SURFACE PRESSURE MEASUREMENTS ON A ROTATING CONTROLLED
DIFFUSION BLADE
By
Andrew Falck Cawood

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Mechanical Engineering
2012

ABSTRACT
SURFACE PRESSURE MEASURMENTS ON A ROTATING CONTROLLED
DIFFUSION BLADE
By
Andrew Falck Cawood
A method for quantifying fluctuating pressure magnitudes on the surface of a Controlled Diffusion (CD) blade was utilized to identify characteristics of the flow structure of
the boundary layer at various streamwise and spanwise locations. This research effort
explored the fundamental aspects of an axial fan flow field - a supplement to and a continuation of the work performed by Douglas Neal (2010). The work is driven by the motivation to identify flow structures within the boundary layer of the CD blade and characterize
similarities between the results for rotating and stationary blades. These results are
expected to be valuable for those working in the areas of aeroacoustics and noise prediction. Streamwise and spanwise surface pressures were measured along the surface of a stationary and a rotating CD blade. Additional trailing edge velocity wake surveys identify
boundary layer features of the rotating and stationary blades. Surface pressure statistics,
spectral characteristics, and correlations distinguish elements of the boundary layer from
the stationary and rotating CD blades. Further results from correlations and spectral analysis on the airfoil trailing edge region identify spatial and temporal decay rates as well as
parameters useful for trailing edge aeroacoustic noise prediction. An off-design operating
point was evaluated to expand the context of this experiment; significant differences were
observed. The fluctuating wall pressure observations are consistent with those made of a
flat plate boundary layer (Willmarth and Woolridge 1962) and of the CD geometry by
Moreau and Roger (2005).

Copyright by
Andrew Falck Cawood
2012

Many men go fishing all their lives not knowing it is not fish they are after.
— Henry David Thoreau —
To my father, the man who taught me to fish

iv

ACKNOWLEDGMENTS
On account of being at the MSU for so many years, as a student and as a local resident, I have encountered so many wonderful friends, colleagues, and strangers.
I would first like to extend thanks to Dr. John Foss and my committee members.
Dr. Foss had the audacity to pick me up as an undergraduate and hired me in his laboratory. Most impressive, two years later he convinced me to stay at MSU for my Masters
and managed to coax the following document out of me (“on-schedule”). Dr. Foss has
been a truly outstanding mentor though the years - I have had the pleasure of working
closely with him and experiencing the finest of his anecdotal advice.
I have also had the pleasure of working with so many great friends and colleagues.
The invaluable advice and guidance from Prof. Scott Morris of Notre Dame and Prof.
Stéphane Moreau of The Université de Sherbrooke made most of this work possible. I was
fortunate enough to follow the work of Dr. Douglas Neal; I served as his assistant through
the latter part of his Ph.D. and learned more than I ever imagined about the nuances of
experimental fluid mechanics, coffee, and kayaking. Doug laid the groundwork for the
present work and provided me many hours of feedback and guidance on a personal and
professional level. I had the opportunity to work with many other great people: To Al,
Kyle, Rohit, and Jason, thank you for the unending support. Your technical and practical
skills made this work possible. Your friendships made it gratifying. I owe a big thanks to
Daniel Höwer, without whom I would still be looking for answers. Daniel was critical for
the computational development of the boundary layer results.
A special thanks goes to the Craigs of the ME department - The two people in the
department who always have the right answers. I always knew who to go to if I needed

v

advice, guidance, or say, last-minute tech support for a thesis defense. Gunn: Modern
usage of the Oxford comma and ad hoc racquetball games - I’m still confidant I know the
rules of neither - Thanks! Somerton: You couldn’t be more correct - It’s the little things
that make lasting impressions! To my machine shop friends, Roy and Ken for providing a
haven from the insanity of school (as well as arguably the worlds best chili recipe)! To the
friends I’ve met through baja - these are probably the only people that can tolerate me
driving them across Montana blaring calypso music that “only old people like”.
To those who have given me a great deal of support through my undergraduate and
graduate years. The secretarial staff, without whom I would cease to function: To Aida and
Mary who somehow managed to wade through the mountains of paperwork I’ve caused
and to Jill and Suzanne who have always been willing to lend a helpful ear.
To my friends, you’ve been more important to me than you’ll ever know. From
karaoke nights, long bike rides, and tolerating me talking through any movie, you’ve kept
me (somewhat) sane through this process. For those who I have neglected and for total
strangers, know that your kindness was not forgotten. A special thanks goes to my team of
editors who laughed with me at my egregious and often obvious mispellings.
And finally, I want to send the most sincerest of thanks to my family. You have literally been there every step of the way. Thank you for your unending support!
On a personal level: For those that will continue reading this document, note that
this is merely the quantitative aspect my degree program. This has been a truly formative
journey - a special thanks is needed for those who kept me motivated and on task but knew
when I needed a break - I am in debt to all of you beyond words. I wish you all the best in
your future adventures. Remember: work hard, be kind, and amazing things will happen!

vi

TABLE OF CONTENTS

LIST OF TABLES........................................................................................................... viii
LIST OF FIGURES ........................................................................................................... ix
KEY TO SYMBOLS AND ABBREVIATIONS ........................................................... xvii
CHAPTER 1.0: INTRODUCTION .....................................................................................1
1.1 Motivation................................................................................................................1
1.2 Pressure Fluctuations in Turbulent Boundary Layers..............................................2
1.3 Aeroacoustics...........................................................................................................6
1.4 Definitions of the Computational Methods Utilized in this Document .................12
CHAPTER 2.0: EXPERIMENTAL EQUIPMENT AND PROCEDURE ........................16
2.1 RCDB Fan Development .......................................................................................16
2.2 Experimental Configuration/Facility .....................................................................21
2.2.1 Axial Flow Research and Development (AFRD) Facility............................21
2.2.2 Rotating Controlled Diffusion Blade (RCDB) Assembly ............................23
2.2.3 Remote Microphone Probes (RMPs) ............................................................34
2.2.4 Hot-Wire Measurements...............................................................................47
2.3 Experimental Parameters/Procedure......................................................................50
2.3.1 Experimental Operating Conditions .............................................................50
2.3.2 Data Acquisition ...........................................................................................53
2.4 Stationary Experiments..........................................................................................54
CHAPTER 3.0: EXPERIMENTAL RESULTS AND DISCUSSION ..............................55
3.1 Pressure Coefficient ...............................................................................................56
3.2 Statistics .................................................................................................................59
3.3 Pressure Spectra .....................................................................................................77
3.4 Space-Time Correlations .....................................................................................101
3.5 Integral Length Scales .........................................................................................112
3.6 Spectral Processing: Coherence and Phase..........................................................115
3.7 Hot-wire measurements .......................................................................................122
CHAPTER 4.0: SUMMARY AND CONCLUSIONS....................................................129
Appendix A: Remote Microphone Probe Development..................................................134
Appendix B: Optimal Noise Cancellation .......................................................................143
Appendix C: Additional Figures......................................................................................144
REFERENCES ................................................................................................................164

vii

LIST OF TABLES
Table 2.1: Suction side taps (y/l=0.5) ....................................................................26
Table 2.2: Pressure side taps (y/l=0.5) ...................................................................26
Table 2.3: Trailing edge spanwise taps (x/c=0.978) ..............................................26
Table 3.1: Cross-Correlation Results ...................................................................128

LIST OF FIGURES
Figure 1.1 Airfoil boundary layer (Gerakopulos and Yarusevych 2012) ............................3
Figure 1.2 Streamwise space-time correlation (Willmarth and Woolridge 1962)...............4
Figure 1.3 Airfoil self-noise (Brooks et al. 1989)................................................................7
Figure 1.4 Spectral characteristics of a turbulent boundary layer .......................................9
Figure 2.1 Flow geometry schematic (left - rotating; right - stationary) ...........................16
Figure 2.2 Modular blade section ......................................................................................18
Figure 2.3 RCDB 5-blade CAD rendering (left - upstream; right - downstream).............19
Figure 2.4 RCDB Cp comparison for several blade configurations ..................................20
Figure 2.5 Final RCDB Cp configuration and CD airfoil ..................................................20
Figure 2.6 The Axial Fan Research and Development (AFRD) Facility ..........................22
Figure 2.7 RCDB assembly installed in AFRD Facility....................................................23
Figure 2.8 RCDB coordinate systems (left - r−θ−z; right - x-y-z) ....................................24
Figure 2.9 RCDB instrumented pressure taps ...................................................................25
Figure 2.10 Surface pressure tap locations ........................................................................25
Figure 2.11 Data acquisition boards (photos: www.mccdaq.com)....................................27
Figure 2.12 Slip ring assembly (Neal 2010) ......................................................................28
Figure 2.13 MEMS pressure transducer (Neal 2010) ........................................................29
Figure 2.14 Pressure transducer schematic (Neal 2010)....................................................31
Figure 2.15 Amplifier and microphone system .................................................................32
Figure 2.16 Hub instrumentation package .........................................................................33
Figure 2.17 Data path schematic (RCDB) .........................................................................33
Figure 2.18 RMP microphone ...........................................................................................36

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Figure 2.19 RMP tube splicing ..........................................................................................36
Figure 2.20 RMP manifold schematic ...............................................................................36
Figure 2.21 Installed RMP manifolds................................................................................37
Figure 2.22 Pneumatic circuit of RMP system ..................................................................38
Figure 2.23 Wavetube experiment and schematic .............................................................39
Figure 2.24 RMP manifold and reference spectra .............................................................40
Figure 2.25 Larson-Davis comparison (in situ method) ....................................................41
Figure 2.26 RCDB in situ microphone calibration rig.......................................................42
Figure 2.27 Transfer functions for RMP24 using in situ method ......................................43
Figure 2.28 Original and calibrated RMP 9 spectra ..........................................................44
Figure 2.29 AFRD acoustic noise......................................................................................45
Figure 2.30 Schematic and image of SN hot-wire.............................................................46
Figure 2.31 Hot-wire traverse ............................................................................................47
Figure 2.32 Hot-wire calibration unit ................................................................................49
Figure 2.33 Flow geometry Case 1 ....................................................................................52
Figure 2.34 Flow geometry Case 2 ....................................................................................52
Figure 2.35 Flow geometry velocity triangles (Case 1 and Case 2) ..................................52
Figure 2.36 Performance curves for RCDB Case 1 and Case 2 ........................................53
Figure 3.1 Cp distribution CD and RCDB Case 1 .............................................................58
Figure 3.2 Cp distribution RCDB Case 1 and Case 2 ........................................................58
Figure 3.3 Sample time series RCDB Case 1 ....................................................................60
Figure 3.4 RCDB-CD Case 1 suction side PDFs...............................................................62
Figure 3.5 RCDB-CD Case 1 suction side PDFs continued..............................................63

x

Figure 3.6 RCDB-CD Case 1 trailing edge spanwise PDFs..............................................64
Figure 3.7 RCDB-CD Case 1 pressure side PDFs.............................................................65
Figure 3.8 RCDB Case1-Case2 suction side PDFs ...........................................................67
Figure 3.9 RCDB Case1-Case2 suction side PDFs continued...........................................68
Figure 3.10 RCDB Case1-Case2 trailing edge spanwise PDFs.........................................69
Figure 3.11 RCDB Case1-Case2 pressure side PDFs........................................................70
Figure 3.12 RCDB-CD Case 1 suction side moments.......................................................74
Figure 3.13 RCDB Case1-Case2 suction side moments....................................................74
Figure 3.14 RCDB-CD Case 1 pressure side moments .....................................................75
Figure 3.15 RCDB Case 1 - Case 2 pressure side moments..............................................75
Figure 3.16 RCDB-CD Case 1 trailing edge spanwise moments ......................................76
Figure 3.17 RCDB Case1 - Case2 trailing edge spanwise moments.................................76
Figure 3.18 RCDB and CD Case 1 (Moreau) spectra: Tap1 (x/c=0.013) .........................80
Figure 3.19 RCDB and CD Case 1 (Moreau) spectra: Tap2 (x/c=0.03) ...........................80
Figure 3.20 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap3 (x/c=0.052).......80
Figure 3.21 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap5 (x/c=0.087).......81
Figure 3.22 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap6 (x/c=0.149).......81
Figure 3.23 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap7 (x/c=0.403).......81
Figure 3.24 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap9 (x/c=0.534).......82
Figure 3.25 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap11 (x/c=0.679).....82
Figure 3.26 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap21 (x/c=0.858).....82
Figure 3.27 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap22 (x/c=0.881).....83
Figure 3.28 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap23 (x/c=0.899).....83

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Figure 3.29 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap24 (x/c=0.922).....83
Figure 3.30 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap25 (x/c=0.978).....84
Figure 3.31 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap25A (y/l=0.45).....84
Figure 3.32 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap25B (y/l=0.48).....84
Figure 3.33 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap25C (x/c=0.58) ....85
Figure 3.34 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap4 (x/c=0.067).......85
Figure 3.35 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap8 (x/c=0.400).......85
Figure 3.36 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap10 (x/c=0.530).....86
Figure 3.37 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap12 (x/c=0.675).....86
Figure 3.38 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap29 (x/c=0.929).....86
Figure 3.39 RCDB Case 1 and Case 2 spectra: Tap1 (x/c=0.013) ....................................87
Figure 3.40 RCDB Case 1 and Case 2 spectra: Tap2 (x/c=0.030) ....................................87
Figure 3.41 RCDB Case 1 and Case 2 spectra: Tap3 (x/c=0.052) ....................................87
Figure 3.42 RCDB Case 1 and Case 2 spectra: Tap5 (x/c=0.087) ....................................88
Figure 3.43 RCDB Case 1 and Case 2 spectra: Tap6 (x/c=0.149) ....................................88
Figure 3.44 RCDB Case 1 and Case 2 spectra: Tap7 (x/c=0.403) ....................................88
Figure 3.45 RCDB Case 1 and Case 2 spectra: Tap9 (x/c=0.534) ....................................89
Figure 3.46 RCDB Case 1 and Case 2 spectra: Tap11 (x/c=0.679) ..................................89
Figure 3.47 RCDB Case 1 and Case 2 spectra: Tap21 (x/c=0.858) ..................................89
Figure 3.48 RCDB Case 1 and Case 2 spectra: Tap22 (x/c=0.881) ..................................90
Figure 3.49 RCDB Case 1 and Case 2 spectra: Tap23 (x/c=0.899) ..................................90
Figure 3.50 RCDB Case 1 and Case 2 spectra: Tap24 (x/c=0.922) ..................................90
Figure 3.51 RCDB Case 1 and Case 2 spectra: Tap25 (x/c=0.978) ..................................91

xii

Figure 3.52 RCDB Case 1 and Case 2 spectra: Tap25A (y/l=0.45) ..................................91
Figure 3.53 RCDB Case 1 and Case 2 spectra: Tap25B (y/l=0.48) ..................................91
Figure 3.54 RCDB Case 1 and Case 2 spectra: Tap25C (y/l=0.58) ..................................92
Figure 3.55 RCDB Case 1 and Case 2 spectra: Tap4 (x/c=0.067) ....................................92
Figure 3.56 RCDB Case 1 and Case 2 spectra: Tap8 (x/c=0.400) ....................................92
Figure 3.57 RCDB Case 1 and Case 2 spectra: Tap10 (x/c=0.530) ..................................93
Figure 3.58 RCDB Case 1 and Case 2 spectra: Tap12 (x/c=0.675) ..................................93
Figure 3.59 RCDB Case 1 and Case 2 spectra: Tap29 (x/c=0.929) ..................................93
Figure 3.60 RCDB Case 1 fore region...............................................................................96
Figure 3.61 RCDB Case 1 mid-chord region ....................................................................96
Figure 3.62 RCDB Case 1 aft region .................................................................................96
Figure 3.63 RCDB Case 1 trailing edge spanwise region .................................................97
Figure 3.64 RCDB Case 1 pressure side............................................................................97
Figure 3.65 CD fore region................................................................................................97
Figure 3.66 CD mid-chord region......................................................................................98
Figure 3.67 CD aft region ..................................................................................................98
Figure 3.68 CD trailing edge spanwise region ..................................................................98
Figure 3.69 CD pressure side.............................................................................................99
Figure 3.70 RCDB Case 2 fore region...............................................................................99
Figure 3.71 RCDB Case 2 mid-chord region ....................................................................99
Figure 3.72 RCDB Case 2 aft region ...............................................................................100
Figure 3.73 RCDB Case 2 trailing edge spanwise region ...............................................100
Figure 3.74 RCDB Case 2 pressure side..........................................................................100

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Figure 3.75 Typical space-time correlation (200-500Hz)................................................102
Figure 3.76 Controlled Diffusion surface tap geometry ..................................................102
Figure 3.77 Space-time correlations: RCDB Case 1 .......................................................105
Figure 3.78 Space-time correlations: CD ........................................................................106
Figure 3.79 Space-time correlations: RCDB Case 2 .......................................................107
Figure 3.80 Coherent motion convection velocity...........................................................108
Figure 3.81 Willmarth and Woolridge (1962) decay.......................................................110
Figure 3.82 RCDB: Case 1 decay ....................................................................................110
Figure 3.83 CD decay ......................................................................................................111
Figure 3.84 RCDB: Case 2 decay ....................................................................................111
Figure 3.85 Decay comparison ........................................................................................111
Figure 3.86 Longitudal integral length scale ...................................................................113
Figure 3.87 Transverse integral length scale ...................................................................114
Figure 3.88 Coherence functions .....................................................................................117
Figure 3.89 Phase RCDB Case 1 .....................................................................................118
Figure 3.90 Phase CD blade ............................................................................................119
Figure 3.91 Phase RCDB Case 2 .....................................................................................120
Figure 3.92 Convection velocity comparison CD - RCDB Case 1 .................................122
Figure 3.93 Schematic of hot-wire measurements...........................................................123
Figure 3.94 Wake Statistics RCDB: Case 1 (R=rθ) ........................................................124
Figure 3.95 Wake Statistics RCDB: Case 2 (R=rθ) ........................................................125
Figure 3.96 Power spectra (u,p) RCDB: Case 1 ..............................................................126
Figure 3.97 Power spectra (u,p) RCDB: Case 2 ..............................................................126

xiv

Figure 3.98 Pressure-velocity cross-correlation RCDB: Case 1......................................127
Figure A.2.1 PWT schematic and specifications .............................................................136
Figure A.2.2 Linear input-output system.........................................................................138
Figure A.2.3 Larson Davis transfer function ...................................................................138
Figure A.2.4 in situ verification rig .................................................................................139
Figure A.2.5 in situ verification.......................................................................................140
Figure A.3.6 RMP microphone specifications.................................................................141
Figure A.3.7 Larson Davis manufacturer specifications .................................................142
Figure B.1 Sample ONC filter applied to preliminary CD data ......................................143
Figure C.1.1 Band-filtered Rpp correlations (CD blade) .................................................144
Figure C.1.2 Band-filtered Rpp correlations (CD blade) .................................................145
Figure C.1.3 Band-filtered Rpp correlations (CD blade) .................................................146
Figure C.1.4 Band-filtered Rpp correlations (CD blade) .................................................147
Figure C.1.5 Band-filtered Rpp correlations (RCDB Case 1) ..........................................148
Figure C.1.6 Band-filtered Rpp correlations (RCDB Case 1) ..........................................149
Figure C.1.7 Band-filtered Rpp correlations (RCDB Case 1) ..........................................150
Figure C.1.8 Band-filtered Rpp correlations (RCDB Case 1) ..........................................151
Figure C.1.9 Band-filtered Rpp correlations (RCDB Case 2) ..........................................152
Figure C.1.10 Band-filtered Rpp correlations (RCDB Case 2) ........................................153
Figure C.1.11 Band-filtered Rpp correlations (RCDB Case 2) ........................................154
Figure C.1.12 Band-filtered Rpp correlations (RCDB Case 2) ........................................155
Figure C.2.1 Coherence and Phase (RCDB Case1: Tap1-Tap2) .....................................156
Figure C.2.2 Coherence and Phase (RCDB Case1: Tap2-Tap3) .....................................156

xv

Figure C.2.3 Coherence and Phase (RCDB Case1: Tap3-Tap5) .....................................157
Figure C.2.4 Coherence and Phase (RCDB Case1: Tap5-Tap6) .....................................157
Figure C.2.5 Coherence and Phase (RCDB Case1: Tap6-Tap7) .....................................158
Figure C.2.6 Coherence and Phase (RCDB Case1: Tap7-Tap9) .....................................158
Figure C.2.7 Coherence and Phase (RCDB Case1: Tap9-Tap11) ...................................159
Figure C.2.8 Coherence and Phase (RCDB Case1: Tap11-Tap21) .................................159
Figure C.2.9 Coherence and Phase (RCDB Case1: Tap21-Tap22) .................................160
Figure C.2.10 Coherence and Phase (RCDB Case1: Tap22-Tap23) ...............................160
Figure C.2.11 Coherence and Phase (RCDB Case1: Tap23-Tap24) ...............................161
Figure C.2.12 Coherence and Phase (RCDB Case1: Tap24-Tap25) ...............................161
Figure C.2.13 Coherence and Phase (RCDB Case1: Tap25-Tap25A) ............................162
Figure C.2.14 Coherence and Phase (RCDB Case1: Tap25-Tap25B) ............................162
Figure C.2.15 Coherence and Phase (RCDB Case1: Tap25-Tap25C) ............................163

xvi

KEY TO SYMBOLS AND ABBREVIATIONS
ABBREVIATIONS
AFRD.........................Axial Fan Research and Development
CD ..............................Controlled Diffusion
RCDB.........................Rotating Controlled Diffusion Blade
ROMAN SYMBOLS
A(f), B(f) .....................Fourier transform of random sinusoidal processes a(t), b(t)
A(x), B(x)....................Random variable
Cp ...............................Static pressure coefficient
Cl ................................Lift coefficient
Kurt ............................Kurtosis - 4th moment (1.16)
L11,L22 ........................Integral length scale at t=0 of R11 and R22
P(x).............................Probability density function (1.12)
R11,R22 .......................Streamwise and spanwise autocorrelation
Raa, Rbb ......................Cross-correlation of random processes a(t), b(t) where a(t)=b(t)
Rab ..............................Cross-correlation of random processes a(t), b(t) (1.20)
Rpp ..............................Cross-correlation of pressure (1.1)
Rec ..............................Chord based Reynolds Number ( Re c = u ∞ c ⁄ υ )
Skew ...........................Skewness - 3rd moment (1.15)
Tij ................................Lighthill’s tensor (1.6)
U.................................RCDB angular velocity (Figure 2.1)
Uc ...............................Convection velocity (1.2, 3.2)
V .................................RCDB inlet velocity in the stationary reference frame (Figure 2.1)

xvii

V a ⁄ b ...........................Velocity of air with respect to the blade
V a ⁄ g ...........................Velocity of air with respect to the ground
V b ⁄ g ...........................Velocity of the blade with respect to the ground

