ABSTRACT
Title of dissertation: DEVELOPMENT OF TWO-POINT
FOCUSED LASER
DIFFERENTIAL INTERFEROMETRY
FOR APPLICATIONS IN HIGH-SPEED
WIND TUNNELS
Andrew Paul Ceruzzi
Doctor of Philosophy, 2022
Dissertation directed by: Professor Christopher P Cadou
Department of Aerospace Engineering
Focused laser differential interferometry (FLDI) and its relative two-point fo-
cused laser differential interferometry (2pFLDI) are completely non-intrusive (i.e.
seedless) optical techniques for measuring density fluctuations and velocity respec-
tively that offer high frequency response (> 10MHz). Developed in the 1970s, FLDI
is receiving renewed attention today for its potential usefulness in measuring tur-
bulent fluctuations and velocity in hypersonic flows. In the technique, two focused,
closely-spaced (? 100?m), orthogonally-polarized beams pass through a region of
interest and are subsequently combined and focused onto a photodetector. Differ-
ences in refractive index between the two focal volumes cause a phase shift, thus
interference, between the beams which is measured by the detector. In this way the
instrument is sensitive to the gradient in refractive index along a line between the
two focal volumes perpendicular to the beams (?n/?x). Since gradients in index
of refraction arise from gradients in density (in homogeneous flows), fluctuations in
the FLDI signal are proportional to local fluctuations in density. If the fluctuations
are due to localized eddies convecting through the FLDI measurement volume, then
the cross-correlation of the FLDI signal with a that from a second FLDI instrument
located a known distance downstream of the first provides a measure of convection
velocity (2pFLDI). The ability to measure density fluctuations and velocity simulta-
neously and at the same point in the flow is critical because it enables one to relate
the temporal scales measured by the instrument to the spatial scales present in the
flow.
In spite of the technique?s age, a unified theory for the FLDI operation and
sensitivity limits which is simple and easy to use does not exist so the first objec-
tive of this thesis is to develop such a theory. It does so using transfer functions
that enable one to isolate the effects of focusing, beam separation, and disturbance
frequency on the performance (i.e. sensitivity and spatial resolution) of the instru-
ment. While the transfer functions have been previously proposed by others, an
application of these functions which accounts for velocity variation in space (uc(z))
and frequency (uc(f)) is unique to this work.
The theory is validated via comparison to experimental measurements in a
canonical turbulent jet where the distributions of velocity and density fluctuations
are well known. Measurements made using different FLDI instruments collapse when
the differences between them are accounted for, indicating that the unified theory
is correctly capturing the effects of instrument parameters like beam separation and
beam diameter. FLDI response to the jet is also modeled by substituting the velocity
distribution for a dispersion relation, uc(f), measured by 2pFLDI. The advantage
of the latter procedure is it allows for signal interpretation in flows where historical
measurements are unavailable. This is demonstrated by comparing modeled FLDI
response to experimental measurements in the flow downstream of a ramp in a small
(6.4cm square) Mach 3 wind tunnel.
The second objective of this thesis is to demonstrate 2pFLDI in other industrially-
relevant flows. To this end, density fluctuations and convection velocities are mea-
sured in the near-wall flow in a 61cm square Mach 4 wind tunnel (Ludwieg tube)
and in the free-stream flow of a 1.5m diameter Mach 18 tunnel. In each case,
a method for estimating the spatiotemporal resolution using transfer functions is
demonstrated. The spatiotemporal resolution of the instrument was not well under-
stood prior to this work so quantifying it is an important contribution. Achieving
acceptable signal/noise at Mach 18 was difficult because densities were so low. How-
ever, convection velocities of ? 75 ? 80% of the freestream velocity are measured
above 200kHz in two runs. Spatiotemporal analysis suggests these measurements
are the result of freestream disturbances; the first measurement of its kind in a Mach
18 flow.
DEVELOPMENT OF TWO-POINT FOCUSED LASER
DIFFERENTIAL INTERFEROMETRY FOR APPLICATIONS IN
HIGH-SPEED WIND TUNNELS
by
Andrew Paul Ceruzzi
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
2022
Advisory Committee:
Professor Christopher Cadou, Chair/Advisor
Dr. Stuart Laurence
Dr. Kenneth Yu
Dr. Christoph Brehm
Dr. Christopher Davis
? Copyright by
Andrew Paul Ceruzzi
2022

Acknowledgments
First, a huge thank you to my advisor, Prof. Christopher Cadou, for many
years of guidance, mentorship, and advice. I greatly appreciate his commitment
not just to his students, but to teaching and to his family; he is a role model I
look up to. Next, thank you to Prof. Stuart Laurence who has been my un-official
co-advisor. He has included me in his research group, provided endless advice, and
is an important mentor in my life. Thank you to Prof. Kenneth Yu for many years
of encouragement and support as well. Thanks as well to the rest of my committee,
Prof. Christoph Brehm and Prof. Chris Davis, for advice and perspective.
Thank you to the many other faculty who I have interacted with at the Uni-
versity of Maryland, including Prof. Pino Martin, Prof. James Baeder, and Prof.
Christopher Lobb. Thanks to the Aerospace Engineering department staff, they are
a team of all-stars.
A massive thanks goes to my numerous lab-mates and class-mates over the
years: Between Cadou-lab, HAPL, and everyone else there are too many of you to
name. Thank you for advice, friendship, and camaraderie.
Next, I want to thank the team at AEDC Hypervelocity Wind Tunnel 9. Their
partnership with the University of Maryland gave me access to resources, mentor-
ship, and a world-class facility. Thank you Mike Smith, Brianne McManamen, John
Lafferty, Dan Marren, Eric Marineau, AJ Spicer, tunnel operators, engineers, and
staff. Thank you also to Tunnel 9 interns (at the time) Christoph Nieses, Dan
Weber, and Braeden Callis who helped a tremendous amount with experiments.
ii
A big thank you to John Schmisseur and Mark Gragston for inviting me to
the University of Tennessee Space institute. I greatly appreciate the opportunity
to collaborate with and learn from other institutions, and I had a great time in
Tennessee. Thanks as well to Lauren Lester and Jack Coburn.
Thanks to the Air Force Office of Scientific Research support through the
University of Maryland Hypersonic Center of Testing Excellence program (Grant
Number: FA9550-17-1-0085), and the University of Maryland Clark Doctoral Fel-
lowship Program.
Finally, thank you to my Mom and Dad, the rest of my family, and my friends.
Your support means the world and I love you all!
iii
List of Publications
1. Ceruzzi, A. P., McManamen, B., & Cadou, C. P. ?Demonstration of Two-
Point Focused Laser Differential Interferometry (2pFLDI) in a Mach 18 flow,?
AIAA Scitech 2021 Forum. January, 2021. https://doi.org/10.2514/6.2021-
0983
2. Ceruzzi, A. P., Neisess, C., McManamen, B., & Cadou, C. P. ?Investiga-
tion of Focused Laser Differential Interferometry (FLDI) Sensitivity Function,?
AIAA Scitech 2021 Forum. January, 2021. https://doi.org/10.2514/6.2021-
1299
3. Ceruzzi, A. P., Callis, B. O., Weber, D. C., Cadou, C. P. ?Application of
Focused Laser Differential Interferometry (FLDI) in a Supersonic Boundary
Layer,? AIAA Scitech 2020 Forum. January, 2020. https://doi.org/10.2514/6.2020-
1973
4. Ceruzzi, A., & Cadou, C. P. ?Simultaneous Velocity and Density Gradient
Measurements using Two-Point Focused Laser Differential Interferometry,?
AIAA Scitech 2019 Forum. January, 2019. https://doi.org/10.2514/6.2019-
2295
5. Ceruzzi, A., & Cadou, C. P. ?Turbulent Air Jet Investigation using Focused
Laser Differential Interferometry,? 53rdAIAA/SAE/ASEE Joint Propulsion
Conference. July, 2017. https://doi.org/10.2514/6.2017-4834
6. Gillespie, G., Ceruzzi, A., & Laurence, S. ?Multi-point Focused Laser Differ-
ential Interferometry for Noise Measurements in High-Speed Tunnels,? AIAA
Aviation 2021 Forum. August, 2021.
iv
List of Contributions
1. Demonstrated the efficacy of the transfer function approach for interpreting
FLDI signals.
2. Demonstrated the efficacy of 2pFLDI for making non-intrusive measurements
of velocity.
3. Developed a methodology for identifying the limits of FLDI?s spatiotemporal
response based on Signal to Noise Ratio.
4. Demonstrated the ability to measure the velocity dispersion relation within
these limits.
5. Developed and demonstrated methods for converting between frequency and
wavenumber in FLDI measurements.
6. Improved our understanding of the capabilities and limitations of FLDI.
7. Reported first ever non-intrusive measurements of turbulence intensity and
convection velocity in a Mach 18 flow.
v
Table of Contents
Acknowledgements ii
Publications iv
Contributions v
List of Tables ix
List of Figures x
List of Abbreviations xiv
List of Symbols xv
1 Introduction 1
1.1 Boundary Layer Transition . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Free Stream Turbulence in Ground Test Facilities . . . . . . . . . . . 3
1.3 Measurement Techniques for Tunnel Noise . . . . . . . . . . . . . . . 4
1.4 Focused Laser Differential Interferometry . . . . . . . . . . . . . . . . 5
1.5 Objectives and Approach . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Laser Differential Interferometry Theory 12
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 LDI Diagnostic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Relation between voltage and phase difference . . . . . . . . . 14
2.2.2 Relation between phase difference and density . . . . . . . . . 16
2.2.3 Modeling the density field . . . . . . . . . . . . . . . . . . . . 17
2.2.4 Reduced analytical relation between phase and density . . . . 19
2.3 Focused Laser Differential Interferometry (FLDI) . . . . . . . . . . . 22
2.3.1 Spatiotemporal resolution . . . . . . . . . . . . . . . . . . . . 27
2.3.2 Wavenumber spectra . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.3 A general sensitivity function . . . . . . . . . . . . . . . . . . 30
2.4 Two-Point FLDI (2pFLDI) . . . . . . . . . . . . . . . . . . . . . . . . 31
vi
3 FLDI model validation experiments 35
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Turbulent Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 FLDI & 2pFLDI equipment and data reduction . . . . . . . . 38
3.2.2 Adjustment and measurement of beam separation ?x1 . . . . 41
3.2.3 Measurement of focal spot radius, w0 . . . . . . . . . . . . . . 42
3.2.4 Measuring beam-pair separation, ?x2 . . . . . . . . . . . . . . 45
3.2.5 Experiment and Modeling . . . . . . . . . . . . . . . . . . . . 45
3.2.6 Uncertainty and sensitivity analysis . . . . . . . . . . . . . . . 48
3.2.7 Results using modeled velocity distribution, uc(z) . . . . . . . 49
3.2.7.1 Sensitivity to beam separation (?x1) . . . . . . . . . 49
3.2.7.2 Sensitivity to position along the optical axis (z) . . . 54
3.2.7.3 Comparison to uniform velocity assumption . . . . . 57
3.2.8 Measuring the dispersion relation . . . . . . . . . . . . . . . . 59
3.2.9 Results using measured dispersion relation, uc(f) . . . . . . . 60
3.2.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Mach 3 Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.1 FLDI and 2pFLDI equipment and data reduction . . . . . . . 67
3.3.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3.3 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.5 Comparison to uniform velocity assumption . . . . . . . . . . 75
3.4 Chapter Conclusion and Takeaways . . . . . . . . . . . . . . . . . . . 77
4 2pFLDI Applications 79
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 UTSI Mach 4 Ludwieg Tube . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.1 2pFLDI equipment and data reduction . . . . . . . . . . . . . 81
4.2.2 Experiment, Modeling and Spatiotemporal resolution . . . . . 82
4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2.3.1 Convection Velocity . . . . . . . . . . . . . . . . . . 87
4.2.3.2 Wavenumber Spectra . . . . . . . . . . . . . . . . . . 89
4.2.3.3 Boundary Layer profiles . . . . . . . . . . . . . . . . 93
4.2.4 Conclusion and Takeaways . . . . . . . . . . . . . . . . . . . . 96
4.3 AEDC T9 Mach 18 Freestream . . . . . . . . . . . . . . . . . . . . . 97
4.3.1 2pFLDI equipment and data reduction . . . . . . . . . . . . . 98
4.3.2 Experiment and Modeling . . . . . . . . . . . . . . . . . . . . 99
4.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.3.1 Convection Velocity . . . . . . . . . . . . . . . . . . 104
4.3.3.2 Wavenumber Spectra . . . . . . . . . . . . . . . . . . 107
4.4 Chapter Conclusions and Takeaways . . . . . . . . . . . . . . . . . . 112
vii
5 Summary and Future Work 114
5.1 Interpreting FLDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Validating the FLDI model . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3 Demonstrations of 2pFLDI . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A Derivations 119
A.1 Derivation of FLDI transfer functions, Eq.2.24 . . . . . . . . . . . . . 119
A.1.1 Evaluation of integral over beam radius, r . . . . . . . . . . . 121
Bibliography 122
viii
List of Tables
2.1 Constants used to convert from index of refraction to density. . . . . 17
3.1 Selected parameters of FLDI experiments in this chapter. See Fig.
2.3 for reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Selected parameters of 2pFLDI experiments in this chapter. See Fig.
2.6 for reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Parameter typical value and uncertainty . . . . . . . . . . . . . . . . 48
3.4 Free stream conditions at measurement location in wind tunnel. . . . 67
4.1 Parameters of 2pFLDI experiments in this chapter. See Fig. 2.6 for
reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Facility calculated free stream (subscript ?) conditions. . . . . . . . 98
4.3 2pFLDI High Pass (HP) filter settings, and convection velocity aver-
aged across all frequencies . . . . . . . . . . . . . . . . . . . . . . . . 107
ix
List of Figures
1.1 Paths to transition in boundary layer. From Reshotko [1]. . . . . . . 2
1.2 Schematic of LDI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Schematic of FLDI in a wind tunnel. . . . . . . . . . . . . . . . . . . 6
1.4 Schematic of 2pFLDI in a wind tunnel. . . . . . . . . . . . . . . . . . 6
2.1 Schematic illustration of a Laser Differential Interferometer. (P1u/P1d)
upbeam/downbeam birefringent prism with splitting angle (?1), (FLu/FLd)
upbeam/downbeam field lens with focal length fFL, (?x1) beam sep-
aration, (LSR) length of most sensitive region. . . . . . . . . . . . . . 13
2.2 Relationship between measured voltage (V ) and phase difference be-
tween beams (??) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Schematic illustration of a Focused Laser Differential Interferometer.
Labels for the various components are as follows: (OL) objective lens,
(P1u/P1d) upbeam/downbeam birefringent prism with splitting an-
gle (?1), (FLu/FLd) upbeam/downbeam field lens with focal length
fFL, (Pol1) polarizer rotated 45
o from x-axis, (Lo) distance from ob-
jective lens focus to field lens, (Li) distance from field lens to focus,
(?x1) beam separation. . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Variation of the product of FLDI transfer functions H?xHw with
wavenumber and z for ?0 = 633nm, w0 = 7.3?m . . . . . . . . . . . . 25
2.5 Illustration of regions and sensitivity functions used to compute spa-
tiotemporal resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Schematic illustration of a Two-Point Focused Laser Differential In-
terferometer. Additional components and parameters beyond single-
point FLDI are: (P2) birefringent prism with splitting angle (?2),
(Pol2) polarizer, (CL) collimating lens with focal length fCL, (?x2)
separation between beam pairs . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Diagram of air nozzle. d = 0.125? (3.2mm), Ln = 2.5?, Dp = 1.9?,
P0 is controlled with the regulator upstream of the plenum . . . . . . 39
3.2 Comparison of beam profiles at the focus for three setups. ?x1 =
290?m is produced using the Wollaston prism. The other two beam
spacings are produced using the Sanderson prism. . . . . . . . . . . . 42
3.3 Beam intensity profiles along the z axis . . . . . . . . . . . . . . . . . 43
x
3.4 Measurements of the beam radius parameter at various points along
the z axis with the fit used to determine w0 (the beam radius at the
focus). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 Results from knife-edge test used to determine beam pair separation
(?x2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6 Schematic of turbulent jet in relation to FLDI beams . . . . . . . . . 46
3.7 Raw FLDI phase difference spectra for three beam separations ac-
quired with the instrument focus located at the jet centerline (y/d = 0
and z0/d = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.8 FLDI sensitivity function to the jet (Eq. 3.3) for three beam sepa-
rations at jet centerline (y/d = 0) and instrument focus (z0/d = 0).
