despite the
lack of second-mode peak in the energy spectrum. The waves disperse upon sepa-
ration before amplifying significantly at reattachment, with energy extending well
away from the wall. The top contour (90 kHz) reveals no evidence of shear-generated
disturbances, showing instead the dramatic growth of low-frequency content just up-
stream of reattachment (at X = 420 mm). Comparable growth is observed at this
same location for frequencies above the second mode in the 318 kHz mode shape.
Each one of these eigenvalue contours is consistent with the immediate transition of
the boundary layer on the flare.
5.5 +15 ?Extension
As evidenced by the streamline spectra of figures 4.2 and 4.4, the large sepa-
ration bubble created by the +15? configuration results in significant content under
70 kHz but little-to-no high frequency content for conditions Re33 and Re45. This is
102
Y [mm]
0 0
10 10
-2 -2
10 10
0 50 100 150 200 0 50 100 150 200 250
Frequency [kHz] Frequency [kHz]
0
10
-2
10
0 50 100 150 200 250 300
Frequency [kHz]
Figure 5.17: SPOD mode energy for the +15? extension at (top-left) condition Re33,
(top-right) condition Re45 and (bottom) condition Re52.
reflected in the SPOD energy spectra, where the dominant peak at these conditions
now corresponds to the shear-generated disturbances around 40? 50 kHz and there
is no hint of a second-mode peak at any condition. The highest-rank modes are
also elevated slightly around 80? 90 kHz. The condition Re33 peak at 130 kHz is a
flickering mode.
The SPOD mode shapes presented in figure 5.18 showcase the structure of
the shear-generated disturbances at condition Re33. The 41 kHz mode closely re-
sembles the shear-generated disturbance identified for the +10? configuration and
appears strongest just upstream of reattachment from X=420 ? 430 mm. The 48
and 58 kHz modes both develop further upstream within the separation region and
103
Energy [Arb.]
Energy [Arb.]
Energy [Arb.]
7.5
5
2.5
0
7.5
5
2.5
0
7.5
5
2.5
0
7.5
5
2.5
0
7.5
5
2.5
0
380 390 400 410 420 430 440
X [mm]
Figure 5.18: SPOD mode shapes for the +15? configuration at condition Re33 for
frequencies of (top to bottom) 14 kHz, 41 kHz, 48 kHz, 58 kHz, 83 kHz.
radiate energy away from the shear layer. Significant energy is also seen in the 83 kHz
mode as the separation region compresses, appearing as longitudinal streaks ema-
nating from the boundary layer at X=420 mm. It is notable that this corresponds
approximately to the harmonic frequency of the shear-generated disturbances. Very
low-frequency (14 kHz), braided structures are observed amplifying rapidly in the
immediate vicinity of reattachment in the uppermost contour of figure 5.18. These
features likely correspond to the lower band of the N-factor peak in figure 4.7.
The +15? configuration is still largely characterized by low-frequency modal
structures at condition Re45, which are illustrated by figure 5.19. The first SPOD
104
Y [mm]
contour, computed at 26 kHz, is concentrated around the downstream end of the
separation bubble with a different structure than the 14 kHz mode from condition
Re33. Notably, this is approximately the subharmonic of the shear-mode, which
will be discussed further in chapter 6. The modal structure at 47 kHz matches the
expected braided structure of the shear-generated disturbance and again amplifies
starting directly above the cone/flare junction, while the 60 kHz mode shows content
developing further upstream with significant radiation away from the shear layer.
Modal content at 99 and 146 kHz is comprised almost entirely of radiation along the
compression waves ahead of reattachment and it is notable that these frequencies
correspond closely to the first and second harmonics of the shear-generated distur-
bances. The potential nonlinear development of these structures will be studied
further in the next chapter. High-frequency (223 kHz) features are observed down-
stream of reattachment within the severely thinned boundary layer in the lowermost
contour of figure 5.19.
Figure 5.20 shows only two mode shapes computed at condition Re52. As
with the +10? configuration, there is little evidence of shear-generated disturbances
as this condition, demonstrated by the 62 kHz contour. This contour does, however,
show substantial amplification of turbulent structures along the flare. Second-mode
structures are captured upstream of separation by the 286 kHz mode, but these
quickly break down within the separation region and appear as turbulent structures
along the flare.
105
7.5
5
2.5
0
7.5
5
2.5
0
7.5
5
2.5
0
7.5
5
2.5
0
7.5
5
2.5
0
7.5
5
2.5
0
380 390 400 410 420 430 440
X [mm]
Figure 5.19: SPOD mode shapes for the +15? configuration at condition Re45 for
frequencies of (top to bottom) 26 kHz, 47 kHz, 60 kHz, 99 kHz, 146 kHz, and 223 kHz.
7.5
5
2.5
0
7.5
5
2.5
0
380 390 400 410 420 430
X [mm]
Figure 5.20: SPOD mode shapes for the +15? configuration at condition Re52 for
frequencies of (top) 62 kHz and (bottom) 286 kHz.
106
Y [mm] Y [mm]
5.6 Propagation Speeds
Although the disturbance propagation speed is not required for spectral analy-
sis in the present work, it is still an important disturbance property and is of interest
for comparing to computations. In this section we present two methods for deter-
mining the propagation speeds, Up, of the bounday-layer disturbances: one based on
the SPOD analysis and another, more conventional technique for comparison. The
conventional formulation, referred to as the correlation method, is related to the
technique employed in refs. [13], [14], and [15], but relies upon the pixel-intensity
time series rather than spatially distributed data, and thus allows for improved spa-
tial resolution. First, for a given test, the time series are bandpass-filtered around
the second-mode frequency. For each streamwise station, we compute the cross-
correlation coefficient between the station of interest and each downstream pixel
assuming time lags of 1, 2 and 3 frames. The streamwise shift which produces the
maximum cross-correlation is identified and a second-order polynomial is fit to the
correlation coefficients of the ?optimal? streamwise shift and the two neighboring
pixels in order to determine the location of the peak shift with sub-pixel accuracy.
This process is repeated backwards in time (by computing the cross-correlation with
upstream pixels) and the results are averaged at each station, yielding propagation
speeds which are central-differenced in space. On average, the structures traverse
approximately 18 pixels between frames, which could result in substantial structural
modulation and poor correlation, particularly in the vicinity of the corner.
The second method is based upon the SPOD modes and provides propagation
107
speeds that vary with both streamwise location and frequency. For each frequency in
the highest-ranked SPOD mode, the real component of the mode shape is extracted
along the pseudo-streamline. Spatial power spectra are then computed along the
streamline using a sliding window of 128 pixels, allowing the dominant wavelength,
?, to be identified at each station. For frequency fi, the propagation speed can then
be computed according to
Up,i = fi ? ?(fi). (5.1)
Sample results obtained using the SPOD method are given for each configura-
tion at condition Re33 in figure 5.21. This method demonstrates poor performance
outside of the second-mode range, giving unrealistically high or low propagation
speeds. This is attributed largely to the lack of coherent flow structures at these
frequencies, but is exacerbated by the fact that the pseudo-streamlines were tailored
to the propagation trajectories of high-frequency, second-mode disturbances which
may not be appropriate for other disturbances. It is likely for the latter reason
that the shear-generated disturbances do not appear for the separated configura-
tions. This could potentially be addressed by averaging the spatial spectrum in
the pseudo-streamline-normal direction. Note that unphysical results are obtained
even within the second-mode range for the +15? configuration because of the lack
of wavepackets.
A comparison of the two methodologies is provided for each condition in fig-
ure 5.22 with the expansion configuration in the left column and the +0? extension
in the right column. The second-mode propagation speed for the SPOD method
108
300
250 1300 1300
250
1200 1200
200
200
1100 1100
150
150
1000 1000
100
100
900 900
50
800 50 800
0 700 0 700
410 420 430 440 360 380 400 420 440
X [mm] X [mm]
400 400
1300 1300
1200 1200
300 300
1100 1100
200 1000 200 1000
900 900
100 100
800 800
0 700 0 700
420 430 440 450 460 390 400 410 420 430
X [mm] X [mm]
200 1300
1200
150
1100
100 1000
900
50
800
0 700
390 400 410 420
X [mm]
Figure 5.21: Propagation speeds computed at condition Re33 using the SPOD-
based methodology; (top-left) -5? configuration, (top-right) +0? configuration,
(middle-left) +5? configuration, (middle-right) +10? configuration, and (bottom)
+15? configuration.