Var ..............................Variance - 2nd moment (1.14)
W ................................RCDB inlet velocity in the rotating reference frame (Figure 2.1)
a(t), b(t) ......................Random sinusoidal process
c..................................Chord length
co ................................Speed of sound in air
d .................................Microphone diameter
d+ ...............................Microphone diameter in wall units
f ..................................Frequency
f+ ................................Frequency in wall units
k .................................Characteristic wavenumber of turbulent motions
·
m ................................Mass flow rate

p(x1,x2) .......................Pressure along airfoil surface
p ∞ ..............................Atmospheric pressure

r,θ,z.............................Radial coordinates
t ..................................Time
ui,uj .............................Fluctuating velocity vector
u ∞ ..............................Freestream velocity: Stationary - CD

uτ ................................Friction velocity
w ∞ ..............................Freestream velocity: Rotating - RCDB

x,y,z.............................Cartesian coordinates

xviii

x1, x2 ...........................Blade surface streamwise and spanwise directions

Δx1, Δx2 ......................Blade surface streamwise and spanwise tap separations (see h1, h2)
GREEK SYMBOLS

Λ11,Λ22 .......................Integral length scale at t=0 of R11 and R22
Φaa,Φbb ......................Power spectral density for random processes a(t), b(t) (1.17)
Φab .............................Cross spectral density
Φpp .............................Wall pressure power spectral density (1.8)
Φp,rad..........................Radiated acoustic power spectral density (1.9)
Ω ................................Rotational speed
α .................................Geometric angle of attack
αc................................Camber angle
αi ................................Aerodynamic angle of attack
γab ...............................Coherence function (1.18)
δ..................................Boundary layer thickness
δ*................................Displacement thickness
δij ................................Kronecker delta
η .................................Directionless correlation distance
η1,η2 ..........................Streamwise and spanwise correlation lengths
λ .................................Wavelength
λw ...............................Eddy wavelength (1.4) - Charactistic wavelength of turbulent
motions

μ .................................Mean - 1st moment (1.13)
υ .................................Kinematic viscosity of air
ρ .................................Density of air
xix

σ .................................Standard deviation (1.14)
σij ...............................Reynolds stress tensor
t ..................................Correlation time delay

Δt ................................Time delay between correlation peaks
τw................................Wall shear stress
f ..................................Phase factor (1.19)

ω .................................Angular frequency
ϖ ................................Characteristic frequency of turbulent motions

xx

1.0 Introduction
1.1 Motivation
This research effort explored the fundamental aspects of an axial fan flow field - a
supplement to and continuation of the work performed by Douglas Neal (2010). A planar,
Controlled Diffusion (CD) airfoil was cast into a rotating axial fan blade and attached to
the rotating hub of the Axial Fan Research and Development (AFRD) facility at Michigan
State University. The acronym: RCDB (Neal 2010) - Rotating CD Blade - is used to
describe the experimental configuration. The RCDB results are compared with a stationary CD blade by way of mean and fluctuating surface pressures. Fluctuating pressure measurements can be used to comparatively infer boundary layer properties by implementing
statistical, spectral, and correlative analysis. The purpose is to identify properties of the
turbulent boundary layer of the fan blade surface and compare the rotating fan to the stationary airfoil at matched flow conditions. Additionally, the fan is operated at an offdesign flow condition to better quantify boundary layer behavior. Ultimately, this project
is part of a larger study whose goal is the reduction of noise from axial fans and similar
devices. This work provides experimental verification of the rotating analog and baseline
measurements to the aeroacoustic prediction community.
Aeroacoustics has received expanded attention in the last few decades and has
become an important discipline within aeronautics. Modern commercial demands require
larger and faster vehicles, bigger structures and higher efficiencies. The environmental
and societal impact of these structures and vehicles has been under more serious investigation as population growth increases; the development of on-shore wind turbines has been
limited due to noise issues with those residing in populated areas (Dassan et al. 1997).
Noise abatement and the design of quieter air vehicles and structures are essential to the
1

continued growth of these industries. A characteristic self-noise-generation condition that
occurs in these application areas is the low Mach number turbulent boundary layer that is
formed over and separates from a surface. Lifting and control surfaces are of particular
importance in this regard.
1.2 Pressure Fluctuations in Turbulent Boundary Layers
This document is focused on characterizing the boundary layer along the surface
of a controlled diffusion airfoil. A typical boundary layer on the airfoil suction surface will
begin in the laminar state and transition to turbulence some distance downstream (see
Figure 1.1). The controlled diffusion airfoil is known to have an early transition to turbulence and a well established boundary layer near the trailing edge.
It is instructive to discuss some of the governing aspects of boundary layer pressure fluctuations and to introduce more modern computational methods. Previous experimental surface pressure boundary layer studies focused primarily on simplified
geometries; a carefully developed flat plate boundary layer ensures a well developed and
stable flow field. The pressure fluctuations of a flat boundary layer are well discussed by
Willmarth and Woolridge (1962), Bull (1967), and Gravante et al. (1998). An excellent
summary of previous research is provided by Bull (1996).
Willmarth defines several data reduction methods and parameters that are useful
for characterizing the boundary layer; he identifies the wall pressure fluctuations as a stationary random variable that is a function of time and position. Since the pressure fluctuations are statistically stationary, Willmarth proposes expressing the fluctuations along the
surface in terms of their root-mean-square (RMS) pressure; he identifies an increase in
RMS pressure as an

2

Figure 1.1 Airfoil boundary layer (Gerakopulos and Yarusevych 2012)
indicator for a boundary layer transition1. Further interpretations of the boundary layer are
accomplished through the implementation of space-time cross-correlations.
The cross-correlation of a pressure signal is defined as
p ( x 1, x 2, t ) ⋅ p ( x 1 + Δx 1, x 2 + Δx 2, t + τ )
R pp ( x 1, x 2, t ) = ------------------------------------------------------------------------------------------------------------2 ( x , x , t ) ⋅ p2 ( x + Δx , x + Δx , t + τ )
p 1 2
1
1 2
2

1.1

where x1 and x2 are the spanwise and streamwise pressure tap coordinates, Δx1 and Δx2 are
the streamwise and spanwise pressure tap separations, and τ is the correlation time delay
between pressure measurements.
Willmarth used the space-time cross-correlation to “track” motions of selected frequency bands. The thought behind filtering the fluctuating pressure signal into frequency
bands is to specify frequency regions and identify their role and impact on the boundary
layer. Figure 1.2 shows the streamwise space-time correlation of a high and low frequency
band. The ordinate shows the correlation coefficient and the abscissa shows the time delay
1. It is noteworthy to mention that the RMS is equal to the standard deviation when
the mean of the signal is zero (the square root of the 2nd statistical moment), as
it is for microphone measurements.
3

Figure 1.2 Streamwise space-time correlation (Willmarth and Woolridge 1962)
Solid line: 300Hz < f < 700Hz
Dashed line: 3000Hz < f < 5000Hz
normalized by the free-stream velocity ( u ∞ ) and boundary layer displacement thickness
(δ*). Each parabola shape in the figure represents the peak segment of the cross-correlation for a given a downstream pressure tap and an upstream reference tap. Note that the
numbers along the peaks identify the location of the downstream pressure tap (x1/δ*, reference tap at: x1=0).
Willmarth proposed using the gross time delay between correlation peaks (Δτ) to
define the convection velocity Uc of the boundary layer
Δx
U c = ----Δτ

1.2

Convection velocity is an implicit function of frequency when the fluctuating pressure signal is filtered following Willmarth’s method. Willmarth suggests defining an average frequency and an average wavenumber for a given frequency band
ω low + ω high
ω
k = ------ = --------------------------------Uc
2U c
The idea of a characteristic wavelength follows:

4

1.3

2U c
2π
λ w = ----- = --------------------------------ω low + ω high
k

1.4

The eddy wavelength is a descriptor of the length scale of the convecting pressure fluctation.
Gravante et al. (1998) identify several key parameters for accurate measurements
at low frequencies and in noisy facilities. Additionally, they explored the effects of sensor
averaging on attenuation at high frequencies. It is ideal to have a very small sensor diameter for measuring the fluctuating pressures; if the sensor diameter is too large, the smallest
scales of the flow cannot be resolved since frequency scales are the inverse of the wavelength. It is suggested that to avoid spectral attenuation for resolving frequencies up to
+ fυ
+ du τ
f = ----- = 1 a pinhole sensor with dimensions d = -------- < 18 should be used. In the
υ
u2
τ
+ du τ
present experiment, a pinhole d=0.5mm corresponds to 6.5 < d = -------- < 12.5 in the
υ
trailing edge region. The sensor diameter is well within the acceptable range for a maximum frequency of ~35kHz (well beyond the frequency range of interest).
More recent experiments have the advantage of more computational power and
improved transducers with respect to the pioneering work of the 1960s. Boutilier and
Yarusevych (2012) considered the surface pressure fluctuations as did Willmarth but used
a more complex geometry: a NACA0018 airfoil. Boutilier combines simultaneous hotwire and surface pressure measurements, spectral analysis, space-time correlations (pressure-pressure and pressure-velocity), and lower-order statistical moments to characterize
the airfoil boundary layer. Moreau and Roger (2005), using the same stationary controlled
diffusion airfoil considered in this report, quantified the boundary layer in terms of pressure spectra, coherence, and phase with applications to noise prediction.

5

1.3 Aeroacoustics
Airfoil self-noise results from an unsteady-flow interaction with the airfoil, the
interaction is often with its own boundary layer and/or wake. Five mechanisms for producing airfoil self-noise are shown in Figure 1.3 (Brooks et al. 1989):
•Trailing edge bluntness – vortex shedding noise: Vortex shedding noise resulting
from a small separated region aft of the trailing edge
•Tip vortex formation noise: Noise resulting from highly turbulent vortices that are
shed from the lateral edge of an airfoil blade.
•Separation/stall noise: At higher angles of attack, the boundary layer can separate
near the trailing edge, producing noise from the shed turbulent vorticity. At
even higher angles of attack, full stall can occur, or large-scale separation. This
typically produces low-frequency noise similar to what a bluff body produces
in a similar flow
•Turbulent boundary layer – trailing edge noise: At high chord Reynolds numbers,
a turbulent boundary layer forms over most of the airfoil and produces noise as
it passes over the trailing edge of the airfoil.
•Laminar boundary layer – vortex shedding noise: At low chord Reynolds numbers, laminar boundary layers form across most of the airfoil and typically produce a Von-Karman vortex street aft of the trailing edge. This contributes
mostly to tonal noise.
The primary flow geometry and associated noise mechanism of interest for the
present study is the turbulent boundary layer – trailing edge noise. It has been demonstrated that if sufficient pressure information is known about the convecting turbulent

6

Figure 1.3 Airfoil self-noise (Brooks et al. 1989)
boundary layer, the trailing edge noise pattern can be predicted from acoustical models
(Roger and Moreau 2004).
As the principal airfoil noise contribution in homogeneous stationary flows, trailing edge noise is a matter of particular interest when discussing noise production from
fans, airfoils, and turbines. A demand exists within industrial applications for reliable, but
realistic, prediction tools of noise intensity with respect to frequency and radiation angles.
Analytical models, in the context of incompressible flow computations and experiments,
allow for this noise prediction with varying degrees of accuracy according to the acoustic
analogy (Singer et al. 2000).
The most practical aeroacoustic analysis relies on Lighthill’s ‘acoustic analogy’ –
a novel method pioneered by M. J. Lighthill in 1951. The Lighthill analogy makes a connection between flow-physics and acoustics by casting the Navier-Stokes equations into
an inhomogeneous wave equation. The Lighthill analogy considers a free flow where the
non-stationary fluctuations are represented by quadrupole sources (Oberai et al. 2002). the
wave equation in an undisturbed medium at rest or, Lighthill’s equation (Howe 1978) is

7

2

2
∂ T ij
 ∂ 2 ρ' 
∂ ρ'
--------- – c 2  --------------- = --------------2
0  ∂x ∂x 
∂x i ∂x j
∂t
i i

1.5

where Tij, Lighthill’s Tensor, is given as
T ij = ρu i u j – σ ij + ( p – ρc 2 )δ ij
0

1.6

where ρu i u j is a Reynolds Stress term, σ ij describes the sound generated by shearing,
and ( p – ρc 2 )δ ij describes non-linear processes associated with internal energy. It is
0
typical to assume σ ij = 0 , neglecting the effects of viscosity and heat transfer; then,
2
 ∂ T ij 
T ij ≈ ρu i u j . The quadrupole source term  --------------- accounts for the noise generated by the
 ∂x i ∂x j
non-linearities in the flow (Howe 2001).
Lighthill’s work was modified by Curle in 1955 to include a dipole (i.e., a loudspeaker) source term, which takes into account the noise generated by the interaction of
the fluid with a non-moving boundary. Ffowcs, Williams and Hawkings corrected the
Lighthill and Curle formulations in 1969 further to include a monopole (i.e., a siren) noise
source to predict the noise generated by the interaction of the fluid with a moving boundary. Monopole noise is generated due to unsteady volume displacement of the fluid volume. Curle’s formulation and Ffowcs Williams and Hawkings corrections yield (Howe
2001)
2
· )
2
2
·
∂ ρ'
∂ ρ'
∂m ∂ ( f i + m u i - ∂ T ij
2  --------------- = ------ – -------------------------- + ----------------------- – c 
2
0  ∂x ∂x 
∂t
∂x i
∂x i ∂x j
∂t
i j

1.7

·
∂ ( fi + m ui )
·
∂m
where -------------------------- is the additional dipole source term and ------ is the additional monopole
∂x i
∂t
source term. It is important to note that Lighthill’s equation, Curle’s formulation, and
Ffowcs Williams and Hawkings corrections are exact for the conditions they describe.

8

The Ffowcs Williams and Hawkings method is the most general approach of the
acoustic analogy and is therefore the most suitable for computing the trailing edge acoustic field, noting that the equations are derived directly from the continuity and momentum
equations and formally solved using a half-plane Greens function (Roger and Moreau
2004)
Computational solutions to the far field fluctuating pressure field can be computationally expensive as they are primarily calculated in conjunction with direct numerical
simulations. It is often simpler to compute a large-eddy simulation or perform experimental measurements to determine the fluctuating wall pressure spectrum and apply a semiempirical model to predict far-field noise.

Figure 1.4 Spectral characteristics of a turbulent boundary layer
(Hwang et al. 2009)

9

The ideal semi-empirical model describes the fluctuating wall pressure field
beneath a turbulent boundary layer using practical data and theoretical knowledge. There
is no uniform single scaling law that collapses experimental data for all frequencies,
rather, there exists a multitude of models each optimized for a specific purpose. The goal
of the semi-empirical model is to combine the following scaling ranges into one model:
Low frequency range: Characterized by a spectral scaling of ω2
Mid frequency range: Characterized by a maximum in the spectra
Overlap range: Characterized by a spectral scaling of ω-(0.7~1.5)
High frequency range: Characterized by ω-5, particularly at high frequencies
Figure 1.4 shows the different scales used to collapse the data from different frequency ranges. These spectral characteristics combine to define the descriptive model
used for calculating the frequency spectrum. The following discussion describes the application of unsteady pressure data for predictive purposes and draws from semi-empirical
models that have been fitted to the descriptive model empirically, but with theoretical
guidance. The Goody model represents the most recent developments in spectral modeling, benefitting from recent measuring techniques and experimental data previously
unavailable (Hwang et al. 2009).
2

Φ pp ( ω )u ∞
C 2 ( ωδ ⁄ u ∞ )
-------------------------- = ----------------------------------------------------------------------------------------------------------------------------------3.7
3.7
τ2 δ



– 0.57 
0.75
w
+ C 1  +  ( ωδ ⁄ u ∞ ) ⋅ C 3 R T
 ( ωδ ⁄ u ∞ )






1.8

where RT is the ratio of the outer to inner boundary layer time scale and C1, C2, and C3 are
empirical constants with recommended values of 0.5, 3, and 1.1 respectively. The ratio of
– 0.57

C1 and C 3 R T

determines the size of the overlap region, which may be very thin at low

10

Reynolds numbers since R T ∝ ( uδ ⁄ υ ) . The Goody model is valid for a large range of
Reynolds numbers, 1400 to 23400, and be extrapolated to higher Reynolds numbers due
to the Reynolds number dependent factor RT (Hwang et al. 2009).
It is intuitively reasonable that as more experimental measurements become available, empirical constants can be calculated more accurately, leading to better predictions.
The semi-empirical models are developed further for use in the aeroacoustic community;
predictive capabilities expand to predicting far-field pressure spectra, the acoustic production from the airfoil. This is accomplished by combining concepts from Lighthill’s analogy to the semi-empirical model. Blake’s model for trailing edge noise prediction is a
common model used in the aeroacoustic community (Blake 1986)
Φp

Uc L2 Λ2
2 θ
ωδ∗
( ω ) = -------------------- Φpp ---------  cos  -- sin φ
u 
 2
rad
2π2 co r 2
∞

1.9

where Φ p ( ω ) is the power spectra of the acoustic radiation, L2 and r are measured
rad
lengths associated with the spanwise length of the trailing edge and the distance from the
trailing edge to the location of predicted acoustic radiation, co is the acoustic wavespeed,
and Uc and Λ2 are calculable quantities that represent the convection velocity and the
spanwise correlation length scale. Uc and Λ2 are typically estimated since accurate measurements that enable their calculation are often unavailable. Λ2 is defined by Pope (2000)
as the integral of the autocorrelation function
∞

Λ 2 =  R 22 ( η 2, t ) dr
0

1.10

ωδ∗
Φ pp  ---------  is the local non-dimensional wall pressure spectra, directly measured by fastu 
∞

response surface pressure transducers. θ and φ are directivity variables which describe the

11

pattern of the far-field acoustic radiation. Note the appearance of the cardioid pattern
cos2(θ/2). The cardioid can be seen as a bounding envelope for the directivity patterns,
emphasizing that trailing edge sources have dipole characteristics.
The contributions from this document to the aeroacoustic community are not
within the scope of defining new prediction methods. Rather, the focus is to characterize
the boundary layer using high-speed pressure sensors to resolve the fluctuating pressures
along the airfoil surface. Additionally, this document will present, demonstrate, and discuss the capability of the current experiment for the future implementation into noise
models.
1.4 Definitions of the Computational Methods Utilized in this Document
For use within this report it is instructive to define a few computational methods
for random data analysis. A background in random process theory is needed to accurately
assess conditions of the turbulent boundary layer of the present study. The following
methods quantify random processes as defined by Munson et al. (1998) and Bendat &
Piersol (1986).
The static pressure coefficient (Cp), a nondimensional form of pressure, is defined
as the ratio of surface static pressure and free stream dynamic pressure (Equation 1.11). Cp
is a useful expression for quantifying the static pressure along an airfoil independent of
body size.
[ p ( x 1, x 2 ) – p ∞ ]
C p = --------------------------------------1 2
-- ρu ∞
2

12

1.11

The probability density function (PDF) of the fluctuating pressure is a expression
that describes the relative intensities as a function of their probability. The PDF P(x), for a
random variable x(k), where k is a set of possible outcomes such that x(k)<x, is
P(x) =

Prob [ x < x ( k ) ≤ ( x + Δx ) ]
--------------------------------------------------------------Δx
Δa → 0
lim

1.12

Statistical moments are identified to quantify the shape of the PDF with a singular
variable; the calculation of moments allows for a simplified representation of a complex
pressure field. Moments one through four are of interest in this experiment. The first
moment is the mean of the signal for a function A(x)
∞

μ =  xA ( x ) dx
–∞

1.13

The second moment is the variance of the signal is defined by Equation 1.14. Variance is a
measure of the ‘spread’ of the PDF.
∞

2

Var ( A ) =  ( x – μ ) fA ( x ) dx = σ
–∞

2

1.14

where σ is the standard deviation. The third moment is the skewness of the signal is
defined by Equation 1.15. Skewness is a measure of the bias (positive or negative) of the
PDF.
∞

3

Skew ( A ) =  ( x – μ ) A ( x ) dx
–∞

1.15

The fourth moment is the kurtosis of the signal is defined by EQ. Kurtosis is a measure of
the ‘flatness’ or ‘peakedness’ of the PDF (Equation 1.16). A normal distribution has a kurtosis equal to 3.
∞

4

Kurt ( A ) =  ( x – μ ) A ( x ) dx
–∞

13

1.16

The power spectral, or autospectral, density (PSD) function of a fluctuating pressure signal describes the average frequency content of a random process a(t). By use of
Fourier transform methods, the PSD is given as
2

Φ aa = 2E [ A ( f ) ]

1.17

where E is an ensemble average for fixed frequency f.
The coherence function is a measure of the relationship between two signals as the
square of the absolute value of the cross-spectral density function to the product of the
autospectral density functions of the two signals.
2
Φ ab ( f )
2
γ ab ( f ) = ---------------------------------Φ aa ( f )Φ ( f )
bb

1.18

2

where 0 ≤ γ ab ( f ) ≤ 1 .
The phase factor, or simply, phase, is a measure of the phase lag of the representative sinusoidal signal calculated using Fourier transfer methods. Defined as the angular
component φ of the Fourier transform
e

iφ

= cos φ + i sin φ

1.19

where i is the imaginary unit.
The cross-correlation function is the measure of the product of two sinusoidal signals a(t) and b(t) at time t and time (t+τ) for an averaging time T
1 T
R ab ( τ ) = --  a ( t )b ( t + τ ) dt
T0
The autocorrelation is a special case where a(t)=b(t).