w0 = 7.3?m, and ?o = 632.8nm. . . . . . . . . . . . . . . . . . . . . . 51
3.9 Average amplitude spectra interrogated by FLDI, found from Eq.
3.2. Measurements are acquired at the jet centerline (y/d = 0) and
the instrument focus (z0/d = 0) . . . . . . . . . . . . . . . . . . . . . 51
3.10 Difference between amplitude spectra for three beam separations,
normalized by amplitude from ?x1 = 35?m. Measurements are
acquired at the jet centerline (y/d = 0) and the instrument focus
(z0/d = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.11 Percent uncertainty in Hjet . . . . . . . . . . . . . . . . . . . . . . . . 53
3.12 Raw FLDI phase difference spectra acquired at jet centerline (y/d =
0) for six jet positions along the optical axis (z0) . . . . . . . . . . . . 54
3.13 FLDI sensitivity function at jet centerline (y/d = 0) for six jet posi-
tions along the optical axis (z0) . . . . . . . . . . . . . . . . . . . . . 56
3.14 Average amplitude spectra interrogated by FLDI for jet centerline
(y/d = 0) and for six jet positions along the optical axis (z0) . . . . . 56
3.15 Difference between amplitude spectra recovered for six jet positions
along the optical axis (z0), normalized by amplitude at z0 = 0 . . . . 57
3.16 Percent uncertainty in Hjet . . . . . . . . . . . . . . . . . . . . . . . . 57
3.17 Comparison of amplitude spectra computed using uniform velocity
profile (a & b) and Gaussian velocity profile (c) for x/d = 30, y/d = 0,
and six z0 positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.18 Comparison of amplitude spectra difference for uniform velocity pro-
file (a & b) and Gaussian velocity profile (c) for x/d = 30, y/d = 0,
and six z0 positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.19 Frequency-dependent convection velocities along the centerline (y =
0, z0 = 0) of a round jet. Computed from the cross-spectra of 2pFLDI
measurements (Eq. 2.48-2.50). . . . . . . . . . . . . . . . . . . . . . . 61
3.20 2pFLDI measured velocity vs. frequency acquired at jet centerline
(y/d = 0) and instrument focus (z0 = 0) . . . . . . . . . . . . . . . . 62
3.21 Phase difference spectra vs. wavenumber acquired at jet centerline
(y/d = 0) for six jet positions along the optical axis (z0) . . . . . . . 63
3.22 Sensitivity function vs. wavenumber at jet centerline (y/d = 0) for
six jet positions along the optical axis (z0) . . . . . . . . . . . . . . . 63
xi
3.23 Amplitude spectra vs. wavenumber at jet centerline (y/d = 0) for six
jet positions along the optical axis (z0) . . . . . . . . . . . . . . . . . 64
3.24 Difference between amplitude spectra for six jet positions along the
optical axis (z0) normalized by amplitude at z0 = 0 . . . . . . . . . . 65
3.25 Percent uncertainty in Hjet vs. wavenumber . . . . . . . . . . . . . . 65
3.26 Diagram of the M3CT flow path showing the location of the FLDI
measurement volume, ramp, and the coordinate directions. . . . . . . 67
3.27 Illustration of the location of M3CT (disturbance generator) within
the FLDI optical path. . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.28 Dispersion relation in M3CT flow field downstream of the ramp ?dis-
turbance generator? measured with 2pFLDI . . . . . . . . . . . . . . 71
3.29 Dispersion relation in M3CT plotted against log scale . . . . . . . . . 71
3.30 Phase difference spectra vs. frequency in M3CT for nine tunnel po-
sitions along optical axis (z0). Flow off measurement taken prior to
z0 = 48cm run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.31 Phase difference spectra vs.wavenumber in M3CT for nine tunnel
positions along optical axis (z0) . . . . . . . . . . . . . . . . . . . . . 73
3.32 Sensitivity function vs.wavenumber in M3CT for nine tunnel positions
along optical axis (z0) . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.33 Amplitude spectra vs. wavenumber in M3CT for nine tunnel posi-
tions along optical axis (z0) . . . . . . . . . . . . . . . . . . . . . . . 74
3.34 Difference between amplitude spectra for nine tunnel positions along
optical axis (z0) normalized by amplitude at z0 = 0 . . . . . . . . . . 75
3.35 Comparison of amplitude spectra computed using uniform velocity
profile (a) and using measured frequency-dependent velocity (b) for
nine z0 positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.36 Comparison of amplitude spectra difference for uniform velocity pro-
file (a) and frequency-dependent velocity (b) . . . . . . . . . . . . . . 77
4.1 The UTSI Mach 4 Ludwieg tube. . . . . . . . . . . . . . . . . . . . . 81
4.2 Schematic of experiment in UTM4 floor turbulent boundary layer . . 83
4.3 Estimate of g(z) in UTM4 floor turbulent boundary layer . . . . . . . 85
4.4 Estimate of FLDI signal to noise ratio for a 2Lc = 44cm depth of
focus in UTM4 boundary layer . . . . . . . . . . . . . . . . . . . . . . 86
4.5 Phase difference vs. time from FLDI (Ch.A) for six wall normal
positions (y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6 Phase difference vs. frequency from FLDI (Ch.A) for six wall normal
positions (y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.7 Phase difference vs. time from 2pFLDI (Ch.A & Ch.B) for y/? = 0.4 89
4.8 2pFLDI convection velocity vs. frequency for six wall normal posi-
tions (y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.9 2pFLDI convection velocity vs. frequency normalized by mean con-
vection velocity over all frequencies for six wall normal positions (y) . 90
4.10 Phase difference vs. wavenumber from FLDI (Ch.A) for six wall nor-
mal positions (y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
xii
4.11 Sensitivity and transfer functions for uniform disturbance in UTM4 . 92
4.12 Amplitude spectra in UTM4 for six wall normal positions (y) . . . . . 93
4.13 Turbulence intensity profile in UTM4 floor BL for three wavenumber
bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.14 Velocity profile in UTM4 floor BL computed using three wavenumber
bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.15 Schematic illustration of experiment in AEDC T9 Mach 18 test section100
4.16 Estimate of g(z) in AEDC T9 Mach 18 test section . . . . . . . . . . 101
4.17 Estimate of FLDI signal to noise ratio for a 2Lc = 73cm depth of
focus in T9M18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.18 Phase difference vs. time from FLDI (Ch.A) for five tunnel runs . . . 103
4.19 Phase difference vs. frequency from FLDI (Ch.A) for five runs. flow
off taken prior to run. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.20 Phase difference vs. time from 2pFLDI (Ch.A & Ch.B) for Run 4671 105
4.21 2pFLDI convection velocity normalized by freestream velocity, U? =
200m/s, vs. frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.22 2pFLDI convection velocity vs. f , normalized by average value over
all f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.23 Phase difference vs. wavenumber from FLDI (Ch.A) . . . . . . . . . . 109
4.24 Sensitivity function for uniform disturbance in T9M18 . . . . . . . . 109
4.25 Amplitude spectra in T9M18 for four runs . . . . . . . . . . . . . . . 110
4.26 Amplitude spectra in T9M18 normalized by freestream density for
four runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
xiii
List of Abbreviations
LDI Laser Differential Interferometry
FLDI Focus Laser Differential Interferometry
2pFLDI Two-point Focus Laser Differential Interferometry
AEDC Arnold Engineering Development Center
M3CT Mach 3 Calibration Tunnel
UTSI University of Tennessee Space Institute
UTM4 UTSI Mach 4 Ludwieg Tube
T9 AEDC Hypervelocity Wind Tunnel 9
T9M18 T9 Mach 18 test section
FL Field lens
OL Objective lens
Pol Polarizer
P1 FLDI Prisms
P2 2pFLDI Prism
CL Collimating lens
xiv
List of Symbols
z optical axis
f frequency
n Index of refraction
? Density of fluid
? Dimensionless Gladstone Dale constant
K Gladstone Dale constant with units of volume/mass
?0 Wavelength of laser
?? Phase difference between FLDI interferometer legs
V Voltage output of photodiode
I Intensity of light
?1 Beam splitting angle of FLDI prisms
?2 Beam splitting angle of 2pFLDI prism
?x1 Separation distance between FLDI interferometer legs
?x2 Separation distance between 2pFLDI beam pairs
w Beam radius parameter
w0 Beam radius at focus
uc Disturbance convection velocity
U mean flow velocity
g(z) non-dimensional amplitude of density fluctuations
? Standard deviation
A(f) Amplitude of density fluctuations
H FLDI transfer functions
kx x-axis wavenumber component
? Disturbance phase
F Fourier transform
RAB Cross-correlation
?AB Cross-spectrum
xv
Chapter 1: Introduction
1.1 Boundary Layer Transition
Next generation hypersonic vehicles such as long range cruise missiles, space
planes and reentry vehicles are critical technologies for our military and offer ex-
citing opportunities for space exploration and civil transport. One of the largest
challenges facing designers of such vehicles is predicting the location of laminar to
turbulent boundary layer transition on the surface of the vehicle. In hypersonic
boundary layers, transition to turbulence is associated with significant increases in
skin friction and heat transfer [2], which in turn affects predictions of drag, engine
inlet conditions, and thermal protection requirements. Uncertainty in the location
of transition leads to over-designed thermal protection systems and unpredictable
performance.
An active area of study for several decades, hypersonic boundary layer tran-
sition research involves the study of laminar boundary layer receptivity to out-
side disturbances, the growth of disturbances inside the boundary layer, and the
breakdown of a transitioning boundary layer to fully turbulent flow. Fig. 1.1, A
popular flowchart, first produced by Morkovin et al. [3] and reproduced here from
Reshotko [1] outlines the ways in which laminar boundary layers transition. Read-
1
Figure 1.1: Paths to transition in boundary layer. From Reshotko [1].
ing from left to right, as outside disturbance amplitude increases, the paths to
transition go from slow modal growth (A), to transient growth (B-D), to bypass
mechanisms (E). For a detailed explanation of these paths, the reader is directed to
Reshotko [1,4] and the references within. For the purpose of this thesis, we note that
understanding the disturbance environment and receptivity mechanisms is critical
to understanding which regime of transition a boundary layer will enter. Thus it is
critical for predicting transition.
2
1.2 Free Stream Turbulence in Ground Test Facilities
Ideal boundary layer transition experiments would involve flows at design al-
titudes with turbulence levels matching those in the atmosphere. The best way to
approximate such conditions is via free flight experiments but these are very expen-
sive. As a result, most transition experiments are conducted in ground test facilities
like wind tunnels. While tunnels can simulate free-flight Mach numbers, their free-
stream turbulence environments are often much harsher than experienced at high
altitudes. As a result, it has become widely accepted that free stream turbulence,
also called ?tunnel noise?, is a principal initiator of transition in wind tunnels [5].
It has also been shown that transition Reynolds numbers measured at the same
conditions on the same geometry often vary widely between facilities [6]. Therefore,
no study of transition is complete without a firm understanding of the tunnel noise
environment.
The origins of tunnel noise include vorticity and entropy fluctuations from the
reservoir as well as acoustic waves radiating from the turbulent boundary layers
along the tunnel walls [7, 8]. Laufer [9?11] investigated the effect of tunnel noise
on transition in supersonic ground facilities. One important takeaway is that the
effects of reservoir vorticity are negligible in comparison to the noise field radiated
from the walls at M ? 2.5. This observation led to the development of ?quiet?
supersonic and hypersonic tunnels. An overview of quiet tunnels and the role they
play in transition prediction is given by Schneider [12]. Building quiet tunnels is
expensive, and while weaker free stream turbulence better simulates free flight, a
3
proper characterization of the free stream is still essential for meaningful studies of
transition in ground test facilities.
1.3 Measurement Techniques for Tunnel Noise
The most common instruments for characterizing tunnel noise are hot wire
anemometers (HWA) and pitot pressure probes. Early examples of HWA studies
include those refrenced in the previous section by Morkovin [8] and by Laufer [10].
Stainback and Wagner [13] used both HWA and pitot probes, providing a useful
comparison of the techniques and a method for interpreting the pitot data. Modern
measurements of tunnel noise using these intrusive probes include work by Lafferty
and Norris [14], Bounitch et al. [15], Masutti et al. [16], Gromyko et al. [17], Schilden
et al. [18], and Chaudhry and Candler [19], and many others [20]. Some takeaways
from these measurements include:
? Hot-wires are fragile an may not withstand large forces in many Hypersonic
facilities.
? Pitot measurements are dependent on the probe geometry, complicating anal-
ysis and increasing uncertainty [15].
? Both techniques are intrusive and can not be placed directly upstream of a
transition experiment.
These limitations motivate the use of non-intrusive techniques for characterizing
tunnels; many are reviewed by Danehy et al. [21]. Most techniques reviewed in
4
this reference can measure mean flow quantities, but few can measure fluctuations,
i.e. tunnel noise. Fluctuations relevant to hypersonic boundary layer instabilities
such as the 2nd mode or Mack mode are high frequency (> 400 kHz in some cases
[22]) and weak (< 0.5% of free stream in some cases [23]) requiring an instrument
with a unique combination of high bandwidth, sensitivity, and spatial resolution.
These requirements motivate the application of a non-intrusive diagnostic tool called
Focused Laser Differential Interferometry (FLDI).
1.4 Focused Laser Differential Interferometry
Focused Laser Differential Interferometry (FLDI) is a diagnostic tool that
has great potential to advance understanding of tunnel noise and transition. Its
relative, Laser Differential Interferometry (LDI), is a method of measuring density
differences in fluid flows first developed and documented in the 1970s by Smeets
and George [24?27]. The LDI signal is proportional to the difference in average
density along the optical paths of two closely-spaced laser beams as illustrated in
Fig. 1.2. While very sensitive and capable of operating at bandwidths limited only
by the detector, it is an integrated line of sight measurement and thus offers no
spatial resolution along the optical (z) axis. Smeets and George sought to correct
this deficiency by introducing lenses that brought the two beams to focus (Fig. 1.3)
in such a way that the beams overlap everywhere except in a small region near
the beams? foci. This modification is called FLDI and has the effect of reducing
sensitivity to density differences away from the focal region. Smeets and George also
5
Figure 1.2: Schematic of LDI.
Figure 1.3: Schematic of FLDI in a wind tunnel.
imagined that it would be possible to measure the convection velocity of disturbances
by computing the correlation between signals from two FLDI measurement volumes
a known distance apart. We call this technique two-point FLDI (2pFLDI) and
provide a diagram of the instrument in Fig. 1.4.
The modern advancement and accessibility of lasers with long coherence lengths
Figure 1.4: Schematic of 2pFLDI in a wind tunnel.
6
and low noise and data acquisition technology with > 10MHz bandwidth has led to
renewed interest in FLDI as diagnostic for high speed flows. This FLDI ?revival? was
spurred on by Parziale et al. [23,28?30] in the early 2010s, who demonstrated mea-
surements of free stream disturbances and 2nd-Mode instabilities in a shock tunnel.
In the ten years following Parziale?s first paper, FLDI has been employed by many
others. Notable early applications were by Fulghum [31], Settles and Fulghum [32],
and Schmidt and Shepherd [33] who sought to characterize the instrument?s transfer
functions, recognizing that uncertainty in these functions would hamper usefulness
of the technique.
Other recent applications of FLDI include work by Benitez et al. [34], who
modified optics to measure 2nd-mode instability amplitudes inside a quiet tunnel
with contoured windows, Houpt et al. [35], who employed a cylindrical FLDI in order
to probe disturbances near the surface of flat walls, and Birch et al. [36] who used
shrouds to minimize the effect of sidewall turbulence on the FLDI signal. Recent
applications of 2pFLDI include those by Jewell et al. [37], who used a Koester
prism to multiplex the FLDI and probe a circular jet, Bathel et al. [38], who used
a Nomarski prism to double the FLDI and measure the disturbance speed of a 2nd
mode wave in a Mach 6 flow, and by the author [39?41] (results to be discussed
in this work). Also conceived by Smeets and George [26], multiplexing the FLDI
offers an attractive method for simultaneously probing a larger spatial domains and
making two-dimensional disturbance velocity measurements. Gragston et al. [42,43]
demonstrated a method to achieve this using a diffracting optic. Weisberger et
al. [44] demonstrated a multi-point ?line FLDI? which uses ovoid beam profiles that
7
are discretely sampled by a photodiode array.
However, and despite all of these applications, there remains no simple, widely
accepted method for quantitatively interpreting the FLDI signal and the disturbance
velocity measured by 2pFLDI in turbulent flows. In most of the applications de-
scribed above, FLDI spectra are presented in arbitrary units or there are large
uncertainties in the density fluctuations reported. Parziale?s measurements of free
stream fluctuations and 2nd mode waves are an exception but they rely on a sep-
arate experiment on a small jet to characterize the FLDI?s response function [30].
While this approach is reasonable for this particular application, it is not clear that
it would work in other flow geometries and it is not clear how the parameters of
the jet and of the FLDI affect the calibration. Nevertheless, there have been many
excellent efforts to characterize the response of the instrument over the past eight
years [31?33,45?48]. Fulghum?s dissertation [31] is filled with insights on the FLDI?s
response including the use of transfer functions to describe how the instrument filters
an arbitrary density field. Schmidt and Shepherd also explored similar analytical
methods for interpreting the signal [33] but ultimately concluded that full simulation
via ray tracing is the best way to quantitatively interpret the signal.
The most comprehensive, modern review of FLDI is Lawson?s dissertation [49],
published less than one year ago. Lawson thoroughly validates the ray tracing model
introduced by Schmidt and Shepherd. He then introduces an analytical method for
interpreting the FLDI response to acoustic waves of arbitrary orientation, in an effort
to model the noise environment in hypersonic facilities. Using this method good
agreement between FLDI and pitot measurements is observed, representing a major
8
step forward in the technique?s practicality. This work serves as a complimentary
piece to Lawson?s and builds off of the framework, terminology and notations used
there. His work gives confidence that the analytical transfer function methods can
be successful, leading to reliable methods for interpreting the FLDI signal which are
computationally cheap and easy to use.
In summary, previous work illustrates the need for a simple, easy to use pro-
cedure for interpreting the FLDI instrument. Analytical transfer functions derived
from ray tracing Gaussian beams appear to be a good candidate for this procedure
but validation of these functions in complex turbulent flows is minimal [32,33,48,49].
2pFLDI provides a direct measure of convection velocity, a critical parameter of
the transfer functions. However, an investigation of how these measured veloci-
ties can reduce the uncertainty in the single-point FLDI measurement has yet to
be conducted because applications of 2pFLDI are relatively novel compared with
single-point FLDI. The need for these procedures, validations, and investigations
motivates the goals of this work, which are outlined next.
1.5 Objectives and Approach
The objectives of this work are to:
1. develop a simple quantitative theory for interpreting the single point FLDI
signal
2. validate the theory via comparison to experiments in complex, flows such as
jets and wind tunnels
9
3. quantify the FLDI?s spatiotemporal resolution
4. expand our understanding of the 2pFLDI velocity measurement
5. apply 2pFLDI in industrial high-speed facilities.
To accomplish these objectives, we begin in Chapter 2 by reviewing LDI theory
and proposing a density fluctuation model for turbulent flows. The model assumes
density fluctuations are sinusoidal, convect parallel to the flow, and amplitude vari-
ations along the optical (z) axis and with frequency (f) are independent of one
another. This model is chosen for simplicity: It allows the governing equation
proposed by Schmidt and Shepherd [33] to reduce analytically into a tractable ex-
pression. The key difference between this model and previous work is we allow the
disturbance convection velocity, uc, to vary with z and f , i.e we accommodate ve-
locity gradients and dispersion. We propose a method for determining the FLDI
spatiotemporal resolution to an arbitrary flow using the model.
Next, in Chapter 3 we quantify our ability to remove instrument effects from
the signal measured in a turbulent air jet and the flow downstream of a ramp in
a Mach 3 wind tunnel. In the jet, two approaches for determining the convec-
tion velocity are explored: (1) model velocity profiles uc(z) based on historical jet
measurements, and (2) measure dispersion relation uc(f) using 2pFLDI. The latter
approach can be conducted in flows where historical measurements are unavailable
or uncertain, such as in the Mach 3 tunnel. In many cases, our model matches
measured amplitudes within ?50%, which we take as sufficient validation.
In Chapter 4 we demonstrate applications of 2pFLDI in a Mach 4 Ludwieg
10
tube and Mach 18 blowdown facility. Using our method for determining spatial-
temporal resolution, results from each of these facilities are put in proper context.
The latter facility represent first-of-their-kind application of 2pFLDI.
Finally, in Chapter 5 we conclude, review implications of our results, and
discuss future directions for research and development.
11
Chapter 2: Laser Differential Interferometry Theory
2.1 Introduction
This chapter is organized as follows: first we discuss the Laser Differential
Interferometer (LDI) including how it works, what a set-up looks like, and how to
relate the output signal measured to the index of refraction field probed by the
instrument. Next we move to the Focused Laser Differential Interferometer (FLDI)
and explain the instrument?s set-up and methods for interpreting the signal. Much
of the FLDI set-up and interpretation builds off of ideas introduced by the LDI, so
proper understanding of both instruments should begin with a firm understanding
of the LDI. Finally, we explain the two-point FLDI and how it improves upon the
capabilities of a single-point FLDI.
2.2 LDI Diagnostic
The Laser Differential Interferometry (LDI) signal is proportional to the finite-
difference approximation of the index of refraction gradient (?n/?x) integrated along
the line of sight. Fig. 2.1 shows the basic LDI set-up which provides the foundation
for FLDI and 2pFLDI. The light source is a monochromatic continuous wave (CW)
12
Figure 2.1: Schematic illustration of a Laser Differential Interferometer. (P1u/P1d)
upbeam/downbeam birefringent prism with splitting angle (?1), (FLu/FLd) up-
beam/downbeam field lens with focal length fFL, (?x1) beam separation, (LSR)
length of most sensitive region.
laser linearly polarized at 45? with respect to the axis that defines the gradient of
interest. A circularly polarized beam is also acceptable. If the laser does not output
a linearly polarized beam at 45? or a circularly polarized beam, a polarizer [28] or
wave-plate [38] is required up-beam of the first birefringent prism (P1u). This prism
splits the beam into two equally intense orthogonally polarized beams that diverge
at a small angle (?1). A field lens (FLu) located one focal length (fFL) down-beam
of the prism arrests the divergence and fixes the distance between the two beams
(?x1). This distance is given by:
(? )1
?x1 = 2fFL tan . (2.1)
2
The beams propagate through a region of interest of length (LSR) where index of
refraction gradients create an optical path difference - and thus a phase difference
(??) - between the beams. Beyond the region of interest, the orthogonally polarized
beams are recombined by a field lens (FLd) and birefringent prism (P1d) identical
to FLu and P1u but operating in reverse. The recombined beams pass through a
13
polarizer aligned with the laser polarization (45? from the splitting axis) allowing
the beams to interfere.