109
Frequency [kHz] Frequency [kHz]
Frequency [kHz]
Frequency [kHz] Frequency [kHz]
is calculated here by averaging over the second-mode frequency range. The aver-
age speeds computed upstream and downstream of the corner using the correlation
method are included as dashed lines and are also provided in table 5.2 together
with theoretically determined edge velocities. The latter were determined from the
Taylor-Maccoll solution, which, for the downstream region on the expansion config-
uration, was passed through a Prandtl-Meyer expansion fan (the resulting value will
be appropriate immediately downstream of the corner and will subsequently relax
towards the freestream velocity). Note that the downstream average for condition
Re52 (for the -5? extension) was terminated at X = 440 mm due to the anomalous
drop in propagation speed caused by a loss of coherence.
As expected, the propagation speeds remain approximately constant over the
cone surface for the undisturbed boundary layer, with good agreement between the
two methodologies. The correlation method demonstrates several anomalous drops
and spikes, but the SPOD method behaves poorly far upstream at condition Re33,
producing unphysically high values. This is likely caused by poor resolution of the
second-mode disturbances within this region of the field of view, which can be seen
in figure 5.2. Both methods show the propagation speed of the second-mode dis-
turbances increases modestly through the expansion, though the normalized value,
Up/Ue, remains approximately constant due to the acceleration of the mean flow.
The two methodologies demonstrate good agreement for conditions Re33 and Re45,
but deviate significantly at Re52. This is likely caused by turbulent contamination,
which may obscure the correlation between pixel stations but seems to have less
impact on the well-resolved flow structures computed by SPOD. Thus, the SPOD
110
1300 1300
1200 1200
1100 1100
1000 Correlation method 1000
SPOD method
900 900
390 400 410 420 430 440 450 460 340 360 380 400 420 440 460
X [mm] X [mm]
1300 1300
1200 1200
1100 1100
1000 1000
900 900
390 400 410 420 430 440 450 460 360 380 400 420 440
X [mm] X [mm]
1300 1300
1200 1200
1100 1100
1000 1000
900 900
400 410 420 430 440 450 460 350 400 450
X [mm] X [mm]
Figure 5.22: Propagation speeds computed along pseudo-streamline for the (left)
-5? and (right) +0? configurations: (top) condition Re33, (middle) condition Re45,
and (bottom) condition Re52.
methodology appears more robust in determining propagation speeds for weakly
disturbed boundary layers, but at the expense of reduced spatial resolution. It is
worth noting that the upstream propagation speeds differ between the two configu-
rations by approximately 10%, though they lie within the range of speeds observed
previously within HyperTERP by Hameed et al. [43] using FLDI.
Results for the +5? (left) and +10? (right) compression-corners are shown in
figure 5.23. The +5? configuration causes a clear deceleration of the disturbances
downstream of the corner, with the mean propagation speed dropping by 3% at
condition Re52. Unfortunately, there is insufficient upstream information for condi-
111
-5? Extension +0? Extension
Re33 Re45 Re52 Re33 Re45 Re52
Upstream Up [m/s] 1055 1052 1051 1121 1139 1155
Upstream Ue [m/s] 1255 1255 1255 1255 1255 1255
Upstream Up/Ue 0.84 0.84 0.84 0.90 0.91 0.92
Downstream Up [m/s] 1078 1069 1068 1143 1157 1165
Downtream Ue [m/s] 1274 1274 1274 1255 1255 1255
Downstream Up/Ue 0.85 0.84 0.84 0.91 0.93 0.93
Table 5.2: Upstream and downstream propagation speeds computed with the cor-
relation method.
tons Re33 and Re45 to quantify the deceleration for these tests with any confidence.
Oddly, the SPOD method computes propagation speeds at condition Re33 which
trace along the top of the localized jumps which occur with correlation method.
Overall, the separated case creates results that are more difficult to interpret, with
the propagation speeds varying substantially using both methods. The SPOD
method performs particularly poorly in the vicinity of separation and reattachment,
displaying large spikes in the propagation speeds. That said, the results consistently
demonstrate substantial deceleration just upstream of reattachment.
112
1300 1300
1200 1200
1100 1100
1000 Correlation method 1000
SPOD method
900 900
410 420 430 440 450 460 470 380 390 400 410 420 430 440
X [mm] X [mm]
1300 1300
1200 1200
1100 1100
1000 1000
900 900
410 420 430 440 450 460 470 380 390 400 410 420 430 440
X [mm] X [mm]
1300 1300
1200 1200
1100 1100
1000 1000
900 900
350 400 450 380 390 400 410 420 430 440
X [mm] X [mm]
Figure 5.23: Propagation speeds computed along pseudo-streamline for the (left)
+5? and (right) +10? configurations: (top) condition Re33, (middle) condition Re45,
and (bottom) condition Re52.
113
Chapter 6: Bispectral Analysis
The high framerate employed throughout this work also facilitates the use of
higher-order spectral techniques such as bispectral analysis. Computation of the
normalized bispectrum, or bicoherence, allows one to identify the primary nonlinear
growth mechanisms at work. More accurately, the bicoherence provides a measure
of the degree of Quadratic Phase Coupling (QPC) present between three frequencies
f1, f2 and f1 + f2. The bispectrum for a frequency triplet is defined by
B(f1, f2) = E[X(f1)X(f2)X(f
?
1 + f2) ], (6.1)
where E is the expectation operator, and X(f) refers to the Fourier transform
component at frequency f for the time series of interest. The squared bicoherence
is computed as
|B(f , f )|2| 1 2b(f1, f 22)| = , (6.2)
S(f1)S(f2)S(f1 + f2)
which utilizes the normalization method of Brillinger [44] where S(f) is the PSD at
frequency f . There is some disagreement in the literature as to the proper normal-
ization method for bispectral analysis, with some researchers (e.g., refs. [45] and
[3]) opting for the definition given by Kim and Powers [46]. The Kim and Powers
114
method of normalization, however, has received scrutiny from ref. [47] for ?artifi-
cially? bounding the result between 0 and 1 and in some cases destroys evidence of
nonlinearity within experimental datasets. In the context of the present work, the
Kim and Powers normalization has been found to ruin the symmetries inherent to
the bispectrum and reduce the prominence of expected resonant interactions. There
also exists precedent for the use of the Brillinger normalization for boundary-layer
transition studies, its having been applied previously by Chokani [4] and Chokani
et al. [48]. Welch?s method was employed for these computations using windows of
width 64.
Before exploring results from the present data set, additional background on
bispectral analysis and its application to boundary-layer transition is presented. The
bicoherence spectra is defined by two triangular regions, illustrated in figure 6.1,
which will be referred to as the inner-triangle (IT) and the outer-triangle (OT).
The inner-triangle is defined by the vertices [0,0], [Fs ,Fs ], [Fs ,0] while the outer-
4 4 2
triangle is defined by [Fs ,Fs ], [Fs ,0], [Fs ,Fs ]. Any information outside of this region
4 4 2 3 3
is a redundant reflection. The outer-triangle is distinct in that it is non-redundant
but relies upon aliased information, i.e., for any frequency pair [f1,f2] within OT,
f1 + f2 will lie above the Nyquist sampling frequency of the signal. Information
within this region thus requires special scrutiny. A particularly useful observation is
that any signal which lies upon the right edge of OT at some frequency f1 may also
correspond to a reflection of the resonant interact at [f1,f1]. To demonstrate this, let
us consider the point [5Fs ,Fs ] which is located on this edge. The resonant frequency
12 6
in this case would be 5Fs/6, which aliases down to Fs/6. It is also worth emphasizing
115
OT
IT
Figure 6.1: Illustration of frequency domain for bicoherence spectra.
that bispectral analysis is incapable of distinguishing between sum and difference
interactions and thus some physical intuition is required for proper interpretation
of the results. As will be shown, SPOD modes are particularly useful in developing
this intuition.