14

1.20

Note that all the mentioned methods are valid only when applied to a stationary
stochastic process. It it determined that the data presented in this document are statistically
invariant with respect to translations in time. These data are sufficiently well converged so
that they are considered stationary and stochastic.

15

2.0 Experimental Equipment and Procedure
2.1 RCDB Fan Development
The Rotating Controlled Diffusion Blade (RCDB) is derived as a rotating analog
of a stationary Controlled Diffusion airfoil; the present work is a continuation of Neal
(2010). The intent of the RCDB investigation is to study fundamental flow properties by
making a direct translation - between stationary and rotating reference frames - of a
known airfoil geometry. Given the equivalent operating conditions, valuable physical
information on the effects of rotation can be inferred from flow-field measurements. See
Neal (2010) for further details.
To reduce complexities and appropriately cast the CD airfoil into a rotating frame,
the RCDB was designed without common fan and turbomachinery features that would
have improved performance, efficiency, noise production, etc. It is important for several
parameters to remain constant between the stationary and rotating experiments: a constant
chord of c=133.9mm across the entire blade span, a geometric angle of attack α of 8 ° , a
5

chord Reynolds number ( Re c = 1.5 ×10 ) based on the relative incident velocity
w ∞ = 16m ⁄ s and an equivalent aerodynamic loading.

w∞

αc α

U
V

αi
W

u∞

Figure 2.1 Flow geometry schematic (left - rotating; right - stationary)

16

It is instructive to note the difference between geometric and aerodynamic angle of
attack for this geometry (Figure 2.1). The geometric angle of attack (α) is defined by the
chord line with respect to the incident flow direction. The aerodynamic angle of attack
(αi) is defined as the tangent of the mean camberline at the leading edge of the airfoil referred to as the camber angle (αc) (Lakshminarayana 1996).
The RCDB was designed with a blade twist so that the angle of the incident velocity V a ⁄ b was maintained across the leading edge of the blade; consequently, the magnitude increases linearly with respect to the radius as a result of the increased tangential
velocity (rΩ) toward the tip. The relation between absolute inlet velocity and incident tip
velocity is best expressed as:
Va ⁄ b = Va ⁄ g – Vb ⁄ g

2.1

where a/g, a/b, and b/g are the relative velocities of air with respect to a fixed reference
frame (ground), air with respect to the rotating blade, and the rotating blade to the fixed
reference frame. It is common to rewrite Equation 2.1 in more specific turbomachinery
terms where V b ⁄ g is written as the tangential velocity: U = Ω × r , and V a ⁄ b is written
as the incident velocity W. Inlet velocity can be rewritten as:
W = V –U

2.2

Schematics of the incident velocity vectors for the CD airfoil and the RCDB are shown in
Figure 2.1. Consequently, the Reynolds number is non-constant radially; the Reynolds
number is matched to the CD airfoil at the midspan of the RCDB where
w ∞ = u ∞ = 16 m/s.

17

Figure 2.2 Modular blade section
Great care was taken to ensure proper matching of the CD airfoil and RCDB flow
conditions as well as to minimize three-dimensional effects. The design target of the
RCDB was first generated with RANS CFD simulations (Neal 2010). As shown in
Figure 2.3, a converging inlet nozzle and centerbody create an annulus of uniform velocity
V = 8 m/s. Additionally, the exit radial velocity is minimized to nearly 3% of the relative
inlet velocity (approximately 0.5m/s). Due to added three-dimensional and tip effects,
compromises in the blade design and minor corrections in the operating conditions (mass
flowrate and rotational speed) are required (Section 2.3.1). Additional trade-offs and
details of the design iteration are discussed further in the Neal 2010 dissertation.
The final design of the RCDB experiment consists of a modular hub, which was
designed and fabricated such that the rotor can accommodate any combination between 2
and 9 blades. Experiments were run to determine the Cp distribution for a range of blade
solidity conditions. The final configuration was selected based on which solidity condition
matched the blade loading of the CD airfoil.

18

Figure 2.3 RCDB 5-blade CAD rendering (left - upstream; right - downstream)
Neal (2010)
For interpretation of the references to color in this and all other figures, the
reader is referred to the electronic version of this thesis
It is noteworthy that the Cp distribution for the stationary airfoil is highly dependent on the inlet nozzle width; it is suggested that the solidity of the rotating fan system is
analogous to the nozzle width of the CD airfoil (Neal 2010). To determine which configuration had the best agreement with the stationary CD airfoil, physical experiments were
conducted on several blade configurations (Figure 2.4). The blade loading was matched
relatively closely between the CD airfoil and RCDB with three blades attached to the
rotating hub. A consequence of the inherently different boundary conditions was apparent
in the leading edge separation region and the general pressure distribution on the pressure
side of the blade. While imperfect, the 3-blade RCDB configuration had the best agreement with the CD airfoil (Figure 2.5).

19

Figure 2.4 RCDB Cp comparison for several blade configurations
Neal (2010)

Figure 2.5 Final RCDB Cp configuration and CD airfoil
Neal (2010)

20

2.2 Experimental Configuration/Facility
2.2.1 Axial Flow Research and Development (AFRD) Facility
The Axial Research and Development (AFRD) facility was used for the RCDB
experiments. Originally designed to measure integral quantities of volume flowrate, pressure rise, and input power as well as wake measurements, the AFRD has since been used
for a variety of low-speed axial fan experiments (Morris and Foss 2001, Neal and Foss
2007, Dusel 2005, Neal 2010).
The AFRD facility, shown in Figure 2.6, is a vertical wind tunnel that draws atmospheric air into an upper receiver (D). The RCDB fan assembly (A) and prime mover (K,
Chicago Design 36 SISW SQA Airfoil) are responsible for the flow through the facility.
Air is moved from the upper receiver into the lower receiver (H) through a set of delivery
nozzles (E) and turning vanes (F). The 90-degree turning action results in a net momentof-momentum flux that is proportional to the mass flow rate; this net flux is balanced by
the moment of a force transducer (G) about a pivot point. The flow inlet (I) to the prime
mover is located in the lower receiver. The flow rate is controlled with a throttle plate (J)
at the exhaust of the prime mover. A wall tap in the upper receiver of the AFRD facility is
used to measure the pressure differential across the test fan.
The RCDB fan assembly was mounted on a vertical shaft cantilevered 60cm above
a bearing support to avoid any aerodynamic blockage of the wake from support members.
The shaft was driven by a Reliance Electric 11.2 kW (15 HP) electric D.C. motor. An optical encoder provided an angular location and rotational speed. A slip ring is used to make
an electrical connection from the stationary frame to the rotating assembly. A built-in

21

traverse system (C) allows for radial, axial, and azimuthal positioning of hot-wire and
similar probes downstream of the fan.
The integral flow metering apparatus was calibrated separately using both metering nozzles and the inlet nozzle for the RCDB. Knowing the discharge coefficient and the
pressure differential across the N nozzle(s), it is a simple matter to determine the mass air
flow following Equation 2.3.
2
·
m = ρ × N × A nozzle × C d -- ( p atm – p receiver )
ρ

Figure 2.6 The Axial Fan Research and Development (AFRD) Facility
Neal (2010)

22

2.3

The AFRD inlet was retrofitted to accommodate the modular hub, aerodynamic
centerbody, and an inlet contraction. These components were designed to meet the RCDB
and CD design targets as discussed in Section 2.1. The final installation is shown in
Figure 2.7
2.2.2 Rotating Controlled Diffusion Blade (RCDB) Assembly
Salient features of the RCDB are shown in Figure 2.7. The RCDB assembly has a
hub-tip ratio of 0.655 (hub diameter of 480mm), a 4mm tip clearance, and a shroud diameter of 740mm. One of the three blades in the current RCDB configuration is instrumented
with 21 surface pressure taps at the midspan of the blade (Figure 2.9).
The particularly large hub section makes the RCDB well suited for onboard instrumentation and data acquisition. All measurements of the RCDB occur in the rotating hub,
necessitating an intricate instrumentation package of OEM and custom transducers and
components. The details of the blade and instrumentation package required to support the
present experiment are outlined in the following sections.
Hub

Driven Shaft

Inner Shroud

Instrumentation

Outer Shroud

Blade

Figure 2.7 RCDB assembly installed in AFRD Facility

23

w
u
Figure 2.8 RCDB coordinate systems (left - r−θ−z; right - x-y-z)
(Neal 2010)

2.2.2.1 Coordinate System
The RCDB assembly is described and data are collected in the traditional polar turbomachinery coordinate system: r-θ-z (Figure 2.8), where r is the radial line normal to the
axis of rotation z and θ is the polar angle or azimuth. Typically, the coordinate system is
recast into one similar to the CD blade: a Cartesian u-v-w. This transform allows for direction comparison with the CD airfoil. However, due to limited trailing edge surveys, wake
data are presented in polar coordinates within this document. See Figure 2.8 for a visual
representation of the difference.
2.2.2.2 Instrumented Blade
Due to the thin profile of the RCDB and the need for high spatial resolution, it is
impractical to embed pressure transducers directly in the surface of the blade. Instead,
1mm OD, 0.5mm ID stainless steel tubing was embedded flush into small channels milled
in the blade surface. Each tube had a small pinhole drilled at the midspan of the blade and
at a unique chord location as shown in Figure 2.9. Eighteen streamwise taps and three
trailing edge spanwise taps exist in the current experiment; each tap is assigned a unique

24

number shown in Figure 2.10 and Tables 2.1, 2.2, and 2.3(Moreau and Roger 2005, Neal
2010). Note that the gaps in the numbering sequence exist to maintain consistent numbering with previous investigations.
The 21 taps will be used for mean and unsteady surface pressure measurements.
See further discussion in Section 2.2.2.6 and Section 2.2.3.

Figure 2.9 RCDB instrumented pressure taps
Left: Suction side (top); Right: Pressure side (bottom)

Figure 2.10 Surface pressure tap locations

25

Table 2.1 Suction side taps (y/l=0.5)
Tap

1

2

3

5

6

7

9

11

21

22

23

24

25

x/c

.013

.030

.052

.087

.149

.403

.534

.679

.858

.881

.899

.922

.978

Table 2.2 Pressure side taps (y/l=0.5)
Tap

4

8

10

12

29

x/c

.067

.400

.530

.675

.929

Table 2.3 Trailing edge spanwise taps (x/c=0.978)
Tap
y/l

25A
.454

25B

25

25C

.477

.500

.577

2.2.2.3 Data Acquisition
Measurements of the RCDB experiment were digitally sampled by two synchronized analog-to-digital (A/D) acquisition boards. The primary A/D (IOTech Inc. WaveBook/516E) measures variables in the stationary reference frame and provides a sync
signal used to clock the secondary A/D. The secondary A/D (IOTech Inc. DaqBoard/
3035USB) is rigidly mounted in the rotating hub of the RCDB. The stationary A/D can
sample up to 16 differential 16-bit channels at 1 MHz and has the capability for advanced
TTL triggering and external clocking, making it well suited for high-speed, downstream,
stationary measurements, as well as serving as a master clock. The rotating board can
sample 64 single-ended 16-bit channels at 1MHz; it is clocked externally via a sync signal
from the stationary A/D.

26

Figure 2.11 Data acquisition boards (photos: www.mccdaq.com)

Its compact design (15cm x15cm), as well as its simple interface (power and data transfer
via USB 2.0), make it particularly suitable for implementation in the RCDB assembly.
2.2.2.4 Signal Transmission
It is inherently difficult to maintain electrical connections between rotating and
stationary reference frames. To transfer power and data between the RCDB and lab environment, a slip ring assembly was mounted on the end of the rotating fan shaft
(Figure 2.12). The slip ring (SR20M by Michigan Scientific Corporation) allows 20 channels of data to be transferred from the rotating environment. Each channel has a maximum
current rating of 500mA, a typical contact resistance of 0.05 ohms, and contact life of 200
million revolutions. The SR20M is ideally suited to this experiment because of its low
noise characteristics and low maintenance requirements.
As discussed in Section 2.2.2.3, the onboard data acquisition board requires only
USB 2.0 for power and data transmission. In addition to USB 2.0, a digital sync signal and
DC power for the instrumentation package is transferred through the slip ring.

27

Figure 2.12 Slip ring assembly (Neal 2010)

The digital signals of USB 2.0 and the sync signal are not as susceptible to noise effects
from the slip ring assembly as are analog signals; and, therefore, they are well suited for
uncorrupted data transfer. DC power ( ± 24 V) is split between multiple channels to meet
the current ratings and conditioned with onboard electronics to remove any electrical
noise.
2.2.2.5 Temperature Measurements
Temperature measurements were made possible by two IC amplifiers with integrated cold junction compensators (Analog Devices AD595). The fast response type-T
thermocouples are used for temperature compensation of hot-wire measurements as well
as temperature monitoring of the instrumentation cluster. The ICs were mounted in a custom circuit board that interfaced directly with the rotating A/D (see Figure 2.17 for an
instrumentation and data path schematic).

28

Figure 2.13 MEMS pressure transducer (Neal 2010)

2.2.2.6 Time-Mean Pressure Measurements
Pressure measurements are made in the rotating reference frame by 22 small
MEMS pressure transducers (model-type 1-INCH-D2-4V-MINI manufactured by All
Sensors, Figure 2.13) installed in the hub of the RCDB assembly. These relatively slow
response transducers are responsible for measuring the time-mean surface pressure. All 22
transducers share a common reference pressure P atm . Measuring P atm in a rotating reference frame presents a challenge. An isobaric reference chamber provides a known, and
equal, reference for all 22 pressure transducers (Figure 2.16). The isobaric chamber is
open so that the pressure in the chamber is the same as the AFRD upper receiver pressure.
The pressure transducers are susceptible to adverse rotational effects due to their
internal construction: the inner diaphragm may be displaced by centrifugal forces leading
to a false pressure reading. To avoid this, great care was taken in mounting the transducers
so that the diaphragm would be normal to the axis of rotation, preventing any unwanted
displacement. In addition to centrifugal forces acting on the transducer membrane, issues

29

arise from the centrifugal forces acting on the air that is present in the connecting tube.
The measured pressure will be different due to the radial separation between the transducer and the pressure tap (Figure 2.14). To account for this rotation-induced pressure gradient, a simple correction of the measured differential pressure Δ P measured to the actual
differential pressure [ P taps – P ref ] is proposed as follows (Neal 2010):
2

2
2
ρω ( r taps – r sensor )

Δ P corr1 = -------------------------------------------------2

2.4

where Δ P corr1 is the correction between the pressure transducer ( r sensor ) and the pressure taps on the blade ( r taps = 303 mm). Equation 2.4 corrects the positive side of the
differential pressure transducer which is connected to the surface taps.
2

2
2
ρω ( r sensor – r ref )

Δ P corr2 = ----------------------------------------------2

2.5

where Δ P corr2 is the correction between the isobaric reference chamber ( r ref = 0 mm)
and the pressure transducer ( r sensor ). Equation 2.5 corrects the negative (reference) side
of the differential pressure transducer. Applying the corrections, the expression for the
actual pressure [ P taps – P ref ] becomes
[ P taps – P ref ] = Δ P measured + Δ P corr1 + Δ P corr2

2.6

or
2 2

ρω r taps

[ P taps – P ref ] = Δ P measured + ----------------------2

2.7

Correcting to atmospheric pressure where P ref = P receiver ,
2 2

ρω r taps

[ P taps – P atm ] = Δ P measured + ----------------------- + [ P receiver – P atm ]
2

30

2.8

-

+

Figure 2.14 Pressure transducer schematic (Neal 2010)

Equation 2.8 shows the reduced relationship between the measured differential pressure
and the actual differential pressure. Note that the r sensor term is eliminated; accordingly,
the pressure transducer can be placed at an arbitrary radial location within the hub.
2.2.2.7 Constant Temperature Hot-Wire Anemometry
Two hot-wire channels were installed in the RCDB instrumentation package for fast
response ( f r > 20kHz ) velocity measurements (Figure 2.16). TSI 1750 anemometers were
chosen for their small size and sufficient frequency response, with the custom probes used
in the rotating reference frame measurements. See Section 2.2.4 for further discussion of
the hot-wire technique.
2.2.2.8 Microphone Amplifiers
A series of microphone amplifiers are mounted in the instrumentation package.
These IC operational amplifiers (Texas Instruments OPA4134) support 21 electret condenser microphones (Knowles FG-23329) used for unsteady surface pressure measure-

31

ments. Figure 2.15 shows a schematic of the amplifier and microphone system. See
Section 2.2.3 for further information on microphone development and implementation
2.2.2.9 Hub Instrumentation Package
The final assembly of the instrumentation package is shown in Figure 2.16. Relevant features are:
A) Level 1: A/D board
B) Level 2: Mezzanine level board which houses thermocouples, pressure transducers, and microphone amplifiers. This is a non-OEM part.
C) Level 3: Hot-wire constant temperature anemometers and power conditioning
circuitry
D) Level 4: Isobaric pressure reference chamber
A detailed schematic of the instrumentation package and signal paths is shown in
Figure 2.17.

FFT

pRMP,i
Calibration

( )
( )

,

A/D

pref

FFT

H(f)
Measurement

psurface

Tube
Attenuation

Knowles
(Pa to V)

pRMP,i

A/D

Figure 2.15 Amplifier and microphone system

32

pRMP,calibrated

Level 4

Level 3

Level 2

Level 1

Figure 2.16 Hub instrumentation package

CAT-5e

Figure 2.17 Data path schematic (RCDB)

33

2.2.3 Remote Microphone Probes (RMPs)
Information regarding the pressure fluctuations, in addition to the time-mean pressures, is important in representing the flow over the stationary and rotating CD airfoils.
The latter ( p ) measurements are obtained by connecting the open, static taps to the
MEMS pressure transducers. For these data, it is assumed that the time-mean voltage output represents the time-mean pressure at the tap. The former ( p' ( t ) = p ( t ) – p ) measurement requires that a “fast-response transducer”, a microphone, is connected to the static
tap in such a manner that the amplitude and phase of its response can be processed to yield
an acceptably accurate representation of p’(t) at the airfoil surface.
The thin airfoil and the desire for good spatial resolution make it imperative that
the microphones are located at some physical (remote) distance from the tap itself. The
associated challenges to fabricate, install, and calibrate the RMP system are presented in
the following sections.
2.2.3.1 Technique
Pérennès and Roger (1998) developed a unique RMP for measuring surface pressures using capillary tubes in thin airfoils. High spatial resolution and a thin airfoil mandated the remote measurement of the surface pressures with the bulky microphone
transducers. The RMP concept evolved for use in the work of Moreau and Roger (2005), a
study where the RMPs were installed in the CD airfoil geometry - the stationary analog of
the RCDB.
The present experiment is equipped with 21 embedded taps, 18 of which are
located at a common radial coordinate and 4 of which are located at the same chord location (Tap 25, x/c=0.9776) but distributed spanwise along the trailing edge. The taps are

34

drilled into stainless steel tubing that is embedded in the blade surface. The present form
of the RMP is predicted to have a maximum frequency response of 20-25kHz but has been
modified procedurally and physically from the previous applications of Pérennès and
Roger to accommodate new techniques and technologies, updated hardware, and packaging requirements.
2.2.3.2 Fabrication/Implementation
The embedded stainless steel capillary tubing (Section 2.2.2.2) provides the transmission path for unsteady pressure propagations between the blade surface and the microphone transducers. The acoustic channel of stainless tubing is terminated with a small
section of PVC tubing that extends to the MEMS pressure transducers (schematic shown
in Figure 2.22). This PVC tubing acts to prevent reflections by attenuating high frequency
pressure fluctuations and matching impedances to avoid acoustic resonances (Hoarau
2006).
The precise dimensions of this tubing allows for leak-proof splicing between segments to be made as shown in Figure 2.19. Specifically, a splice for the installed 0.5mm
ID, 1.0mm OD tubing is accomplished by using a 1.0mm ID “sleeve” tube to mate the
smaller steel tubes and heat-shrink tubing to seal the combined sections.
The microphones (Knowles FC-23329-C05: 2.56mm physical diameter, 0.76mm
sensor diameter) are chosen for their small size, sensitivity, and cost (see Appendix A for
specifications). The microphones are mounted in a support member that is placed between
the surface tap and the MEMS pressure transducer. Since there is zero flow in the 0.5mm
passage, the fluctuating pressure signal is transmitted from the open surface tap (and past)

35

the opening that exposes the microphone element to the fluctuating pressure in the passage.

Figure 2.18 RMP microphone

Heat-shrink Tubing

Acoustic Channel

Acoustic Channel

“Sleeve” Tubing
Figure 2.19 RMP tube splicing

Microphone
Acoustic ‘tee’
Completed Manifold
To Pressure
Transducers

Figure 2.20 RMP manifold schematic

36

To Blade
Surface

Figure 2.20 identifies salient features of the RMP support member. For ease of
installation and packaging purposes, the RMPs were fit into a small PVC support manifold
and easily spliced into the existing stainless steel tubing as shown in Figure 2.19. The
PVC structure acts to passively damp vibrations and other external noise sources. This
contributes to clearer and better resolved microphone data. The PVC manifold houses 6
RMP microphone assemblies; see Figure 2.21. It is necessary to realize that the acoustic
properties of the microphones and tubing connected to each RMP are inherently unique.
Each RMP additionally had a unique, unknown, electrical response. This mandate a routine to identify a known RMP response/transfer function between pressures measured at
the microphone and pressures acting on the surface of the airfoil.