2.2.1 Relation between voltage and phase difference
The intensity of light (IPD) incident on the photodiode is a function of the
phase difference as follows:
IPD = 2Ib(1 + cos(??)), (2.2)
where (Ib) is the intensity of a single beam (p. 259 of [50]). In practice, perfect
interference contrast is not achievable and the intensity will not reach zero when
?? = ?. Also, high-frequency signals will be attenuated by the photodiode circuit?s
response. Thus, the voltage output from the photodiode (V ) is expressed as:
1
V = [Vmax + Vmin + (Vmax ? Vmin) cos(??)HPD] , (2.3)
2
where HPD is a transfer function that accounts for the photodiode circuit?s response.
Voltage is plotted as a function of phase difference in Fig. 2.2. Using the same
terminology as Parziale [28], we define
Vmax + Vmin
V0 = . (2.4)
2
Adjusting the phase by translating P1u or P1d along the x-axis so that V = V0
ensures that the instrument?s response is approximately linear over its sensitive
region i.e. V ? V (t = 0) ? ?? ? ??(t = 0). This adjustment is critical because
it maximizes sensitivity, simplifies interpretation of the signal, and minimizes the
likelihood of phase ambiguity which occurs when the signal reaches Vmax or Vmin
14
Figure 2.2: Relationship between measured voltage (V ) and phase difference be-
tween beams (??)
during an experiment. The relationship between voltage measured over time and
phase difference can now be written as
V (t) = V0 + V0 sin(??(t))HPD. (2.5)
Fulghum measured the response of the same detector system used in this work (a
Thorlabs DET36 photodiode with a 1k? resistor) to a LED pulsed square wave for
frequencies to 500kHz [31] and found it to be flat. Thus, we assume
HPD = 1 for f < 500kHz (2.6)
which means that we can ignore the response of the photodiode circuit for frequencies
below 500kHz. This allows us to write
( )
V (t)
??(t) = arcsin ? 1 , (2.7)
V0
which is a common assumption in other FLDI work [23,33,34,36].
15
2.2.2 Relation between phase difference and density
For two beams traversing paths s1 and s2 respectively, the phase difference at
the detector face (s = D) associated with the index of refraction field n(x) is given
by ? ? [?
2? D
? D ]
??(t) = I0 n(x1, t)ds1 ? n(x2, t)ds2 d?d?, (2.8)
?0 D S S
where ?0 is the laser wavelength, S is the location of the light source, xi(si) =
(xi, yi, zi), and I0(?, ?) is the intensity on the face when there is no phase difference
between the two beams. ? and ? are coordinates which map the detector surface.
Eq. 2.8 describes a general interferometer and is taken directly from Schmidt and
Shepherd [33] who derived it from Born and Wolf [50]. The relationship between
index of refraction and gas density is given by:
?
n = 1 + ? , (2.9)
?s
where ?, a dimensionless constant, is a function of gas composition and light wave-
length (p. 154 of [51]) and ?s is density at 0
?C and 760mmHg. ? is often shorted to
the Gladstone-Dale constant, K = ?/?s [52,53]. Table 2.1 gives the constants used
in this work. Thus, we can express Eq. 2.8 in terms of the density field, ?(x, t):
? ? [? ? ]
2?K D D
??(t) = I0 ?(x1, t)ds1 ? ?(x2, t)ds2 d?d?. (2.10)
?0 D S S
One can solve for the exact phase change provided a complete, well-resolved solution
for ?(x, t) is available. However, there is no way to solve Eq. 2.10 in the other
direction ie. for ?(x, t) from ??(t). This is because many density fields can result
16
Parameter Air Nitrogen Units
? 2.91 2.96 10?4 (-)
?s 1.29 1.25 kg/m
3
K 2.25 2.37 10?4 m3/kg
Table 2.1: Constants used to convert from index of refraction to density.
in the same measured phase difference. Thus, in order to invert Eq. 2.10 and solve
for some aspect of the density field, we must make assumptions about the field itself.
2.2.3 Modeling the density field
Previous work [33] has modeled disturbances as sinusoidal waves with frequency-
dependent amplitudes that vary in time and in space but only along the flow direc-
tion (x). Lawson [49] extends this analysis to account for waves propagating at any
angle. While this may be appropriate for homogeneous free-stream turbulence and
acoustic noise, it may be less so for flows with substantial variations along the beam
direction (z). Settles and Fulghum [32] suggested modeling an amplitude variation
with z, such as a Gaussian model for a jet. Building on these previous works, we
propose the following model for the density field
? ?
?(x, y, z, t) = g(z) A(f) exp {i[kxx? 2?ft+ ?]} df (2.11)
??
where kx is the disturbance wavenumber oriented parallel to the flow direction (x),
? = ?(f) is the disturbance phase, A(f) is the disturbance amplitude (which de-
pends on the frequency f), and g(z) is a unit-less function which models the distur-
17
bance amplitude variation with z, averaged across all frequencies. Wavenumber is
defined as
2?f
kx = . (2.12)
uc
where uc is the streamwise convection velocity of a disturbance. Thus, Eq. 2.11 can
also be expressed as
? ?
?(x, y, z, t) = g(z) A(f) exp {i [kx (x? uct) + ?]} df. (2.13)
??
While it is not explicitly clear in Eq. 2.13, allowing the disturbance convective
velocity to vary with z and f , i.e. uc = uc(z, f) differentiates this model from those
used in previous work. In other words, we are allowing for velocity gradients and
dispersion.
For comparison, we also include the model used by Lawson [49] (Eq. 5.45 in
his dissertation), given by
? ?
?(x, y, z, t) = A(f) exp {i [k ? x? 2?ft+ ?]} df. (2.14)
??
k = (kx, ky, kz) is the full wavevector with magnitude k and direction defined by
angles ? and ? such that
kx = k cos? cos ? (2.15)
ky = k sin? (2.16)
kz = k cos? sin ?. (2.17)
A similar model is also explored by Hameed and Parziale [48]. Note that this model
does not include g(z), while our model does not include ky and kz. In other words,
18
Lawson assumes disturbance magnitudes are uniform along the optical axis and we
assume disturbances are oriented in the x-direction. While accounting for the full
wavevector is more physically accurate, it also leads to a more complex equation.
Our analysis will use the density field given by Eq. 2.11 because our goal is to find
a concise reduction of Eq. 2.10 that leads to a simple model of the instrument.
2.2.4 Reduced analytical relation between phase and density
In order to reduce Eq. 2.10, we also need to model the intensity distribution
in the beam. We use a Gaussian profile given(by )
2 2r2
I0(r, ?) = exp ? , (2.18)
?w2 w2
where w is the Gaussian beam radius parameter [54]. The polar coordinates r and
? are related to the Cartesian coordinates
x = r cos ? (2.19)
y = r sin ?. (2.20)
If we restrict ourselves to the region between field lenses FLu and FLd, the paths
of complimentary rays in the interferometer can be parameterized by the z-axis and
written as ( )
?x1
x1(z) = (x+ , y, z) (2.21)2
?x1
x2(z) = x? , y, z . (2.22)
2
Returning to Eq. 2.10, re-writing the 2-D integral in polar coordinates, and exchang-
ing the order of integration yields the following expression for the phase difference
19
measured by the instrument in terms of the local density field:
? [? ?
2?K LSR/2 2? ?
??(t) = I0?(x1, t)rdr
?0 ?LSR/2 0 0 ?d??2? ? ? ]
I0?(x2, t)rdrd? dz. (2.23)
0 0
Introducing the model of the density field (Eq. 2.13), Eq. 2.18-Eq. 2.22, and
simplifying the integrand analytically leads to:
? L /2 [ ?2?K SR ?
??(t) = g(z) 2A(f)?
?0 ?LSR/2 ??( ) ( ) ]
k2? xw
2 kx?x1
exp sin e?2?iftiei?df dz. (2.24)
8 2
The details of this derivation are presented in Appendix A.1. This form is useful
because it isolates the effects of the beam radius (w) and the beam spacing (?x1)
on the signal through the exponential and sinusoidal functions in the integrand.
It is convenient to represent these dependencies as transfer functions H?x and Hw
respectively where ( ) ( )
kx?x1 ?f?x1
H?x = sin = sin , (2.25)
2 uc
and ( ) ( )
k2 2 2 2 2
H x
w ? w f
w = exp ? = exp ? (2.26)
8 2u2c
These functions have been derived in previous works [23, 31, 33, 45] using slightly
different definitions and terminologies. For example, H?x is equivalent to the cor-
rection factor c in [28] and (H i??xs ) in [33]. Schmidt and Shepherd refer to this2cr
as the differentiation performed by the instrument. Hw describes the spatial filter-
ing caused by integrating over the local beam intensity profile I0(r, ?). Fulghum
20
and Schmidt and Shepherd [31, 33] derive the version of Eq. 2.26 based on the
disturbance wavenumber (kx) and beam radius parameter (w). The version of Eq.
2.26 that uses the disturbance frequency (f) and its convective velocity (uc) is not
presented in these works, though it is implied. All approaches assume that the dis-
turbances are sinusoidal. A concise validation of Eq. 2.26 was recently performed
by Lawson et al. [45] using acoustic waves.
Returning to Eq. 2.24, we exchange the order of integration again and use the
transfer functions as short-hand to write:
? ? ?4?K LSR/2
??(t) = A(f) g(z)H?xH
?2?ift i?
wdze ie df (2.27)
?0 ?? ?LSR/2
Note the similarity of Eq. 2.27 to the Fourier transform of a time dependent signal:
? ?
F [ ] = [ ]e2?iftdt. (2.28)
??
Therefore, taking the Fourier transform of Eq. 2.27 eliminates the e?2?ift term. If
we restrict our interest to the magnitude of disturbances (i.e the magnitude of the
Fourier transform) and make the additional assumption that phases of the distur-
bances ?(f) are random and uncorrelated - which is a reasonable assumption in
many turbulent flows (see Chapter 5 of [49]) - then |iei?| = 1 and 2.27 becomes:
?
F 4?K
LSR/2
[??] = A(f) |g(z)H?xHw|dz. (2.29)
?0 ?LSR/2
Eq. 2.29 is a concise description of the response of the LDI instrument to a flow
whose density field can be modeled by Eq. 2.11. The left-hand-side is the Fourier
transform of the measured signal. The right hand side is the response of the instru-
ment which is a function of three instrument parameters:
21
1. ?0 - the laser wavelength
2. w - the local beam radius
3. ?x1 - the beam separation
and three flow field parameters:
1. g(z) - frequency-averaged density disturbance amplitude variation with z.
2. uc(z, f) - convective (x-axis) disturbance velocity variation with z or f .
3. A(f) - density disturbance amplitude spectra.
All instrument parameters are constant across the interval ?LSR/2 ? z ? LSR/2
in the LDI system illustrated in Fig. 2.1. Thus, any variation in sensitivity along
the optical axis is caused by flow parameters only. This poses a problem in wind
tunnel applications where strong fluctuations associated with turbulence along the
tunnel walls can overwhelm the signals from the weaker fluctuations in the core flow
that are of interest. However, this is a problem that affects all line of sight optical
techniques.
2.3 Focused Laser Differential Interferometry (FLDI)
FLDI is a variant of LDI designed to overcome the line of sight problem. It
works by expanding the interferometer beams to large diameters and then focusing
them such that they overlap along most of their optical paths save for small regions
near the focal points. This has the effect of limiting the sensitive region (LSR) to a
22
Figure 2.3: Schematic illustration of a Focused Laser Differential Interferometer.
Labels for the various components are as follows: (OL) objective lens, (P1u/P1d)
upbeam/downbeam birefringent prism with splitting angle (?1), (FLu/FLd) up-
beam/downbeam field lens with focal length fFL, (Pol1) polarizer rotated 45
o from
x-axis, (Lo) distance from objective lens focus to field lens, (Li) distance from field
lens to focus, (?x1) beam separation.
small area around the focal points. A schematic illustration of the FLDI instrument
is presented in Fig. 2.3. An objective lens (OL) with short focal length (fOL) located
at a distance fOL + Lo up-beam of the field lens expands the beams so that they
overlap substantially reaching a max diameter (DFL) at the field lens. The field lens
focuses the beams at a distance Li down-beam of the field lens. The relationship
between Lo, Li, and fFL is given by the thin lens equation:
1 1 1
+ = . (2.30)
Lo Li fFL
The only difference between LDI and FLDI is that the beam diameter is now
a function of distance along the optical axis, i.e. w = w(z). This function is given
by [ ( ) ]2
?0z
w(z)2 = w20 1 + , (2.31)?w20
23
where w0 is the beam radius at the focus (z=0) [54]. w0 can be determined by
measuring the beam diameter at any known position z and solving Eq. 2.31 for
w0. Several measurements are better than one for determining w0. Substituting Eq.
2.31 into Eq. 2.26 gives:
( ) ( )
k2w2 k2?2z2 ?2f 2w2 f 2?2z2
Hw = exp ? x 0 ? x 0 = exp ? 0 ? 0 (2.32)
8 8?2w2 2u2 2 20 c 2w0uc
With this context, let us re-examine Eq. 2.29 applied to the region between the field
lenses of an FLDI:
?
F 4?K
Li
[??] = A(f) |g(z)H?xHw|dz. (2.33)
?0 ?Li
The right hand side describes the attenuation of the true density fluctuation spec-
trum by the instrument. H?x captures the effect of beam separation which at-
tenuates response as the disturbance wavenumber moves away from ?/?x1. Hw
captures the effect of focusing which causes high wavenumbers to be attenuated
faster than smaller ones as one moves away from the focus. The combined effects of
H?x and Hw on the response of one FLDI set-up used in this work are illustrated
in Fig. 2.4 which is a contour plot of the response of the instrument as a function
of distance along the optical axis (z-direction) and the wavenumber of the distur-
bance. The spatial resolution in the z-direction (taken as the full-width half max
of H?xHw) is ? ?w0mm for kx = ?/?x1 and improves slightly as wavenumber in-2?0
creases (wavelength decreases). As wavenumber decreases below ?/?x1, resolution
degrades while the overall response decreases. This has the effect of reducing the
range of wavenumbers to which the instrument is sensitive far from the focus.
24
Figure 2.4: Variation of the product of FLDI transfer functions H?xHw with
wavenumber and z for ?0 = 633nm, w0 = 7.3?m
25
One objective of this work is to understand if the combined transfer function,
H?xHw accurately models the FLDI?s response to an arbitrary flow field. To do
this, we define the ?FLDI-flow-field?sensitivity function?:Li
Hflow = |gflowH?xHw|dz. (2.34)
?Li
Hflow has units of length, and is proportional to FLDI signal strength. In this way
it is similar to an ?effective? integration length of the instrument. However, it does
not represent a physical integration length or spatial resolution, because it models
signal sensitivity to beam separation in addition to sensitivity along the optical axis.
Thus, two FLDIs with different beam separations could have the same Hflow, but
would not necessarily have the same spatial resolution. Determining the FLDI?s
spatiotemporal resolution is discussed in section 2.3.1.
To evaluate this integral, we need estimates of gflow(z) and uc,flow(z, f) for
the flow field. While these may be readily available for well-studied flows such as
turbulent jets, they are far less so for complex environments such as high-speed wind
tunnels. We will investigate the accuracy of this model for both examples. Hflow
is evaluated by plugging in the instrument parameters, ?0, w0, ?x1, as well as the
flow models, gflow(z) and uc,flow(z, f) into the right side of Eq. 2.34. If the flow
models are simple, the integral may be evaluated analytically. More generally, the
integral can be evaluated numerically. The result will be a function of frequency,
i.e. Hflow(f), and once found Eq. 2.33 can be re-arranged to give an expression for
the density fluctuation spectrum in the flow:
?0 F [??](f)
Aflow(f) = (2.35)
4?K Hflow(f)
26
Aflow(f) is the amplitude of the density fluctuation at frequency f . Eq. 2.35
deconvolves the effects of beam size and beam separation from the signal so Aflow
may also be thought of as the spectrum measured by an ?ideal? LDI with a beam
radius and beam separation of zero. Note that by deconvolving the spatial filtering
associated with beam size, we are reversing the focusing effects of the instrument.
Therefore, Eq. 2.35 alone is not enough to interpret the FLDI signal when spatial
resolution along the optical axis is desired.
2.3.1 Spatiotemporal resolution
While the transfer functions H?x and Hw, plotted in Fig. 2.4, comprehensively
describe FLDI?s response to disturbance wavenumbers modeled by Eq. 2.11, they
do not tell us which frequencies in the signal can be resolved at what depth of focus,
i.e. the spatiotemporal resolution. In order to determine this, we need to know
how wavenumbers in the flow map to frequencies in the signal (from the convection
velocity uc(z, f)) and whether signals from disturbances far from the focus can
overwhelm signals from disturbances near the focus (from g(z) and the transfer
functions). The following is a method for determining spatiotemporal resolution
when uc and g are known:
1. Choose a desired depth of focus with length 2Lc.
2. Break the flow field sensitivity function integral (Eq. 2.34) into three parts
giving: ? ?L ?c L ?c Li
Hflow = |gH?xHw|dz + |gH?xHw|dz + |gH?xHw|dz. (2.36)
?Li ?Lc Lc
27
Define the following sensitivity functions as illustrated in Fig. 2.5:
? Lc
Hm = ? |gH?xHw|dz (2.37)?Lc?Lc
Hrl = ? |gH?xHw|dz (2.38)?LiLi
Hrr = |gH?xHw|dz (2.39)
Lc
such that Eq. 2.36 is expressed as
Hflow = Hrl +Hm +Hrr, (2.40)
Hm represents sensitivity to the region of the flow field you want to measure
while and Hrl and Hrr represent the sensitivity to regions of flow you want to
reject (to the left and right of the focus, respectively). Abbreviate further by
defining Hr = Hrl +Hrr as the sensitivity to both regions of flow you want to
reject. The overall FLDI-flow-field sensitivity function is now:
Hflow = Hm +Hr. (2.41)
3. Compute the ratio Hm/Hr as a function of frequency. This is a signal to
noise ratio where the numerator represents the contribution to the signal from
disturbances within the desired depth of focus (?Lc < z < Lc) and the de-
nominator represents the contribution to the signal from disturbances outside
of the depth of focus. We achieve our desired spatial resolution to disturbance
frequencies when Hm/Hr ? 1.
This process give us a band (or bands) of frequencies over which we are able to
achieve the desired resolution, 2Lc. In other words, it provides the spatiotemporal
28
Figure 2.5: Illustration of regions and sensitivity functions used to compute spa-
tiotemporal resolution
resolution of the instrument. In Chapter 4 we will demonstrate how this method is
applied in real flows. Finally, because FLDI rejects high wavenumbers (and thus high
frequencies) more rapidly than low wavenumbers, the band of frequencies resolved
at a desired depth of focus is likely to be centered at higher frequencies than the
bands which are not resolved. Put another way, low frequency signals are more
likely than high frequency signals to be corrupted by disturbances away from the
focus.
2.3.2 Wavenumber spectra
A consequence of allowing for velocity gradients in our model (i.e. uc(z)) is
that there is not a one-to-one relationship between wavenumber and frequency. This
is because two different disturbance wavenumbers can convect at different speeds
thereby mapping to the same frequency and vise versa. In this case, the expression
for H?x and Hw in terms of frequency must be used and the sensitivity function,
29
Hflow, and amplitude spectra, Aflow, cannot be expressed in terms of wavenumber.
Wavenumber spectra may be more desirable because they are easier to compare
to turbulence models and spatial features in the flow. These can only be obtained
from FLDI measurements by neglecting velocity gradients (assuming ?uc/?z = 0).
In this case, the convection velocity can be constant, or dispersive, i.e. uc(f) and
the frequencies in the signal can be converted directly to wavenumbers per Eq. 2.12.