Bispectral analysis was first applied to the study of high-speed boundary-
layer transition by Kimmel and Kendall [3] to study nonlinear interactions involving
second-mode disturbances. The data was obtained using hot-wire anemometry for
the Mach 7.95 tests described by Stetson et al. [10] on a 7? half-angle cone. The
bicoherence spectrum observed at a single streamwise station for these experiments
is shown in the left plot of figure 6.2. They observed strong phase-coupling be-
tween the fundamental second-mode frequency (approximately 88 kHz) and the first
harmonic, indicated by region (a). This led to their conclusion that the harmonic
was developed through self-resonance of the fundamental. A weak peak is also ob-
served in region (b), attributed to an interaction between the fundamental and the
116
Figure 6.2: Bicoherence spectrum for a single streamwise station taken from (left)
Kimmel and Kendall [3] and (right) Chokani [4].
first harmonic to produce a second harmonic. Regions (c) and (d) correspond to
coupling between low-frequency content and the fundamental and first harmonic,
respectively. Kimmel and Kendall conclude that low-frequency structures modulate
the second-mode content and transfer energy to the sidebands.
Similar interactions were observed by Chokani [4] for hot-wire measurements
obtained on a flared cone model in the NASA LaRC Mach 6 quiet facility. First and
second harmonics were observed at upstream stations as a result of fundamental res-
onance (now at 300 kHz) and sum interactions between the fundamental and the first
harmonic, illustrated by figure 6.2. Coupling was again seen between low frequencies
and the fundamental sidebands, and was characterized as low-frequency modulation
of the high-frequency fundamental and harmonics. These low-frequency interactions
became dominant immediately preceding breakdown of the laminar boundary layer.
In addition to these fundamental resonance interactions, Shiplyuk et al. [49] first
observed potential subharmonic resonance within the boundary layer over a 7? cone
117
at Mach 5.95, again using hot-wire anemometry.
6.1 +0 ?Extension
We begin by examining the bicoherence spectra observed in the +0 ? configuration
to discern the undisturbed nonlinear behavior of the wavepackets. Representative
bicoherence spectra for condition Re33 are shown in figure 6.3 for three distinct
streamwise stations, annotated with red lines in the top meanflow image. Each of
these regions corresponds to 21 pixels located along the pseudo-streamline. Well up-
stream of the corner, the dominant nonlinear interaction is fundamental resonance
at 210 kHz, as expected based on prior literature. This resonance shifts to lower
frequencies downstream to match the fundamental second-mode peak frequency.
Early on the extension, illustrated by the middle contour plot of figure 6.3,
the bicoherence spectrum reaches elevated levels around [f1, f2] = [165 kHz, 30 kHz].
Such peaks which occur in the vicinity of the second mode and involve low fre-
quencies are typically attributed to sideband interactions, though it is potentially
significant that 30 kHz roughly corresponds to the braided structures observed in
the SPOD analysis. The final contour of figure 6.3 demonstrates that these resonant
and sideband interactions persist far along the extension, where the instantaneous
images have show that the boundary layer tends to be only intermittently transi-
tional.
We may better visualize the spatial development of these interactions as in fig-
ure 6.4, which shows the rms bicoherence that has been computed within two select
118
regions of the frequency plane (annotated in figure 6.3 with dashed boxes) for each
pixel within the field of view. The bottom image corresponds to fundamental reso-
nance, and we see the interaction is largely concentrated along the pseudo-streamline
and amplifies along the frustum. The energy signature of this interaction begins to
disperse slightly over the extension where the boundary-layer is intermittently tran-
sitional, but maintains its presence until the end of the field of view. Note that
the resonant peak shifts to frequencies too low to be captured by this contour far
downstream. The top image of figure 6.4 illustrates the spatial distribution of the
low-frequency coupling, which appears to be concentrated along the boundary-layer
edge and only appears along the pseudo-streamline at approximately X = 415 mm.
This concentration above the pseudo-streamline matches the peak intensity of the
structures seen in the top SPOD contour of figure 5.2.
At condition Re45, fundamental resonance (at approximately 260 kHz) remains
the primary interaction of note far upstream, as seen in the first contour of figure 6.5.
However, spectra for the downstream regions demonstrate a reduction in fundamen-
tal resonance and, at the furthest downstream location, there are no relevant peaks
in the bicoherence spectrum; the boundary layer at this location is no longer well-
described by modal interactions. The spatial development of fundamental resonance
is illustrated in figure 6.6 where the interaction appears along the pseudo-streamline
until approximately X = 400 mm and again around X = 430 mm. Oddly, the in-
teraction mimics the appearance of second-mode wavepackets; this may be because
the interaction is resolved near the upper vertex of OT in the bicoherence spectra.
Due to aliasing, this vertex corresponds to QPC between the triplet [1f 1s, fs,
1fs].3 3 3
119
350 360 371 381 391 402 412 423 433 444 454
X [mm]
0.1
200
0.08
150
0.06
100
0.04
50 0.02
0 0
0 100 200 300 100 200 300 100 200 300
f  [kHz] f  [kHz] f  [kHz]
1 1 1
Figure 6.3: (Top) Time-averaged, flow-on image for the +0? configuration at condi-
tion Re33 with red circles denoting distinct regions for computing the bicoherence.
(Bottom) Bicoherence spectra for each annotated region with the contours ordered
stream-wise from left to right.
4 0.05
2
0 0
4 0.05
2
0 0
340 360 380 400 420 440 460
X [mm]
Figure 6.4: Spatial bicoherence contours at condition Re33 for frequency triplets of
(top) [175,20,195], and (bottom) [190,190,380].
120
Y [mm]
f  [kHz]
2
367 378 388 399 409 419 430 440
X [mm]
0.1
250
0.08
200
0.06
150
0.04
100
50 0.02
0 0
0 100 200 300 100 200 300 100 200 300
f  [kHz] f  [kHz] f  [kHz]
1 1 1
Figure 6.5: (Top) Time-averaged, flow-on image for the +0? configuration at condi-
tion Re45 with red circles denoting distinct regions for computing the bicoherence.
(Bottom) Bicoherence spectra for each annotated region with the contours ordered
stream-wise from left to right.
0.1
2 0.05
0 0
360 370 380 390 400 410 420 430 440
X [mm]
Figure 6.6: Spatial bicoherence contours at condition Re45 for the frequency triplet
[250,250,500].
The lower-frequency resonance which appears around 80 ? 100 kHz has little spa-
tial coherence and is believed to be unphysical. Note that the turbulent nature
of the downstream boundary layer at condition Re52 results in bicoherence spec-
tra dominated by noise and thus results will not be presented; this holds for all
configurations.
121
Y [mm]
f  [kHz]
2
397 405 414 422 430 439 447 456
X [mm]
200 0.1
0.08
150
0.06
100
0.04
50
0.02
0 0
0 100 200 0 100 200 0 100 200
f  [kHz] f  [kHz] f  [kHz]
1 1 1
Figure 6.7: (Top) Time-averaged, flow-on image for the -5? configuration at condi-
tion Re33 with red circles denoting distinct regions for computing the bicoherence.
(Bottom) Bicoherence spectra for each annotated region with the contours ordered
stream-wise from left to right.
6.2 -5 ?Extension
As may be expected from its effect on the boundary-layer disturbances, the
expansion acts largely to dampen nonlinear interactions. Upstream of the corner at
condition Re33 (figure 6.7), the primary peaks correspond to fundamental resonance
and sideband interactions/low-frequency coupling. Downstream these interactions
diminish substantially, with no new peaks appearing in the bicoherence spectra.
The top contour of figure 6.8 was computed for interactions within the region
[f1,f2] = [190 ? 20 kHz,190 ? 20 kHz], corresponding to fundamental resonance.
122
f  [kHz]
2
4 0.2
2
0 0.1
-2
-4 0
390 400 410 420 430 440 450 460
4 X [mm] 0.1
2
0 0.05
-2
-4 0
390 400 410 420 430 440 450 460
X [mm]
Figure 6.8: Spatial bicoherence contours at condition Re33 for frequency triplets of
(a) [190,190,380] and (b) [190,15,205].