Tubing
splice

Figure 2.21 Installed RMP manifolds (Note spliced steel tubing)

37

P = p + p'

MEMS Transducer

rtap

rsensor
Pref

rRMP

RMP

Figure 2.22 Pneumatic circuit of RMP system

2.2.3.3 Evaluation/Calibrations
It is apparent that a calibration will be required to relate the fluctuating pressure at
the surface tap to the fluctuating pressure experienced by the microphone. That is, both the
amplitude and the phase of the recovered signal must be related to those of a known signal
at the surface tap via calibration data. The calibration will account for attenuation effects
from the acoustic channel, including imperfections and blemishes in assembly and manufacturing, as well as accounting for individual microphone responses. For further details
on the development of the calibration procedure, the reader is referred to Appendix A.
A plane wave tube (PWT) was fabricated with removable inserts at an x/L location
of roughly 0.4 (Figure 2.23). The PWT is used to generate a uniform acoustic pressure signal along a given cross-section sufficiently downstream of the tube’s opening (AED-1ID1991). The PWT was used to evaluate and validate calibration routines and RMP manifold
geometries.

38

Speaker Driver

Larson Davis
Reference Mic
Crosssection
(x/L=0.4)
Knowles Test
Mics:
RMP Manifold

Wavetube
(0.75”x0.75”)
Figure 2.23 Wavetube experiment and schematic

The apparatus in Figure 2.23 allows for the simultaneous testing and subsequent
calibration of five microphones within a given RMP manifold. Phase and magnitude transfer functions for each of the RMPs were obtained by comparing the five microphone signals to the concurrent output from the Larson Davis reference microphone (specifications
in Appendix A). Calibrating the RMP microphones (i.e., applying the calculated transfer
functions) accomplishes the effect of “matching” the RMPs to a known magnitude and
phase response (measured by the Larson Davis reference microphone). The calibration
algorithms and RMPs are tested on their ability to accurately recover known spectral properties when subjected to a known acoustic pressure field. It is evident in Figure 2.24 that
the calibration functions provide a quite acceptable representation of the acoustic signal
with exceptions near 6000Hz and 8000Hz. Additional measurements (not presented here see Appendix A) suggest that the seemingly unreliable transfer functions are an artifact of
the speaker-wavetube configuration.

39

Figure 2.24 RMP manifold and reference spectra
While useful for characterization and testing, the PWT is not a practical device to
calibrate microphones installed in the RCDB due to the inhibiting blade camber and
sweep. Accordingly, an in situ method, where the reference microphone is held normal to
the airfoil surface, offset roughly by 1mm and centered on a pressure tap, is used in lieu of
the PWT. A sufficiently downrange set of speakers (low-range and mid-range) is used to
generate a broad-spectrum acoustic pressure field - exciting the reference and RMP microphones simultaneously. The calculation of phase and magnitude transfer functions proceeds using the same analysis used for the PWT.

40

Figure 2.25 Larson-Davis comparison (in situ method)

This method is accurate to frequencies whose wavelengths are of order of the separation distance and microphone diameter (f>>10kHz, λ<<0.03m). This is verified in
Figure 2.25 where two phase-matched Larson Davis reference microphones were tested in
a simulated experiment. It is clear that there is excellent coherence across the frequency
range of interest; this is validation that the in situ is appropriate to use for RMP calibrations.
The in situ method is time consuming; the microphones cannot be calibrated in
bulk; therefore, a calibration rig containing indexed positions for all 21 microphones was
developed to meet the needs of the calibration procedure. Accuracy, precision, and repeatability are critical for consistent calibrations (Figure 2.26).

41

Figure 2.26 RCDB in situ microphone calibration rig

Each RMP calibration produces a unique set of transfer functions (Figure 2.27).
The data of Figure 2.27 reveal two important aspects of the experimental procedures to
recover fluctuating pressure information at the open taps on the RCDB surface. The first
immediately clear message is that calibrations are very consistent with few day-to-day
variations; this is a result of a very stable calibration routine. The second message relates
to the identification of the physical processes at work in the p’(t) measurement scheme.
This second message is based upon the properties of the “magnification-factor” and the
phase as a function of the frequency of the incident sound waves. The magnification-factor and phase combined represent the response difference between the reference microphone, the literal representation of the pressure at the surface tap, and a given RMP,
including the microphone response and the attenuation effects of the acoustic channel.

42

a)

Magnitude Calibration RMP24 to Reference (LD1)

0.3

06_01_2012
06_07_2012
06_15_2012

Magnification Factor

0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0

b)

2000

4000
6000
Frequency (Hz)

8000

10000

Phase Calibration RMP24 to Reference (LD1)

4
3

Phase Delay (rad)

2
1
0
-1
-2
-3
-4
0

2000

4000
6000
Frequency (Hz)

8000

10000

Figure 2.27 Transfer functions for RMP24 using in situ method

43

Figure 2.28 Original and calibrated RMP 9 spectra

Figure 2.28 compares the power spectrum of the reference microphone and RMP9,
raw/original signal and calibrated signal, using a magnitude and phase transfer function
method produced using the in situ calibration technique. Agreement between the reference
and corrected RMP9 microphone is considered to be excellent.
2.2.3.4 Facility Noise Concerns
Due to the nature of the unsteady pressure measurement sensors, it is important to
address radiated far-field acoustic noise produced by the AFRD flow facility. Since the
flow-facility (AFRD) in use is non-anechoic, it was expected that vibrations and extraneous noise sources within the facility would pollute surface pressure sensors. In particular,
the RMP microphones employed in the present experiment are traditionally used as acous-

44

tic sensors and are thus optimized for human sound perception; additionally, the sensors
are particularly sensitive to secondary noise phenomena beyond the isolated mechanical
wave propagation within the acoustic channel.
It was considered that the hydrodynamic pressures of the boundary layer might be
of similar magnitude to the noise sources identified in the AFRD facility. A ground-level
Larson-Davis microphone was used to characterize the strength of the ambient noise; the
microphone was located within the upper receiver of the AFRD. Typical power spectra of
the acoustic pressure (measured by the Larson Davis) and the hydrodynamic pressure
(measured by an airfoil surface tap) are shown in Figure 2.29.

Figure 2.29 AFRD acoustic noise

45

It can be seen that the hydrodynamic pressure is approximately 100 times more
powerful than the acoustic noise produced by the facility. It is well known that hydrodynamic pressures are typically orders of magnitude higher than acoustic pressures - acoustic pressures are a lower intensity, whereas hydrodynamic are higher intensity and local
phenomena. This confirms that the microphone sensors are appropriate for accurately
measuring the blade hydrodynamic surface pressures in a “noisy” environment; the measured intensity is well above the noise floor.
Additionally, correlated tonal noise between the ground-level ambient microphones and RMPs is attenuated using the Optimal Noise Cancellation (ONC) method suggested and provided by Professor Ahmed Naguib (see Appendix B). The ONC method is
another way to ensure the integrity of the collected data in an otherwise noisy environment.

Figure 2.30 Schematic and image of SN hot-wire

46

2.2.4 Hot-Wire Measurements
Single sensor hot-wire probes (Straight-Normal: SN) were used to measure mean
and fluctuating velocity magnitudes in the wake of the RCDB. (Figure 2.30). A special
traverse was developed to support the SN probe in the rotating reference frame, enabling
the measurement of time-resolved wake data and velocity spectra. TSI 1750 constant-temperature anemometers were used to drive the hot-wire probes. The anemometers were
tuned to a typical frequency response greater than 20kHz at a flow velocity of 16m/s. Postprocessing temperature compensation of the hot-wires was accomplished using measurements from onboard thermocouple amplifiers (Analog Devices AD595).

see (b)

hot-wire
probe
pressure tap

scale
(mm)
b)

a)

probe active
region

traverse
mechanism

hot-wire
support arms

Figure 2.31 Hot-wire traverse
a) Overall view (not pictured: counter balance)
b) Trailing edge hot-wire location

47

2.2.4.1 Traverse
The “flying” hot-wire rig was designed to position the probe at the midspan of the
blade and traverse azimuthally (θ) while maintaining a constant radial position. Limited
adjustment exists radially and vertically. The rig was balanced so that it was dynamically
stable, minimizing vibrations and flutter as well as maintaining the rotational balance of
the entire RCDB assembly. Through a fine lead screw, the rig is capable of traversing with
a resolution of r Δθ = 0.4 mm at a radius of 303mm. (Figure 2.31).
Hot-wire wake surveys were performed in three r-θ planes downstream of the airfoil at z = 1mm, 3mm, 8mm . Each survey had ten azimuthal locations.
2.2.4.2 Probe Construction/Calibration
A custom hot-wire probe was developed specifically for use within the rotating
frame. The active region of the tungsten wire sensor (D=5um) was L=1mm. L/D>200 satisfies the condition of a minimized end-conduction over the active region (Champagne et
al. 1967). This probe is shown in Figure 2.30. An overheat ratio of 1.6 was used for all
measurements.
A hot-wire calibration facility was used to produce a low disturbance and similar
magnitude velocity to the expected experimental flow field. Laboratory air is drawn into
the calibration facility where the disturbance level is minimized through a layer of filter
material before passing through a well-characterized nozzle. The calibration facility is
capable of holding a probe at angles of ( ± 36 )° from center in 6° increments (SN is calibrated at one angle: 90 ° ). A schematic of the facility is shown in Figure 2.32.

48

Figure 2.32 Hot-wire calibration unit
The calibration velocity is defined as the “Bernoulli velocity”. Velocity is obtained
by measuring the static pressure difference across the low disturbance nozzle and calculated following Equation 2.9, where air density ρ was calculated using the ideal gas law.
V cal =

2 [ p atm – p cal ]
---------------------------------ρ

2.9

The calibration was made across a range of velocities using a “quasi-steady-state” condition where the velocity varies as a function of time but changes slowly enough such that
the transients are within the response of the calibration transducers (Hellum 2006). A pre
and post-calibration of the hot-wires was performed to identify variances, or drift, that

49

may have occurred during hot-wire data acquisition. The typical drift was within 2%
between pre and post-calibration.
It became evident during experiments that there was an inherent time lag between
the on-board hot-wire anemometer channels and the rest of the data acquisition channels.
The hot-wire was delayed on the order of 3000 time samples, or 0.075 seconds. This is
identified to be a phenomenon of the electronics rather than an anomaly of the measured
flow field. Hot-wire measurements are corrected for this by identifying the distance
between the trailing edge and the hot-wire and subtracting the time-lag using a known
wake velocity and convection velocity from the trailing edge pressure taps.
2.3 Experimental Parameters/Procedure
2.3.1 Experimental Operating Conditions
As previously discussed, the primary operating point (see Figure 2.33) was established to match the loading conditions of the experimental work performed by Moreau and
Roger (2005). This target primary condition is defined by an incident velocity of 16m/s
and a geometric angle of attack of 8 ° . Note that all stationary CD results presented in this
report correspond to this operating condition.
In order to evaluate the behavior of the rotating CD blade, a new operating condition was defined that maintained similar features to the stationary blade. However, due to
the complexities of the flow geometry and intricacies with casting a stationary 2D airfoil
to a rotating reference frame, the experimental flow field of the RCDB is not identical to
that of the CD blade.
Calculating the actual operating condition is limited by how well the angle of the
twisted rotating blade can be identified (this was computed using CAD drawings) and how

50

well the components of the inlet velocity triangle can be computed (angular and axial
velocity). Presently, the angular velocity is fixed to a specific RPM for a given operating
condition and the axial velocity is varied to match the target flow condition (angle and
magnitude of the incident velocity). The axial velocity is varied by throttling the main
blower in the AFRD facility; a Pitot tube measures the axial velocity just upstream of the
fan to ensure the correct operating point is met.
Two operating points were established for the RCDB study: Case 1 and Case 2.
Case 1 corresponds to a near-identical inlet condition to the CD blade with an incident
velocity of 16.25m/s, a geometric angle of attack of 8 ° , and a mass flow-rate of 2.305kg/s
See Figure 2.33 for a schematic of the geometry and Figure 2.35 for details of the inlet
velocity triangle. Note that the velocity is within 2% of the stationary experiment flow
field.
The second operating condition (Case 2) is defined by an incident velocity of
16.4m/s, a geometric angle of attack of 15 ° , and a mass flow-rate of 1.799kg/s. This operating condition was targeted to match an alternate flow field used in Moreau and Roger
(2005). For purposes of this report, Case 2 represents an “off-design” condition; the “offdesign” flow field provides an additional parameter to identify details of the RCDB
boundary layer, specifically, new separation/reattachment regimes and different statistical
behaviors. See Figure 2.34 for a schematic of the geometry and Figure 2.35 for details of
the inlet velocity triangle.

51

w∞
8°
u∞

a) RCDB

8°

b) Stationary CD airfoil
Figure 2.33 Flow geometry Case 1

w∞
15°
u∞

a) RCDB

15°

b) Stationary CD airfoil
Figure 2.34 Flow geometry Case 2

8°

16.25m/s

15°

16.4m/s

8.5m/s
rΩ=13.87 m/s

Case1

7.3m/s
rΩ=14.72 m/s

Case2

Figure 2.35 Flow geometry velocity triangles (Case 1 and Case 2)

52

30.0
Case1: 31 May 2012
Case2: 31 May 2012
Target Values

20.0

P (Pa)

10.0
Case 1
0.0

-10.0

Case 2

-20.0
-30.0
-40.0
0

0.5

1

1.5
Mass Flow Rate (kg/s)

2

2.5

Figure 2.36 Performance curves for RCDB Case 1 and Case 2

Fan performance curves were derived experimentally to identify the associated
pressure drop from the atmosphere to the upper receiver for a given flow rate, establishing
facility parameters to match the operating conditions of the CD experiment. Figure 2.36
shows the performance curves for both fan conditions. The operating point of the AFRD
facility is inferred from the known mass flow rate across the fan. The target operating
point is marked on each curve. Each case has a unique RPM condition.
2.3.2 Data Acquisition
As mentioned previously, the rotating A/D board is clocked and triggered from the
stationary A/D board. This ensures accurate and simultaneous data acquisition. The system samples at 40000Hz for 30 seconds, which is long enough to converge higher-order
statistics. Due to the unsteady nature and sensitive operating point of the fan, a series of
53

data files are taken for a particular target operating point where the main blower is throttled through various flow conditions. During post-processing, the file with the calculated
operating conditions nearest to the target is selected for further post-processing.
2.4 Stationary Experiments
All data presented by Höwer from stationary experiments follow with similar procedures and equipment. Höwer performed his experiments in parallel to the RCDB experiment. Further procedures are well discussed in Höwer’s Masters Thesis (2012).
In addition to stationary experimental data from Höwer, Moreau was kind enough
to forward CD airfoil data from 2005. See Moreau and Roger (2005) for further reading.

54

3.0 Experimental Results and Discussion
This section focuses on identifying and describing the nature of the convecting
pressure patterns along the airfoil surface as well as calculating and presenting the usefulness of parameters of interest to the aeroacoustic noise prediction community. For purposes of this report, it is useful to first recognize specific flow regimes associated with the
airfoil boundary layer - identified by Moreau and Roger (2005) and Neal (2010). These
are (for the suction side):
•

Leading edge separation - identified as the region at the leading edge where the
boundary layer separates from the surface. It is characterized by a very low
pressure coefficient. Estimated to occur between x/c=0 and x/c=0.1.

•

Reattachment - the region following the separated zone where the boundary
layer transitions and reattaches as a turbulent boundary layer. Estimated to
occur between x/c=0.05 and x/c=0.15.

•

Mid-chord - the region where pressure increases (an effect of airfoil shape) and
the turbulent boundary layer thickens as in an adverse pressure gradient flat
plate boundary layer. Estimated to occur between x/c=0.2 and x/c=0.8.

•

Trailing edge - the aft region of the airfoil where the boundary layer is similar
to a well developed flat-plate boundary layer.

•

Trailing edge (spanwise) - This region consists of 4 spanwise distributed taps
at x/c=0.98. The region is an extension of the trailing edge region with the
added purpose to investigate radial effects. Estimated to occur between
x/c=0.8 and x/c=1.0.

The pressure side:
•

The pressure side of the airfoil is treated as one region since the flow has laminar characteristics at and between the measurement locations. This “viscous
dominated region” is distinguished with a quiescent pressure field. The last tap
(Tap 29 x/c=0.93) begins to show evidence of a trailing-edge interaction.

55

With the lack of full-field data and the inherent unsteadiness of the geometry/flow
field, it is difficult to identify the exact locations of the aforementioned regions. As mentioned previously, the exploration and quantification of the boundary layer properties
necessitate the need for surface embedded steady and unsteady pressure transducers.
It is clear from known Cp data that there exist discrete flow regions along the airfoil - these can be further identified and studied through the application of mean and fluctuating surface pressure measurements. The following methods and techniques quantify
the flow regions along the airfoil surface.
3.1 Pressure Coefficient
It is useful to first consider the mean pressure quantities on the fan blade. The
blade loading is expressed in the form of a pressure coefficient (Cp), of which the integral
will yield a non-dimensional lift coefficient. As discussed in Section 2.1, the pressure
coefficient profile is critical for matching the blade loading conditions between the rotating and stationary Controlled Diffusion experiments.
Figure 3.1 shows the Cp distribution between the rotating and stationary experiments for the Case 1 operating condition. It can be seen that the general shape of the
curves match well, but there is lower lift on the pressure side of the rotating airfoil. The
non-dimensional lift coefficient, following Equation 3.1, yields 0.83 and 0.74 for the stationary and rotating blades respectively.
Cl =

1

x

0 ( Cp, lower – Cp, upper ) d-c

3.1

There is more total lift produced by the stationary blade, mostly from additional contributions from the pressure side. Note that there is an inherent unsteadiness in the leading-edge

56

laminar separation and turbulent reattachment region; it was expected to see a strong difference between the two cases. As discussed previously, Figure 3.1 confirms similar operating conditions between the rotating and stationary experiments.
Figure 3.2 shows a comparison of the Cp distribution between Case 1 and Case 2
operating conditions of the RCDB. It is clear from the pressure distribution that the two
cases have distinctly different flow conditions. The blade in Case 2 is more highly loaded
as is evident from the integral (Equation 3.1) of the profile: 1.0 for Case 2 versus 0.74 for
Case 1. There exists a higher magnitude pressure at the leading edge (x/c<0.2), but less
pressure building across the midspan (0.2<x/c<0.8) of the blade for Case 2. The Cp distribution of RCDB Case 2 does not follow the characteristic curvature of the CD blade as the
stationary and RCDB Case 1 does, this suggests that there exists a very thick boundary
layer or a separation condition is occurring across the suction side of the Case 2 blade.
While the RCDB Case 2 blade is more highly loaded, it is speculated from the distribution
that a full or partial stall may be occurring. It is evident that the pressure side of the airfoil
contributes more significantly to the integral loading of the blade. Note that the pressure
coefficient along the trailing edge of the suction side of the blade is similar for each case.
It is clear the mean pressure profiles do not adequately describe the details of the
near-wall pressure field. Further similarities and differences of the wall pressure field
between the rotating and stationary CD blades are explored further through the measurement of the local fluctuating pressures.

57

1
0.5
0

Cp

-0.5
-1
-1.5

RCDB: Case1
RCDB: Case2

-2
0

0.2

0.4

x/c

0.6

0.8

1

Figure 3.1 Cp distribution CD and RCDB Case 1
1
0.5
0

Cp

-0.5
-1
-1.5

RCDB: Case1
RCDB: Case2

-2
0

0.2

0.4

x/c

0.6

0.8

Figure 3.2 Cp distribution RCDB Case 1 and Case 2

58

1

3.2 Statistics
Mean pressure quantities do not adequately identify and characterize the details of
the near-wall region pressure field. High speed - fast response - microphone transducer
measurements are particularly well suited for quantifying some aspects of the boundary
layer near to the airfoil surface by measuring the fluctuating pressures.
Regions of quiescence and regions of disorder are visibly identifiable along the
airfoil surface from time-series data. Figure 3.3 shows a sample collection of time-series
data for several locations on the blade surface. It is seen that taps in the leading edge
region (Figure 3.3a) of the blade have stronger fluctuations than those in other regions.
The separation region (Figure 3.3b) shows high magnitude fluctuations with rather large
structures (evident by the intermittency and long time delay between peaks/valleys in the
time-series). As the flow reattaches and transitions to a turbulent boundary layer (x/c=0.1),
the flow maintains the larger structures and intermittency while evolving to include high
frequency, intense fluctuations. The separation/reattachment region shows the strongest
fluctuations on the blade surface. The fluctuations decay farther downstream and become
more regular. The mid-chord region (Figure 3.3c) shows evidence of sporadic low frequency structures and evolve into a more uniform signal downstream near the trailing
edge (Figure 3.3d). These are characterized by high frequency, low magnitude fluctuations. The pressure side mid-chord (Figure 3.3e) time-series is typical of all the measurement locations on that side of the blade; as suggested by Moreau and Roger (2005), the
low magnitude and rolling signal is indicative of a viscous dominated boundary layer with
minimal disturbances.

59

100

a) Leading edge
x/c=0.013

0
-100

2000

4000

6000

8000

10000

100

b) Separation region
x/c=0.052

0

Pressure (Pa)

-100

2000

4000

6000

8000

10000

100

c) Mid-chord
x/c=0.53

0

-100

2000

4000

6000

8000

10000

100

d) Trailing-edge
x/c=0.98

0
-100

2000

4000

6000

8000

10000

100

e) Pressure side
x/c=0.53

0
-100

2000

4000

6000

8000

10000

samples

Figure 3.3 Sample time series RCDB Case 1
The collected unsteady pressure data are assumed to represent a continuous random variable. The probability density function (PDF)1 supports further conclusions from
the pressure field. The probability density functions represent the statistical likelihood of a
random variable to take on a particular value; the PDF for a particular pressure tap reflects
the statistical description of the wall pressure field. Several statistical quantities can be

1. It is recognized that the histograms of this thesis may not be fully converged as
are required to fully justify the use of the term: probability density function. The
latter is used for convenience with the adequate argument that the distributions
are relatively smooth.
60

calculated from the PDF and used to infer and support conclusions on the condition of the
flow field above the airfoil surface.
The PDF of the fluctuating pressure signal visually demonstrates the statistical distribution. Generally, a narrow PDF is indicative of a viscous dominated boundary layer. A
wide PDF suggests the presence of a turbulent boundary later. There are varying degrees
of distributions.
Figure 3.4 and Figure 3.5 show the PDFs of the suction side pressure taps for
RCDB and CD Case 1. It is seen that the CD airfoil generally has a more narrow distribution across the entirety of the airfoil. The magnitude and narrowness of the leading-edge
PDFs, Taps 1 and 2 (Figure 3.4a and Figure 3.4b), suggest a strong viscous dominated
region. The PDF distributions are typically dissimilar between the RCDB and CD blades
except the trailing edge spanwise region (Figure 3.6), where the acoustic production is significant (as discussed in Section 1.3). The pressure side PDFs (Figure 3.7) show the high
peakedness of the pressure signal, particularly with the CD airfoil. The inconsistency with
CD Tap 29 (Figure 3.7e) is attributed to a non-physical corruption of the microphone data.