The governing equation can then be expressed as
?0 F [??](kx)
Aflow(kx) = , (2.42)
4?K Hflow(kx)
where Aflow(kx) is the wavenumber spectra.
2.3.3 A general sensitivity function
Up to this point, our analysis has assumed that the flow field can be modeled
by Eq. 2.11. As mentioned in section 2.2.3, an alternative model of acoustic waves
convecting at an arbitrary angle (Eq. 2.14) is explored by Lawson [49]. In this case
the governing FLDI equation (Eq. 2.8) can be reduced to the same form as Eq. 2.35
such that we can write
?0 F [??]
Aaco = (2.43)
4?K Haco
where Aaco is the amplitude spectra of acoustic waves spanning a length, L, and
Haco is the FLDI sensitivity function to those acoustic waves, given by
?? { [ ]}2 2?3/2w w20 16?2k2H = H exp ? 0 k2 + k2 + zaco ?
k2 2
?x
+ k ? 8 x{[ y (k2 + k2x y 0 x y)?0 ]}
(k2x + k
2
y)?L?20 + i ? 4?2k w2R ? z 0 . (2.44)
2 2? k2 2x + ky?0w0
30
Eq. 2.44 is adapted from Eq. 5.21 in Lawson?s dissertation and readers are directed
there for more details on the derivation [49]. The bigger takeaway is that a general
form exists. The FLDI signal can be modeled as
F [??] = A ?H. (2.45)
where H represents a general sensitivity function that is not necessarily described
by Hflow (Eq. 2.34) or Haco (Eq. 2.44). Alternative sensitivity functions are found
by altering the flow field model (see section 2.2.3) and then analytically reducing
the governing equation (Eq. 2.8) in a process similar to the one described in this
chapter. We find the flow-parallel disturbance model (Eq. 2.11) and resulting
sensitivity function, Hflow, strikes an appropriate balance between simplicity and
accuracy (accuracy is evaluated in Chapter 3). While further investigation of a more
general form is intriguing, it is considered outside of the scope of this work.
2.4 Two-Point FLDI (2pFLDI)
Fig. 2.6 is a schematic illustration of a typical 2pFLDI used in this work. Many
components are shared with, and in the same location as in the single point FLDI,
shown in Fig. 2.1. Down-beam of the objective lens (OL), an additional birefringent
prism (P2) splits the beam to create a second FLDI ?channel?. A polarizer (Pol2) (or
a quarter-wave-plate rotated at 45? or a half-wave plate rotated at 22.5?) is needed
to reset the polarization before the up-beam prism (P1u). Many variants of 2pFLDI
set-ups exist: P2 can be placed up-beam of the objective lens (OL) [55], or replaced
with a Koester prism [37], Nomarski prism [38], or diffracting optic [43]. For the
31
Figure 2.6: Schematic illustration of a Two-Point Focused Laser Differential Inter-
ferometer. Additional components and parameters beyond single-point FLDI are:
(P2) birefringent prism with splitting angle (?2), (Pol2) polarizer, (CL) collimating
lens with focal length fCL, (?x2) separation between beam pairs
32
set-up diagrammed here, P2 should be as close to P1u as possible (with room for
the polarizer/wave plate in between).
The desired result is a set of beam pairs that come to focus along one line on the
x-axis as shown in Section C-C of Fig.2.6. Down-beam of the focal volumes, another
optical element like a collimating lens (CL) is required to divert the two FLDI beams
to separate photodetectors. We now have two independent FLDI signals A & B with
phase differences ??A(t) and ??B(t) respectively. The beam pair foci are separated
by a distance (?x2).
The normalized cross-correlation between signals A & B is given by:
? T ???A(t)???B(t+?t)RAB(?t) = dt, (2.46)
0 |?? |2 |?? |2A B
where ?t is the time shift between signals. In general, convection velocities are found
from cross (also called space-time) correlations between signals in turbulent flow [56].
The convective velocity (uc, in the direction of the x-axis) is found by dividing the
beam separation by the time shift which maximizes the cross-correlation:
? ? ?x2uc = . (2.47)
?t[max(RAB)]
The use of angled brackets, ??, emphasizes that this measurement is a time-averaged
velocity over the sample period T . Taylor famously hypothesized and later showed
[57] that turbulent structures in flows with low turbulence intensity (u?/U ? 1) and
negligible mean shear (kxU ? dU ) [58] such as the those produced behind a grid indy
a wind tunnel convect with the mean flow velocity such that uc = U regardless of
probe separation, ?x2.
33
In shear-generated turbulent flows like those found in boundary layers and
jets, the convection velocity is more complex and varies with probe separation as
well as turbulent length and time scales [59, 60]. Wills [61] discusses the ambi-
guity of defining a convection velocity in shear flows and suggests that either a
wavenumber-dependent velocity, uc(kx), or a frequency-dependent velocity, uc(f),
is more physically meaningful. The 2pFLDI can measure frequency-dependent con-
vection velocity either by bandpass filtering signals A and B [40] or by computing
the cross-spectrum of A and B [62]. The cross-spectrum is given by the Fourier
transform of the cross-correlation:
? ?
? (f) = R (?t)e?2?if?tAB AB d?t. (2.48)
??
The phase of the the cross-spectrum is given by
[ ]
I(?AB)
??(f) = arctan , (2.49)R(?AB)
where I and R denote the imaginary and real parts, respectively. The frequency-
dependent convection velocity is given by:
2??x2f
uc(f) = . (2.50)
??
Both methods (ie. ?uc? and uc(f)) are used to compute the convective velocity from
the 2pFLDI signals.
34
Chapter 3: FLDI model validation experiments
3.1 Overview
In this chapter, we seek to quantify the accuracy of our governing equation
(Eq. 2.33 which is derived from our FLDI model, Eq. 2.8) and our flow field model,
Eq. 2.11. The FLDI model has been validated by others [33,49,63] but our flow field
model has not. Thus we expect the results obtained in this chapter to be a reflection
of the accuracy of the latter. To accomplish this, we apply FLDI in two flow fields:
(1) a round transonic turbulent jet and (2) the wake behind an 24 degree ramp inside
a small Mach 3 calibration wind tunnel (M3CT). In both the jet and the tunnel, we
vary the location of the flow along the instrument?s optical axis (in our coordinate
system this is the z-axis) from the focus (z = 0) to some maximum value, zmax. In
the jet, we also vary the single-point FLDI beam separation, ?x1. The right-hand
side of Eq. 2.33 models how instrument parameters affect the measurement while
the left-hand side, Aflow, should be independent of instrument parameters like ?x1
and w (the local beam radius) which varies along the z-axis according to Eq. 2.31.
In the following experiments, we will check to see if this is the case. The instrument
parameter we do not vary in this work is the laser beam wavelength, ?0. All lasers
in this work are Helium Neon (HeNe) with ?0 = 632.8nm. Checking to see that the
35
Parameter Jet M3CT Units
max velocity 300 600 m/s
flow width 3.2 - 25 64 mm
laser power 0.8 21 mW
fOL 15 9 mm
fFL 260 400 mm
?x1 35, 100, 290 470 ?m
Li 51 140 cm
zmax 22 48 cm
w0 7.3 14 ?m
w(zmax) 6 7 mm
Table 3.1: Selected parameters of FLDI experiments in this chapter. See Fig. 2.3
for reference
model can also account for variations in ?0 is a good subject for future work.
Table 3.1 gives an overview of selected flow and FLDI parameters for the
experiments in this chapter. For the max velocity of the jet and tunnel we use the
velocity at the nozzle exit and the freestream velocity, respectively. For the jet,
we take measurements at several locations downstream of the nozzle, which has a
diameter d = 3.2mm. Since the jet broadens as one moves downstream, the width
of the flow increases from 3.2mm to approximately 2?jet,? = 25mm at the furthest
downstream position measured (50 jet diameters downstream).
As discussed in the previous chapter, the convection velocity, uc, is a critical
36
Parameter Jet M3CT Units
fOL 15 9 mm
fFL 125 400 mm
?x1 145 470 ?m
Li 17 140 cm
w0 3 14 ?m
?x2 1.2 1.3 mm
Table 3.2: Selected parameters of 2pFLDI experiments in this chapter. See Fig. 2.6
for reference
parameter in our governing equation. For the jet, we investigate two approaches
for determining uc. The first is to model a velocity distribution, uc(z), based on
historical measurements by others [64?67]. The second is to measure the dispersion
relation, uc(f), using 2pFLDI. For the M3CT, only the dispersion relation is used.
Table 3.2 gives an overview of the 2pFLDI parameters used in both experiments.
Note that the 2pFLDI in the jet is different than the single point FLDI while the
M3CT uses the exact same set-up (detailed in table 3.1).
3.2 Turbulent Jet
The flow discussed in this section is produced by the round jet illustrated
in Fig. 3.1. Compressed air enters a 48mm (1.9?) ID plenum and accelerates
through a smoothly converging nozzle with an exit diameter of 3.175mm (0.125?).
A diaphragm-type pressure regulator equipped with a strainer to trap debris and a
37
Bourdon tube gauge (accuracy ?2%) maintains a constant pressure of 12 ? 1psig
(26.7 psia, 1.8 atm, 184 kPa) in the plenum. This pressure is chosen to maximize
Reynolds number while avoiding supersonic flow at the jet exit. The resulting
nozzle exit velocity, Mach number, and Reynolds number based on jet diameter
are 306? 7m/s, 0.96? 0.01, and 8.55? 0.34? 104 respectively assuming isentropic
flow through the nozzle.
The air jet is oriented vertically and mounted on a motorized 3-axis Velmex
Bi-Slide stage. This enables the jet to be moved in three dimensions through the
FLDI measurement volume which remains fixed. The x-axis is the centerline of the
jet and parallel to the line connecting the centers of the four focal volumes. Positive
x points up with respect to the laboratory and downstream with respect to the jet
flow. The y and z axes are perpendicular to the x-axis with z parallel to the FLDI
optical axis and y perpendicular.
3.2.1 FLDI & 2pFLDI equipment and data reduction
The FLDI discussed in this section uses a 0.8mW HeNe laser, N-BK7 lenses
(OL, FL) from ThorLabs, Polarizers from Thorlabs (LPVISE100-A), and a 13mm2,
14ns rise-time photodetector from ThorLabs (DET36A). For beam splitting/recombining
(P1u & P1d) we alternate between Wollaston prisms from United Crystals with a
splitting angle of 4arcmin and custom made stress-birefringent ?Sanderson? prisms
[31,32,68] which allow splitting angles below 4arcmins.
The 2pFLDI uses much of the same equipment as the single point FLDI. All
38
Figure 3.1: Diagram of air nozzle. d = 0.125? (3.2mm), Ln = 2.5?, Dp = 1.9?, P0
is controlled with the regulator upstream of the plenum
39
beam splitting is accomplished with custom made quartz Wollaston prisms (P1u,
P1d with 4arcmin divergence angle & P2 with 10
o divergence angle) from United
Crystals and the optical axis length of the set-up is shortened to increase the con-
vergence angle of the beam thereby improving rejection of disturbances away from
the focus. P2 is located 15mm up-beam of OL. Tables 3.1 & 3.2 give all relevant
parameters for both set-ups.
The photodetector signals are terminated at 1 k? and digitized by a 16-bit
Picoscope 5444A. The sampling frequency varies but is always between 1 and 4
MHz. Raw voltages are downloaded from the scope and processed using MATLAB.
Voltage is converted to phase difference using Eq. 2.7.
Fourier transforming the phase difference fluctuations is required for the signal
analysis described in chapter 2. This is accomplished by taking the square-root of
the power spectral density, which we estimate using Welch?s method [69] via the
MATLAB function pwelch with 3.3ms Hamming windows and 50% overlap. This
window length is chosen in order to have a sufficiently large number windows to
average spectra over, while still maintaining resolution to ? 1kHz time scales.
For 2pFLDI, the MATLAB function xcorr computes the normalized cross-
correlation (Eq. 2.46) between two signals. If the correlation peak is less than 0.1,
the signals are considered noise-dominated and velocity is not computed. This pro-
cess reduces the appearance of non-physical velocities (say, much larger than the
local mean velocity or less than zero) which could arise from weak correlation be-
tween unknown noise sources. Otherwise, the index of maximum correlation divided
by the sampling frequency gives the time lag (?t) between the two signals enabling
40
the disturbance velocity, ?uc?, to be calculated using Eq. 2.47. For the dispersion
relation (Eq. 2.50), the MATLAB function cpsd computes the cross-spectrum (Eq.
2.48). Like the power spectrum, this method also uses Welch?s method with 3.3ms
windows and 50% overlap. The cross spectra phase (Eq. 2.49) can only take val-
ues from ?? to ?. This will give accurate velocity measurements for kx < ?/?x2.
Phase ambiguity will occur for wavenumbers above this range. This is corrected by
assuming that the velocity variation with wavenumber is smooth and continuous.
In practice, as frequency increases (hence wavenumber increases), 2? is added to
the phase every time there is a discontinuous jump from ? to ??. This process is
described in more detail by Buxton et al. [70].
3.2.2 Adjustment and measurement of beam separation ?x1
Beam separation is varied by replacing the Wollaston prisms (whose splitting
angles are fixed) with variable-angle stress-birefringent Sanderson Prisms [68]. FLDI
measurements are made using three values of ?x1. Beam separation is measured
by imaging the beams at the focus using a beam profiler camera (Newport LBP2)
with a 3.69?m? 3.69?m pixel size. Sample images are presented in Fig. 3.2. Two
Gaussian distributions (one for each beam) are fit to the intensity distributions in
the x-y plane. The difference between the centers of each Gaussian distribution
is the beam separation. The uncertainty is approximately equal to the pixel size
(3.7?m) resulting in an uncertainty of ? 10% for the smallest beam separation.
Note that the Wollaston prism (?x1 = 290?m) produces beams that are much more
41
3500
150 150 150
3000
100 100 100
2500
50 50 50
2000
0 0 0
1500
-50 -50 -50
1000
-100 -100 -100
500
-150 -150 -150
0
-30 0 30 -30 0 30 -30 0 30
Figure 3.2: Comparison of beam profiles at the focus for three setups. ?x1 = 290?m
is produced using the Wollaston prism. The other two beam spacings are produced
using the Sanderson prism.
uniform/Gaussian-like than those produced by the Sanderson prism (?x1 = 100?m
and ?x1 = 35?m).
3.2.3 Measurement of focal spot radius, w0
An accurate measure of the beam spot size at the focus is needed to compute
the FLDI sensitivity function as well as the spot size at any z-location via. Eq. 2.31.
Unfortunately, the pixel size of the beam profiler camera is too large to directly
measure the spot size at the focus. However, since the beam diameter increases
linearly with z far from z = 0, we can use the beam profiler camera to measure w(z)
at seven non-uniformly spaced distances ranging from z = 3.175mm to z = 25.4mm
42
Figure 3.3: Beam intensity profiles along the z axis
and infer the beam radius at the focus (w0) using Eq. 3.1 with ?0 = 633nm.
?0
w(z) ? z, (3.1)
?w0
Examples of beam intensity profiles measured at three locations are presented in Fig.
3.3. Gaussian intensity distributions fit to each image are used to find the Gaussian
beam radius parameter w where w = 2? and ? is the standard deviation of the
Gaussian distribution [54]. Note that the up beam prism is temporarily removed
so a single beam is being measured, not two beams split by 290?m as in the actual
setup. This makes no difference since the divergence angles of the two beams are
the same as that of the original.
The results of the beam radius parameter measurements are plotted in Fig.
3.4. A linear fit correlates the measurements extremely well (R2 = 0.999) and
enables us to determine that the beam radius at the focus is 7.3? 0.3?m (? 5%).
43
0.8
0.6
0.4
0.2
0
0 10 20 30
Figure 3.4: Measurements of the beam radius parameter at various points along the
z axis with the fit used to determine w0 (the beam radius at the focus).
15 6
5
10 4
3
5 2
1
0 0
0 1 2 3 0 1 2 3
(a) Mean voltage vs. knife edge position. (b) Voltage difference and Gaussian fits.
Figure 3.5: Results from knife-edge test used to determine beam pair separation
(?x2)
44
3.2.4 Measuring beam-pair separation, ?x2
The 2pFLDI beam pair separation (?x2) can be estimated using paraxial
ray-tracing [71] or ray-tracing software such as Code-V or ZEMAX but it is more
accurate to measure it. This is best accomplished using a beam profiler camera
as done in section 3.2.2. Unfortunately, this camera was not available when the
2pFLDI experiments were performed so the beam separation was determined by
translating a knife-edge in the x-direction through the focal region while measuring
both detectors? output voltage at 33 equally spaced points over 3.175mm. Fig. 3.5.a
shows the results. The knife edge initially blocks both signals so the output voltages
are both zero. As the knife edge is moved away, first detector A and then detector
B are uncovered. Fig. 3.5.b, shows the derivative of the signals dV/dx with respect
to x. Gaussian functions are fit to the derivatives of each signal in order to reduce
uncertainty in locating the peak. The distance between the Gaussian peaks is taken
to be the beam spacing. The results show that ?x2 = 1.18? 0.1mm.
3.2.5 Experiment and Modeling
The density fluctuation in the turbulent jet is given (via Eq. 2.35) by:
?0 F [??]
Ajet = , (3.2)
4?K Hjet
where ? Li
Hjet = |gjetH?xHw|dz. (3.3)
?Li
45
Figure 3.6: Schematic of turbulent jet in relation to FLDI beams
We need estimates of gjet and uc,jet to evaluate Hjet. These are readily available
in the literature for turbulent jets which is why we use a turbulent jet to validate
our theory. As mentioned in the chapter overview, two approaches are explored for
determining uc,jet. The approach which models uc(z) based on historical measure-
ments is detailed next and the results are presented in section 3.2.7. The approach
which uses the measured dispersion relation, uc(f), from 2pFLDI is presented in
section 3.2.9.
The schematic illustration of the FLDI probe beam relative to the round tur-
bulent jet presented in Fig. 3.6 is useful for understanding the relationship between
the FLDI and the parameters describing the turbulent jet. The Reynolds number of
the jet we will use to validate the instrument function is 8.55?104. Measurements by
Hinze in a round turbulent jet at similar Reynolds number (Red = 6.7 ? 104, [64])
show that the radial density profiles at different downstream distances x follow
Gaussian distributions with radial distance away from the jet centerline. Thus, the
46
density distribution in the jet is given by: ( )
(z ? z 20)
gjet(x, z, z0) = exp ? . (3.4)
2?2jet,?(x)
where ?jet,? is the jet spreading angle based on density. Similarly, measurements by
Hinze, Wygnanski & Fiedler and others [64, 65, 67] show that axial velocities also
follow Gaussian distributions about the jet axis. Therefore, the velocity distribution
in the jet is given by: ( )
(z ? z )20
uc(x, z, z0) = UCL exp ? , (3.5)
2?2jet,u(x)
where ?jet,u(x) is the jet spreading angle based on velocity and UCL(x) is the velocity
distribution along the jet center line. Hinze?s measurements showed that ?jet,?(x) =
0.080x and ?jet,u(x) = 0.075x in the Red = 6.7?104 turbulent jet. The jet centerline
velocity distribution is found using Witze?s velocity decay model [66], which is based
on historical pitot and hot-wire measurem(ents. This mod)el is expressed as
1
UCL(x) = 1? exp ? , (3.6)
?(x/d)?Xc
where Xc, a universal parameter describing core length, takes a value of 0.7 and ?,
a parameter describing decay rate based on jet exit Mach number and ratio of jet
exit density to ambient density, takes a value of 0.13 for a Mach number of 0.96 and
a density ratio of 0.84 [66].