This interaction amplifies until the corner before gradually diminishing along with
the second-mode waves. While this interaction is largely concentrated along the
pseudo-streamline, there also appears to be a band of elevated bicoherence above
the streamline upstream of the corner. It is along this upper band that the lower
contour of figure 6.8, [f1,f2] = [190? 10 kHz,15? 10 kHz], demonstates significant
intensity. This corresponds to low-frequency coupling with the second-mode dis-
turbances and appears similar to the phenomenon observed along the stright cone
in figure 6.4. It is worth emphasizing again that bispectral analysis is incapable of
distinguishing between sum and difference interactions. It is thus ambiguous as to
whether this interaction corresponds to deconstructive interference between second-
mode structures at 190 kHz and 205 kHz to produce low-frequency content or if con-
tent at approximately 15 kHz is facilitating energy exchange between second-mode
structures; the latter explanation seems more likely, however, given the presence
of the 13 kHz structures upstream of the bicoherence peak revealed by the SPOD
analysis.
This dampening effect is further exaggerated at condition Re45, as seen in
123
Y [mm] Y [mm]
the spectra of figure 6.9. Upstream of the corner we see what are likely aliased
second-mode interactions around [220 kHz,120 kHz]. As a reminder, interactions
which occur along the right edge of the triangle may also be mapped to resonant
interactions at [f1,f1]. However, the spatial contour for this interaction (lower image
of figure 6.10) demonstrates limited coherence. This interaction occurs alongside
low-frequency resonance at 80? 90 kHz with corresponding low-frequency coupling.
Immediately downstream of the corner these interactions all vanish, leaving only
a peak at 10 kHz of dubious significance. Probing the spatial distribution of the
80 kHz resonance reveals something surprising in that the bicoherence intensity is
concentrated in a fashion which resembles the fundamental resonance at condition
Re33. These interactions may thus relate to the first harmonic of the second mode,
as content at 80?90 kHz would also correspond to aliased content from 460?470 kHz
and, although no coherent flow structures were observed in the SPOD mode shapes
within this frequency range, harmonic structures were revealed by the bandpass-
filtered images.
124
397 405 414 422 430 439 447 456
X [mm]
200 0.1
0.08
150
0.06
100
0.04
50
0.02
0 0
0 100 200 100 200 100 200
f  [kHz] f  [kHz] f  [kHz]
1 1 1
Figure 6.9: (Top) Time-averaged, flow-on image for the -5? configuration at condi-
tion Re45 with red circles denoting distinct regions for computing the bicoherence.
(Bottom) Bicoherence spectra for each annotated region with the contours ordered
stream-wise from left to right.
0.05
4
2
0
-2
0
0.05
4
2
0
-2
0
0.05
4
2
0
-2
0
390 400 410 420 430 440 450 460
X [mm]
Figure 6.10: Spatial bicoherence contours at condition Re45 for frequency triplets
of (top to bottom) [85,10,95], [85,85,170], and [220,120,340].
125
Y [mm] Y [mm] Y [mm]
f  [kHz]
2
6.3 +5 ?Extension
The compression corner configurations generally lead to the development of
new nonlinear exchange mechanisms. Figure 6.11 demonstrates this point for the
+5? configuration at condition Re33, for which the increased framerate of the test
(822 kHz) facilitates better resolution of interactions. The top-left contour veri-
fies the presence of fundamental resonance near 200 kHz as well as low-frequency
coupling in the immediate vicinity of the corner. From the spatial contour of fig-
ure 6.12(d) we see that this resonance enters from upstream of the field of view con-
centrated along the pseudo-streamline, and persists downstream of the corner until
approximately X = 430 mm, at which point the interaction dissipates and moves
closer to the wall. Indeed, the second streamwise station in figure 6.11 demonstrates
the continued presence of 200 kHz resonance, but it is absent at the third station.
Additional resonant interactions develop at the second station of figure 6.11
around frequencies of 80 kHz and 160 kHz, which happen to be a harmonic pair.
There also appears to be an interaction between these frequencies, with a peak oc-
curing at [f1, f2] = [160 kHz, 70 kHz], and a large degree of low-frequency coupling
is seen in the vicinity of [160 kHz,10 kHz]. As seen in figures 6.12(b) and (c), these
interactions peak sharply from X = 420? 430 mm. This region corresponds closely
to where the 77 kHz and 161 kHz SPOD mode shapes presented in figure 5.10 first
develop, with the 160 kHz structure appearing slightly further upstream. The in-
teraction at [160 kHz,70 kHz] may thus be explained as deconstructive interference
between second-mode wavepackets at 230 kHz and the low-frequency content to pro-
126
412 423 433 444 454 464
X [mm]
300 0.1 300 0.1
250 250
0.08 0.08
200 200
0.06 0.06
150 150
0.04 0.04
100 100
0.02 0.02
50 50
0 0 0 0
0 100 200 300 400 0 100 200 300 400
f  [kHz] f  [kHz]
1 1
300 0.1 300 0.1
250 250
0.08 0.08
200 200
0.06 0.06
150 150
0.04 0.04
100 100
0.02 0.02
50 50
0 0 0 0
0 100 200 300 400 0 100 200 300 400
f  [kHz] f  [kHz]
1 1
Figure 6.11: (Top) Time-averaged, flow-on image for the +5? configuration at condi-
tion Re33 with red circles denoting distinct regions for computing the bicoherence.
(Bottom) Bicoherence spectra for each annotated region with the contours ordered
stream-wise from top-left to bottom-right.
127
f  [kHz] f  [kHz]
2 2
f  [kHz] f  [kHz]
2 2
0.2
5
0.1
a) 0 0
410 420 430 440 450 460 470
X [mm]
0.1
5
0.05
b) 0 0
0.1
5
0.05
c) 0 0
410 420 430 440 450 460 470
X [mm] 0.1
5
0.05
d) 0 0
410 420 430 440 450 460 470
X [mm]
0.1
5
0.05
0 0
e) 410 420 430 440 450 460 470
X [mm]
Figure 6.12: Spatial bicoherence contours for the +5? flare at condition Re33 for fre-
quency triplets of (a) [75,75,150], (b) [160,10,170], (c) [160,70,230], (d) [120,120,240],
and (e) [200,200,400].
duce additional low-frequency structures. This conclusion seems far more likely than
the alternative that the second-mode waves are amplified by interaction of separate,
lower-frequency disturbances. Notably, the interaction at [160 kHz,10 kHz] actually
emanates energy from just upstream of the corner; this likely corresponds to the
corner shock, as mean flow interactions are captured by this contour.
The resonant interaction at 120 kHz which peaks at the third pixel station has
the same general spatial distribution (seen in figure 6.12d) as the preceding inter-
actions, but shifted downstream. This interaction can be matched to the SPOD
structure at 125 kHz, which peaks in amplitude within the vicinity of X = 440 mm.
At the furthest downstream station, the bicoherence spectra has developed a grid of
128
Y [mm] Y [mm] Y [mm] Y [mm] Y [mm]
412 423 433 444 454 464
X [mm]
300 0.1
250
0.08
200
0.06
150
0.04
100
0.02
50
0 0
0 100 200 300 400 100 200 300 400 100 200 300 400
f  [kHz] f  [kHz] f  [kHz]
1 1 1
Figure 6.13: (Top) Time-averaged, flow-on image for the +5? configuration at condi-
tion Re45 with red circles denoting distinct regions for computing the bicoherence.
(Bottom) Bicoherence spectra for each annotated region with the contours ordered
stream-wise from left to right.
peaks with a spacing of approximately 60 kHz, suggesting there may exist some key
structure at this frequency which facilitates energy transfer. That said, this station
lies far enough downstream that the flow is consistently transitional and intermit-
tently turbulent, which may contaminate the spectra and produce less meaningful
results. In fact, it should be noted that the spatial bicoherence distributions of
all interactions illustrated by figure 6.12 appear dominated by turbulent artifacts
downstream of X = 450 mm.