61

Tap1 x/c=0.0127
2

Tap2 x/c=0.0299
1

RCDB 1
CD 1

1

0
-100

a)

RCDB 1
CD 1

0.5

-50

0
Pressure (pa)

50

100

0
-100

b)

-50

Tap3 x/c=0.0522
0.06

RCDB 1
CD 1

0.03

-50

0
Pressure (pa)

0.2

50

100

0

-100
d)

-50

0
Pressure (pa)

50

100

Tap7 x/c=0.403
0.8

RCDB 1
CD 1

0.1

e)

100

RCDB 1
CD 1

Tap6 x/c=0.1493

0
-50

50

Tap5 x/c=0.0873
0.06

0.03

0
-100
c)

0
Pressure (pa)

RCDB 1
CD 1

0.4

-25

0
Pressure (pa)

25

50

f)

0
-20

-10

0
Pressure (pa)

Figure 3.4 RCDB-CD Case 1 suction side PDFs

62

10

20

Tap9 x/c=0.5336

Tap11 x/c=0.6791

0.4

0.4

RCDB 1
CD 1

0.2

0
-20
a)

RCDB 1
CD 1

0.2

-10

0
Pressure (pa)

10

20

b)

0
-30

-15

Tap21 x/c=0.8582
0.2

0

RCDB 1
CD 1

RCDB 1
CD 1

0.1

-25

0
Pressure (pa)

0.2

25

50

0

d)-25

-12.5

0
12.5
Pressure (pa)

25

Tap24 x/c=0.9216
0.2

RCDB 1
CD 1

0.1

e)

30

Tap22 x/c=0.8806

Tap23 x/c=0.8993

0
-25

15

0.2

0.1

c) -50

0
Pressure (pa)

RCDB 1
CD 1

0.1

-12.5

0
12.5
Pressure (pa)

25

f)

0
-50

-25

0
Pressure (pa)

25

Figure 3.5 RCDB-CD Case 1 suction side PDFs continued

63

50

Tap25A y/l=0.4538
0.1

Tap25B y/l=0.4769
0.2

RCDB 1
CD 1

0.05

a)

0
-25

0.1

-12.5

0
12.5
Pressure (pa)

25

b)

0
-25

Tap25 y/l=0.5
0.2

0

-12.5

0
12.5
Pressure (pa)

25

Tap25C y/l=0.5769
0.2

RCDB 1
CD 1

0.1

c) -20

RCDB 1
CD 1

RCDB 1
CD 1

0.1

-10

0
Pressure (pa)

10

20

0
-25

d)

-12.5

0
12.5
Pressure (pa)

Figure 3.6 RCDB-CD Case 1 trailing edge spanwise PDFs

64

25

Tap4 x/c=0.0672
2

Tap8 x/c=0.3993
3

RCDB 1
CD 1

1

0

a)-50

RCDB 1
CD 1

1.5

-25

0
Pressure (pa)

25

50

0

b)-50

-25

0

33

RCDB 1
CD 1

50

RCDB
RCDB 1 1
CD
CD 1 1

1.5
1.5

1.5

c)-50

25

Tap12 x/c=0.6754
Tap12 x/c=0.6754

Tap8 x/c=0.3993
3

0
Pressure (pa)

-25

0
Pressure (pa)

25

50

00

-50
d)-50

-25
-25

00
2525
Pressure (pa)
Pressure (pa)

Tap29 x/c=0.9291
0.4

RCDB 1
CD 1

0.2

0

e)-200

-100

0
100
Pressure (pa)

200

Figure 3.7 RCDB-CD Case 1 pressure side PDFs

65

5050

Figure 3.8 and Figure 3.9 show the suction side PDFs of RCDB Case 1 and Case 2.
From these distributions, it is clear that there are similarities and differences for the design
point flow field and the off-design flow field. Most notably, there are wider PDF distributions for the off-design condition for most of the airfoil surface - this suggests a more
energetic wall pressure field for the RCDB Case 2 flow field. Following stationary CD
boundary layer survey results from Neal (2010), the more energetic wall pressure fluctuations of the RCDB Case 2 are inferred to be indicative of a more turbulent boundary layer.
Additionally, the PDFs for the leading edge taps are skewed positively for the off-design
condition - there is a higher tendency for negative pressures. Considering the trailing edge
spanwise and pressure side PDFs (Figure 3.10 and Figure 3.11) the similar conclusion
holds: the design-point condition (Case 1) is characterized by a more stable boundary
layer with fewer fluctuations than the off-design condition.

66

Tap2 x/c=0.0299

Tap1 x/c=0.0127
0.03

0.03

RCDB 1
RCDB 2

0.02

0.02

0

a) -100

0
Pressure (pa)

100

0

b)-100

Tap3 x/c=0.0522
0.03

0

RCDB 1
RCDB 2

0.03

0
Pressure (pa)

100

0
-100
d)

Tap6 x/c=0.1493

0
Pressure (pa)

100

Tap7 x/c=0.403
0.2

RCDB 1
RCDB 2

0.04

e)

100

Tap5 x/c=0.0873
RCDB 1
RCDB 2

0.08

0
-100

0
Pressure (pa)

0.06

0.02

c) -100

RCDB 1
RCDB 2

RCDB 1
RCDB 2

0.1

0
Pressure (pa)

100

0

f) -75

0
Pressure (pa)

Figure 3.8 RCDB Case1-Case2 suction side PDFs

67

75

Tap9 x/c=0.5336
0.2

Tap11 x/c=0.6791
RCDB 1
RCDB 2

0.1

0
-50
a)

0.1

0.05

0
Pressure (pa)

0
-50

50

b)

Tap21 x/c=0.8582
0.08

0

0.1

RCDB 1
RCDB 2

0
Pressure (pa)

0.08

0

50

RCDB 1
RCDB 2

d)-50

0
Pressure (pa)

50

Tap24 x/c=0.9216
0.1

RCDB 1
RCDB 2

0.04

e)

50

0.05

Tap23 x/c=0.8993

0
-50

0
Pressure (pa)
Tap22 x/c=0.8806

0.04

c) -50

RCDB 1
RCDB 2

RCDB 1
RCDB 2

0.05

0
Pressure (pa)

50

f)

0
-50

0
Pressure (pa)

Figure 3.9 RCDB Case1-Case2 suction side PDFs continued

68

50

Tap25A y/l=0.4538
0.1

Tap25B y/l=0.4769
RCDB 1
RCDB 2

0.05

0

a) -50

0.1

0.05

-25

0
Pressure (pa)

25

0
50 b) -50

Tap25 x/c=0.9776
0.2

0

-25

0
Pressure (pa)

25

50

Tap25C y/l=0.5769
0.1

RCDB 1
RCDB 2

0.1

c) -50

RCDB 1
RCDB 2

RCDB 1
RCDB 2

0.05

0
Pressure (pa)

0
50 d) -50

-25

0
Pressure (pa)

Figure 3.10 RCDB Case1-Case2 trailing edge spanwise PDFs

69

25

50

Tap4 x/c=0.0672
0.2

Tap8 x/c=0.3993
0.2

RCDB 1
RCDB 2

0.1

0
-50

a)

RCDB 1
RCDB 2

0.1

-25

0
Pressure (pa)

25

50

b)

0
-50

-25

Tap8 x/c=0.3993
0.2

0

25

50

Tap12 x/c=0.6754
0.2

RCDB 1
RCDB 2

0.1

c) -50

0
Pressure (pa)

RCDB 1
RCDB 2

0.1

-25

0
Pressure (pa)

0
50 d) -50

25

-25

0
Pressure (pa)

Tap29 x/c=0.9291
0.4

RCDB 1
RCDB 2

0.2

e)

0
-50

-25

0
Pressure (pa)

25

50

Figure 3.11 RCDB Case1-Case2 pressure side PDFs

70

25

50

In order to quantify time series data with standard statistical representations, the
method of moments is applied to the unsteady pressure data to identify the parameters:
mean, variance, skewness, and kurtosis (1st through 4th statistical moments - see
Section 1.4). These statistical quantities numerically quantify the distribution of the PDF.
The mean pressure is measured by the onboard pressure MEMS transducers where higher
order statistics are calculated from the unsteady microphone measurements (note that the
microphones only measure fluctuating pressures - i.e., the mean of microphone data is
zero). The variance, 2nd moment, represents the width, or spread, of the fluctuating pressure amplitude. Skewness, the 3rd moment, indicates the bias of the pressure pattern whether the fluctuations are biased positively or negatively. A skewed distribution often
results if the variable is bounded on the high end (skewness is positive) or the low end
(skewness is negative) of the mean. Kurtosis, the 4th moment, is commonly known as
‘peakedness’ - an expression of how peaked or flat the distribution is. A Gaussian distribution has a flatness of 3.0 (Bendat and Piersol 1986). This allows for a simple comparison
to a Gaussian distribution.1
Figure 3.12 shows the 1st through 4th moments of the RCDB and CD Case 1 suction side. The agreement in the 1st moment (Figure 3.12a) Cp has been demonstrated previously, but it is worth comparing alongside higher moments for further understanding of
the flow regimes. The suction side of the airfoil is characterized by a leading edge boundary layer transition. The separation and reattachment of the boundary layer (0<x/c<0.15)
causes significant unsteadiness in the pressure field; the variance is high and the skewness

1. Note that the connecting lines in the following figures (Figure 3.12 to
Figure 3.17) do not imply trends. The lines exist purely to identify and connect
data points from adjacent measurement locations.
71

and kurtosis do not show a trend in the streamwise direction. Variance is highest at x/
c=0.05, the expected peak of the separated region. The flow reattaches downstream
(x/c>0.15), characterized by a decrease in the magnitude of Cp and variance as well as an
increase in the relative spatial uniformity of higher order statistics from 0.15<x/c<1. Following separation, the mid-chord flow statistics remain relatively constant - supporting the
Moreau and Roger (2005) idea of a boundary layer region analogous to an adverse pressure gradient flat-plate boundary layer. The trailing edge also exhibits similarly uniform
streamwise statistics.
Figure 3.13 compares the moments for RCDB Case 1 and Case 2. The statistics of
the off-design point suggest that there exists an energetic pressure field aft of the typical
reattachment region (x/c=0.15). The variance (Figure 3.13b) is high across the entire surface, supporting the claim that the boundary layer is more energetic in Case 2 than Case 1.
The off-design flow field shows a strong positive skewness (Figure 3.13c) near the leading
edge of the airfoil (x/c<0.15). The skewness is generally higher across the suction side of
RCDB Case 2 compared to RCDB Case 1. It is noteworthy to mention that the 4th
moment is uniform across the majority of the airfoil surface while the other lower-order
moments tend to vary more.
Figure 3.14 and Figure 3.15 show the pressure side statistics for RCDB-CD Case 1
and RCDB Case 1-Case 2, respectively. The pressure side has been characterized by a predominantly viscous dominated boundary layer. This is supported by the general consistency and low magnitudes for the 1st, 2nd, and 3rd moments. The lower order moments of
Figure 3.14 show spatial homogeneity within their respective flow fields and show similar
magnitudes and trends between the two flow fields for x/c<0.7. The high magnitude of the

72

4th moment in RCDB Case 1 is recognized but the interpretation is subject to uncertainty.
It is noteworthy to mention the similarity of the pressure side moments between Case1 and
Case 2 (Figure 3.15) compared to the differences shown in Figure 3.13. This suggests that
the RCDB Case 2 flow field has a similar viscous dominated boundary layer.
It has been demonstrated previously (Neal 2010) that the stationary and rotating
CD airfoils have minimal spanwise velocity components; therefore, it is expected that the
stationary and rotating CD airfoils will have constant statistics in the spanwise direction.
A comparison of the RCDB and CD Case 1 spanwise moments (Figure 3.16) show similar
statistical properties. The moments show limited spatial variation along the span of each
blade and the similar magnitudes demonstrate good agreement between the rotating and
stationary blades. Additionally, a comparison of Case 1 and Case 2 spanwise statistics
(Figure 3.17) demonstrates a similar consistency in the spatial variation of each flow field,
but agreement between Case 1 and Case 2 statistics is poor. The relative spatial homogeneity of the spanwise boundary layer moments show that the wall pressure field fluctuations are not a strong function of local radial position; the flow field across the surface is
fairly uniform in the radial direction.

73

Kurtosis-3 Skewness Var/.5*ρ*W

2

-Cp

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

RCDB: Case1

0.9

1

CD: Case1

1
0
0.2

b)

0.1
0
0.5

c)

0

-0.5
20

d)

10
0
0

2

-Cp

2

Kurtosis-3 Skewness Var/.5*ρ*W

a)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/c
Figure 3.12 RCDB-CD Case 1 suction side moments

RCDB: Case1

RCDB: Case2

a)

1
0
0.2

b)

0.1
0
1

c)

0
-1
5

d)

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c
Figure 3.13 RCDB Case1-Case2 suction side moments

74

0.9

1

Cp

0.4

0

0.1

0.2

0.3

2

0.5

0.6

0.7

0.8

0.9

1

a)

0.2
RCDB: Case1

Kurtosis-3 Skewness Var/.5*ρ*W

0.4

CD: Case1

0
0.4

b)

0.2
0
2

c)

0

-2
100

d)

50
0
0

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/c
Figure 3.14 RCDB-CD Case 1 pressure side moments

a)

RCDB: Case2

0.5

0
0.04

Kurtosis-3 Skewness Var/.5*ρ*W

2

Cp

RCDB: Case1

b)

0.02
0
2

c)

0
-2
100

d)

50
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x/c
Figure 3.15 RCDB Case 1 - Case 2 pressure side moments

75

0.9

1

-Cp

0.44
0.2

0.46

0.48

RCDB: Case1

0.5

0.52

0.54

0.56

0.58

a)

CD: Case1

0.1

2

0
0.04

Kurtosis-3 Skewness Var/.5*ρ*W

0.6

b)

0.02
0
0.5

c)

0
-0.5
2

d)

1
0
0.44

0.44
0.2

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

y/l
Figure 3.16 RCDB-CD Case 1 trailing edge spanwise moments

0
0.1

Kurtosis-3 Skewness Var/.5*ρ*W

-Cp

a)

RCDB: Case2

0.1

2

RCDB: Case1

b)

0.05
0
0.5

c)

0
-0.5
2

d)

1
0
0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

y/l
Figure 3.17 RCDB Case1 - Case2 trailing edge spanwise moments

76

0.6

3.3 Pressure Spectra
Continuing to characterize the boundary layer characteristics, a power spectral
density of the wall pressure fluctuations (PSD, Φpp) is calculated to identify how the
boundary layer pressure fluctuations vary as a function of frequency. The spectral calculations are generally repeatable between experiments within 2dB for all taps across the airfoil with the exception of the leading edge taps (Taps 1, 2, and 3) where there is an
inherent instability associated with the separation/reattachment and transition of the
boundary layer. Before comparing boundary layer regimes for a particular flow condition,
it is useful to compare the wall pressure spectra of the Case 1 and Case 2 flow fields on a
tap-by-tap basis. Figure 3.18 through Figure 3.59 show the wall pressure spectra for each
surface tap and their corresponding operating conditions (RCDB: Case1, Case 2 and CD:
Case 1).
Consider a comparison between RCDB Case 1 and CD Case 1 flow conditions
(Figure 3.18 to Figure 3.38)1. The direct comparison of the stationary and rotating analog
for a similar flow condition is instructive for demonstrating the properties of the fluctuations of surface pressure field. It is evident that the leading edge taps (Figure 3.18 to
Figure 3.20) have widely varying spectral properties. This is attributed to the separation
region unsteadiness and to fundamental differences in the flow geometry. Figure 3.18, representative of the furthest forward measurement location (x/c=0.013), shows a significant
difference in spectral magnitude for frequencies ωc ⁄ u ∞ < 20 . The exact behavior of the
separation/reattachment region is unknown between the rotating and stationary CD
blades; the difference shown in Figure 3.18 demonstrates the inherent difference the two

1. Note that Höwer data is high pass filtered with a pass-band: ωc ⁄ u ∞ > 6
77

geometries. The interpretation is subject to some uncertainty at this time. Note that it was
expected that the Höwer and Moreau results be identical due to their similar geometries.
Moreau and Höwer results agree better as the boundary layer develops farther downstream. The differences near the leading edge are associated to instabilities in the leading
edge boundary layer separation/reattachment and flow facility differences (see Höwer
2012 for a more thorough analysis).
Following separation, the flow reattaches and becomes established as a turbulent
boundary layer. The pressure spectra of the stationary and rotating CD blades become
more uniform at measurement locations farther downstream. Mid-chord (0.2<x/c<0.8)
measurements (Figure 3.18 to Figure 3.25) show strong spectral agreement for frequencies ωc ⁄ u ∞ > 20 . Low frequencies along the mid-chord indicate that the RCDB Case 1
contains more energy in larger flow structures than the CD blade. Power spectra of the
trailing edge (0.1<x/c<1) indicate that energy from large structures is transferred to small
structures. Stationary and rotating CD spectra in Figure 3.26 to Figure 3.30 show a systematic decrease in the low frequency ( ωc ⁄ u ∞ < 10 ) energy on order of 5-10dB from Tap
21 (x/c=0.858) to Tap 25 (x/c=0.978) where the mid-frequency ( 10 < ωc ⁄ u ∞ < 100 )
shows an increase in energy along the same distance. Trailing edge spanwise spectra
(Figure 3.30 to Figure 3.33) show similar differences to those seen in the trailing edge
streamwise taps. Generally, the spectra of RCDB Case w are similar to CD Case 1 with the
notable exception that the RCDB spectra is more biased toward lower frequencies.
The pressure side (Taps 4, 8, 10, 12, and 29) spectra show similar spectral shape
between the stationary and rotating CD blades, noting that the RCDB Case 1 had a magnitude shift of approximately 10-20dB. There is a higher power in the RCDB pressure signal

78

across the entire spectra; while typical of most of the measurements taken for the CD and
RCDB geometries, it is worth considering the magnitude of the difference. Following
Moreau and Roger (2005), the spectral shape seen on the pressure side is indicative of a
viscous dominated boundary layer; the measurements are particularly susceptible to
broadband noise due to the weak hydrostatic pressure signal. Moreau notes that the sensors on the pressure side of the blade likely act as acoustic sensors. Moreau’s measurements reach the floor of the transducer’s resolution at higher frequencies.
Consider a comparison between RCDB Case 1 and RCDB Case 2 flow conditions
(Figure 3.39 to Figure 3.59). The comparison of the two operating conditions supports a
deeper understanding of the RCDB boundary layer characteristics. Generally, the power of
the Case 2 flow field is higher than Case 1 - this is a confirmation of the statistical measurements from Section 3.2, which suggested a more energetic disturbance in the Case 2
flow field. The leading edge measurements (Figure 3.39 to Figure 3.41) show similar
spectral shapes in both cases. Following reattachment, toward the mid-chord of the airfoil,
the Case 2 spectra develop more energy in all frequencies. Near the trailing edge
(Figure 3.47 to Figure 3.51), the Case 2 spectra develops a more consistent profile, characterized by high power in the low frequencies and a steady, shallow decay. The low frequency content suggests generally larger flow structures due to a larger integral scale of
the boundary layer. A comparison of the pressure side spectra (Figure 3.55 to Figure 3.59)
yields little new information; the Case 2 power spectra have a generally constant magnitude shift relative to the Case 1 spectra.