Hjet is evaluated by substituting Eq. 3.5 into Eq. 2.25 and Eq. 2.32, and then
plugging those and Eq. 3.4 into Eq. 3.3 and evaluating the integral numerically
using trapezoidal integration (MATLAB function trapz) with 1001 linear-spaced z
values between z0 ? 4?jet,? and z0 + 4?jet,?.
47
Parameter Index, i Typical Value, Pi Uncertainty, bP Unitsi
?x1 1 290 3.7 ?m
w0 2 7.3 0.3 ?m
?jet,? 3 6.4 0.1 mm
?jet,u 4 6 0.1 mm
UCL 5 100 2 m/s
Table 3.3: Parameter typical value and uncertainty
3.2.6 Uncertainty and sensitivity analysis
The uncertainty analysis uses standard methods outlined by the American
Society of Mechanical Engineers [72]. The measurable parameters, Pi, of Hjet are
the beam separation ?x1 and the beam radius at the focus w0. The free parameters
are the jet standard deviation based on density, ?jet,?, the jet standard deviation
based on velocity ?jet,u, and the jet centerline velocity UCL (which is based on two
free parameters per Eq. 3.6). Table 3.3 gives indices, typical values (taken from
x/d=25), and the systematic (or bias) uncertainty bP in these parameters. Thei
sensitivity of Hjet to parameter Pi is estimated by computing a first order finite
difference as follows:
?Hjet ? Hjet(Pi +?Pi)?Hjet(Pi) , (3.7)
?Pi ?Pi
where ?Pi/Pi = 0.03, or 3% is chosen for the approximation. The total systematic
uncertainty bH to N parameters is found by summing the uncertainties associatedjet
48
with each parameter in a root mean??squar(e manner:???N )2?HjetbH = bjet P . (3.8)?P i
1 i
3.2.7 Results using modeled velocity distribution, uc(z)
3.2.7.1 Sensitivity to beam separation (?x1)
As its name suggests, the differential interferometer performs a finite difference
on the index of refraction field. Dividing the signal by the beam separation (?x1)
yields a signal proportional to the first order approximation of the density gradient.
However, the instrument is also sensitive to wavenumber as explained in the previous
section and by Parziale in much of his FLDI work (see for example Fig. 3 in
[23]). This is accounted for here using a transfer function developed for sinusoidal
variations (Eq. 2.25). Lawson et al. [45] uses FLDI simulations to illustrate the
effect of varying the beam separation when the disturbance field is static. They
note that the signal magnitude increases and x-axis spatial resolution decreases as
?x1 increases. These effects are qualitatively consistent with those predicted by
H?x (Eq. 2.25). The question we seek to answer in this section is to what extent
Eq. 2.25 models the FLDI?s response to the dynamic, turbulent flow field produced
by the jet.
In each experiment, the turbulent jet is aligned with the beam?s focus (z0 = 0)
and is translated in the streamwise direction along the x-axis. Measurements are
made at five uniformly spaced locations (ranging from x/d=5 to x/d=25) where the
FLDI signal is sampled for 20ms at 2MHz. Measurements of signal strength (which
49
-4 -4 -4
10 10 10
-6 -6 -6
10 10 10
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.7: Raw FLDI phase difference spectra for three beam separations acquired
with the instrument focus located at the jet centerline (y/d = 0 and z0/d = 0)
is proportional to phase difference) at x/d=15, 20 and 25 are presented in Fig. 3.7
for three different values of ?x1. The results show that the phase difference increases
with beam separation at low frequencies. This can be explained by thinking of the
signal magnitude divided by the beam separation as proportional to the density
gradient, i.e. ??/?x1 ? ??/?x. Thus for a fixed density gradient magnitude,
phase difference increases linearly with beam separation. We observe this trend
for frequencies below ? 105Hz. Above ? 104Hz phase difference decreases with
frequency. Above ? 105Hz, This decline becomes more rapid for the largest beam
separation, ?x1 = 290?m. This behavior represents spatial filtering of wavenumbers
which exceed ?/?x1. This disturbance scale corresponds to a range of frequencies
because the convection velocity varies through the jet. Note that spectral content
below ?? ? 2 ? 10?6 is assumed to be noise and is removed prior to subsequent
data processing. This ?noise floor? is evident at the highest frequencies in Fig. 3.7
where the spectra flatten and coalesce with spectra measured when the flow is off
(not shown here).
50
1 1 1
10 10 10
0 0 0
10 10 10
-1 -1 -1
10 10 10
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.8: FLDI sensitivity function to the jet (Eq. 3.3) for three beam separations
at jet centerline (y/d = 0) and instrument focus (z0/d = 0). w0 = 7.3?m, and
?o = 632.8nm.
-4 -4 -4
10 10 10
-6 -6 -6
10 10 10
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.9: Average amplitude spectra interrogated by FLDI, found from Eq. 3.2.
Measurements are acquired at the jet centerline (y/d = 0) and the instrument focus
(z0/d = 0)
51
1 1 1
0.5 0.5 0.5
0 0 0
-0.5 -0.5 -0.5
-1 -1 -1
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.10: Difference between amplitude spectra for three beam separations, nor-
malized by amplitude from ?x1 = 35?m. Measurements are acquired at the jet
centerline (y/d = 0) and the instrument focus (z0/d = 0)
Fig. 3.8 shows the overall transfer function Hjet as a function of frequency
and beam spacing for the measurements presented in Fig. 3.7. The filtering behav-
ior is apparent as Hjet increases linearly with frequency until the frequency of the
disturbance approaches the point of maximum sensitivity. The true average density
fluctuation spectrum is recovered by dividing the signal by Hjet as per Eq. 3.2. The
results are presented in Fig. 3.9. The collapse of all spectra indicates Hjet (Eq. 2.34)
and thus H?x (Eq. 2.25) is a good model for the effects of beam separation on this
flow field. While results from measurements at only three locations are presented
here, similar collapses are observed at all jet locations.
Fig. 3.10 shows the difference between amplitude spectra (Ajet) measured
at the three beam separations normalized by the value from ?x1 = 35?m as a
function of frequency. The amplitudes recovered from ?x1 = 100?m & 35?m differ
by ? 25% over the band 5kHz < f < 200kHz while the amplitudes recovered from
?x1 = 290?m & 35?m differ by ? 50% over the same band. The effect of beam
52
50 50 50
40 40 40
30 30 30
20 20 20
10 10 10
0 0 0
3 4 5 6 3 4 5 6 3 4 5 6
10 10 10 10 10 10 10 10 10 10 10 10
Figure 3.11: Percent uncertainty in Hjet
separation appears to be independent of x/d. Notice also that there appears to be
no consistent trend in the difference between amplitude spectra as ?x1 is increased.
The increase in error at the highest frequencies can be explained by the un-
certainty and the relatively small values of Hjet at these frequencies. Percent uncer-
tainty in Hjet is plotted in Fig. 3.11, found from Eq. 3.8. For example, at x/d = 15
and ? 400kHz, Hjet drops rapidly to small values. To find Ajet, we divide by these
small values, thus any uncertainty will be magnified. This is illustrated by the rapid
increase of uncertainty in Fig. 3.11.a at 400kHz.
In the next section we show that errors associated with varying the local beam
diameter are significantly smaller (? 10?20%). This, as well as the fact that there is
no consistent trend in the difference between amplitude spectra as ?x1 is increased,
suggests that the differences observed here are mainly due to changes in the optical
setup required to change ?x1 and not ?x1 itself or discrepancies between our model
(Eq. 2.11) and the true density gradient fluctuations in the flow.
53
-4 -4 -4
10 10 10
-5 -5 -5
10 10 10
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.12: Raw FLDI phase difference spectra acquired at jet centerline (y/d = 0)
for six jet positions along the optical axis (z0)
3.2.7.2 Sensitivity to position along the optical axis (z)
In this section, we investigate how FLDI sensitivity changes along the opti-
cal axis (z-direction) which is important for evaluating the improvement in spatial
resolution brought about by focusing. A single beam separation of ?x1 = 290?m
(which provides the largest overall signal strength) is considered in order to iso-
late the effect of z. The turbulent jet is translated in the streamwise (x-axis) and
spanwise (z-axis) directions. Data are collected at ten uniformly spaced x- loca-
tions ranging from x/d=5 to x/d=50 and 21 non-uniformly spaced locations in the
z-direction ranging from z0/d = ?5 (z0 = ?15.9mm) to z0/d = 70 (z0 = 222.2mm).
At each location, the FLDI signal is sampled for 400ms at 1MHz. A sketch of the
experiment is shown in Fig. 3.6.
Phase difference spectra are presented in Fig. 3.12 for eighteen measurement
locations (x/d=20, 30, 40 and six z0 locations). The results show that signal strength
decreases as the jet moves away from the instrument?s focus (increasing z0). Higher
54
frequencies are more strongly affected by the movement of the jet away from the
focus because these frequencies correspond to small wavelength disturbances which
are spatially averaged to a greater extent than relatively larger wavelengths (lower
frequencies). Fig. 3.13 shows the values of Hjet corresponding to the eighteen mea-
surement locations plotted in Fig. 3.12. Prior to calculating Hjet, frequencies which
contain spectral content below ?? ? 2?10?6 are assumed to be noise and removed.
Hjet is not evaluated or plotted at these frequencies in order to de-clutter the plots.
These curves predict the instrument?s sensitivity to the jet at different values of z0.
As in the previous section, we use Eq. 3.2 to recover the true average density fluc-
tuation spectra. The results, presented in Fig. 3.14, show that measurements at all
z-locations collapse indicating that Hjet has correctly captured the effect of position
along the optical axis. While data for only three streamwise and six z-axis positions
are presented here - for clarity - collapse is observed for all frequencies measured
(f > 500Hz) and for all streamwise positions above x/d=15. Below x/d=15, the
assumption that disturbances are normally distributed breaks down and the model
is not expected to work.
Once again, we assess the model?s effectiveness - i.e. its ability to predict and
compensate for the effects of disturbances away from the focal volume - by computing
the difference between amplitudes recovered for the various z-axis positions and the
amplitude at the instrument?s focus (z0 = 0) as illustrated in Fig. 3.15. The results
show that the error is less than ?20% for almost all frequencies and cases presented
here. The error only exceeds 20% for the higher frequencies at x/d = 20. In fact,
for x/d=30 & 40 and f < 10kHz, the error is below ?10%. This is a significant
55
1 1 1
10 10 10
0 0 0
10 10 10
-1 -1 -1
10 10 10
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.13: FLDI sensitivity function at jet centerline (y/d = 0) for six jet positions
along the optical axis (z0)
-4 -4 -4
10 10 10
-6 -6 -6
10 10 10
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.14: Average amplitude spectra interrogated by FLDI for jet centerline
(y/d = 0) and for six jet positions along the optical axis (z0)
56
1 1 1
0.5 0.5 0.5
0 0 0
-0.5 -0.5 -0.5
-1 -1 -1
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.15: Difference between amplitude spectra recovered for six jet positions
along the optical axis (z0), normalized by amplitude at z0 = 0
50 50 50
40 40 40
30 30 30
20 20 20
10 10 10
0 0 0
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.16: Percent uncertainty in Hjet
improvement over the error associated with varying the beam separation (25-50%).
The larger errors for higher frequencies at x/d=20 can be explained by the
relatively small values of Hjet at these high frequencies, which we divide by to find
Ajet, causing large increases in uncertainty. This behavior is illustrated in Fig. 3.16
3.2.7.3 Comparison to uniform velocity assumption
A significant difference between this work and previous is the inclusion of
a modeled convection velocity distribution (given by Eq. 3.5). To illustrate how
57
-4 -4 -4
10 10 10
-6 -6 -6
10 10 10
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.17: Comparison of amplitude spectra computed using uniform velocity
profile (a & b) and Gaussian velocity profile (c) for x/d = 30, y/d = 0, and six z0
positions.
this approach differs from one where velocity is assumed constant Fig. 3.17 &
3.18 compare the amplitude spectra and amplitude spectra difference, respectively,
between a uniform velocity assumption and the Gaussian distribution model at
x/d=30. For each figure, (a) shows the results when convection velocity is equal to
the local jet centerline velocity everywhere, (b) shows the results when convection
velocity is equal to the local jet centerline velocity divided by the square root of
two, and (c) shows the results using the Gaussian distribution, which are the same
as those presented in the previous section, repeated here to aid comparison.
The difference between collapsed amplitude spectra in Fig. 3.17.a, Fig. 3.17.b,
& Fig. 3.17.c is hard to discern. Amplitude appears to decrease slightly with z0 in
Fig. 3.17.a where the convection velocity is equal to the centerline mean velocity,
indicating the result is not independent of z0. Better agreement is observed in
Fig. 3.17.b where the convection velocity is equal to the centerline mean velocity
?
divided by 2. This collapse looks comparable to Fig. 3.17.c. Difference between
58
1 1 1
0.5 0.5 0.5
0 0 0
-0.5 -0.5 -0.5
-1 -1 -1
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.18: Comparison of amplitude spectra difference for uniform velocity profile
(a & b) and Gaussian velocity profile (c) for x/d = 30, y/d = 0, and six z0 positions.
amplitudes are easier to see when not plotted on a log scale as in Fig. 3.18. Fig.
3.18.a shows a clear trend of decreasing amplitude with z0 where differences exceed
50% near 10kHz for the larger z0 plotted. Less variation in amplitude with z0 is
?
observed in Fig. 3.18.b, compared to Fig. 3.18.a, indicating uc = UCL/ 2 is a better
choice than uc = UCL for a uniform convection velocity in the flow. The best collapse
is illustrated in Fig. 3.18.c where the convection velocity is modeled as Gaussian
with z. Differences between amplitudes are near 10%, compared with 20-40% in
Fig. 3.18.b. Amplitudes are clearly independent of z0 in Fig. 3.18.c. This affirms
the importance of accurately modeling convection velocities when interpreting the
FLDI signal.
3.2.8 Measuring the dispersion relation
Fig. 3.19 shows frequency-dependent convection velocities computed from our
2pFLDI measurements using Eq. 2.48-2.50. Measurements are made at the jet
59
centerline using 13ms of data (collected at 3MHz) at each position downstream
(x/d). The figure shows that velocity increases with frequency, decreases with x for
f > 5kHz, and becomes independent of x as frequency goes to zero. This behavior
is the result of two phenomena:
1. The scale dependence of FLDI?s sensitivity discussed in the previous section
causes disturbances with low wavenumbers (corresponding to low frequency
signals) to be more averaged across the span of the jet than disturbances with
high wavenumber (high frequencies) causing the latter to be closer to the local
centerline value.
2. low wavenumber disturbances convect slower than high wavenumbers along
the jet centerline [56, 61, 65] because the larger scales carry more momentum
from the outer regions of the jet than the smaller scales.
The purpose of these measurements is to evaluate the accuracy of our model for
FLDI?s response when the dispersion relation measured by 2pFLDI is used in place
of the modeled velocity distribution, i.e when substituting these measurements in
place of Eq. 3.5. Further discussion of these results is considered outside the scope
of this work.
3.2.9 Results using measured dispersion relation, uc(f)
Taken from Fig. 3.19, convection velocities vs. frequency for x/d=20, 30 and
40, corresponding to the locations discussed previously in section 3.2.7.2, are re-
plotted in Fig. 3.20. These velocities are used to convert frequency to wavenumber,
60
150
100
50
0
0 20 40 60 80 100
Figure 3.19: Frequency-dependent convection velocities along the centerline (y = 0,
z0 = 0) of a round jet. Computed from the cross-spectra of 2pFLDI measurements
(Eq. 2.48-2.50).
61
120 120 120
100 100 100
80 80 80
60 60 60
40 40 40
20 20 20
0 0 0
3 4 5 3 4 5 3 4 5
10 10 10 10 10 10 10 10 10
Figure 3.20: 2pFLDI measured velocity vs. frequency acquired at jet centerline
(y/d = 0) and instrument focus (z0 = 0)
kx, (Eq. 2.12) for the data presented previously in Fig. 3.12. This is presented in
Fig. 3.21. This conversion allows us to use the versions of the transfer functions, H?x
and Hw which use wavenumber rather than frequency (Eq. 2.25 and 2.32). Thus,
Hjet is computed using Eq. 3.3 in terms of wavenumber and plotted in Fig. 3.22 for
x/d=20, 30 and 40. As discussed in section 2.3.2, we can now find the amplitude
of disturbances vs. wavenumber which is more useful than frequency for comparing
to turbulence models and spatial features in the flow. We are also investigating to
what extent wavenumber spectra, Ajet(kx), are independent of jet position along the
optical axis, z0. Ajet is plotted in Fig. 3.23, which shows the recovered amplitudes
are independent of z over the wavenumber band [0.5 < k < 5]mm?10 x for x/d=20,
30, and 40. At x/d=40 and z0 = 0, a linear fit to the log of Ajet vs the log of kx
yields a slope of -1.45, which is plotted as a black line in Fig. 3.23.c.
To quantify the degree to which recovered amplitude wavenumber-spectra are
independent of z0, we compute the percent difference between Ajet(z0) and its value
at z0 = 0, shown in Fig. 3.24. This is a measure of the accuracy of our model
62
-4 -4 -4
10 10 10
-5 -5 -5
10 10 10
0 1 0 1 0 1
10 10 10 10 10 10
Figure 3.21: Phase difference spectra vs. wavenumber acquired at jet centerline
(y/d = 0) for six jet positions along the optical axis (z0)
1 1 1
10 10 10
0 0 0
10 10 10
-1 -1 -1
10 10 10
0 1 0 1 0 1
10 10 10 10 10 10
Figure 3.22: Sensitivity function vs. wavenumber at jet centerline (y/d = 0) for six
jet positions along the optical axis (z0)
63
-5 -5 -5
10 10 10
-6 -6 -6
10 10 10
0 1 0 1 0 1
10 10 10 10 10 10
Figure 3.23: Amplitude spectra vs. wavenumber at jet centerline (y/d = 0) for six
jet positions along the optical axis (z0)
for FLDI?s response to the jet when the 2pFLDI-measured dispersion relation is
used to convert frequencies in the signal to wavenumbers. Less than 20% error
is observed over the wavenumber band [0.5 < k < 5]mm?1x when the jet is not
too far from the focus z0 = 12.7mm and z0 = 25.4mm for all x/d. At x/d=40,
error is below 20% for z0 = 57mm and z0 = 82mm also. For smaller x/d, larger
z0, and larger wavenumbers, error increases rapidly, as observed in Fig. 3.24.a for
z0 ? 57mm. This behavior might be explained by the uncertainty in Hjet when
the magnitude of Hjet becomes small. Percent uncertainty in Hjet for the same
18 jet positions is plotted in Fig. 3.25. It shows that model uncertainty increases
correspond to the same conditions in which the largest errors are observed: smaller
x/d, larger z0, and larger kx. However, the fact that larger errors are consistently
positive (over-predicting Ajet) and consistently occur for larger distances away from
the focus suggest our model is under-predicting the value of Hjet for large z. This
could indicate a breakdown of our flow-parallel disturbance model (Eq. 2.11).
64
1 1 1
0.5 0.5 0.5
0 0 0
-0.5 -0.5 -0.5
-1 -1 -1
0 1 0 1 0 1
10 10 10 10 10 10
Figure 3.24: Difference between amplitude spectra for six jet positions along the
optical axis (z0) normalized by amplitude at z0 = 0
50 50 50
40 40 40
30 30 30
20 20 20
10 10 10
0 0 0
0 1 0 1 0 1
10 10 10 10 10 10
Figure 3.25: Percent uncertainty in Hjet vs. wavenumber
65
3.2.10 Conclusions
The response of a single-point FLDI to a round turbulent jet is modeled by ray
tracing (Eq. 2.8) through flow parallel disturbances whose magnitude and velocity
vary with z and f (Eq. 2.11). These equations, when combined, reduce analytically
to Eq. 2.35 (re-written as Eq. 3.2 for the jet) which allows us to solve for the distur-
bance amplitude spectra in the jet. The fluctuation magnitude spatial variation is
modeled as a Gaussian based off of historical jet studies (Eq. 3.4). Two approaches
for extracting the disturbance velocity are explored: (1) modeling velocity variation
in space using historical studies (Eq. 3.5), and (2) measuring velocity variation with
frequency using 2pFLDI (section 3.2.8). The advantage of the second method is that
it eliminates several free parameters in the governing equation and enables FLDI
signal interpretation in flows where velocity data are not readily available.