Figure 6.13 shows bicoherence spectra computed for pixel stations on the
+5? flare at condition Re45. Right at the corner, the primary interaction is still
129
f  [kHz]
2
0.1
5
0.05
a) 0 0
410 420 430 440 450 460 470
X [mm]
0.2
5
0.1
b) 0 0
410 420 430 440 450 460 470
X [mm]
0.2
5
0.1
c) 0 0
410 420 430 440 450 460 470
X [mm]
0.1
5
0.05
d) 0 0
410 420 430 440 450 460 470
X [mm]
0.1
5
0.05
e) 0 0
410 420 430 440 450 460 470
X [mm]
0.2
5
0.1
0 0
f) 410 420 430 440 450 460 470
X [mm]
Figure 6.14: Spatial bicoherence contours for the +5? flare at condition Re45 for fre-
quency triplets of (a) [40,40,80], (b) [80,80,160], (c) [150,150,300], (d) [230,230,460],
(e) [135,10,145], and (f) [340,140,480].
fundamental resonance at 230 kHz, though there appear to be additional resonances
at 75 and 150 kHz. Slightly downstream, strong peaks appear at [135 kHz, 0 kHz]
and [340 kHz, 140 kHz]. These interactions along with the second-mode resonance
dissipate quickly, with a new resonant interaction at 165 kHz appearing in the spec-
trum of the third station. The majority of high-frequency interactions observed at
the most downstream pixel station were found to have little spatial coherence and
are believed to be turbulent artifacts.
Spatial bicoherence distributions for the primary interactions mentioned above
130
Y [mm] Y [mm] Y [mm] Y [mm] Y [mm] Y [mm]
are given in figure 6.14. The first four contours correspond to resonant interactions
at 40 kHz, 80 kHz, 150 kHz, and 230 kHz, respectively. The dominant feature for all
these interactions is a band of substantially elevated signal which emanates away
from the boundary layer. This band originates from X = 415? 435 mm depending
on the resonant frequency and likely corresponds to turbulent features. Resonance
of the second-mode fundamental at 230 kHz (figure 6.14d) is seen along the pseudo-
streamline at the beginning of the field of view but rapidly drops off downstream
of the corner. There is some evidence of continued fundamental resonance within
the flare boundary layer, but it is difficult to assign physical significance to the
elevated signal here given the obvious turbulent contamination. The other interac-
tions spike in amplitude dramatically from 410 ? 420 mm, with much of the signal
concentrated above the streamline. Synthesizing the results for both conditions,
the +5? compression results in a spike in nonlinear interaction within a small region
early on the flare which may alter the transition process significantly.
6.4 +10 ?Extension
The appearance of the separation bubble and the shear-generated disturbances
with the +10? flare has a significant impact on the bicoherence computed along
the pseudo-streamline. That said, figure 6.15 shows that fundamental resonance
is still captured upstream of the separation point at [220 kHz, 220 kHz] at condi-
tion Re33. Low-frequency coupling is simultaneously observed at approximately
[230 kHz, 20 kHz], corresponding to energy exchange with the upper sideband of the
131
382 393 403 414 424 435
X [mm]
300 0.1
250
0.08
200
0.06
150
0.04
100
0.02
50
0 0
0 100 200 300 400 100 200 300 400 100 200 300 400
f  [kHz] f  [kHz] f  [kHz]
1 1 1
Figure 6.15: (Top) Time-averaged, flow-on image for the +10? flare at condition
Re33 with red circles denoting three distinct regions for computing the bicoherence.
(Bottom) Bicoherence spectra for each annotated region with the contours ordered
stream-wise from left to right.
second-mode disturbances. Within the separation bubble, these peaks drop off and
there is little evidence of nonlinearity along the pseudo-streamline. Downstream
of reattachment there is a resurgence in fundamental resonance, though the peak
now lies at [205 kHz, 205 kHz]. There is also weak evidence of a new, low-frequency
resonant interaction at [45 kHz, 45 kHz].
The disappearance of second-mode interactions downstream of separation is
further demonstrated by the bottom two spatial contours in figure 6.16 which corre-
spond to [220? 20 kHz,20? 10 kHz] and [220? 20 kHz,220? 20 kHz]. Fundamental
resonance amplifies up until X = 392 mm, at which point the peak diminishes
132
f  [kHz]
2
0.1
5
0.05
0 0
0.1
5
0.05
0 0
0.1
5
0.05
0 0
380 390 400 410 420 430 440
X [mm]
Figure 6.16: Spatial bicoherence contours for the +10? flare at condition Re33 for fre-
quency triplets of (top) [45,45,90], (middle) [220,20,260] and (bottom) [220,220,440].
rapidly. This is mimicked by the interaction at [220 kHz,20 kHz], which is again
concentrated above the pseudo-streamline. The top contour of figure 6.16 reveals a
coherent structure to the 45 kHz resonance which is obscured when restricting our
analysis to the pseudo-streamline. This interaction appears along the upper edge
of the separation bubble and peaks in the vicinity of reattachment. We will see a
similar structure for resonance of the shear mode with the +15? configuration.
Bicoherence spectra within the separation bubble and in the vicinity of reat-
tachment are given in figure 6.17 for condition Re45. There is a significant increase
in nonlinearity within the separation region relative to condition Re33. There now
exist two distinct resonant interactions at approximately [140 kHz, 140 kHz] and
[70 kHz, 70 kHz]. There are also prevalent high-frequency interactions in the vicin-
ity of [320 kHz, 10 kHz] and [320 kHz, 180 kHz]. These interactions likely correspond
to harmonic content aliased down from 500 kHz. Just upstream of reattachment, we
see the development of significant resonances near 55 kHz and 160 kHz. The upper
133
Y [mm] Y [mm] Y [mm]
spatial contour of figure 6.18 shows that the 55 kHz interaction peaks within a con-
fined spatial region along the pseudo-streamline. From the SPOD analysis we can
conclude that this behavior corresponds to the lower-frequency shear mode observed
at 61 kHz (in figure 5.15). As with the +5? configuration, the streamline curvature
appears to create a concentrated region close to the corner in which the nonlinear
interactions occur. Resonance of the high-frequency, shear-generated structures at
[75 kHz, 75 kHz] appears in the middle contour of figure 6.18, and is visible further
upstream but does not peak as significantly in the vicinity of reattachment. By
contrast, the 160 kHz interaction in the lower contour demonstrates little spatial
coherence and appears largely due to turbulence. The significance of interactions
which appear in the third pixel station of figure 6.17, just downstream of reattach-
ment, is likewise unclear.
134
386 396 406 417 427 438
X [mm]
300 0.1
250
0.08
200
0.06
150
0.04
100
0.02
50
0 0
0 100 200 300 400 100 200 300 400 100 200 300 400
f  [kHz] f  [kHz] f  [kHz]
1 1 1
Figure 6.17: (Top) Time-averaged, flow-on image for the +10? flare at condition
Re45 with red circles denoting three distinct regions for computing the bicoherence.
(Bottom) Bicoherence spectra for each annotated region with the contours ordered
stream-wise from left to right.
0.1
5
0.05
0 0
0.1
5
0.05
0 0
0.2
5
0.1
0 0
380 390 400 410 420 430 440
X [mm]
Figure 6.18: Spatial bicoherence contours for the +10? flare at condition Re45 for
frequency triplets of (top to bottom) [55,55,110], [75,75,150], and [160,160,320].
135
Y [mm] Y [mm] Y [mm]
f  [kHz]
2
6.5 +15 ?Extension
The reduced framerates employed with this configuration (440 kHz at condi-
tion Re33 and 550 kHz at Re45) render it impossible to properly resolve nonlinear
interactions involving second-mode disturbances without significant aliasing. For-
tunately, the preceding sections have demonstrated that the boundary layer in this
configuration is largely dominated by lower-frequency structures which are still well-
resolved. At condition Re33, as with the +10? configuration, there are no discernible
peaks in the bicoherence through much of the upstream portion of the separation
bubble. The first region presented in figure 6.19 is located slightly upstream of
the corner and is near where the shear-generated disturbances are first observed
instantaneously. At this point, the bicoherence spectrum shows weak resonant in-
teractions at 53 kHz and 117 kHz and a stronger, though still mild, interaction at
[72 kHz, 12 kHz]. The physical significance of this interaction is unclear. Within the
second region, now downstream of the corner, a strong peak develops at [40 kHz,
40 kHz] corresponding to self-resonance of the shear-generated disturbances. The
upper spatial contour of figure 6.20 shows this interaction is concentrated near the
pseudo-streamline, though not aligned with it, from X = 418 ? 425 mm as the
boundary layer recompresses. This correlates well with the peak interaction region
seen for the +10? configuration in figure 6.18. Interestingly, there is an additional
band of elevated bicoherence for this interaction above the pseudo-streamline; this
band is seen as far upstream as 385 mm but peaks in the vicinity of reattachment.