79

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap1: RCDB Case1
Tap1: CD Case1 (Moreau)
1
2
101 ωc/u
102
10
10
∞
Figure 3.18 RCDB and CD Case 1 (Moreau) spectra: Tap1 (x/c=0.013)

0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap2: RCDB Case1
Tap2: CD Case1 (Moreau)
1
2
101 ωc/u
102
10
10
∞
Figure 3.19 RCDB and CD Case 1 (Moreau) spectra: Tap2 (x/c=0.03)

0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap3: RCDB Case1
Tap3: CD Case1 (Hower)
B
Tap3: CD Case1 (Moreau)

2
1011 ωc/u
102
10
10
∞
Figure 3.20 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap3 (x/c=0.052)
0
100
10

80

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap5: RCDB Case1
Tap5: CD Case1 (Hower)
B
Tap5: CD Case1 (Moreau)

1
101 ωc/u
1022
10
10
∞
Figure 3.21 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap5 (x/c=0.087)
0
100
10

80
2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

Tap6: RCDB Case1
B
Tap6: CD Case1 (Hower)
Tap6: CD Case1 (Moreau)

60
40
20
0

1
101 ωc/u
1022
10
10
∞
Figure 3.22 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap6 (x/c=0.149)
0
100
10

80
2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

Tap7: RCDB Case1
B
Tap7: CD Case1 (Hower)
Tap7: CD Case1 (Moreau)

60
40
20
0

1
101 ωc/u
1022
10
10
∞
Figure 3.23 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap7 (x/c=0.403)
0
100
10

81

80
2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

Tap9: RCDB Case1
B
Tap9: CD Case1 (Hower)
Tap9: CD Case1 (Moreau)

60
40
20
0

1
2
101 ωc/u
102
10
10
∞
Figure 3.24 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap9 (x/c=0.534)
0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap11: RCDB Case1
Tap11: CD Case1 (Hower)
B
Tap11: CD Case1 (Moreau)

1
101 ωc/u
1022
10
10
∞
Figure 3.25 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap11 (x/c=0.679)
0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap21: RCDB Case1
Tap21: CD Case1 (Hower)
B
Tap21: CD Case1 (Moreau)

2
1
101 ωc/u
102
10
10
∞
Figure 3.26 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap21 (x/c=0.858)
0
100
10

82

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap22: RCDB Case1
Tap22: CD Case1 (Hower)
B
Tap22: CD Case1 (Moreau)

1
2
101 ωc/u
102
10
10
∞
Figure 3.27 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap22 (x/c=0.881)
0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap23: RCDB Case1
Tap23: CD Case1 (Hower)
B
Tap23: CD Case1 (Moreau)
0
100
10

1
101
10

ωc/u

∞

2
102
10

Figure 3.28 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap23 (x/c=0.899)

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap24: RCDB Case1
Tap24: CD Case1 (Hower)
B
Tap24: CD Case1 (Moreau)

1
101 ωc/u
1022
10
10
∞
Figure 3.29 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap24 (x/c=0.922)
0
100
10

83

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap25: RCDB Case1
Tap25: CD Case1 (Hower)
B
Tap25: CD Case1 (Moreau)

1011 ωc/u
1022
10
10
∞
Figure 3.30 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap25 (x/c=0.978)
0
0

10
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap25A: RCDB Case1
Tap25A: CD Case1 (Hower)
B
Tap25A: CD Case1 (Moreau)

1
2
101 ωc/u
102
10
10
∞
Figure 3.31 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap25A (y/l=0.45)
0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap25B: RCDB Case1
Tap25B: CD Case1 (Hower)
B
Tap25B: CD Case1 (Moreau)

1
2
101 ωc/u
102
10
10
∞
Figure 3.32 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap25B (y/l=0.48)

0
100
10

84

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap25C: RCDB Case1
Tap25C: CD Case1 (Hower)
B
Tap25C: CD Case1 (Moreau)

1
101 ωc/u
1022
10
10
∞
Figure 3.33 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap25C (x/c=0.58)
0
100
10

80
2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

Tap4: RCDB Case1
B
Tap4: CD Case1 (Hower)
Tap4: CD Case1 (Moreau)

60
40
20
0
0
100
10

1
101
10

ωc/u

∞

1022
10

Figure 3.34 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap4 (x/c=0.067)
80
2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

Tap8: RCDB Case1
B
Tap8: CD Case1 (Hower)
Tap8: CD Case1 (Moreau)

60
40
20
0
0
100
10

1
101
10

ωc/u

∞

2
102
10

Figure 3.35 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap8 (x/c=0.400)

85

80
2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

Tap10: RCDB Case1
B
Tap10: CD Case1 (Hower)
Tap10: CD Case1 (Moreau)

60
40
20
0

1
2
101 ωc/u
102
10
10
∞
Figure 3.36 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap10 (x/c=0.530)
0
100
10

80
2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

Tap12: RCDB Case1
B
Tap12: CD Case1 (Hower)
Tap12: CD Case1 (Moreau)

60
40
20
0
0
100
10

1
101
10

ωc/u

∞

2
102
10

Figure 3.37 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap12 (x/c=0.675)

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap29: RCDB Case1
Tap29: CD Case1 (Hower)
B
Tap29: CD Case1 (Moreau)

1
2
101 ωc/u
102
10
10
∞
Figure 3.38 RCDB and CD Case 1 (Höwer and Moreau) spectra: Tap29 (x/c=0.929)

0
100
10

86

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap1: RCDB Case1
Tap1: RCDB Case2
1
2
101 ωc/u
102
10
10
∞
Figure 3.39 RCDB Case 1 and Case 2 spectra: Tap1 (x/c=0.013)

0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap2: RCDB Case1
Tap2: RCDB Case2
2
1
101 ωc/u
102
10
10
∞
Figure 3.40 RCDB Case 1 and Case 2 spectra: Tap2 (x/c=0.030)
0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap3: RCDB Case1
Tap3: RCDB Case2
2
1
102
101 ωc/u
10
10
∞
Figure 3.41 RCDB Case 1 and Case 2 spectra: Tap3 (x/c=0.052)
0
100
10

87

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap5: RCDB Case1
Tap5: RCDB Case2
1
2
101 ωc/u
102
10
10
∞
Figure 3.42 RCDB Case 1 and Case 2 spectra: Tap5 (x/c=0.087)

0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap6: RCDB Case1
Tap6: RCDB Case2
0
100
10

1
101
10

ωc/u

∞

2
102
10

Figure 3.43 RCDB Case 1 and Case 2 spectra: Tap6 (x/c=0.149)

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap7: RCDB Case1
Tap7: RCDB Case2
2
1
101 ωc/u
102
10
10
∞
Figure 3.44 RCDB Case 1 and Case 2 spectra: Tap7 (x/c=0.403)
0
100
10

88

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap9: RCDB Case1
Tap9: RCDB Case2
1
2
101 ωc/u
102
10
10
∞
Figure 3.45 RCDB Case 1 and Case 2 spectra: Tap9 (x/c=0.534)
0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap11: RCDB Case1
Tap11: RCDB Case2
1
2
101 ωc/u
102
10
10
∞
Figure 3.46 RCDB Case 1 and Case 2 spectra: Tap11 (x/c=0.679)

0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap21: RCDB Case1
Tap21: RCDB Case2
2
1
102
101 ωc/u
10
10
∞
Figure 3.47 RCDB Case 1 and Case 2 spectra: Tap21 (x/c=0.858)
0
100
10

89

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap22: RCDB Case1
Tap22: RCDB Case2
1
2
101 ωc/u
102
10
10
∞
Figure 3.48 RCDB Case 1 and Case 2 spectra: Tap22 (x/c=0.881)
0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap23: RCDB Case1
Tap23: RCDB Case2
2
1
102
101 ωc/u
10
10
∞
Figure 3.49 RCDB Case 1 and Case 2 spectra: Tap23 (x/c=0.899)
0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap24: RCDB Case1
Tap24: RCDB Case2
1
2
101 ωc/u
102
10
10
∞
Figure 3.50 RCDB Case 1 and Case 2 spectra: Tap24 (x/c=0.922)

0
100
10

90

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap25: RCDB Case1
Tap25: RCDB Case2
1
2
101 ωc/u
102
10
10
∞
Figure 3.51 RCDB Case 1 and Case 2 spectra: Tap25 (x/c=0.978)

0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap25A: RCDB Case1
Tap25A: RCDB Case2
1
2
101 ωc/u
102
10
10
∞
Figure 3.52 RCDB Case 1 and Case 2 spectra: Tap25A (y/l=0.45)
0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap25B: RCDB Case1
Tap25B: RCDB Case2
0
100
10

1
101
10

ωc/u

∞

2
102
10

Figure 3.53 RCDB Case 1 and Case 2 spectra: Tap25B (y/l=0.48)

91

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap25C: RCDB Case1
Tap25C: RCDB Case2
1
2
101 ωc/u
102
10
10
∞
Figure 3.54 RCDB Case 1 and Case 2 spectra: Tap25C (y/l=0.58)

0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap4: RCDB Case1
Tap4: RCDB Case2
0
100
10

1
101
10

ωc/u

∞

2
102
10

Figure 3.55 RCDB Case 1 and Case 2 spectra: Tap4 (x/c=0.067)

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap8: RCDB Case1
Tap8: RCDB Case2
2
1
101 ωc/u
102
10
10
∞
Figure 3.56 RCDB Case 1 and Case 2 spectra: Tap8 (x/c=0.400)
0
100
10

92

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap10: RCDB Case1
Tap10: RCDB Case2
1
2
101 ωc/u
102
10
10
∞
Figure 3.57 RCDB Case 1 and Case 2 spectra: Tap10 (x/c=0.530)

0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap12: RCDB Case1
Tap12: RCDB Case2
2
1
102
101 ωc/u
10
10
∞
Figure 3.58 RCDB Case 1 and Case 2 spectra: Tap12 (x/c=0.675)
0
100
10

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
60
40
20
0

Tap29: RCDB Case1
Tap29: RCDB Case2
2
1
101 ωc/u
102
10
10
∞
Figure 3.59 RCDB Case 1 and Case 2 spectra: Tap29 (x/c=0.929)
0
100
10

93

Boundary layer trends across the airfoil surface are better identified by re-representing the previous figures as a collection of regions along the blade. Figure 3.60 shows
the forward region of the airfoil - this includes separation and reattachment. The large difference of spectral energy magnitude between Taps 1 and 2 and Taps 3 and 4 from
20 < ω c ⁄ u ∞ < 600 is indicative of the laminar-to-turbulent boundary layer transition; the
transition begins near Tap 2. The long decay of Taps 1 and 2 ( 5 < ω c ⁄ u ∞ < 600 )suggests
a viscous dominated boundary region during the transition process. The wall pressure
spectrum stabilizes and the streamwise spatial agreement between taps improves in the
mid-chord region (Figure 3.61). The shape begins to scale more appropriately to that suggested by Hwang (see Section 1.3) at measurement locations farther downstream. The collapse of the spectra near the trailing edge (Figure 3.62) suggests that the boundary layer is
homogenous in the streamwise direction. This observation is of particular use to the aeroacoustic community, where predictions are often based on the spatial averaging of the wall
pressure spectra near an airfoil’s trailing edge (Blake 1986). spanwise statistics
(Figure 3.63) suggest that there is some variation of the boundary layer in the radial direction since the spectral power difference from Tap 25A to Tap 25C is on the order of
~10dB. The pressure side power spectra (Figure 3.64) show a consistent boundary layer
across the entire pressure side with a very clear decay rate of ω-2 - a viscous dominated
boundary layer.
There are a few clear decay rates present among the spectra: the inner scale (high
frequency), overlap (mid-range frequencies), and the outer scale (low frequencies). Each
scale is characterized by a typical decay rate: an almost constant intensity for the outer
scale, a slight decay in the overlap region, and a steep decay within the inner scale. ω-5 is

94

a typical decay rate of the inner scale for turbulent boundary layers of any pressure gradient (Hwang et al. 2009). The outer scale and overlap region have decay rates that are
dependent of the streamwise pressure gradient. An adverse pressure gradient will cause
the pressure fluctuations to decay at a higher rate (Goody and Simpson 2000).
Consider the decay rates of RCDB Case 1 (Figure 3.60 to Figure 3.64), CD Case 1
(Figure 3.65 to Figure 3.69), and RCDB Case 2 (Figure 3.70 to Figure 3.74). It is interesting to note that the spectral decays follow similar trends between the three flow conditions/geometries for corresponding pressure taps. The leading edge region has a large
variation between Taps 1-5 for Case 1 flow fields (a function of the boundary layer instability/unpredictability as it transitions from laminar to turbulent); decay rates are similar
between corresponding taps of CD and RCDB Case 1 flow fields. The mid-chord region
shows similar spectral matching of CD and RCDB Case 1. Tap 6 shows a decay of ~ω-2.5
and Taps 7, 9, and 11 show a decay of ω-5 for 200 < ω c ⁄ u ∞ < 600 . The trend continues
through the aft region to the trailing edge, where the spectral decay rates and magnitudes
match very well for frequencies 50 < ω c ⁄ u ∞ < 600 . RCDB Case 1 has a higher energy
for low frequencies ( 0.5 < ω c ⁄ u ∞ < 50 ) compared to CD. The similarity of the wall pressure spectra across the suction and pressure sides of RCDB and CD Case 1 suggests that a
similar wall pressure field exists - further confirmation of matched boundary conditions
between the stationary and rotating analog.
It is unclear why the spectra of Case 2 (Figure 3.70 to Figure 3.74) all similarly
decay at of a rate ~ω-2 - the interpretation is subject to uncertainty when considering only
spectral processing.

95

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80
-0.5

ω

60

-2.5

ω

-3

ω

40

20

0

-5

ω

Tap1 (x/c=0.013)
Tap2 (x/c=0.030)
Tap3 (x/c=0.052)
Tap5 (x/c=0.087)
1
101 ωc/u
10
∞

0
100
10

2
102
10

Figure 3.60 RCDB Case 1 fore region

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

60
-2.5

ω

40

20

0

-5

Tap6 (x/c=0.149)
Tap7 (x/c=0.403)
Tap9 (x/c=0.534)
Tap11 (x/c=0.679)

ω

1
101 ωc/u
10
∞

0
100
10

2
102
10

Figure 3.61 RCDB Case 1 mid-chord region

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

60

ω-1.5

40
-5

20

0

ω

Tap21 (x/c=0.858)
Tap22 (x/c=0.881)
Tap23 (x/c=0.899)
Tap24 (x/c=0.922)
Tap25 (x/c=0.978)

100
10
0

1
101 ωc/u
10
∞

Figure 3.62 RCDB Case 1 aft region

96

2
102
10

60
2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

50
40

-2

ω

30
20
10
0

-10

ω-5

Tap25A (y/l=0.454)
Tap25B (y/l=0.477)
Tap25 (y/l=0.500)
Tap25C (y/l=0.978)
0
100
10

1
101 ωc/u
10
∞

2
102
10

Figure 3.63 RCDB Case 1 trailing edge spanwise region

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

60

40
-2

ω

20

0

Tap4 (x/c=0.067)
Tap8 (x/c=0.400)
Tap10 (x/c=0.530)
Tap12 (x/c=0.675)
Tap29 (x/c=0.929)
1
101 ωc/u
10
∞

0
100
10

2
102
10

Figure 3.64 RCDB Case 1 pressure side
70
2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

60
-2.5

ω

50
40
-3

ω

30
Tap1 (x/c=0.013)
Tap2 (x/c=0.030)
Tap3 (x/c=0.052)
Tap5 (x/c=0.087)

20
10
0

0
100
10

-5

ω

1
101 ωc/u
10
∞

Figure 3.65 CD fore region

97

2
102
10

50
-2.5

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

ω

40
30
20
10

ω-5

Tap6 (x/c=0.149)
Tap7 (x/c=0.403)
Tap9 (x/c=0.534)
Tap11 (x/c=0.679)
1
101 ωc/u∞
10

0
100
10

2
102
10

Figure 3.66 CD mid-chord region
50
-1.5

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

ω

40
30
20
10
0

-5

ω

Tap21 (x/c=0.858)
Tap22 (x/c=0.881)
Tap23 (x/c=0.899)
Tap24 (x/c=0.922)
Tap25 (x/c=0.978)
1
101 ωc/u∞
10

0
100
10

2
102
10

Figure 3.67 CD aft region
50
0.3

ω

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

-2

ω

40
30
20
10

-5

ω

Tap25A (y/l=0.454)
Tap25B (y/l=0.477)
Tap25 (y/l=0.500)
Tap25C (y/l=0.978)

1000
10

1
101 ωc/u∞
10

1022
10

Figure 3.68 CD trailing edge spanwise region

98

50
-3.5

ω

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

40
30
20
10
0

-10
-20

-2

Tap4 (x/c=0.067)
Tap8 (x/c=0.400)
Tap10 (x/c=0.530)
Tap12 (x/c=0.675)
Tap29 (x/c=0.929)

ω
-1

ω

101 ωc/u
10
∞

100
10
0

102
10

1

2

Figure 3.69 CD pressure side
80

ω-2

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

70
60
50
40
30
20
10

Tap1 (x/c=0.013)
Tap2 (x/c=0.030)
Tap3 (x/c=0.052)
Tap5 (x/c=0.087)

-5

ω
1
101 ωc/u
10
∞

0
100
10

1022
10

Figure 3.70 RCDB Case 2 fore region
80
2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

70
60

-2.5

ω

50
40
30
20
10

Tap6 (x/c=0.149)
Tap7 (x/c=0.403)
Tap9 (x/c=0.534)
Tap11 (x/c=0.679)
0
100
10

-5

ω
1
101 ωc/u
10
∞

2
102
10

Figure 3.71 RCDB Case 2 mid-chord region

99

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

80

60
-2

ω

40

20

0

Tap21 (x/c=0.858)
Tap22 (x/c=0.881)
Tap23 (x/c=0.899)
Tap24 (x/c=0.922)
Tap25 (x/c=0.978)

-5

ω

1
101 ωc/u
10
∞

0
100
10

2
102
10

Figure 3.72 RCDB Case 2 aft region
70
-2

ω

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

60
50
40
30
20
10
0

Tap25A (y/l=0.454)
Tap25B (y/l=0.477)
Tap25 (y/l=0.500)
Tap25C (y/l=0.978)

ω-5
1
101 ωc/u
10
∞

0
100
10

2
102
10

Figure 3.73 RCDB Case 2 trailing edge spanwise region

2 2
∞

Φ pp (dB re [1/2×ρ×u ] )

60

40

-2

ω

20

0

Tap4 (x/c=0.067)
Tap8 (x/c=0.400)
Tap10 (x/c=0.530)
Tap12 (x/c=0.675)
Tap29 (x/c=0.929)
0
100
10

1
101 ωc/u
10
∞

2
102
10

Figure 3.74 RCDB Case 2 pressure side

100

3.4 Space-Time Correlations
A space-time cross-correlation (Rpp) of the fluctuating pressure field between surface taps allows for the “tracking” of a coherent motion. Following the convention of
Willmarth and Woolridge (1962) introduced in Section 1.2, this section is designated to
calculate some of the useful parameters for predicting acoustic trailing edge noise production: convection velocity, length scales.
The space-time correlations are performed in the trailing edge region only. This is
due to geometric constraints - favorable tap spacing in the trailing edge region - and a well
established boundary layer in the aft region. Figure 3.75 shows a typical correlation from
Tap 21-21, Tap 21-22, Tap 21-23, Tap 21-24, and Tap 21-25. The correlations in
Figure 3.75 demonstrate the time-lag (τ) of the boundary layer coherent motions, band-filtered from 200-500Hz (yielding an average frequency of 350Hz). As the streamwise correlation distance (η) is increased, the cross-correlation peak has a time-lad and magnitude
decrease. The correlation distance η, defined in Figure 3.76, is the streamwise or spanwise
distance associated with the cross-correlation of any two given pressure taps. The speed of
the coherent motion is calculated by considering the time lag and the spatial separation this velocity is known as the convection velocity (Uc). The convection velocity is identified for a set of frequency ranges, each range identifies an eddy of a corresponding wavelength (Willmarth and Woolridge, 1962). Figures 3.77, 3.78, and 3.79 show a small
collection of band-filtered space-time correlations for RCDB Case 1, CD, and RCDB
Case 1. A more complete set of frequency ranges can be found in Appendix C.1.

101

Tap21

Tap22

Tap23

Tap24

Tap25

1

pp

0.5

R

0

-0.5

-1
-200

-100
0
100
Time Delay (t/40000)

200

Figure 3.75 Typical space-time correlation (200-500Hz)
Toward tip

Leading
edge

Trailing
edge

η
y
x
Toward hub
Figure 3.76 Controlled Diffusion surface tap geometry

Figure 3.77 shows the frequency band-filtered correlations of RCDB Case 1 for (a)
400-600Hz, (b) 800-1100Hz, (c) 1300-1700Hz, and (d) 1900-2300Hz. This yields eddy
wavelengths ( λ w ) following Equation 1.4 of 0.0155m, 0.0091m, 0.0065m, and 0.0043m
using average convection velocities inferred from the time-lag of the first three spatial correlations. The convection velocity is averaged spatially from x/c=0.858 to x/c=0.922. Simply, each plot represents the correlation of a given “structure” characterized by the eddy

102

wavelength λ w . It is seen that correlations are stronger at lower frequencies and deteriorate at higher frequencies as well as at high spatial separation between correlated taps. The
weak correlations at high frequency lead to a break-down of Uc calculations; the maximum correlation is no longer in the appropriate range of accepted time delays (note that
the typical delay for the correlation of Taps 21-25 is: 0.16 ≤ tu ∞ ⁄ c ≤ 0.18 ). Convection
velocity is most accurately calculated for lower frequencies and small tap separation.
U c ⁄ u ∞ varies as a function of frequency and has a typical range of 0.5 to 0.7.
Figure 3.78 shows a set of cross-correlations for the CD Case 1 geometry. As in
the RCDB Case 1 geometry, the correlations are generally stronger and more stable for
smaller tap separations. Convection velocities are higher for higher frequency bands and,
because correlations are much weaker over longer separations, the convection velocity is
identified as the spatial average over Taps 21-24 (x/c=0.858 to x/c=0.922). The associated
λ w for each plot is 0.0192m, 0.0113m, 0.0073m, and 0.0055m. The stationary airfoil
boundary layer’s coherent pressure fluctuations travel faster across the airfoil compared to
RCDB Case 1; this directly leads to higher eddy wavelengths for the same frequency
band. It is interesting to consider that the decay of the pressure fluctuation cross-correlation is a function of convection velocity in addition to the clear frequency dependence there is a similar trend in the decay of Rpp for similar λ w between the RCDB and CD Case
1. However, it is expected that the stationary airfoil boundary layer has a more stable
structure compared to the rotating geometry with similar flow conditions.
The cross-correlations of the off-design operating condition (RCDB Case 2,
Figure 3.79) show dramatically different results. From previous spectral information, it is
known that the wall pressure field is not consistent with the Case 1 flow field. It is clear

103

that the correlations are much weaker, particularly at high frequencies. The weak correlations make it much more difficult to accurately infer convection velocities and, subsequently, eddy wavelength.