Both approaches illustrate how the signal attenuates as the jet is translated
away from the focus. Errors are on the order of 20% when the sensitivity function,
Hjet, is sufficiently large (roughly > 1mm in this example). Next, we investigate to
what extent Eq. 2.35 can describe the FLDI signal in a complex, ?wind-tunnel-like?
disturbance environment produced in a Mach 3 facility.
3.3 Mach 3 Tunnel
AEDC?s Mach 3 atmospheric temperature/pressure in-draft calibration tun-
nel (M3CT) is used evaluate the performance of the FLDI instrument in a ?wind
tunnel-like? disturbance environment. Some of this work has been presented at con-
66
Figure 3.26: Diagram of the M3CT flow path showing the location of the FLDI
measurement volume, ramp, and the coordinate directions.
M? (-) p? (Pa) T (K) ? (kg/m
3) Reunit? ? ? (1/m) U? (m/s)
2.62 4165 123 0.118 8? 106 580
Table 3.4: Free stream conditions at measurement location in wind tunnel.
ferences [41]. The tunnel produces approximately 10 seconds of steady flow. More
information about the tunnel itself is available elsewhere [73]. Conditions for all
tunnel runs are provided in table 3.4. The probe volume location is approximately
500mm downstream of the nozzle and 19mm below the tunnel ceiling. Runs are
conducted with the 24?, 19mm tall ramp ?disturbance generator? illustrated in Fig.
3.26 installed 191mm upstream of the probe location. The ramp is chosen simply as
a mechanism for generating a disturbance environment with large amplitudes. The
details of the flow field downstream of the ramp are not considered in this work.
3.3.1 FLDI and 2pFLDI equipment and data reduction
The experimental set-up is shown in a side view in Fig. 3.26 and in a top-
down view in Fig. 3.27. Parameters of the FLDI and 2pFLDI are given in tables 3.1
67
Figure 3.27: Illustration of the location of M3CT (disturbance generator) within
the FLDI optical path.
& 3.2 respectively. The beams are focused over 140cm from a maximum diameter
of 2w = 40mm to a minimum diameter of 2w0 = 28?m at the focus [54]. 48cm
from the focus, the beam diameter (2w) is approximately 14mm. The FLDI uses
a 21mW HeNe laser (THORLABS HNL210LB), a 20X microscope objective lens,
(OL), large plano-convex condenser lenses (FL) with 100mm diameter and a 400mm
focal length (Edmund Optics 27-503), Polarizers from Thorlabs (LPVISE100-A),
and a 13mm2, 14ns rise-time photodetector from ThorLabs (DET36A). Wollaston
prisms from United Crystals with splitting angle of 4 arcmin are used for beam
splitting/recombining (P1u & P1d). The photodetector signal is terminated at 1
k? and digitized by a 16-bit Picoscope 5442D. The sampling frequency is 4 MHz.
Raw voltages are downloaded from the scope and processed using MATLAB.
Voltage is converted to phase difference using Eq. 2.7. All analyses are performed
on 4 ? 106 data points acquired starting at t = 4s where t = 0 is the time the
68
value opens to start the run. The flow in the tunnel is steady during this interval.
Once again, Fourier transforms are computed by taking the square-root of the power
spectral density which we estimate using Welch?s method [69] and the MATLAB
function pwelch with 125?s Hamming windows and 50% overlap.
Two trials are conducted. In each trial, 17 runs are conducted at 17 equally
spaced z-axis positions ranging from z0 = 0 (tunnel centerline at focus) to z0 = 48cm.
The tunnel is fixed and the FLDI equipment is translated to vary z0. Both trials
yielded similar results so only one trial is discussed here.
3.3.2 Modeling
Re-expressimg Eq. 2.35 for the M3CT gives
?0 F [??]
AM3CT = , (3.9)
4?K HM3CT
where ? Li
HM3CT = |gM3CTH?xHw|dz. (3.10)
?Li
The schematic illustration of the FLDI probe beam relative to the tunnel presented
in Fig. 3.27 is useful for understanding the relationship between the FLDI and the
width of the tunnel, L = 6.4cm. We assume disturbances are uniform across the
tunnel width such that
??????1 for ?L < z ? z0 < +LgM3CT (z) = ???0 elsewhere.
69
This allows us to solve the integral in Eq. 3.10 analytically giving:
? ( )
2?3/2w0 w
2k2
H 0 xM3CT = H{?x exp ? ??0kx [ 8 ] [ ]}
??0kx ? ??0kxerf (z0 + L) erf (z0 ? L) . (3.11)
2 2?w0 2 2?w0
Similar expressions for modeling FLDI response to uniform disturbances which are
symmetrically distributed with respect to the focus were explored by Fulghum [31]
and Schmidt and Shepherd [33]. Our expression allows for disturbances offset from
the focus by distance z0. We measure uc,M3CT (f) using 2pFLDI to convert wavenum-
bers to frequencies and vise versa. This approach was motivated by its success in
the turbulent jet.
3.3.3 Dispersion relation
The 2pFLI experiment used to measure uc,M3CT is conducted with the in-
strument?s focus located at the tunnel centerline (z0 = 0). The two measurement
volumes are separated by approximately 1.3mm along the x-axis (streamwise) and
centered approximately around the location of the single FLDI probe illustrated
in Fig. 3.26. 4 ? 106 points of data are acquired (with the ramp present in the
tunnel) from t = [4 ? 5]s just like all other single point FLDI experiments. The
dispersion relation (a plot of disturbance velocity vs. frequency) is computed from
the cross-spectra of the FLDI signals, i.e. Eq. 2.48-2.50. Fig. 3.28 shows that
the disturbance velocity is near U? = 580m/s over most of the frequency range
(> 100kHz). Plotting this on a logarithmic frequency scale in Fig. 3.29 shows
that velocity is near 350m/s at 1kHz, decreases to ? 180m/s at 4kHz, and then
70
600
400
200
0
0 2 4 6 8 10
5
10
Figure 3.28: Dispersion relation in M3CT flow field downstream of the ramp ?dis-
turbance generator? measured with 2pFLDI
600
400
200
0
3 4 5 6
10 10 10 10
Figure 3.29: Dispersion relation in M3CT plotted against log scale
increases with frequency until ? 300kHz.
3.3.4 Results
Phase difference spectra are presented in Fig. 3.30 for nine optical axis posi-
tions (z0) of tunnel centerline with respect to FLDI focus. z0 = 0 represents tunnel
centerline coincident with the focus. For the entire frequency band measured here
(4?1000kHz), signal strength decreases as the 6.4cm-wide tunnel moves away from
71
-4
10
-5
10
-6
10
4 5 6
10 10 10
Figure 3.30: Phase difference spectra vs. frequency in M3CT for nine tunnel posi-
tions along optical axis (z0). Flow off measurement taken prior to z0 = 48cm run
the focus until z0 = 30cm. This trend continues beyond z0 = 30cm for all fre-
quencies except ? 10 ? 20kHz, where signal strength becomes independent of z0.
Fluctuations observed for z0 ? 18cm and 100kHz < f < 250kHz are also present
in the signal taken when the tunnel is not running (labeled ?flow off? and plotted
with a dashed line). Thus, these fluctuations are not considered relevant to the flow
features under investigation.
Phase difference spectra are plotted in Fig. 3.31. Frequencies are converted
to wavenumbers using the convection velocities plotted in Fig. 3.29. Wavenumbers
?
which contain spectral content below 5?10?6rads/ Hz are assumed to be noise and
removed. Fig. 3.31 shows the same trends as Fig. 3.30, but with a spatial, rather
than temporal, perspective. It illustrates that signal strength becomes independent
of z0 for 0.3mm
?1 < kx < 0.35mm
?1 which corresponds to a wavelength of ? 19mm.
19mm is the height of the ramp as well as the distance between the tunnel ceiling
and FLDI probe.
72
-4
10
-5
10
0 1
10 10
Figure 3.31: Phase difference spectra vs.wavenumber in M3CT for nine tunnel po-
sitions along optical axis (z0)
Our model for the instrument?s response (HM3CT , Eq. 3.11) is plotted for nine
z0 values in Fig. 3.32. Recall that these curves enable us to deconvolve instrument
effects from the true disturbance amplitudes present in the flow (AM3CT ) via Eq.
3.9. The measured disturbance amplitudes are plotted in Fig. 3.33. The spectra
generally collapse over the 0.3mm?1 < kx < 0.6mm
?1. The collapse extends to
k ? 3mm?1x when the tunnel is not far (z0 < 18cm) from the focus. This collapse
demonstrates the general efficacy of the model in compensating for the effect of zo
although it is not as good as in the jet (see figure 3.2.1).
The degree to which AM3CT is independent of z0 is quantified by computing
the difference between AM3CT (z0) and its value at the focus, AM3CT (z0 = 0). This
is illustrated in Fig. 3.34 which shows that all amplitudes are within 50% of one-
another for 0.2mm?1 < kx < 0.65mm
?1 and for 0.2mm?1 < kx < 3mm
?1 when
z0 < 18cm. Rapid increases in error for larger z0 are reminiscent of trends observed
in the jet (see Fig. 3.24 for example). Once again, this could be explained by
73
2
10
1
10
0
10
-1
10
0 1
10 10
Figure 3.32: Sensitivity function vs.wavenumber in M3CT for nine tunnel positions
along optical axis (z0)
-5
10
-6
10
0 1
10 10
Figure 3.33: Amplitude spectra vs. wavenumber in M3CT for nine tunnel positions
along optical axis (z0)
74
1
0.5
0
-0.5
-1
0 1
10 10
Figure 3.34: Difference between amplitude spectra for nine tunnel positions along
optical axis (z0) normalized by amplitude at z0 = 0
uncertainty in the sensitivity function, HM3CT , which is magnified when its value
becomes small (?< 1mm for z0 > 30cm). Like the jet, our model consistently over-
predicts amplitudes thereby under-predicting the sensitivity function in the M3CT
when it is far from the focus and suggesting that this trend is due to bias and not
random error.
3.3.5 Comparison to uniform velocity assumption
Similar to section 3.2.7.3 for the jet, in this section we compare our results to
those found when assuming the convection velocity is uniform. Amplitude spectra
and amplitude spectra difference are plotted in Fig. 3.35 and Fig. 3.36, respectively.
For each, sub-figure (a) shows the results when convection velocity is assumed a
constant value of 550m/s while (b) shows the results when convection velocity varies
with frequency according to the relation measured by 2pFLDI. Thus, Fig. 3.35.b
and Fig. 3.36.b are the same as Fig. 3.33 and Fig. 3.34. 550m/s is chosen because
75
-5 -5
10 10
-6 -6
10 10
0 1 0 1
10 10 10 10
Figure 3.35: Comparison of amplitude spectra computed using uniform velocity
profile (a) and using measured frequency-dependent velocity (b) for nine z0 positions
this is approximately the average velocity measured over 100? 900kHz, i.e. ? 90%
of the bandwidth. Note that 550m/s is ? 94% of the freestream velocity in the
tunnel.
In general, Fig. 3.35 and Fig. 3.36 show that the difference between the
two approaches to modeling velocity are minimal. This is not surprising, because
the velocity is measured as nearly constant over a wide range: 100 ? 900kHz, i.e.
? 90% of the bandwidth. Thus, for frequencies above 100kHz corresponding to
wavenumbers above ? 1mm?1 for uc = 550m/s, the uniform velocity assumption
is a good one. Below 100kHz/1mm?1, we expect to see significant differences in
the plots, because the measured velocity is much lower than 550m/s here (see Fig.
3.28). Indeed, we observe better collapse and smaller amplitude difference in Fig.
3.35.b and Fig. 3.36.b vs Fig. 3.35.a and Fig. 3.36.a for 0.2mm?1 < kx < 0.5mm
?1.
76
1 1
0.5 0.5
0 0
-0.5 -0.5
-1 -1
0 1 0 1
10 10 10 10
Figure 3.36: Comparison of amplitude spectra difference for uniform velocity profile
(a) and frequency-dependent velocity (b)
3.4 Chapter Conclusion and Takeaways
In this chapter, the ability to model FLDI?s response to a turbulent jet and
to the flow downstream of a ramp in a Mach 3 tunnel using transfer functions
derived from a flow parallel disturbance assumption is investigated and quantified.
Particular emphasis is placed on the response as the flow is translated along the
optical axis (z0). Response to FLDI beam separation, ?x1, is also investigated in
the jet.
Methods for measuring FLDI and 2pFLDI instrument parameters ?x1, ?x2,
and w0 are demonstrated. A procedure for estimating uncertainty in the model of
the response in the jet is detailed. A critical parameter needed to relate frequency
to wavenumber is the convection velocity (uc) which is allowed to vary either spa-
tially with z, or temporally with f . Both of these approaches to relating frequency
to disturbance wavenumber in jets are novel and able to capture instrument re-
77
sponse with ? 20% error across a significant band of frequencies/wavenumbers
in the jet: 100kHz when historical measurements are used to predict uc(z) and
4.5mm?1/50kHz when the dispersion relation, uc(f) measured by 2pFLDI is used
instead. Instrument response is captured over a smaller bandwidth with larger error
? 40% when the velocity is assumed constant.
Instrument response to disturbances in the Mach 3 tunnel is captured with
< 50% error over 3mm?1/200kHz when the 6.4cm tunnel is translated within 18cm
of the focus. Further away from the focus, a large band of the FLDI signal is
significantly attenuated. The remaining low wavenumber/low frequency band is
captured by the transfer functions with < 50% error.
In these experiments, ?error? refers to the ability of transfer functions to ac-
count for the location of the disturbance (jet or M3CT) with respect to the FLDI
focus. Errors of 20% to 50% are not surprising, because we are using a simple
disturbance model (Eq. 2.11) for complex flows. These ?error? values quantify the
effectiveness of our approach. No previous work known to the authors has quantified
this approach before in flows similar to a turbulent jet or Mach 3 tunnel, thus the
results are a significant contribution. Improved modeling, model evaluation over
a wider parameter space, and evaluation against other jets and complex flows are
good candidates for future work.
78
Chapter 4: 2pFLDI Applications
4.1 Overview
In this chapter, we apply 2pFLDI in two high speed wind tunnels in order to
demonstrate how to interpret the measurements using the transfer function frame-
work. The first tunnel is the University of Tennessee Space Institute?s Mach 4
Ludwieg tube (UTM4). The second is Arnold Engineering Development Center?s
Hypervelocity Wind Tunnel 9 Mach 18 test section (T9M18). In UTM4 we probe 12
wall-normal locations in the floor boundary layer and one location above the bound-
ary layer. In T9M18 we probe the freestream of the test section at the centerline
at two unit Reynolds numbers. Table 4.1 gives an overview of the conditions and
dimensions in each tunnel and the parameters and specs of each 2pFLDI set-up.
Note that the 2pFLDI used in T9M18 has the same parameter values and specs as
the 2pFLDI used in the M3CT discussed previously in section 3.3. Just as in the
previous chapter all lasers are Helium Neon (HeNe) with ?0 = 633nm.
79
Parameter UTM4 T9M18 Units
Re? (unit) 20 3.8 - 4.7 ?106 m?1
U? (nominal) 670 2000 m/s
?? 140 4 - 5 g/m
3
test section width 61 150 cm
laser power 10 21 mW
fOL -9 9 mm
fFL 200 400 mm
?x1 300 464 ?m
Li 41 140 cm
w0 7 14 ?m
?x2 2.6 1.2 mm
Table 4.1: Parameters of 2pFLDI experiments in this chapter. See Fig. 2.6 for
reference
80
Figure 4.1: The UTSI Mach 4 Ludwieg tube.
4.2 UTSI Mach 4 Ludwieg Tube
The UTSI Mach 4 Ludwieg tube is shown in Fig. 4.1. The planar nozzle ex-
pands into a square test section with a 61 cm x 61 cm (24? x 24?) cross-section and
a length of 1.82 meters (6 ft.). The facility has a run time of roughly 200 ms for the
first wave passage. Of that 200 ms, approximately 130 ms is at steady-state condi-
tions suitable for testing. Per length Reynolds numbers range from approximately
5.6e6 to 4.7e7 m-1. No heating is applied to the driver tube and the stagnation
temperature is room temperature. Optical access in the test-section is provided by
BK7 windows.
4.2.1 2pFLDI equipment and data reduction
The parameters of the optical setup for the 2pFLDI system are given in table
4.1. A 10 mW HeNe laser (Newport) beam expanded using an -9mm focal-length
(fOL = ?9mm) plano-concave lens before passing through a polarizer (Thorlabs
LPVISE100-A). Next, the beam continues through a 30 mm x 30 mm 1.5? double
Wollaston prism (United Crystal) and is split based on polarization. A second
81
polarizer (Thorlabs LPVISE200-A) is then used to change the beam polarization to
45? so that a 50 mm x 50 mm 5 arcminute Wollaston prism (United Crystals) again
splits the beams before the 75 mm diameter f = 200 mm plano convex focusing lens.
The collection optics beyond the focus consist of a 75 mm diameter f = 200 mm plano
convex lens, a 50 mm x 50 mm 5 arcminute Wollaston prism (United Crystals), a
polarizer (Thorlabs LPVISE100-A), a = -50mm focal length (fCL = ?50mm) plano
concave lens, and a neutral density filter. Two photodetectors(Thorlabs PDA36A2)
are used for FLDI monitoring beam intensities. To reduce the effect of vibrations
during tunnel operation, the system was built on a single 95 mm extruded rail. This
also allowed easier translation to the region of interest without much realignment.
Voltages from the photodetectors are sampled with a 16-bit Picoscope 5442D at
20MHz. The max and min voltages are found by translating the up-beam Wollaston
prism (P1u) along the x-axis and then the voltage is set to the mean, V0, prior to
each run. A baseline is established by collecting 0.5sec of data are prior to each
run. Voltages are converted to phase differences via. Eq. 2.7. Phase difference
amplitude spectra are found from the square-root of the power spectral density
which is estimated using Welch?s method with 129?s hamming windows and 50%
overlap.
4.2.2 Experiment, Modeling and Spatiotemporal resolution
Fig. 4.2 is a schematic illustration of the UTSI Mach 4 Ludwieg tube test
section cross section showing the location of the 2pFLDI probe. 13 non-uniformly
82
Figure 4.2: Schematic of experiment in UTM4 floor turbulent boundary layer
spaced wall-normal (y) locations are probed ranging from y = 0.4cm to y = 8.2cm.
The boundary layer thickness is estimated to be 6.3cm using a RANS simulation.
The single point beam separation, beam radius parameter at the focus, and two-
point beam pair separation are found using a beam profiler camera (Newport LBP2)
and the same methods used in the jet (see sections 3.2.2 - 3.2.4). They are ?x1 =
300?m, w0 = 7.1?m, and ?x2 = 2.6mm.
The governing equation for the FLDI signal (Eq.2.35) expressed using symbols
for the UTM4 tunnel is given by
?0 F [??]