Downstream of reattachment, a series of interactions can be seen in figure 6.19
136
to lie approximately along lines corresponding to f1 = 40 kHz and f2 = 40 kHz. It
is also worth noting that these peaks have approximately uniform spacing of ?F =
20 kHz. The interactions at [42 kHz, 20 kHz] and [60 kHz, 40 kHz] are presented in
the lower contours of figure 6.20. Both contours share a similar structure with the
14 kHz SPOD mode in figure 5.18. As with the 40 kHz resonance, these interactions
peak in strength as the boundary layer reattaches, but are visible far upstream. All
this suggests that low-frequency oscillation of the separation bubble and the shear-
generated disturbances couple together to facilitate energy transfer within the shear
layer. This also demonstrates the shortcomings of the pseudo-streamline for this
analysis, as these interactions follow very different trajectories.
At conditon Re45, resonance of the shear-generated instability at [50 kHz,
50 kHz] is observed in the first probed region of figure 6.21 as far upstream as
X = 405 mm. This interaction amplifies downstream and new resonances appear
at 25 kHz and 95 kHz within the second region. These are notable in that they
correspond approximately to the subharmonic and harmonic of the shear-generated
disturbances. The spatial distribution of the shear-mode resonance is shown in
the second contour of figure 6.22 to peak just above the pseudo-streamline as the
boudary layer recompresses, just as it did at condition Re33. However, the bico-
herence peak no longer outlines the entirety of the separation bubble, and a ray of
elevated signal is seen emanating from the shear layer. The bicoherence peak and
ray of elevated signal align well with the SPOD structure displayed in figure 5.19 at
99 kHz. This suggests that the ribbon-like waves of energy which emanate from the
boundary-layer may be caused by nonlinear resonance of the shear waves as they ap-
137
380 388 397 405 413 422 430 439
X [mm]
0.1
160
140 0.08
120
100 0.06
80
0.04
60
40
0.02
20
0 0
0 100 200 0 100 200 0 100 200
f  [kHz] f  [kHz] f  [kHz]
1 1 1
Figure 6.19: (Top) Time-averaged, flow-on image for the +15? configuration at con-
dition Re33 with red circles denoting distinct regions for computing the bicoherence.
(Bottom) Bicoherence spectra for each annotated region with the contours ordered
stream-wise from left to right.
10 0.2
5 0.1
0 0
380 390 400 410 420 430 440
10 0.2
X [mm]
5 0.1
0 0
380 390 400 410 420 430 440
10 0.2
X [mm]
5 0.1
0 0
380 390 400 410 420 430 440
X [mm]
Figure 6.20: Spatial bicoherence contours for the +15? flare at condition Re33 for
frequency triplets of (top to bottom) [40,40,80], [40,20,60] and [60,40,100].
138
Y [mm] Y [mm] Y [mm]
f  [kHz]
2
proach reattachment. The potential subharmonic resonance at 25 kHz is illustrated
by the upper-most contour of figure 6.22 and peaks above the pseudo-streamline
within the same region as the fundamental shear-mode resonance; however, this
interaction also displays elevated signal below the pseudo-streamline. Overall, this
interaction correlates well with the 26 kHz structure resolved by the SPOD analysis
in figure 5.19.
Another interesting feature of the bicoherence spectrum of the second station is
the peak which develops at [100 kHz, 50 kHz], potentially corresponding to an inter-
action between the shear-generated disturbances and its first harmonic. The spatial
distribution of this interaction (second contour of figure 6.22) again appears as a
ray emanating from X = 418 mm and closely aligns with the 146 kHz SPOD mode
shape from figure 5.19, providing further evidence that second-harmonic content is
generated within this interaction region.
Within the vicinity of reattachment, around X = 430 mm, interactions develop
along the band corresponding roughly to f2 = 20 kHz. This band of interactions
persists and strengthens downstream (note the increased contour limits of the final
station). These interactions are likely facilitated by a low-frequency structure simi-
lar to the 17 kHz SPOD mode which was concentrated near reattachment. Notably,
these downstream interactions involve some much higher frequencies. The bottom
contour of figure 6.22 shows that the coupling between [215 kHz, 20 kHz] is concen-
trated entirely within the compressed boundary-layer downstream of reattachment.
This correlates with the high-frequency structures resolved within this region by the
SPOD analysis.
139
380 388 396 405 413 422 430 438
X [mm]
200 0.1 200 0.1
0.08 0.08
150 150
0.06 0.06
100 100
0.04 0.04
50 50
0.02 0.02
0 0 0 0
0 50 100 150 200 250 0 50 100 150 200 250
f  [kHz] f  [kHz]
1 1
200 0.1 200 0.15
0.08
150 150
0.1
0.06
100 100
0.04
0.05
50 50
0.02
0 0 0 0
0 50 100 150 200 250 0 50 100 150 200 250
f  [kHz] f  [kHz]
1 1
Figure 6.21: (Top) Time-averaged, flow-on image for the +15? configuration at con-
dition Re45 with red circles denoting distinct regions for computing the bicoherence.
(Bottom) Bicoherence spectra for each annotated region with the contours ordered
stream-wise from left to right.
140
f  [kHz] f  [kHz]
2 2
f  [kHz] f  [kHz]
2 2
0.2
5 0.1
0 0
0.2
5 0.1
0 0
380 390 400 410 420 430 440
X [mm] 0.1
5 0.05
0 0
380 390 400 410 420 430 440
X [mm] 0.1
5 0.05
0 0
380 390 400 410 420 430 440
X [mm]
Figure 6.22: Spatial bicoherence contours for the +15? flare at condition Re45
for frequency triplets of (top to bottom) [25,25,50], [50,50,100], [100,50,150] and
[215,20,235].
141
Y [mm] Y [mm] Y [mm] Y [mm]
Chapter 7: LASTRAC
A limited computational study of the +10? configuration was carried out using
LASTRAC (Langley Stability and Transition Analysis Code) at condition Re33.
This was meant as a check on the fidelity of the experimental results and an attempt
to computationally validate the appearance of the shear-generated instability. Only
condition Re33 was selected for analysis due to the transitional nature of the flow
at higher Rem, which degrades the applicability of LST and PSE.
LASTRAC is a transition-prediction code developed by NASA which offers
computations based on either quasi-parallel linear stability theory (LST) or the
parabolized stability equations (PSE). The PSE formulation may be either linear
(LPSE) or nonlinear, the latter of which requires knowledge of the incoming distur-
bance amplitude. All of these formulations are built upon a decomposition of the
Navier-Stokes equations into mean and fluctuating components. By subtracting the
mean-flow component, one arrives at the governing equations for disturbances im-
posed on the flow. In the linear formulations, these disturbances are assumed to be
very small relative to the mean quantities, allowing the equations to be linearized.
This means the Fourier modes develop independently and any nonlinear interactions
are ignored. LST relies upon the assumption that the flow is locally quasi-parallel,
142
meaning that streamwise variations are negligible. This results in the disturbance
mode shape (?) formulated as
?(x, y) = ?(y)ei?x, (7.1)
where ? is the local complex wavenumber and ? is the shape function which varies
only with y.
Clearly, this assumption breaks down when considering separated flow over a
compression corner. For this reason, the LST computations of this chapter were
used primarily to identify the dominant instabilities and were supplemented by
LPSE calculations at these discrete frequencies. LPSE accounts for nonparallel
effects by decomposing the disturbances into a wave component described by the
complex wave number, ?, and a shape function, ?, which is now allowed to vary
with x. This formulation is given by
? x
i ?(?)d?
?(x, y) = ?(x, y)e x0 . (7.2)
Note that the history of the wavenumber is preserved within the integral. This
results in upstream dependence, a feature which is absent in LST computations. A
full description of the governing equations and numerical formulation of LASTRAC
are given in the user manual [50].
143
Rem [10
6/m] U? [m/s] P? [Pa] T? [K] ?? [kg/m
3]
3.02 1279 601 113 0.019
Table 7.1: Inflow properties for the STABL2D mean flow computation.