104

a)

b)

400-600Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=7.7757(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

-1

0.25

d)

1300-1700Hz

1

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

0.25

1900-2300Hz

1

Rpp

Rpp

0

0.5

0.5
0

0

-0.5

-0.5
-1

Uc=8.6176(m/s)
∞

∞

c)

0

-0.5

-0.5
-1

800-1100Hz

1

Uc=9.7586(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

0.25

Tap21-22

Uc=8.9855(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

0.25

∞

∞

Tap21-21

-1

Tap21-23

Tap21-24

Tap21-25

Figure 3.77 Space-time correlations: RCDB Case 1

105

Uc/U

∞

a)

b)

400-600Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=9.6127(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

0

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

0.25

1900-2300Hz

1

0.5
Rpp

0.5
Rpp

Uc=10.7357(m/s)
∞

d)

1300-1700Hz

1

-1

0.25

∞

c)

0

0

-0.5

-0.5
-1

0

-0.5

-0.5
-1

800-1100Hz

1

Uc=11.3996(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

0.25

Tap21-22

Uc=11.6203(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

0.25

∞

∞

Tap21-21

-1

Tap21-23

Tap21-24

Tap21-25

Figure 3.78 Space-time correlations: CD Case 1

106

Uc/U

∞

a)

b)

400-600Hz

1

0.5
Rpp

Rpp

0.5
0

0

-0.5

-0.5
-1

800-1100Hz

1

Uc=8.6681(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

-1

0.25

Uc=7.954(m/s)
0

d)

1300-1700Hz

1

Rpp

Rpp

1900-2300Hz

1

0.5

0.5
0

0

-0.5

-0.5
-1

0.25

∞

∞

c)

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

Uc=7.9246(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

0.25

Tap21-22

Uc=11.0052(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u /c]

0.25

∞

∞

Tap21-21

-1

Tap21-23

Tap21-24

Tap21-25

Figure 3.79 Space-time correlations: RCDB Case 2

107

Uc/U

∞

1

U c/u ∞

0.8
0.6
0.4
RCDB: Case 1
CD: Case 1
RCDB: Case 2

0.2
0
0

20

40

60
80
ωc/u ∞

100

120

140

Figure 3.80 Coherent motion convection velocity
Figure 3.80 shows convection velocity as a function of non-dimensional frequency
for RCDB and CD Case 1 and RCDB Case 2. It is clear that Uc is a non-linear function of
frequency. Interestingly, a similar trend occurs between the CD and RCDB Case 1 geometries. Both show a convection velocity that is a similar function of frequency where the
CD Case 1 has a magnitude shift of approximately 0.05 < U c ⁄ u ∞ < 0.1 . The agreement
between the RCDB and CD Case 1 convection velocity is very notable - it further reinforces the similarities between the two geometries. As expected, due to difficulties accurately inferring convection velocity, RCDB Case 2 is significantly different from that of
Case 1 and the stationary airfoil.
The cross-correlation plots suggested a frequency and spatial dependent systematic
decay of the correlated coherent pressure field over the trailing edge region of the airfoil.
To quantify this, the cross-correlation is expressed as a function of the normalized eddy
wavelength. This follows the convention of Willmarth and Woolridge (1962); the rela-

108

tively short aft region can benefit from a similar manipulation that Willmarth gave to his
canonical flat plate boundary layer. Willmarth’s results are reproduced in Figure 3.81. His
flat plate boundary layer is characterized by an exponential decay of 10

– 0.3

. The decay

characterization of the canonical flat plate boundary layer is adapted as a distinguishing
parameter for the three cases explored in this document. It is important to note that the
pressure patterns do not “decay”, rather, they evolve to a pattern that is incoherent after
some distance downstream.
Figure 3.82 shows the collapse of RCDB Case 1 cross-correlation data. The decay
is characterized by a 10

– 0.7

exponential decay, where the pressure pattern evolves to a

pattern which is not correlatable after nominally four wavelengths. CD Case 1
(Figure 3.83) shows a similar decay pattern - characterized by a rate of 10

– 0.6

. The slower

decay of the CD airfoil boundary layer is an expression of the improved stability of the
stationary over the rotating airfoil. Figure 3.84 shows a very steep decay 10

– 1.8

for

RCDB Case 2. RCDB Case 2 sees a decay of the cross-correlation coherent pressure field
after 1-2 wavelengths.

109

10

– 0.3

Figure 3.81 Willmarth and Woolridge (1962) decay

1

Rpp
Rpp

– 0.7 ( x ⁄ λ )

10 !0.7#x/6ww
10

0.75
Rpp

Rpp 0.5
0.25
0
0

0.5

1

1.5

2
x/λw
x/6w

2.5

3

Figure 3.82 RCDB: Case 1 decay

110

3.5

4

1

R pp
Rpp
– 0.6 ( x ⁄ λ )

10 !0.6#x/6ww
10

0.75
Rpp

Rpp 0.5
0.25
0
0

0.5

1

1.5

2
x/λw
x/6w

2.5

3

3.5

4

Figure 3.83 CD Case 1 decay

1

Rpp
Rpp
– 1.8 ( x ⁄ λ )

10
10 !1.8#x/6ww

0.75
Rpp

Rpp 0.5
0.25
0
0

0.5

1

1.5

2
x/λw
x/6w

2.5

3

3.5

4

Figure 3.84 RCDB: Case 2 decay

1

– 0.7 ( x ⁄ λ )

RCDB 1: 10 !0.7#x/6ww
10
– 0.6 ( x ⁄ λ )

CD 1: 10 !0.6#x/6w w
10

0.75

– 1.8 ( x ⁄ λ )

w
RCDB 2: 10 !1.8#x/6w
10

Rpp

Rpp 0.5

– 0.3 ( x ⁄ λ )

Willmarth (1962): 10 !0.3#x/6ww
10

0.25
0
0

1

2

3
x/λw
x/6w

4

Figure 3.85 Decay comparison

111

5

6

Figure 3.85 shows a comparison of the pressure pattern decay rates between Willmarth’s results and the present experiment. It is seen that the RCDB Case 1 and CD Case 1
cross-correlation decay rates are very similar; both wall pressure patterns decay to something incoherent after nominally four wavelengths. The cross-correlation of RCDB Case 2
decays much quicker so that coherent pressure patterns cannot be tracked beyond two
wavelengths. The contrasting decay rate of Case 2 distinguishes Case 1 rotting and stationary wall pressure fields as comparatively similar. Additionally, the contrast of the
present experimental cross-correlation decay rates to the canonical flat plate boundary
layer further reinforces the similarities between RCDB Case 1 and the CD blade as a
bounding condition. This is an excellent verification of the stationary-rotating analog.
3.5 Integral Length Scales
Figure 3.86 shows the longitudal (streamwise) cross-correlation (R11) as a function
of correlation length, calculated by considering all permutations of tap-to-tap cross-correlations in the trailing edge region. At zero time delay, the correlation R11 is purely a function of space; high correlation suggests a wall pressure field that have correlated pressure
patterns between taps separated by length η. The integral length scale (Λ1) is defined as
the area under the R11 curve (Equation 1.10); Λ1 is a length scale that identifies a characteristic length to describe the “largeness” of the pressure pattern. The longitudal integral
length scale is a useful length scale in the prediction of trailing edge aeroacoustic noise
radiation (Blake 1986). It can be seen in Figure 3.86 that there is a slight positive-negative
asymmetry, arguing that the cross-correlation is dependent on the direction of the calculation (positive: fore-to-aft or negative: aft-to-fore). The fore-to-aft length scale is 2.949mm,
1.041mm, and 6.612mm for RCDB Case 1, CD Case 1, and RCDB Case 2, respectively.

112

The aft-to-fore length scale is 3.155mm, 1.074mm and 8.897mm for RCDB Case 1, CD
Case 1, and RCDB Case 2, respectively. There is little asymmetry in RCDB and CD Case
1 where RCDB Case 2 is heavily biased in the negative direction. This implies that the
boundary layer is uniform in the streamwise direction for RCDB and CD Case 1 and nonuniform for RCDB Case 2.

1

RCDB: Case 1
CD: Case 1
RCDB: Case 2

0.8

R11

0.6
0.4
0.2
0
-0.2
-0.15

-0.1

-0.05

0
η/c

0.05

0.1

0.15

Figure 3.86 Longitudal integral length scale
Note that the connecting lines do not imply trends. The lines exist purely
to identify and connect data points from adjacent measurement locations.

113

1

RCDB: Case 1
CD: Case 1
RCDB: Case 2

0.8

R22

0.6
0.4
0.2
0
-0.2
-0.15

-0.1

-0.05

0

2/

0.05

0.1

0.15

Figure 3.87 Transverse integral length scale
Note that the connecting lines do not imply trends. The lines exist purely to
identify and connect data points from adjacent measurement locations.
Figure 3.87 shows the transverse (spanwise) cross-correlation (R22) as a function
of correlation length, calculated by considering all permutations of tap-to-tap cross-corrections at zero time delay of the trailing edge spanwise taps. The transverse integral
length scale (Λ2) is defined as the area under the R22 curve. The transverse integral length
scale is another useful length scale in the prediction of trailing edge aeroacoustic noise
radiation. In addition to the streamwise length scale, Λ2 identifies a length scale associated
with the size of the pressure pattern. Again, it can be seen in Figure 3.87 that there is a
slight positive-negative asymmetry; the cross-correlation is dependent on the direction of
the calculation (positive: hub-to-tip or negative: tip-to-hub). The hub-to-tip length scale is
3.920mm, 2.865mm, and 8.631mm for RCDB Case 1, CD Case 1, and RCDB Case 2,
respectively. The tip-to-hub length scale is 4.202mm, 2.713mm and 7.689mm for RCDB
Case 1, CD Case 1, and RCDB Case 2 respectively. There is little asymmetry in CD Case

114

1, suggesting that the boundary is uniform in the spanwise direction. There is mild asymmetry in RCDB Case 1 and Case 2, implying that the boundary layer is non-uniform in the
streamwise direction; this is not surprising considering the rotational effects on the flow
field.
3.6 Spectral Processing: Coherence and Phase
In addition to the cross-correlation, coherent structures and correlations between
pressure taps can be quantified using a spectral coherence function (Equation 1.18). The
spectral coherence function represents a statistical value between zero and one - expressing the relationship between two known signals as a function of frequency. This operation
produces similar conclusions as the cross-correlation method, but does so with better frequency resolution. The use of the Fast Fourier Transform (FFT) allows calculations to be
made in the frequency domain rather than the time domain. Frequency domain calculations are typically less computationally intensive when calculating spectral information
and do not rely on filtering algorithms (often poorly tuned) for band-passed frequency
results.
Figure 3.88 shows the coherence functions of the trailing-edge pressure taps for
the three flow conditions and geometries: (a) RCDB Case 1, (b) RCDB Case 2, (c) CD
Case 1. There is generally a higher coherence at lower frequencies; it has already been
established that low frequencies propagate farther downstream than high frequencies since
the pressure patterns decay spatially as a function of their eddy wavelength. It is seen that
correlation is typically highest for adjacent pressure taps and decreases as spatial separation increases; this is a direct result of the evolution of the pressure signal as it propagates
downstream. Pressure patterns are created and destroyed across the surface of the airfoil;

115

but, they are coherent, or trackable, for roughly four wavelengths before they grow or
decay into other patterns. RCDB Case 1 shows a plateau from roughly
10 < ( ωc ) ⁄ u ∞ < 40 . This suggests a coherent motion with a strong mid-frequency component; evidence of the plateau is seen, but decays, as spatial separation increases. The CD
blade shows a very long coherence in the frequency domain, suggesting that there is a
coherent high-frequency pressure signal. The CD blade and RCDB Case 1 coherences are
distinguished from RCDB Case 2 by their notably similar spectral decay rates - quantified
in the previous section. RCDB Case 2 shows a sharp high coherence peak at low frequencies. This suggests a wall pressure field with prominent low-frequency structures. This
supported by the spectral plots that show a shallower decay of high frequency
( 100 < ( ωc ) ⁄ u ∞ ) fluctuations compared to RCBD Case 1.
Phase delay is a measure of the delay between sinusoidal components in a random
signal as a function of the frequency of the sinusoidal components (Equation 1.19). The
phase delay is a useful expression for quantifying the effective frequency resolution of the
correlations between pressure taps. Consider Figure 3.89, RCDB Case 1, where the phase
of Taps 21-22 is well resolved past ( ωc ) ⁄ u ∞ = 200 but Taps 21-25, where η is much
larger, the phase factor is only clear to ( ωc ) ⁄ u ∞ = 90 . This is another way to identify the
decay in the pressure pattern after some spatial separation.
Generally, the phase of the flow conditions match well with the conclusions the
coherence function yielded. Low frequency content is better defined, as are low spatial
separation conditions. RCDB and CD Case 1 (Figure 3.89 and Figure 3.90) show similar
phase curves - further reinforcing the similarities between the rotating and stationary
geometries. RCDB Case 2 (Figure 3.91) has a much poorer frequency resolution; phase

116

information for Figure 3.91d Taps 21-25 is well resolved to ωc ⁄ u ∞ ≈ 40 whereas RCDB
Case 1 (Figure 3.89d) and the CD blade (Figure 3.90d) resolve phase to ωc ⁄ u ∞ ≈ 100 .
This matches the results from the coherence function, indicating a stronger coherent highfrequency component of the pressure signal for the CD blade and RCDB Case 1.

RCDB: Case 1

1

0

20

40

60

80

100

120

140

160

180

200

Taps21-22
Taps21-23
Taps21-24
Taps21-25

0.75
0.5

a)

0.25

RCDB: Case 2

0
1

b)

0.75
0.5

0.25
0
1

c)

CD: Case 1

0.75
0.5

0.25
0
0

20

40

60

80

100
ωc/u∞

120

140

Figure 3.88 Coherence functions

117

160

180

200

5

0

20

40

60

80

100

120

140

160

180

200

a) Taps 21-22

0

RCDB: Case 1 (rad)

-5
5

b) Taps 21-23

0
-5
5

c) Taps 21-24

0
-5
5

d) Taps 21-25

0
-5
0

20

40

60

80

100 120
ωc/u∞

140

160

Figure 3.89 Phase RCDB Case 1

118

180

200

5

0

20

40

60

80

100

120

140

160

180

200

a) Taps 21-22

0

CD: Case 1 (rad)

-5
5

b) Taps 21-23

0
-5
5

c) Taps 21-24

0
-5
5

d) Taps 21-25

0
-5
0

20

40

60

80

100 120
ωc/u∞

140

160

Figure 3.90 Phase CD blade

119

180

200

5

0

20

40

60

80

100

120

140

160

180

200

a) Taps 21-22

0

RCDB: Case 2 (rad)

-5
5

b) Taps 21-23

0
-5
5

c) Taps 21-24

0
-5
5

d) Taps 21-25

0
-5
0

20

40

60

80

100 120
ωc/u∞

140

160

Figure 3.91 Phase RCDB Case 2

120

180

200

Convection velocity can be inferred using the time delay associated with the phase
lag of the pressure signals. Linear regions of the phase plot suggest a constant convection
velocity over that frequency range. Considering RCBD and CD Case 1, there is a general
linear trend, but locally there is significant variation.
Figure 3.92 expresses the convection velocity calculated from the cross-correlation
and phase methods for RCDB and CD Case 1. Following Moreau and Roger (2005), the
convection velocity is calculated from the phase by:
2πΔfη
U c = --------------Δφ

3.2

where Δf is the width of the frequency bin of the Fourier transform, η is the separation
between taps and Δφ is the phase change for a particular frequency bin size. As noted previously, similar trends exist between the RCDB and CD airfoils with a moderate magnitude offset. Additionally, there is a strong similarity between the two methods of
calculating convection velocity, particularly with the RCDB. Differences arise from the
inherent averaging process in the cross-correlation method (each point represents the center of the band-pass limits - the average frequency). While both methods represent an
“average” of the convection velocity, the cross-correlation method averages content for a
large frequency range, on the order of 200Hz, whereas the phase calculations average frequency on order of 10Hz. This leads to an unintentional smoothing of the cross-correlation
convection velocity. The phase method quantifies transients better and is able to quantify
higher frequencies more accurately.

121

1

0.8

c

U /u

∞

0.6

0.4
RCDB Case 1: Uc, phase
RCDB Case 1: Uc, xcorr

0.2

CD Case 1: Uc, phase
CD Case 1: Uc, xcorr

0
0

20

40

60

80 100
ωc/u∞

120

140

160

180

Figure 3.92 Convection velocity comparison CD - RCDB Case 1
Note that the connecting lines do not imply trends. The lines exist
purely to identify and connect data points from adjacent measurement
Convection velocity is often a necessary parameter when predicting aeroacoustic
noise production. In general, Uc is taken to be a constant value across all frequencies. It is
clear from the RCDB and CD airfoils that it is more accurate to describe Uc as a non-linear
function of frequency.
3.7 Hot-wire measurements
Trailing edge hot-wire measurements, acquired using the “flying” hot-wire probe,
quantify the velocity magnitudes that describe the RCDB wake. The shape and the locations of the defect and shear layer regions supplement the fluctuating surface pressure data
by providing velocity data that are used to infer boundary layer shape and size. Wake measurements near the trailing edge are representative of the boundary layer on the blade just

122

before the flow separates off the trailing edge1. A schematic of the hot-wire survey and
coordinates is shown in Figure 3.93.
Figure 3.94 shows the velocity magnitude and RMS from the hot-wire wake survey of RCDB Case 1. Note that R=0 is the location of the identifiable airfoil trailing edge
and the center of the wake moves to higher R values as measurements are acquired in
lower z planes - this is a result of non-similar coordinate systems of the hot-wire traverse
and airfoil exit velocity. It can be seen that there is a relatively thin wake; at z=1mm the
width of the wake is nominally 0.6R/c. The center of the shear layer is identified from the
maximum RMS; z=1mm, R/c=0.38; z=3mm, R/c=0.058; z=8mm, R/c=0.088. The width
of the wake grows downstream of the airfoil. Velocities higher than the free stream velocity, seen at high and low R, are a direct effect of the fluid accelerating over the airfoil surface, a more extensive survey would yield velocities nearer to the free-stream.

R=rθ
z

z1=1mm

z1,rms max

z2=3mm

z2,rms max

z3,rms max

z3=8mm

Figure 3.93 Schematic of hot-wire measurements

1. Note that the quantity of hot-wire wake data is limited to three z-rθ surveys. The
flying hot-wire survey is time intensive and requires the facility to be shut down
and restarted between each discrete location.
123

Figure 3.95 shows the velocity magnitude and RMS of the RCDB Case 2 wake. It
is immediately obvious that there are significant differences between the Case 1 and Case
2 flow conditions. The lack of velocity recovery over the measured regimes suggests that
the wake is very thick. A thick wake implies a thick boundary layer at the trailing edge of
the airfoil. The thick boundary layer validates and explains some observations from the
wall pressure fluctuations. It was previously mentioned that the shape of the Cp curve,
suction-side statistics, spectral decay, and correlations were uncharacteristic of the CD
blade. The hot-wire measurement provides validation to this observation.

1.2
1

z = 1mm, (U/u∞ )
z = 3mm, (U/u∞ )

0.8

z = 8mm, (U/u∞ )

0.6

z = 1mm, ( u2 /u∞ )

0.4

z = 3mm, ( u2 /u∞ )

0.2

z = 8mm, ( u2 /u∞ )

0
0

0.05

0.1

R/c

0.15

0.2

0.25

Figure 3.94 Wake Statistics RCDB: Case 1 (R=rθ)
Note: the line connecting the discrete measurements do not imply
intermediate values

124

1.2
1

z = 1mm, (U/u∞ )
z = 3mm, (U/u∞ )

0.8

z = 8mm, (U/u∞ )

0.6

z = 1mm, ( u2 /u∞ )
z = 3mm, ( u2 /u∞ )

0.4

z = 8mm, ( u2 /u∞ )

0.2
0
0

0.05

0.1

R/c

0.15

0.2

0.25

Figure 3.95 Wake Statistics RCDB: Case 2 (R=rθ)
Note: the line connecting the discrete measurements do not
imply intermediate values

It is interesting to compare the pressure spectra and velocity spectra of the trailing
edge pressure fluctuations and the wake. Figure 3.96 and Figure 3.97 show the spectra for
RCDB Case 1 and Case 2 respectively. Note that the velocity spectra were selected based
on the location of the maximum RMS, or the center of the sheared region on the suction
side. Note the similarities between the pressure and velocity spectra. Most interesting is
the spatial homogeneity of each spectrum. The streamwise pressure spectra near the trailing edge collapse on a common curve and the velocity spectra collapse in the sheared
region. Curiously, this spatial homogeneity holds for both RCDB Case 1 and RCDB Case
2. CD data were unavailable to perform a similar comparison.