AUTM4 = , (4.1)
4?K HUTM4
where ? Li
HUTM4 = |gUTM4H?xHw|dz. (4.2)
?Li
For gUTM4, we assume disturbances are uniform magnitude (g = 1) across a 44cm
83
span centered about the y-axis. Outside of that span, but inside the tunnel, (i.e
?30.5cm < z < ?22cm & 22cm < z < 30.5cm) we assume g = 2, representing
disturbances in the corner flow boundary layers (Corner BL in Fig. 4.2). Everywhere
else (outside of the tunnel) g = 0. gUTM4 is symmetric about the y axis and plotted
for half the tunnel width in Fig. 4.3.
These assumptions are intended to be simple and conservative with the goal
of demonstrating a method and procedure for analyzing FLDI. If more accuracy is
desired, a more complex form of g(z) should be chosen based on historical studies, as
is demonstrated previously in the turbulent jet (section 3.2). Historical studies were
not available for this flow, thus uniform assumptions are made. The assumption of
g = 2 in the corner BL is intended to be conservative, representing a worst case
scenario where large, strong disturbances in the tunnels corner could corrupt the
measurement. Next, we chose a desired depth of focus and compute the wavenumber
above which we can confidently attribute signal to disturbances within that depth of
focus. Thus, the details of regions outside of that depth of focus are not important,
only that they represent a worst case scenario.
Following the procedure for determining FLDI spatiotemporal resolution, out-
lined in section 2.3.1, we choose a spatial resolution of 2Lc = 44cm. Now that we
have g(z) and Lc we can compute Hm, the sensitivity to disturbances we want to
measure, and Hr, the sensitivity to disturbances we want to reject, using Eq. 2.36
- 2.41. The ratio Hm/Hr is plotted in Fig. 4.4 vs. wavenumber, which shows that
the ratio is near unity until k = 0.1mm?1x , after which it increases rapidly. The
ratio exceeds five at kx = 0.5mm
?1, thus we take this to be the cut-off wavenumber,
84
3
2
1
0
0 10 20 30
Figure 4.3: Estimate of g(z) in UTM4 floor turbulent boundary layer
above which we achieve our desired depth of focus. The temporal resolution varies
based on the convection velocity of the disturbance, uc, which we will measure with
2pFLDI. As an example, for a disturbance convecting at the freestream velocity
(670m/s, k ?1x = 0.5mm ) corresponds to a cutoff frequency of 53kHz.
For wavenumbers above this cut-off and for uniform disturbances of length 2Lc
centered about the focus:
? ( ) [ ]
2?3/2w 2 20 w k ?0kxLc
HUTM4 = H?x exp ? 0 x erf ? . (4.3)
?0kx 8 2 2?w0
Note that the sensitivity to corner boundary layers is ignored as long as we restrict
ourselves above the wavenumber cut-off. Below this cut-off, signal should not be
analyzed.
85
10
8
6
4
2
0
-2 -1 0
10 10 10
Figure 4.4: Estimate of FLDI signal to noise ratio for a 2Lc = 44cm depth of focus
in UTM4 boundary layer
4.2.3 Results
Time traces of single point FLDI phase difference (Ch.A) are displayed in Fig.
4.5 for six tunnel runs corresponding to six different wall normal (y/?) positions of
the FLDI. Each trace is offset by 2.5 radians so that all traces can be seen on one
plot. For each run, the largest fluctuations are observed from 0 < t < 15ms and
for t > 240ms. These correspond to the periods of tunnel start up and shut-down,
respectively. The period of steady state test time used for subsequent analysis is
130ms < t < 220ms - indicated in the figure as ?good flow used?. During this time,
fluctuations from the run when FLDI is positioned just outside of the boundary
layer at y/? = 1.3 are clearly smaller than those measured inside of the boundary
layer (all other traces). These six runs will be analyzed in detail next.
86
6
4
2
0
-2
-4
-6
0 50 100 150 200 250 300
Figure 4.5: Phase difference vs. time from FLDI (Ch.A) for six wall normal positions
(y)
The amplitude spectral density of phase difference is shown in Fig. 4.6 for the
?good flow? period of each of the six runs. The dashed line shows the amplitude
spectral density of phase difference prior to tunnel start-up in the y/? = 1.3 run.
Below ? 80kHz, humps of increased amplitude are observed for all measurements.
As frequency increases, the slope decreases slightly. This behavior is similar to
predictions of how low frequency signal might appear if it is corrupted by side-wall
boundary layer disturbances by Settles and Fulghum [32]. Near 500kHz, all spectra
decrease rapidly, which resembles behavior seen when wavenumbers exceed ?/?x1.
To interpret these signals further, we need to convert frequencies to wavenumbers
with the convection velocity.
4.2.3.1 Convection Velocity
Fig. 4.7 shows a zoom in of the phase difference time trace for both signals
(Ch.A and B) for the run when the 2pFLDI is located at y/? = 0.4. Ch.B looks like a
87
-4
10
-5
10
4 5 6
10 10 10
Figure 4.6: Phase difference vs. frequency from FLDI (Ch.A) for six wall normal
positions (y)
a copy of Ch.A, shifted in time by ?t, indicating the passage of coherent disturbances
through both measurement volumes. As described in section 2.4, mean convection
velocity, ?uc?, is found from the cross-correlation between Ch.A and Ch.B. (Eq.
2.46). Frequency dependent convection velocity, uc(f), is found from the phase of
the cross-spectra between Ch.A and Ch.B, which is the Fourier transform of the
cross-correlation (Eq. 2.48-2.50).
Convection velocity vs. frequency is plotted for six runs in Fig. 4.8. It shows
velocity that increases rapidly with frequency until 100kHz for all runs. Above
100kHz, velocity increases gradually or is constant and ranges from 500m/s to
630m/s. In this range, velocity increases with y, as is expected in a boundary layer.
The rapid changes observed below 100kHz are likely the result of signal originating
from outside the desired depth of focus per our previous analysis (section 4.2.2). At
88
0.2
0.1
0
-0.1
-0.2
-0.3
175.03 175.04 175.05 175.06 175.07 175.08
Figure 4.7: Phase difference vs. time from 2pFLDI (Ch.A & Ch.B) for y/? = 0.4
frequencies below ? 53kHz, slower moving structures in the corner boundary layers
cause measured velocities to be slower.
Fig. 4.9 shows the same velocities normalized by each run?s respective velocity
computed from the cross-correlation. This appears to collapse the data between
0.2 ? y/? ? 0.9 although slight deviations from the trend are apparent in the
measurements closest to the wall and outside the boundary layer. In general, Fig.
4.9 shows that the velocities for f > 50kHz only deviate from the cross-correlation
velocity by a few percent showing that dispersion is minimal above 50kHz.
4.2.3.2 Wavenumber Spectra
Fig. 4.10 shows the single-point FLDI (Ch.A) amplitude spectral density of
phase difference plotted against wavenumber computed using the measured convec-
tion velocities in Fig. 4.8. The trends observed here are nearly identical to those
89
600
500
400
300
2 4 6 8 10
5
10
Figure 4.8: 2pFLDI convection velocity vs. frequency for six wall normal positions
(y)
1
0.8
0.6
2 4 6 8 10
5
10
Figure 4.9: 2pFLDI convection velocity vs. frequency normalized by mean convec-
tion velocity over all frequencies for six wall normal positions (y)
90
-4
10
-5
10
-6
10
-1 0 1
10 10 10
Figure 4.10: Phase difference vs. wavenumber from FLDI (Ch.A) for six wall normal
positions (y)
observed in Fig. 4.6 because the convection velocities are nearly constant across f .
Fig. 4.10 puts the signal in the context of the spatial features of the flow. This
allows us to determine what parts of the signal may be corrupted by disturbances
outside of the desired depth of focus. Per the analysis in section 4.2.2, we expect sig-
nals with wavenumbers below 0.5mm?1 to be corrupted by the corner flow. For the
one measurement above the boundary layer (y/? = 1.3), this cut-off may be larger
because disturbances within the depth of focus will be relatively weaker than those
within the boundary layer. The ?bump? in the y/? = 1.3 curve at kx = 0.6mm
?1
for the curve corresponding to y/? = 1.3 suggests that our analysis is valid because
significant changes in the spectum are observed near the wavenumber predicted.
Moving forward, we restrict our analysis to wavenumbers above 0.5mm?1
which represent disturbances within the 44cm depth of focus. The sensitivity func-
91
1
10
0
10
-1
10
0 1
10 10
Figure 4.11: Sensitivity and transfer functions for uniform disturbance in UTM4
tion is plotted vs. wavenumber in Fig. 4.11, which shows HUTM4 is between 20cm
and 30cm and nearly constant from 0.5mm?1 < k < 10mm?1x . This is initially sur-
prising, because in general the instrument?s response varies strongly with wavenum-
ber. Plotting the two transfer functions explains the behavior, shown in Fig. 4.11 as
dashed lines. For this particular FLDI setup, the depth of focus and the wavenumber
band the rate at which the transfer function due to beam separation H?x increases
with wavenumber. This is nearly equal to the rate at which the transfer function due
to beam diameter Hw decreases with wavenumber. Thus the product, the overall
transfer function, remains approximately constant.
Fig. 4.12 shows density-based amplitude spectra (found by evaluating Eq. 4.1)
normalized by the freestream density in the tunnel (?? = 0.14kg/m
3). The trends
are similar to those present in the phase difference spectra because the sensitivity
function is nearly constant across this wavenumber band. Above kx = 5mm
?1, a
92
-5
10
-6
10
-7
10
0 1
10 10
Figure 4.12: Amplitude spectra in UTM4 for six wall normal positions (y)
steep roll-off is present in all spectra as well as some scatter in the y/? = 0.07
data. This region occurs before the predicted roll off of ?/?x ? 10mm?11 . It
is not clear whether this behavior is a flow feature or an instrument effect. For
1mm?1 < k < 5mm?1x and y/? = 1.3, the decrease of amplitude with wavenumber
is linear (in a logarithmic scale) according to k?0.9x . For all other measurements in
the boundary layer, the slope is more gradual
4.2.3.3 Boundary Layer profiles
The FLDI-measured turbulence intensity can be found for specific wavenumber
bands by taking the square-root of the power spectral density integrated over the
wavenumber band of interest. This is the same as the root-mean squared value
over a specific band. For a band from k1 to k2, wavenumbers are converted back
to frequencies corresponding to f1 and f2 using the measured convection velocities.
93
The turbulence intensity is given by: ?? f2
RMS(A/??) = (A/??)2df. (4.4)
f1
This quantity is computed using the trapezoidal rule (MATLAB trapz) to evaluate
the integral. This is done for three different wavenumber bands and 13 runs of
the tunnel corresponding to all 13 y positions interrogated with 2pFLDI in this
campaign. The results are plotted in Fig. 4.13 and show that turbulence intensity
increases rapidly from a minimum value at y/? = 1.3 to a maximum at y/? = 0.5.
All bands have similar shapes. With the exception of y/? = 0.5 & 0.6, turbulence
intensity in the boundary layer below y/? = 0.9 is ? 10? larger than the value
measured outside the boundary layer at y/? = 1.3.
For perspective, Zhang and Duan [74] find that the magnitudes of density
based turbulence intensity across all wavenumbers in a Mach 2 and Mach 6 bound-
ary layer simulated by DNS are on the orders of 10 and 20%, respectively. The
magnitudes measured by our FLDI over the 0.5mm?1 < k < 5mm?1x band are
? 0.3% for this Mach 4 boundary layer. We believe the reason for this discrepancy
is that the wavenumbers containing the majority of the energy (< 0.5mm?1) are
not resolved with sufficient depth of focus by the FLDI. A comparison between DNS
and FLDI in the comparable wavenumber bands is intriguing future work.
The mean velocity over a wavenumber band k1 to k2 is found by converting
wavenumbers back to frequencies f1 and f2 and then using the data in Fig. 4.8 to
computing the mean value of uc(f) in the interval f1 to f2. The results are plotted
in Fig. 4.8 for all 13 y positions in Fig. 4.14. The bands used are the same as those
94
1.5
1
0.5
0
0 1 2 3 4 5
-3
10
Figure 4.13: Turbulence intensity profile in UTM4 floor BL for three wavenumber
bands
used to determine turbulence intensity. Also plotted is the cross-correlation velocity
which is the mean value over all wavenumbers (denoted ?all? in the figure) weighted
by signal strength. Error bars represent uncertainty in velocity introduced by small
variations (< 5%) in the beam pair separation ?x2 (measured with the beam profiler
camera) over the tunnel width. The mean velocity boundary layer profile predicted
by a RANS simulation is also included in Fig. 4.14. Convection velocities measured
by 2pFLDI agree with the computations in the region 0.2 < y/? < 0.5. Velocities
measured using 2pFLDI exceed the RANS predictions for y/? < 0.2 and are smaller
than the RANS predictions for y/? > 0.5. All velocities for y/? > 0.2 are between
80 and 95% of the RANS values. 2pFLDI velocity measurements using the high
wavenumber bands (corresponding to structures lying within the depth of focus) are
nearly identical. This suggests that the cross-correlation velocity is not significantly
95
1.5
1
0.5
0
0.5 0.6 0.7 0.8 0.9 1
Figure 4.14: Velocity profile in UTM4 floor BL computed using three wavenumber
bands
corrupted by disturbances away from the instrument?s focus. Measurements in the
freestream of AEDC T9?s Mach 18 wind tunnel presented later in this chapter will
show this is not always the case.
4.2.4 Conclusion and Takeaways
A method for modeling the response and determining the spatiotemporal reso-
lution of 2pFLDI in the floor boundary layer of a Mach 4 Ludwieg Tube is presented.
The method predicts that the FLDI can resolve disturbances with wavenumbers
greater than 0.5mm?1 at 44cm depth of focus. For disturbances convecting near the
freestream velocity, this corresponds to frequencies greater than 50kHz. Evidence
of this wavenumber/frequency cut-off is present in the measured spectra in the form
of increased amplitude ?humps? below the cut-off and in the measured velocities in
96
the form of a rapid roll-off with decreasing frequency below 50kHz. Above the cut-
off, density-based turbulence intensity is recovered using transfer functions which
assume uniform flow parallel waves across the depth of focus. Turbulence intensity
peaks at half the boundary layer thickness and is on the order of 10? stronger in
the boundary layer than outside. Convection velocity profiles measured using only
signal from inside the depth of focus do not significantly deviate from those mea-
sured using the entire signal. Almost all velocities measured are between 80 and
95% of RANS predictions of the freestream velocity except for measurements made
closest to the floor (y/? < 0.2).
4.3 AEDC T9 Mach 18 Freestream
AEDC?s Hypervelocity Tunnel 9 (T9) in White Oak, Maryland is a Nitrogen
gas blowdown facility capable of producing Mach numbers from 7 to 18 in its 1.5m
wide circular test section. For more detail on the wide range of capabilities at T9,
readers are directed to Marren and Lafferty [75]. Mach 18 flows at two nominal
unit Reynolds numbers (1, 1.5 ? 106/ft) are investigated. This high Mach number
flow results in very low free stream density (< 5g/m3) and very large (? 40cm)
sidewall boundary layers thus presenting a unique challenge for FLDI measurements:
density changes are especially small and disturbances away from the focal volume
are especially strong and widespread. Nominal free stream and reservoir conditions
are provided in table 4.2. ?Core size? refers to the approximate width of ?clean?
flow free from sidewall boundary layers. Details on how these facility conditions are
97
Nom. Re? Re? M? p? ?? U? Core Size
1/ft? 106 1/m? 106 (-) (Pa) (g/m3) (m/s) (m)
1.0 3.79 18.2 35.4 3.97 2032 0.737
1.5 4.74 18.3 45.0 4.96 2064 0.737
Table 4.2: Facility calculated free stream (subscript ?) conditions.
found is given here [76,77].
4.3.1 2pFLDI equipment and data reduction
The parameters of the optical setup for the 2pFLDI system are given in table
4.1. The equipment is the same as that used previously in the M3CT (see section
3.3.1). The photodetector signals are terminated at 1k?, amplified and filtered by
Stanford SR560 low-noise pre-amplifiers, and digitized by a 16-bit Picoscope 5442D.
A 1k? as opposed to a 50? resistor is used because the signal strength is low. Vari-
ous gain and high-pass filter settings for the SR560 are used with adjustments made
run-to-run in an effort to reduce noise. The recommended settings for future FLDI
experiments are 20? gain, 100Hz high pass filter, and 50? resistor to reduce high
frequency roll off. The sampling frequency is 7MHz. Raw voltages are downloaded
from the scope and processed using MATLAB. Voltage is converted to phase differ-
ence using Eq. 2.7. V0 is determined experimentally prior to each run by measuring
the maximum and minimum voltage associated with translating P1d along the x-axis
using a remote-operated picomotor.
All spectral analyses are based on 2 seconds of ?good-flow? (defined where
98
the freestream Reynolds number is essentially constant [78]) starting at t = 2s
where t = 0 is the time the start-up trigger signal is received. Fourier transforming
the phase difference fluctuations is computed by taking the square-root of the power
spectral density. The latter is estimated using Welch?s method [69] via the MATLAB
function pwelch with 1ms Hamming windows and 50% overlap.
4.3.2 Experiment and Modeling
Fig. 4.15 is a cross-section of the AEDC T9 Mach 18 test section at the
location where the 2pFLDI probes the freestream flow. The beams are focused
through 10cm diameter windows which set an upper limit on the convergence angle
of the beams toward the focus - and thus the depth of field - achievable. In future
FLDI experiments in T9, we recommend using larger windows and filling them as
much as possible to maximize the convergence angle of the beams. The windows are
in recessed wells where sidewall boundary layers on the tunnel walls become shear
layers denoted ?Turbulent SL? in the schematic illustration. Just as in previous
experiments, the single point beam separation ?x1 = 464?m, the beam radius
parameter at the focus w0 = 14?m, and the two-point beam pair separation ?x2 =
1.22mm are found with a a beam profiler camera (Newport LBP2) using the methods
described in sections 3.2.2 - 3.2.4.
Re-expressing Eq. 2.35 for T9 gives
?0 F [??]
AT9M18 = , (4.5)
4?K HT9M18
99
Figure 4.15: Schematic illustration of experiment in AEDC T9 Mach 18 test section
where ? Li
HT9M18 = |gT9M18H?xHw|dz. (4.6)
?Li
For gT9M18, we assume disturbances are uniform magnitude (g = 1) across the core
flow (73cm) span centered about the y-axis. The size of the core flow is determined
here [76]. Outside of that span, but inside the tunnel, (i.e inside the turbulent shear
layers ?75cm < z < ?37cm & 37cm < z < 75cm) we assume g = 15 representing
disturbances in the shear layers which are 15 times stronger than disturbances in
the freestream. Everywhere else (outside of the tunnel) g = 0. gT9M18 is symmetric
about the y axis and plotted for half the tunnel width in Fig. 4.16.
Once again, g(z), is intended to be conservative. Free stream turbulence mea-
sured with pitot probes in the same facility but at mach numbers of 14 or lower is
on the order of 3-5% [14]. If this is also the case in the Mach 18 test section, our
100
20
15
10
5
0
0 20 40 60 80
Figure 4.16: Estimate of g(z) in AEDC T9 Mach 18 test section
estimate of g(z) would imply 45-75% turbulence intensity in the shear layers. This
is high compared to magnitudes near 45% measured by Zhang and Duan [74] in a
Mach 14 turbulent boundary layer with DNS. It is hard to say how accurate our
assumption is, due to a lack of measurements in the test section shear layers and in
Mach 18 flows generally.