7.1 Mean Flow
Computation of the mean flow was carried out using STABL2D (Stability and
Transition Analysis for hypersonic Boundary Layers), developed by the University
of Minnesota. The flow solver is based on the implicit Data Parallel-Line Relaxation
method developed by Wright et al.[51] and employs a second-order, finite-volume
formulation. A full description of the governing equations and implementation for
2-D/axisymmetric flows is given by Johnson [52]. Although the computations we
performed for reservoir conditions equivalent to Re33, the freestream properties were
instead computed based on the assumption of vibrational equilibrium throughout
the nozzle and are given by table 7.1. The switch to a frozen-flow assumption from
the nozzle throat was made after these computations had been completed. The
mean flow was treated with a calorically perfect gas assumption; this is justified by
the relatively low freestream temperature (113 K). The cone surface was treated as
an isothermal wall at 300 K.
The computational grid was generated in a two-step process. First, the grid
over the nosetip and frustum was generated with STABL?s built-in grid solver in-
cluding a nosetip bluntness of radius 0.1 mm. The flare geometry was then added
within Pointwise. The full grid was dimensioned 1535 x 351 in the streamwise and
wall-normal directions, and is shown in the left image of figure 7.1. Note that only
144
80 6
5
60
4
40 3
2
20
1
0 0
0 100 200 300 400 500 409 410 411 412 413 414 415
X [mm] X [mm]
Figure 7.1: (Left) Full computational domain for mean-flow computations with
reduced resolution for clarity and (right) half-resolution grid in the vicinity of the
corner.
every tenth grid line is displayed to aid visual inspection. Wall-normal stretching
was employed to cluster grid points within the boundary layer and the mean-flow
computations were analyzed to ensure a resolution of at least 150 nodes. The grid
was also stretched in the streamwise direction to increase resolution near the nosetip
and within the separation region. Particular care was taken to keep the grid lines
normal to the flare surface. This was accomplished by modelling the corner junction
with a radius from X = 411?413 mm, as can be seen in the right image of figure 7.1,
where the visible grid resolution has been halved.
Contours of x-velocity for the mean flow computation are shown in figure 7.2.
Overlaid on the contour plot are: the boundary-layer thickness determined from
the computation based on a 99.7% total enthalpy condition (solid line); the pseudo-
streamline for the equivalent experiment (dashed line); and the separation and reat-
tachment locations determined from the computation (solid) and the experiment
(dashed). We see reasonable agreement between the experiment and computation
for all metrics. It is expected that the pseudo-streamline would lie closer to the
145
Y [mm]
Y [mm]
5
0
370 380 390 400 410 420 430 440 450
X [mm]
Figure 7.2: Mean flow contour of x-velocity overlaid with lines representing:
the boundary-layer edge (solid); the pseudo-streamline for the corresponding test
(dashed); the separation and reattachment locations determine from the computa-
tion (solid) and the experiment (dashed).
surface than the boundary-layer edge as the maximum second-mode fluctuations
are typically observed around 80-90% of the boundary-layer thickness.
7.2 LST Results
LST calculations were performed for frequencies from 60 ? 450 kHz over the
length of the model with a quasi-parallel flow assumption. Resulting amplifica-
tion rates and N factors over the parameter space are displayed in figure 7.3; the
computationally determined separation and reattachment locations are included for
reference. Note that within the gray regions, the local eigenvalue search returned
no unstable modes and no data is given. The N factors are integrated from the first
appearance of an unstable mode at each frequency. The second mode is the only
amplifying disturbance over much of the frustum, first appearing at X = 179 mm
around 300 kHz (not pictured) and peaking in amplitude at 222 kHz just prior to
separation. This is a slightly higher peak frequency than that observed experimen-
tally along the pseudo-streamline, but still represents acceptable agreement. Within
the separation region, amplification shifts to lower frequencies, with three distinct
shear-generated modes appearing. The middle mode overlaps significantly with
146
Y [mm]
N
450 80 450 3
400 400
2.5
60
350 350
2
300 300
40
250 250 1.5
200 200
20 1
150 150
0.5
100 0 100
50 50 0
300 350 400 450 500 300 350 400 450 500
X [mm] X [mm]
Figure 7.3: LST contours of (left) amplification rate and (right) N-factor as functions
of streamwise location and frequency.
the second mode, resulting in N factors which likely are not representative of the
shear-generated disturbance. The lower frequency mode is the one which persists
downstream of reattachment, peaking at around 78 kHz. This is also slightly higher
than the 71 kHz peak observed experimentally.
7.3 LPSE Results
Based on the preceding results, non-parallel LPSE computations were per-
formed at frequencies of 222 kHz (the peak second-mode frequency upstream of
separation) and 78 kHz (representing the shear-generated disturbances). Distur-
bance eigenfunctions of pressure, temperature and numerical ?schlieren? (vertical
density gradient) are visualized spatially in figure 7.4 for the 222 kHz disturbance.
Note the visualization domain has been cropped to approximate the field of view in
the experiments. This ?schlieren? contour reveals a spatial structure which closely
matches that observed in the reference-subtracted image sequences and the SPOD
analysis. Upstream of separation, the disturbances are characterized by the typical
147
F [kHz]
F [kHz]
5
0
5
0
5
0
370 380 390 400 410 420 430 440 450
X [mm]
Figure 7.4: Fluctuation eigenvalue contours computed for a 222 kHz disturbance:
(a) pressure, (b) temperature, and (c) numerical schlieren.
rope-like appearance. Traversing the separation bubble, the disturbances flatten out
substantially and radiation of disturbance energy is observed at both separation and
reattachment.
These same eigenfunction contours are shown for the 78 kHz disturbance in
figure 7.5. Encouragingly, the ?schlieren? eigenfunction for this disturbance shows
excellent agreement with the structure of the shear-generated disturbances revealed
in the 71 kHz SPOD contour of figure 5.14. There is also a strong ray of disturbance
energy emanating away from the shear layer at around X = 415 mm. This feature
appears to be acoustic in nature, as it is also apparent in the pressure contour. Al-
though this radiation is cut off in the SPOD mode by the field of view, there is some
elevated signal visible near the top of the frame which correlates well with the LPSE
structure. The shear-layer disturbance is visible in the temperature and ?schlieren?
contours even upstream of separation; however, the pressure disturbances do not be-
come apparent until the immediate vicinity of the corner, where they amplify rapidly
near the model surface. This is consistent with the observations of the disturbance
148
Y [mm] Y [mm] Y [mm]
5
0
5
0
5
0
370 380 390 400 410 420 430 440 450
X [mm]
Figure 7.5: Fluctuation eigenvalue contours computed for a 78 kHz disturbance: (a)
pressure, (b) temperature, and (c) numerical schlieren.
in the spectra of PCB D1. Overall, these experiments show good agreement with
the experimental results.
149
Y [mm] Y [mm] Y [mm]
Chapter 8: Conclusions
A notable gap in the literature surrounding high-speed boundary-layer transi-
tion concerns the interaction of disturbances with the mean flow structures imposed
by sudden angle changes. This study has sought to fill this gap and character-
ize the development of second-mode wavepackets in the vicinity of axisymmetric
compression and expansion corners.
All experiments were conducted at the University of Maryland - College Park
in our HyperTERP reflected-shock facility. For these experiments, HyperTERP was
equipped with a contoured nozzle with a nominal exit Mach number of 6. Pitot rake
characterization studies suggest this is a good approximation of the freestream con-
ditions considering the effects of vibrational nonequilibrium. Three test conditions
were considered with the total specific enthalpy held constant in order to isolate the
effects of Reynolds number. A model consisting of a 5? half-angle, conical forebody
was selected to encourage the dominance of second-mode wavepackets upstream
of the interaction region. Interchangeable afterbody attachments corresponding to
flow deflections of -5? to +15? in 5? increments were considered. High-speed schlieren
photography was employed as the primary means of flow interrogation, with sup-
plemental surface measurements provided by PCB132B38 pressure transducers. A
150
lens calibration was applied to the images to provide quantitative fluctuations in
density gradient.
Reference-subtracted image sequences revealed the time-resolved, qualitative
behavior of wavepackets in the vicinity of the corner. With the +0? extension, the
boundary layer was dominated by the typical, rope-like structure of second-mode
wavepackets. Transitional behavior was observed near the forebody/afterbody junc-
tion and generally moved upstream with increasing Rem. Surface pressure spectra
demonstrated peak second-mode frequencies in the range of 200? 300 kHz. Signif-
icant spectral broadening occured in the PCB spectra within the transitional flow
region.
The primary visible effect of the expansion corner was the suppression of dis-
turbances. This applied to second-mode wavepackets as well as turbulent structures,
and was accompanied by a decrease in the magnitude of surface pressure fluctua-
tions. Interestingly, second-mode energy was observed emanating away from the
boundary layer along the head of the expansion wave. A similar phenomenon was
witnessed in the +5? configuration whereby second-mode wavepackets would radiate
energy along the corner shock. Even this modest compression appeared to encourage
transition of incoming wavepackets, though qualitative alterations to the structure
of the incoming disturbances was difficult to identify.
When the flow deflection reached +10? the boundary layer separated upstream
of the corner, resulting in a recirculation region bounded by separation and reat-
tachment shocks (along which additional second-mode radiation was observed). The
second-mode disturbances would generally lift off the surface at the separation point
151
and propagate largely within the separated shear layer, flattening as they travelled
downstream. At the lowest studied Rem, the boundary layer was found to remain
laminar through both separation and reattachment, though the disturbances ap-
peared significantly modulated over the flare with clearly reduced structure angles.
At the highest Rem, the observed instabilities would break down immediately upon
encountering the separation shock, resulting in a predominantly turbulent boundary
layer on the flare.
The +15? extension produced a much larger separation region which was no
longer characterized by second-mode wavepackets at low Rem. Instead, the corner
region was dominated by shear-generated disturbances which amplified significantly
within the downstream portion of the separation bubble. These disturbances ap-
peared as braided structures in the schlieren images and emanated wavy energy
structures as they reattached on the flare. These structures persisted downstream
of reattachment in the thin, flare boundary layer. The few second-mode wavepack-
ets observed at low Rem generally lost distinction over the corner and appeared
to merge with the shear waves. At the highest-studied Rem, the separation region
shrunk significantly and second-mode disturbances again became dominant. As in
the +10? case, the SWBLI would provoke bypass transition of the disturbances,
leading to a consistently turbulent flare boundary layer. There was some evidence
of unsteadiness in the mean flow field for this high Rem case, as particularly strong
bursts of turbulence would collapse the separation bubble which would then reform
over time.
To facilitate novel image analysis, the camera framerate was kept at or above
152
the Nyquist sampling rate of the dominant second-mode frequencies being studied.
These time-resolved image sequences thus permitted the use of spectral analysis
techniques to study the spatial development of modal content. For the expansion
case, this revealed growth of lower-frequency second-mode content even as the in-
coming disturbances decayed. In all compression cases, the radiation phenomenon
observed instantaneously appeared to cause a small but abrupt drop in second-mode
energy. The +5? compression was shown to amplify not only high-frequency distur-
bances but also low-frequency content below 50 kHz. Complex growth and decay
trends of second-mode disturbances were seen through separation and reattach-
ment in the +10? configuration. The power of the disturbances appeared largely to
freeze within the upstream portion of the bubble before amplifying again near reat-
tachment. This spectral analysis technique also demonstrated the presence of low-
frequency (70? 80 kHz), shear-generated disturbances within the separation region
that were not obvious in the reference-subtracted sequences. The absence of second-
mode wavepackets was confirmed for the +15? configuration and the shear generated
disturbances were assigned dominant frequencies in the range of 40? 60 khz.
Temporally bandpass-filtering the image sequences proved a useful technique
for isolating instantaneous modal disturbances. In all cases, this method could be
used to accentuate the structure of second-mode disturbances. The previously ob-
scured shear-generated disturbances for the +10? configuration were shown to have
a braided structure oriented nearly parallel to its propagation direction, matching
the observations for the +15? tests. By leveraging aliasing effects, this technique was
even proven capable of resolving instantaneous second-mode harmonic structures.
153
A more powerful technique for resolving coherent flow structures was demon-
strated in spectral proper orthogonal decomposition. This technique provides ranked,
orthogonal modes which both optimally capture the flow energy provided by snap-
shots and oscillate at distinct frequencies. While this technique performed well at
demonstrating the development of second-mode frequencies, it was perhaps more
useful for identifying less prominent structures related to amplification peaks from
the spectral analysis. Through SPOD, the amplifying lower-frequency second-mode
waves downstream of the expension corner could be visualized. This technique also
revealed the structure of low-frequency disturbances amplified by the +5? configuration
and provided an additional method by which to observe the shear-generated distur-
bances in the +10? case. For the stronger shear waves generated by the +15? compression,
SPOD resolved the structures emanating from the downstream portion of the sepa-
ration region which appeared to correspond to the first and second harmonics of the
shear disturbances. At high enough Rem, high-frequency structures were observed
within the reattached boundary layer. The capability of determining second-mode
propagation speeds from these SPOD modes was also demonstrated.
Nonlinear interactions were probed through bispectral analysis, which con-
firmed the presence of fundamental resonance and sideband interactions of the sec-
ond mode upstream of the corner junction. Both of these interactions have been
documented by prior researchers. Notably, bicoherence spectra were computed for
each pixel in the field of view, permitting a global view of the spatial distribu-
tion of key nonlinear interactions. The expansion corner caused substantial decay
of nonlinear behavior, although fundamental resonance persisted for some distance
154
downstream. The spatial contours revealed low-frequency coupling between be-
tween second-mode waves and low-frequency content to be concentrated near the
boundary-layer edge. The +5? compression provoked the development of many new
resonant interactions which could be matched to structures resolved by the SPOD
analysis. The streamline curvature imposed by the compression appeared to create
a concentrated region early on the flare which led to a localized spike in nonlin-
ear behavior. At the lowest studied Reyolds number, the separation bubble of the
+10? configuration served largely to damp nonlinear interactions, with fundamental
resonance ceasing immediately downstream of separation. At higher Reynolds num-
bers, significant low-frequency resonance was found to occur within the separation
region, particularly in the vicinity of reattachment, with additional high-frequency
and non-resonant interactions appearing on the flare. Even stronger resonance of the
shear-generated disturbances was observed in the +15? case, again peaking upstream
of reattachment. Spatial contours again revealed a region of elevated nonlinear ac-
tivity which correlated with streamline curvature. These bispectral results suggest
that the energy structures seen emanating from the shear layer near reattachment
in the SPOD analysis are formed due to self-resonance of the shear waves. Evidence
of subharmonic growth related to these shear waves was also presented.
Finally, a limited computational study was performed using LASTRAC to
serve as a first comparison against results compiled for the +10? configuration. The
mean flow, computed using STABL2D, demonstrated reasonable agreement with the
experiments in terms of both boundary-layer thickness and separation length. The
peak second-mode frequency upstream of separation was predicted to be 222 kHz
155
based on linear stability theory, again representing good agreement with experi-
ments. Multiple unstable modes were predicted within the separation region, with
the maximum amplification acheived at 78 kHz. Additional computations were per-
formed for these two frequencies employing the linear parabolised stability equa-
tions to examine their structural development through the SWBLI. Disturbance
eigenfunction contours from the LPSE analysis showed excellent agreement with
the disturbance structures resolved by SPOD, particularly for the shear-generated
disturbances.
Continuation of this work should look first at improving the fidelity of the data
set. The nature of impulse facilities produces a noisy flow environment, which has
manifested itself throughout this work in the form of anomalous turbulent bursts
which contaminate the data and make robust computation of modal structures and
nonlinear interactions difficult. It would thus be greatly beneficial to obtain sim-
ilar measurements in a tunnel following a different operating principle (such as a
Ludwieg Tube) where freestream disturbances are likely to be of lower magnitudes.
Another line of inquiry would involve mounting the cone at various angle of at-
tack, which would introduce crossflow instability. This added instability mechanism
may fundamentally alter the SWBLIs, particularly in the separated cases where the
crossflow may interact with the shear-generated disturbances. The parameter space
could be enhanced further by examining the effects of nosetip bluntness. Boundary-
layer structures not resembling second-mode disturbances have been observed on
cones with sufficient bluntness [18]. It would be interesting to observe these struc-
tures passing through the corner interaction regions presented in this study.
156
157
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