Next, we choose the core flow width as our desired spatial resolution such
that 2Lc = 73cm. The methods outlined in section 2.3.1 are employed to compute
the sensitivity to disturbances we want to measure (Hm) and those we want to
reject (Hr). Changing notation slightly we write Hcore = Hm and HSL = Hr for
the core flow and shear layer respectively. The ratio Hcore/HSL as a function of
wavenumber (graphed in Fig. 4.17) is near zero until kx ? 0.5mm?1, exceeds unity
at ? 0.7mm?1, and exceeds five near 0.9mm?1. Thus we take kx = 0.9mm?1 as
the cut-off wavenumber above which we achieve our desired spatial resolution. To
101
10
5
0
-1 0 1
10 10 10
Figure 4.17: Estimate of FLDI signal to noise ratio for a 2Lc = 73cm depth of focus
in T9M18
give a sense of the corresponding temporal resolution, a kx = 0.9mm
?1 disturbance
convecting at 1600m/s corresponds to a 230kHz signal. Signal above this cut-off
is processed using Eq. 4.5 where HT9M18 = Hcore, and Hcore is equivalent to the
analytic sensitivity function for a uniform, flow parallel disturbance expressed in
Eq. 4.3.
4.3.3 Results
Time traces of phase difference measured by single point FLDI (Ch.A) are
plotted for five tunnel runs in Fig. 4.18. Runs 4659 and 4667 correspond to a unit
Reynolds number of 1? 106/ft and runs 4671, 4658, and 4662 correspond to a unit
Reynolds number of 1.5? 106/ft. The absence of low frequency oscillations in run
4671 may be the result of a 3Hz high pass filter applied by the pre-amplifiers. A
102
1
0.5
0
-0.5
-1
1000 1500 2000 2500 3000 3500 4000 4500 5000
Figure 4.18: Phase difference vs. time from FLDI (Ch.A) for five tunnel runs
100Hz high pass filter is applied for run 4659. The other runs do not employ a high
pass filter which may explain the larger oscillations observed in run 4658. Vary large
amplitude fluctuations are evident in several runs for t > 4.5s which indicates tunnel
shut-down. The steady state test time used in subsequent analysis is indicated as
?good flow used? in the plot.
Fig. 4.19 shows the amplitude spectral density of phase difference from 2 < t <
4s plotted in various colors and symbols for the five runs. Measurements of phase
spectra taken ? 15 minutes prior to each run are plotted in black with symbols
matching those used in the respective run data. For all runs, signal strength is
significantly larger than the flow off noise for 20kHz < f < 200kHz and decreases
with frequency. The amplitudes of the flow off spectra vary significantly between
runs for f > 100kH. It is largest for run 4667 where signal to noise reaches unity
near 300kHz. Noise is smallest for the two runs which employed a high pass filter.
Wave packets are observed above 200kHz for runs 4658 and 4659. These features
103
-4
10
-5
10
-6
10
4 5 6
10 10 10
Figure 4.19: Phase difference vs. frequency from FLDI (Ch.A) for five runs. flow
off taken prior to run.
can not be explained by features in the flow and are assumed to be signal processing
artifacts or electronic noise.
4.3.3.1 Convection Velocity
Time traces of phase difference computed from Ch.A (upstream probe) and
Ch.B (downstream probe) for the 2pFLDI employed in run 4671 are plotted in Fig.
4.20. Ch.B (blue) appears very similar to Ch.A (red) except that it is shifted in time
by a tiny ?t ? 1?s, indicated on the graph. The time shift here is smaller than that
observed in UTM4 because the flow is faster and the beam pair separation ?x2 is
smaller. Next, cross-correlation and cross-spectra are used to compute convection
104
0
-0.01
-0.02
-0.03
-0.04
3000.16 3000.18 3000.2 3000.22 3000.24
Figure 4.20: Phase difference vs. time from 2pFLDI (Ch.A & Ch.B) for Run 4671
velocity.
Fig. 4.21 shows convection velocity normalized by freestream velocity vs.
frequency for the five runs. Free stream velocity is estimated by a methodology
involving pitot pressure probes as well as other measurement techniques detailed
here [76,77]. Velocities range from 50 to 100% of the freestream value for 25kHz <
f < 150kHz and increase with frequency for 100kHz < f < 200kHz in all runs.
For 200kHz < f < 350kHz, velocities measured in run 4671 and 4662 remain
near a constant value of 80% of the free stream value whereas those measured
in runs 4659 and 4658 exceed the free stream value and show a large degree of
variance. SNR in run 4667 was too small to measure velocities associated with
frequencies > 230kHz. Velocities above the freestream value are non physical in a
supersonic flow and the behavior seen in runs 4659 and 4658 may be the result of the
wavepacket noise observed in the spectra at the same frequencies (Fig. 4.19). The
modeling and spatiotemporal resolution analysis presented in section 4.3.2 suggests
105
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4
5
10
Figure 4.21: 2pFLDI convection velocity normalized by freestream velocity, U? =
200m/s, vs. frequency
that signals below 200kHz will be significantly influenced by disturbances in the
shear layers while signals above 200kHz will be largely the result of disturbances in
the freestream.
Table 4.3 gives the convection velocity computed from the cross-correlation
(Eq. 2.46) for each run as well as the high pass filter settings used. The veloci-
ties range from 52 to 84% of the freestream value. No trend with filter setting is
apparent. Fig. 4.22 shows the frequency dependent velocities normalized by the
cross-correlation values which represent a signal-strength-weighted average across
all frequencies. uc/?uc? is near unity at 50kHz and increases with frequency for all
runs. This indicates that the velocities found from the cross-correlation, ?uc? are
associated with disturbances having frequencies around 50kHz. These are likely to
106
Run # Re [1/m? 106] HP filter ?uc? [m/s] ?uc?/U?
4659 3.8 100Hz 1320 0.65
4667 3.8 none 1680 0.83
4671 3.8 3Hz 1060 0.52
4658 4.7 none 1730 0.84
4662 4.7 none 1100 0.53
Table 4.3: 2pFLDI High Pass (HP) filter settings, and convection velocity averaged
across all frequencies
be located in the sidewall shear layers and not within the depth of focus. Therefore,
the key takeaway from these experiments is that the velocities found by taking the
cross-correlation of the 2pFLDI signals may not represent velocities of disturbances
near the focus of instrument.
It appears that only two runs (4671 and 4662) produced physically meaningful
convection velocities for freestream disturbances. Both measured uc/U? ? 0.75 ?
0.82.
4.3.3.2 Wavenumber Spectra
The phase difference amplitude spectral density vs. wavenumber is plotted
in Fig. 4.23 for all five runs. For runs 4671 and 4662 frequencies are converted
to wavenumbers using the measured convection velocities plotted in Fig. 4.21.
For the other three runs, uc = 0.8U? is used for all frequencies. Our modeling
and spatiotemporal resolution analyses predict that signals below a wavenumber
107
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4
5
10
Figure 4.22: 2pFLDI convection velocity vs. f , normalized by average value over
all f
of approximately kx = 0.9 will be corrupted by disturbances outside of the 73cm
depth of focus. For run 4662, there is a change in slope near the predicted cut-off
wavenumber kx = 0.9, indicating the presence of a low wavenumber hump similar to
those seen in the UTM4 measurements and predicted by Settles and Fulghum [32].
This wavenumber region is indicated in Fig. 4.23 as ?sidewall SL signal?. Above
this wavenumber we can be confident that the signal originates from disturbances
within the depth of focus until the point at which the signal and the flow off noise
floor become coincident, i.e. where SNR reaches unity. The region of the spectra
between these limits is circled and labeled ?core flow signal? on the plot. Note that
this corresponds to a small wavenumber band of 0.9mm?1 < kx < 1.3mm
?1. We
will restrict our analysis to this region from now on.
108
-4
10
-5
10
-6
10
-1 0 1
10 10 10
Figure 4.23: Phase difference vs. wavenumber from FLDI (Ch.A)
85
80
75
70
65
60
0.5 1 1.5
Figure 4.24: Sensitivity function for uniform disturbance in T9M18
109
-8
10
-9
10
0.8 0.9 1 1.1 1.2 1.3 1.4
Figure 4.25: Amplitude spectra in T9M18 for four runs
The FLDI sensitivity function to uniform, flow parallel disturbances for this ex-
periment is graphed in Fig. 4.24. Over the wavenumber band of interest (0.9mm?1 <
k ?1x < 1.3mm ), the value of Hcore is constant and near 80mm. This function is used
along with the measured spectra in Eq. 4.5 to recover the true amplitudes of density
fluctuations in the core flow of the Mach 18 test section. Amplitude spectral density,
labeled Acore, is shown in Fig. 4.25 for four of the five runs. The signal measured
during run 4667 does not have sufficient SNR at these high wavenumbers. Note that
the amplitudes are on the order of 10?9kg/m3 which gives an idea of the lower limit
of what FLDI can measure.
A linear (logarthimic scale) trend is observed in the data, and linear fits to
110
-7
10
11
10
9
8
7
6
5
4
3
0.9 1 1.1 1.2 1.3 1.4 1.5
Figure 4.26: Amplitude spectra in T9M18 normalized by freestream density for four
runs
111
the amplitudes measured in runs 4671 and 4662 yield k?2.56 and k?1.38x x , respectively.
These linear fits are illustrated in Fig. 4.25 as solid and dashed lines. Fig. 4.26 gives
the amplitude spectra normalized by respective freestream density, which is equiva-
lent to a density-based turbulence intensity. Turbulence intensity is near agreement
for all runs at k ?1x = 0.9mm . As wavenumber increases, the measurements deviate.
The data set is too small to draw any conclusions about how turbulence intensity
varies with Reynolds number in this Mach 18 facility. Analysis of these amplitudes
using transfer functions which account for acoustic waves at inclination angles with
respect to freestream [49] is a good candidate for future work.
4.4 Chapter Conclusions and Takeaways
In this chapter, methods for interpreting and reducing the signal from 2pFLDI
in high speed facilities are demonstrated. In UTSI?s 61cm Mach 4 Ludwieg tube, the
2pFLDI (w0 = 7?m) can resolve wavenumbers larger than 0.5mm
?1, corresponding
to frequencies larger than ? 50kHz, with 44cm depth of focus. In AEDC?s Mach 18
tunnel, the 2pFLDI (w0 = 14?m) can resolve wavenumbers larger than 0.9mm
?1,
corresponding to frequencies larger than ? 200kHz, with 73cm depth of focus. A
change in spectra slope is observed near these wavenumber and frequency cut-offs in
both flows. Measurements of convection velocity as a function of frequency, i.e. the
dispersion relation, are novel. Minimal dispersion (a few percent) is measured above
50kHz in the Mach 4 turbulent boundary layer and above 200kHz in T9 for runs
4671 and 4662. A freestream disturbance velocity near 75-82% of the mean velocity
112
is measured in the Mach 18 flow. Amplitude spectra of freestream disturbances are
resolved over a 0.9mm?1 < kx < 1.3mm
?1 band using transfer functions based on
the assumption that disturbances are uniform and flow parallel. The procedures
and data reduction methods outlined here serve as a framework for future 2pFLDI
and multi-point FLDI experiments.
113
Chapter 5: Summary and Future Work
In this chapter we review and summarize the contributions of this thesis which
can be grouped into three main categories:
? Development of analytical transfer function methods for interpreting FLDI
signals including a method for determining spatiotemporal resolution.
? Validation and quantification of the accuracy of transfer function methods in
two ?canonical? flows: a turbulent jet and a Mach 3 wind tunnel.
? Demonstration of FLDI and 2pFLDI in high speed facilities including sig-
nal interpretation methods and the first ever nonintrusive measurements of
freestream disturbances in a Mach 18 flow.
After these conclusions, future avenues for FLDI development and application are
suggested.
5.1 Interpreting FLDI
Chapter 2 gives an overview of focused laser differential interferometry theory.
The instrument is modeled by ray tracing the beams. Assuming that the beams are
Gaussian and that disturbances are sinusoidal along a known propagation direction
114
enables significant analytical reduction of the model. Assuming that disturbances
propagate parallel to the flow direction yields two transfer functions, H?x and Hw
which account for the ways in which beam separation and beam size affect the mea-
sured signal, respectively. While the foundational work on FLDI transfer functions
was performed by others [32,33,49], the unique contribution of this work is the de-
velopment of a method for accounting for spatiotemporal variations in disturbance
amplitude and velocity via the parameters g(z) and uc(z, f). Also unique in this
work is a procedure for determining FLDI spatiotemporal resolution from the trans-
fer functions. These methods provide simple and easy to use standards for future
FLDI experiments.
5.2 Validating the FLDI model
Chapter 3 describes experiments in two canonical flows - a turbulent jet and
Mach 3 flow in a small wind tunnel - that quantify the accuracy of the transfer
function model. Specifically, we show how instrument parameters, ?x1 and w(z),
affect the measurement. We also present a method for determining the beam radius
parameter at the focus (w0) using a beam profiler camera and show that accurately
measuring w0 is critical for application of the transfer functions.
For our model of the jet, convection velocity uc is varied spatially (uc(z)) based
on historical jet studies or temporally (uc(f)) based on 2pFLDI measurements.
For the z-dependent case, the z-axis integral of the transfer functions is resolved
numerically. Both cases yield errors between model and measurement on the order
115
of 10-50% when the jet is translated along the optical axis. This suggests we have
a good understanding of the way in which FLDI rejects signal away from its focal
region. The advantage of uc(f) as opposed to uc(z) is that it can be measured
directly by 2pFLDI, enabling signal analysis in flows where historical measurements
are not available. This is demonstrated in the Mach 3 tunnel which also yields
errors between model and measurement on the order of 10-50% when the FLDI is
translated along the optical axis through the tunnel. These successful applications of
the the simple FLDI model in the complex jet and tunnel flows represent a significant
validation of the model.
5.3 Demonstrations of 2pFLDI
Chapter 4 details the application and signal analysis of 2pFLDI measurements
made in two separate campaigns. The first is in University of Tennessee Space In-
stitute?s Mach 4 Ludwieg tube (UTM4) and the second is in AEDC T9?s Mach 18
test section (T9M18). In each campaign, a procedure for determining the FLDI
spatiotemporal resolution and interpreting the signal is described. This procedure
sets an example for future FLDI experiments to follow, thus it is an important con-
tribution. Transfer functions illustrate how FLDI achieves a spatial resolution of
O(50cm) to disturbance wavenumbers greater than O(1mm?1), corresponding to
frequencies greater than O(100kHz) in these facilities. With this context, measure-
ments of turbulence intensity and convection velocity in the floor boundary layer
of the Mach 4 tunnel are presented. Measurements of freestream disturbances in
116
T9?s Mach 18 test section are reported as well. Successfully applying 2pFLDI in
this large-scale industrial setting is a significant achievement, and the results repre-
sent a contribution to the understanding of disturbance convection velocity in this
hypersonic facility.
5.4 Future Research
One avenue of future research is to explore alternatives to the flow parallel
(Eq. 2.11) and acoustic wave (Eq. 2.14) disturbance models employed in this
work and by Lawson [49], respectively. Recent work by Hameed and Parziale [48]
offers an example of a model which accounts for non-uniform disturbances with
non flow parallel inclinations angles. Investigating alternative models will lead to
different forms of the transfer functions which may be more accurate, depending on
the specific flow of interest. Alternative models can be validated using the same
experiments and procedures employed in chapter 3.
The FLDI depth of focus is improved by increasing the convergence angle of
the beams which in turn reduces w0. The smallest w0 achieved in this work is on
the order of 3 microns. In theory, this could be reduced to as small as twice the
laser wavelength [54]. A study which seeks to maximize FLDI?s spatial resolution by
employing very large, short focal length field lenses would illuminate the limitations
of the technique.
Finally, the 2pFLDI velocity measurement does not need to share the signal-
point FLDI measurement spatial resolution. An optical arrangement where the two
117
FLDI optical axes are not parallel, but the focus still coincide along the x-axis could
yield velocity measurements with significantly higher spatial resolution. Similar to
a cross-beam Schlieren set-up [31], a cross-beam FLDI could yield measurements of
dispersion with spatial resolution on the order of the beam pair separation, ?x2.
118
Appendix A: Derivations
A.1 Derivation of FLDI transfer functions, Eq.2.24
Beginning with the argument of the z-axis integral in Eq.2.23, which we will
call Z, we have:
? 2? ? ? ? 2? ? ?
Z = I0(r, ?)?(x1, t)rdrd? ? I0(r, ?)?(x2, t)rdrd? (A.1)
0 0 0 0
Plug in the definition of beam intensity profile I0 (Eq.2.18), the density field model
?(x, y, z, t) (Eq.2.13) and the definitions of the beam paths x1 and x2 (Eq.2.21 and
Eq.2.22):
?Z =2? ? ? ( ) [ ( ( ) )]2 2r2 ? ? ?x1
? ? exp(? ) g(z)? A(f) exp [i(kx(x+ )? 2?ft+ ? dfrdrd??2 20 0 ?w w ?? 22? ? )]2 2r2 ? ?x1
exp ? g(z) A(f) exp i k
2 2 x
x? ? 2?ft+ ? dfrdrd?
0 0 ?w w ?? 2
(A.2)
119
Next, exchange the order of integration and re-arrange some terms:
Z? =? ? ? ( )?2r 2r2 2? [ ( ( ) )]?x1
g ? A ? exp (? )? exp [i (kx (x+ )? 2?ft+ ? d?drdf??w2 2?? 0 w 0 2? ? )]2r 2r2 2? ?x1
g A exp ? exp i kx x? ? 2?ft+ ? d?drdf
?? 0 ?w
2 w2 0 2
(A.3)
Re-arrange more terms and convert from Cartesian to polar coordinates using x =
r cos(?):
? ?
{Z =[g ( (A??? ) )]?
?x ?
( )?
2r 2r2 2?1
exp[ (i k(x + ) ? 2?ft+ ?)]? exp( ? )? exp(ikxr cos ?)d?dr?2 2 20 ?w w 0
?x ? 2r 2r2 2?
}
1
exp i kx ? ? 2?ft+ ? exp ? exp(ikxr cos ?)d?dr df
2 2 20 ?w w 0
(A.4)
Re-arrange terms and evaluate the integral over ?:
? ? ( [ ] [ ])ikx?x1 ?ikx?x1
Z = g A exp [i(?2?ft+ ?)] exp? ?(exp?? 2? )
?
2
2r 2r2
exp ? 2?J0(kxr)drdf (A.5)
?w20 w
2
where J0 is the Bessel function of the first kind. Next, we evalute the subtraction
of the two exponential terms and evaluated the integral over beam radius, r. For
more detail on this integral evaluation, see the next subsection, appendix A.1.1.
? ? ( ) ( )2 2
Z = g A exp [i(? kx?x1 ?kxw2?ft+ ?)] 2i sin exp df. (A.6)
?? 2 8
Finally, re-arrange to get the argument of the z-axis integral in Eq.2.24.
120
A.1.1 Evaluation of integral over beam radius, r
Start with the integral over r in Eq.A.5, which we will call R:
? ? ( )
R 4r 2r
2
= exp ? J0(kxr)dr (A.7)
w2 w20
Using the definition of the Bessel function of[the first kind, J( ) 0]
:
? ?
R 4
?
?2r
2 ? (? 2 2 nk
= r exp x
r /4)
dr (A.8)
w2 20 w (n!)
2
n=0
Expand out the sum:
?
4 ?
( )[ ]
2r2 k2R r
2 k4r4
= r exp ? 1? x + x ? ... dr (A.9)
w2 20 w 4 64
Re-arrange:
[?
4 ?
( ) ( )
2 2 ? ? 2
R = r exp ?2r ? kx 3 ?2rdr r e?xp dr+w2 w2 4 w20 0 ? ( ) ]k4 2x r5 ?2rexp dr ? ... (A.10)
64 20 w
Evaluate the integrals over r:
[ ( ) ( ) ]
4 w2R ? k
2 4 4 6
= x
w k w
+ x ? ... (A.11)
w2 4 4 8 64 8
Finally, we have:
k2w2 4R k w
4
= 1? x + x ? ... (A.12)
8 128
Recognizing the Taylor?s series expansion, we have
( )
R ?k
2
xw
2
= exp . (A.13)
8
121
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