125

Φpp (dB re [1/2×ρ×u 2 ]2) or Φuu (dB re [u ∞]2)
∞

80

60

40

Φuu(z1,rmsmax)
Φuu(z2,rmsmax)

20

0

Φuu(z3,rmsmax)
Φpp(Tap24)
Φpp(Tap25)
1
101
10

0
100
10

ωc/u ∞

2
102
10

Φpp (dB re [1/2×ρ×u 2 ]2) or Φuu (dB re [u ∞]2)
∞

Figure 3.96 Power spectra (u,p) RCDB: Case 1

80

60

40

Φuu(z1,rmsmax)
Φuu(z2,rmsmax)

20

0

Φuu(z3,rmsmax)
Φpp(Tap24)
Φpp(Tap25)

100
10
0

101
10

2
2

1

ωc/u ∞

10
10

Figure 3.97 Power spectra (u,p) RCDB: Case 2

126

0.15
Rpu(z1,rmsmax)
Rpu(z2,rmsmax)

0.1

Rpu(z3,rmsmax)

Rpu

0.05
0

-0.05
-0.1
-0.15
-500 -400 -300 -200 -100 0 100 200 300 400 500
Time shifts (τ)

Figure 3.98 Pressure-velocity cross-correlation RCDB: Case 1
Additionally, a preliminary investigation of pressure-velocity cross-correlations
was explored (Figure 3.98). Using a similar technique as Section 3.4, the velocity fluctuations of the wake (center of sheared region where RMS is a maximum) are correlated to
the trailing edge fluctuating pressure measurements (Tap 25 x/c=0.978). There is a clear
time lag between the trailing edge pressure signal and the wake fluctuations; the strength
of the correlation is not high, but it is clearly present. Note that Figure 3.98 was calculated
over the entire frequency band; filtered frequency bands will yield stronger correlations
for a given set of frequencies.
Table 3.1 presents the results from the pressure-velocity cross-correlation. The
time delay (τmeasured) is measured from the delay in the cross-correlation. A correlation
length scale (ηcalculated) is calculated using the hot-wire velocity as Uc following
Equation 1.2. The distance between the pressure tap and the correlated hot-wire
(ηmeasured) is well resolved from the hot-wire traverse. A time-shift (τcalculated) corre127

sponding to the estimated time delay between the pressure tap and the hot-wire probe is
calculated following Equation 1.2 from the known spatial separation and the hot-wire
velocity (Uc). The similarity between the estimated and measured time-shifts and spatial
separations suggests that the correlation is well-resolved and physical. More detailed measurements would be needed to perform similar analysis as in Section 3.4 and Section 3.6

Table 3.1 Cross-Correlation Results
τmeasured

ηinfered(mm)

ηmeasured(mm)

τinfered

Rpu(z1,rmsmax)

39

9.1

8.6

38

Rpu(z2,rmsmax)

45

11.3

10.85

43.5

Rpu(z3,rmsmax)

68

18.2

17.1

65

128

4.0 Summary and Conclusions
Measurements were made on a rotating analog (RCDB) of a stationary Controlled
Diffusion airfoil. The airfoil shape was reproduced for the rotating blade and the blade’s
twist (hub to tip) provided a constant angle of attack over the blade span. A method for
identifying fluctuating pressures on the surface of the blade was developed using the prior
work of Pérennès & Roger (1998) to identify characteristics of the flow structure of the
boundary layer along various streamwise and spanwise locations. The development of the
the Remote Microphone Probe (RMP) measurements have allowed the similarities and
differences between the CD airfoil flow fields to be identified.
The following conclusions are supported by the results of this study. The conclusions are presented in five categories:
1) The characterization of regions along the RCDB and CD blade surface for an
incident velocity of 16m/s, a geometric angle of attack of 8 ° :
a) The leading edge region ( 0 < x ⁄ c < 0.15 ) is identified by a boundary layer
transition from a laminar to a turbulent state. There is a strong spatial inhomogeneity in the leading edge region due to the boundary layer evolution.
These inhomogeneites are expressed by dissimilarities in the leading edge
statistics (Section 3.2) and large variations between spectral magnitudes
and spectral decay rates (Section 3.3) of the wall pressure field in the
streamwise survey.
b) The mid-chord region ( 0.15 < x ⁄ c < 0.75 ) is identified by adverse pressure
gradient boundary layer growth and increasing spatial homogeneity. Statis-

129

tical moments and spectral magnitudes and decay show modest uniformity
in the streamwise direction (Section 3.2 and Section 3.3).
c) The trailing edge region ( 0.75 < x ⁄ c < 1 ) is characterized by a spatially
homogenous wall pressure field - similar to a well developed flat plate
boundary layer with a mild adverse pressure gradient condition. Space-time
correlations and spectral analyses identify a convection velocity, as
·
0.4 < U c < 0.65 for the RCDB and

·
0.5 < U c < 0.8 for the CD blade

(Section 3.4 and Section 3.6). The convection velocities are a nonlinear
function of the frequency of the fluctuations.
d) The trailing edge spanwise measurements ( x ⁄ c = 0.98, 0.45 < y ⁄ l < 0.58 )
reflect the spanwise uniformity of the boundary layer. Statistical and spectral results demonstrate that the radial variations of the RCDB boundary
layer are minimal (Section 3.2).
e) The pressure side wall pressure field of the RCDB is characterized by uniform pressure statistics and a spectral decay that corresponds to a viscous
dominated boundary layer (Section 3.2 and Section 3.3).
2) The processed results for the fluctuating pressures of the rotating analog
(RCDB) of the Controlled Diffusion blade are found to be in good agreement
with the results for the stationary Controlled Diffusion Blade. This conclusion
was quantified through a comparison of the mean and fluctuating pressure
evaluations (The agreement of the mean values was observed by Neal 2010).
Similarities in the following measurements argue for similarities in the wall
pressure fluctuations and, as an extension, the boundary layer:

130

a) The profile of the pressure coefficient (Cp) is well matched for the suction
side blade loading. There is a slight magnitude shift for the pressure side
(Section 3.1).
b) Fluctuating pressure statistics quantify the agreement of the wall pressures
at corresponding tap locations following reattachment of the boundary
layer (Section 3.2)
c) The magnitude and decay rates of the pressure spectra for corresponding
regions are also well matched (Section 3.3). Space-time correlations and
spectral processing of trailing edge fluctuating pressures show similarities
in the spatial decay rates and coherence functions, the magnitude of the
boundary layer convection velocity is higher for the CD blade but follows
a similar trend to that for the RCDB. Pressure patterns decay to zero correlation after ~4 wavelengths (Section 3.4 and Section 3.6)
3) The trailing edge hot-wire wake survey in the present study confirms wake measurements made of the RCDB and CD blade by Neal (2010); the wake profile
of the RCDB is very thin, with a notable shedding phenomenon around
f=1800Hz. The spectra of the wake survey where the highest velocity RMS
values occurred collapsed onto a single curve for each plane surveyed. Correlation between pressure and velocity, although weak, showed agreement
between calculated and measured time lags τ and correlation distances η
(Section 3.7).
4) The CD/RCDB experiment, when outfitted with unsteady surface pressure sensors, is capable of measuring and calculating pressure spectra and length scales

131

- streamwise and spanwise - along the airfoil trailing edge. These measurements are designed to provide better predictive capabilities for acoustic noise
models as discussed in Section 1.3.
5) An off-design condition (RCDB Case 2), corresponding to an incident velocity
of 16m/s and a geometric angle of attack of 15 ° , was investigated. Principal
results relative to RCDB Case 1 (design point) are:
a) A dissimilar pressure coefficient curve between Case 1 and Case 2
(Section 3.1) - the blade in Case 2 is more highly loaded. The statistical
moments have higher magnitudes at all x/c locations (Section 3.2); there
are more intense and random pressure fluctuations along the surface of the
RCDB Case 2 blade.
b) RCDB Case 2 has higher spectral power for low frequencies and decays, as
a function of frequency, at a much slower rate (Section 3.3).
c) The coherent motions of the pressure field spatially decay more rapidly in
RCDB Case 2. Correlated pressure patterns decay to zero correlation after
approximately 2.5 wavelengths as compared with a decay of approximately
4 wavelengths shown by CD and RCDB Case 1 flow fields. Similarly, the
maximum frequency of correlated pressure patterns is lower (Section 3.4
and Section 3.6).
d) Hot-wire measurements indicate a much wider wake for the off-design
flow field. This suggests that there is a very thick boundary layer at the separation lip of the blade.(Section 3.7).

132

APPENDICES

133

Appendix A: Remote Microphone Probe Development
This appendix will detail the development and implementation of the RMP device.
A.1 RMP Motivation
In order to measure pressure fluctuations with a boundary layer, microphones are
typically flush-mounted to the surface of some bounding geometry. Flush-mounting
requires the size of the microphone sensor to be sufficiently small but with now noise and
high sensitivity. This places limitations on sensor type; conventional condenser-type
microphones are too large. Small capacitive and MEMS-Bases microphones have become
popular in recent years for their size and improved quality. Considering the geometry of
the present experiment, a thin heavily cambered airfoil, it is very difficult to instrument the
surface with adequate resolution and accurate sensors. In order to make the necessary
measurements, it imperative that the microphones are located at some “remote” distance
from the tap itself.
Remote mounting the microphones traditionally embedded in the surface satisfies
the need for high spatial resolution measurements for a thin, heavily cambered airfoil. The
remote microphone probe (RMP) infers the fluctuating surface pressures by measuring the
local pressure fluctuations within a small channel, excited by pressure fluctuations at the
start of the channel: the airfoil surface. The associated challenges to fabricate, install and
calibrate the RMP system are presented in the following sections.
A.2 Development
Pioneering measurements from (Willmarth and Woolridge 1962) demonstrations
the possibilities of using fast-response pressure transducers for characterizing a boundary
layer. Additional work of Sheplak et al. (2001), Moreau and Roger (2005), Gerakopulos

134

and Yarusevych (2012), among others extended the use of microphone transducers for the
use on an airfoil surface. Each of these modern experiments required a method to characterize frequency and magnitude response of the array of test microphones used within the
experiment. Due to the rotating nature of the present experiment, an evolved microphone
technique was needed to accurately and precisely, measure the fluctuating pressures along
the airfoil surface.
A.2.1 Measurements/calibrations/previous work
Following the work of Sheplak et al. (2001), Moreau and Roger (2005), Boerrigter
and Charbonnier (1997), a method to identify frequency response calibrate the microphone sensors was developed. A plane wave tube (PWT) was fabricated following AES
standard AED-1ID-1991, ASTM standard E1050-10, and suggestions from the work of
Magalotti et al. (1999) and Anderson (2003).
A source driver is located at one end of the PWT while the other is left open to the
lab environment. The PWT, with a uniform cross section across the length of the duct,
establishes a one-dimensional sound field within the tube when excited by the source
driver. Under steady state conditions, the one-dimensional sound field condition is satisfied and the acoustic pressure is uniform over a given cross section of the duct. Microphones placed along the wall of the duct at the same cross section will measure identical
acoustic pressures; the microphones measure the field at the same nodal line.
The PWT apparatus in the present experiment is a 12.7mm by 12.7mm square tube
1.5m long. Microphones are located on the same cross section 0.6m from the duct
entrance. Following ASTM E1050-10, it is suggested that the length of the tube be sufficiently long enough to prevent non-plane waves. Non-plane waves typically subside by

135

three tube diameters; as such, it is suggested to place microphones at distances greater
than three diameters. A schematic of the PWT is shown in Figure A.2.1.
The working frequency range of the PWT is limited by a lower and upper limit;
identified by geometric limits. It is recommended that the microphone spacing be greater
than one percent of the wavelength of the lowest frequency. In reality the lowest resolvable frequency is determined by the driving speaker (approximately 200Hz). The high frequency limit is generally determined by the physical dimensions of the device following
Equation A.1.
d < 0.5c ⁄ f

A.1

where fu is the upper frequency limit in hertz, c is the speed of sound and l is the largest
sectional dimension of the tube. The high frequency limit for the present PWT corresponds to approximately 13.5kHz.
The critical aspect of the PWT is that at a given cross section, or nodal line, the
pressure signal is uniform across all surfaces. This idea is fundamental for the use of the
PWT as a calibration device. A broadband speaker (Dayton PA130-8) is driven by a highquality reference amplifier (Behringer A500) to excite the PWT. A broad-spectrum whitenoise signal provides sufficient spectral content to infer microphone sensitivity functions
from approximately 50Hz to 20kHz.

Figure A.2.1 PWT schematic and specifications

136

A.2.2 Signal processing/Calibrations
Each RMP device is identified as a simple single-input/single-output model where
the blade is modeled as a single-input/single-output model with 21 parallel transmissions.
Consider the RMP system (Figure A.2.2), where x(t) and y(t) are the input and output of
the model. y(t) is the fluctuating pressure measured remotely by the RMP microphone and
x(t) is the fluctuating pressure at the surface. The attenuation of the transmitted signal
associated with the tubing, and electrical responses is represented by the system response
function, or, transfer function Hxy(f). Where Hxy(f) is the parameter that defines the relationship between the pressure measured at the remote microphone and the pressure measured at the airfoil surface.
The recommended method for calculating Hxy(f), following Bendat and Peirsol
(1986) is:
G xy ( f )
– jφ ( f )
H xy ( f ) = --------------- = H xy ( f ) e
G xx ( f )

A.2

where Gxx and Gxy are the auto and cross-spectral densities. The magnitude factor and
phase factor can be estimated by
2

2

1⁄2

[ C xy ( f ) + Q xy ( f ) ]
H xy ( f ) = --------------------------------------------------G xx ( f )

A.3

φ ( f ) = atan ( C xy ( f ) ⁄ Q xy ( f ) )

A.4

where Cxy(f) and Qxy(f) are the real and imaginary parts of the cross-spectrum Gxy(f)

137

.

x(t)

Hxy(f)

y(t)

Figure A.2.2 Linear input-output system
a)

Gain

2

1

0
0

phase (rad)

b)

5000
10000
Frequency (Hz)

15000

5000
10000
Frequency (Hz)

15000

4
2
0
-2
-4
0

Figure A.2.3 Larson Davis transfer function
The amplitude and phase response of the system are analogous to the ‘transfer
function’ referred to in the engineering community. A sample function generated from a
calibration is seen in Figure A.2.3. Figure A.2.3 shows the gain and phase factor for two
phase and gain matched reference microphones in the PWT. A gain of 1 and phase of 0
indicates the microphones are paired. Note the breakdown of the calibration around
13.5kHz; a representation of the high frequency limit of the device.

138

As mentioned in the main text, the PWT was not a suitable device to perform the
calibration of the RMP devices. An open-air in situ method was developed to allow for
more complex geometries. To verify the accuracy of the method, a rig (FIGURE) was
developed using known, phase and magnitude matched reference microphones. The reference microphones are held at some distance apart and traversed until their faces are touching at the surface. A calibration routine was performed at each height. Results of the test
are shown in Figure A.2.5.

Figure A.2.4 in situ verification rig
It is clear from Figure A.2.5 that the in situ method is best suited for small microphone separations. Small separations better match wavetube gain and phase response (see
Figure A.2.3. Note that there is much higher frequency resolution with the in situ method.
This is due to a cut-off frequency that is on order or tap separation (fc>20kHz). Implementation of the RMP device is discussed in the main text (Section 2.2.3)

139

a)

5

2.0 in
1.4 in
0.9 in
0.6 in
0.4 in
0.25 in
0.1 in

4
3
2
Gain

1
0
-1
-2
-3
-4
-5
0

b)

3000

6000

9000
12000
Frequency (Hz)

15000

18000

3

2.0 in
1.4 in
0.9 in
0.6 in
0.4 in
0.25 in
0.1 in

2

Phase (rad)

1
0
-1
-2
-3
0

3000

6000

9000

Frequency (Hz)

12000

15000

Figure A.2.5 in situ verification

140

18000

A.3 Technical Data Sheets

Figure A.3.6 RMP microphone specifications (Knowles FG-23329-C05)

141

Figure A.3.7 Larson Davis manufacturer specifications

142

Appendix B: Optimal Noise Cancellation
Correlated tonal noise between the ground-level ambient microphones and RMPs
is attenuated using the Optimal Noise Cancellation method suggested and provided by
Professor Ahmed Naguib from Michigan State University. “Noise” is identified and
removed from the RMPs by considering a correlation between a test microphone (RMP)
and a reference microphone (Larson-Davis). High correlations correspond to a pressure
signal that is shared by both microphones. Since the two microphones are located in stochastically unique fields, any correlated pressure must correspond to a global acoustic
noise source; the signal is attenuated wherever correlations are high (Naguib et al. 1996)
Figure B.1. It is expected that if the correlation between two statistically separate microphones is high, there exists a common producer that both microphones measure - noise.

Figure B.1 Sample ONC filter applied to preliminary CD data

143

Appendix C: Additional Figures
C.1 Cross-correlations
50-250Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=10.5334(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

b) -1

250-450Hz

1

Rpp

Rpp

-0.5

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

350-550Hz

0

-0.5
Uc=9.9812(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

d) -1

450-650Hz

1

Uc=9.6222(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

550-750Hz

1

0.5

0.5
Rpp

Rpp

0

0.5

0

0

-0.5

e) -1

Uc=10.3465(m/s)

1

0.5

c) -1

0

-0.5

-0.5

a) -1

150-350Hz

1

0

-0.5
Uc=9.6928(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-21

Tap21-22

0.25

f) -1

Tap21-23

Uc=9.9837(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-24

Tap21-25

Figure C.1.1 Band-filtered Rpp correlations (CD blade)

144

0.25

Uc/U

∞

650-850Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=10.3745(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

950-1150Hz

0.5
Rpp

Rpp

Uc=10.8716(m/s)

1

0.5
0

0

-0.5

-0.5
Uc=10.7408(m/s)
0

1

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

Uc=10.7357(m/s)
0

1

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

1150-1350Hz

0.5
Rpp

0

0

-0.5

-0.5

e) -1 0

d) -1

1050-1250Hz

0.5
Rpp

b)-1

850-1050Hz

1

c) -1

0

-0.5

-0.5

a)-1

750-950Hz

1

Uc=10.8803(m/s)
0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-21

Tap21-22

0.25

f)-1 0

Tap21-23

Uc=11.0329(m/s)
0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-24

Tap21-25

Figure C.1.2 Band-filtered Rpp correlations (CD blade)

145

0.25

Uc/U

∞

1250-1450Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=11.1837(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

Rpp

Rpp

0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

1550-1750Hz

0

-0.5
Uc=11.4435(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

d) -1

1650-1850Hz

1

Uc=11.4839(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

1750-1950Hz

1

0.5

0.5
Rpp

Rpp

0

0.5

-0.5

0

-0.5

e)

Uc=11.3135(m/s)

1

0.5

-1

b) -1

1450-1650Hz

1

c) -1

0

-0.5

-0.5

a)-1

1350-1550Hz

1

0

-0.5
Uc=11.4839(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-21

Tap21-22

0.25

f) -1

Tap21-23

Uc=11.5844(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-24

Tap21-25

Figure C.1.3 Band-filtered Rpp correlations (CD blade)

146

0.25

Uc/U

∞

1850-2050Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=11.6203(m/s)
0

1

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
2050-2250Hz

0.25

b) -1

Rpp

Rpp

0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

2150-2350Hz

0.5

0

-0.5

0

-0.5
Uc=11.5415(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25 d)

-1

Uc=11.657(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

2250-2450Hz

1
0.5
Rpp

c)

Uc=11.555(m/s)

1

0.5

-1

0

-0.5

-0.5

a)-1

1950-2150Hz

1

0

-0.5

e)

Tap21-21

-1

Uc=11.7245(m/s)
0

Tap21-22

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-23

Tap21-24

0.25

Tap21-25

Figure C.1.4 Band-filtered Rpp correlations (CD blade)

147

Uc/U

∞

50-250Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=9.0024(m/s)
0

1

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
250-450Hz

0.25

Rpp

Rpp

0

Uc=8.5235(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

450-650Hz

0.25

350-550Hz

Uc=7.8844(m/s)
0

1

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
550-750Hz

0.25

0.5
Rpp

Rpp

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0

-1

0.25 d)

0.5
0

0

-0.5

-0.5
-1

0

-0.5

1

e)

Uc=9.1869(m/s)

0.5

-0.5

c)

b) -1

1

0.5

-1

0

-0.5

-0.5

a)-1

150-350Hz

1

Uc=7.8245(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-21

Tap21-22

0.25

f)

-1

Tap21-23

Uc=8.0228(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-24

Tap21-25

Figure C.1.5 Band-filtered Rpp correlations (RCDB Case 1)

148

0.25

Uc/U

∞

650-850Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=8.3141(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

Rpp

Rpp

0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

950-1150Hz

0.5

0

-0.5

0

-0.5

c) -1 0

Uc=8.5968(m/s)
0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

-1

0.25 d)

1050-1250Hz

1

Uc=8.7771(m/s)
0

1

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
1150-1350Hz

0.25

0.5
Rpp

0.5
Rpp

Uc=8.4049(m/s)

1

0.5

0

0

-0.5

-0.5
-1

b) -1

850-1050Hz

1

e)

0

-0.5

-0.5

a)-1

750-950Hz

1

Uc=9.0247(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-21

Tap21-22

0.25

f)

-1

Tap21-23

Uc=9.3964(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-24

Tap21-25

Figure C.1.6 Band-filtered Rpp correlations (RCDB Case 1)

149

0.25

Uc/U

∞

1250-1450Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=9.6058(m/s)
0

1

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
1450-1650Hz

0.25

Rpp

Rpp

0

Uc=9.8119(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
1650-1850Hz

0.25

1550-1750Hz

Uc=9.9301(m/s)
0

1

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
1750-1950Hz

0.25

0.5
Rpp

Rpp

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0

-1

0.25 d)

0.5
0

0

-0.5

-0.5
-1

0

-0.5

1

e)

b)

Uc=9.6864(m/s)

0.5

-0.5

c)

-1

1

0.5

-1

0

-0.5

-0.5

a)-1

1350-1550Hz

1

Uc=10.0925(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-21

Tap21-22

0.25

f)

-1

Tap21-23

Uc=10.2175(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-24

Tap21-25

Figure C.1.7 Band-filtered Rpp correlations (RCDB Case 1)

150

0.25

Uc/U

∞

1850-2050Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=10.2895(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

Uc=8.9545(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

2150-2350Hz

1
0.5
Rpp

0.5
Rpp

b)

-1

2050-2250Hz

1

0

0

-0.5

-0.5
Uc=10.2675(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25 d)

-1

Uc=10.3184(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

2250-2450Hz

1
0.5
Rpp

c) -1

0

-0.5

-0.5

a) -1

1950-2150Hz

1

0

-0.5

e)

Tap21-21

-1

Uc=10.3508(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-22

Tap21-23

0.25

Tap21-24

Tap21-25

Figure C.1.8 Band-filtered Rpp correlations (RCDB Case 1)

151

Uc/U

∞

50-250Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=7.2909(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

Rpp

Rpp

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

350-550Hz

0

-0.5

-0.5
Uc=8.684(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

d)

-1

450-650Hz

1

Uc=8.6412(m/s)
0

1

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
550-750Hz

0.25

0.5
Rpp

0.5
Rpp

0

0.5

0

0

0

-0.5

-0.5

e) -1

Uc=8.3475(m/s)

1

0.5

c)

b) -1

250-450Hz

1

-1

0

-0.5

-0.5

a)-1

150-350Hz

1

Uc=8.6531(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-21

Tap21-22

0.25

f) -1

Tap21-23

Uc=8.0228(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-24

Tap21-25

Figure C.1.9 Band-filtered Rpp correlations (RCDB Case 2)

152

0.25

Uc/U

∞

650-850Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=8.7943(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

0

-0.5
Uc=7.8219(m/s)
0

0.25

950-1150Hz

0

-1

0.25 d)

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
1050-1250Hz

Uc=7.6253(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

1150-1350Hz

1
0.5
Rpp

0.5
Rpp

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

-0.5

1

0

0

-0.5

-0.5

e) -1

0

0.5
Rpp

Rpp

b)

Uc=8.6412(m/s)

1

0.5

c)

-1

850-1050Hz

1

-1

0

-0.5

-0.5

a) -1

750-950Hz

1

Uc=7.726(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-21

Tap21-22

0.25

f)

-1

Tap21-23

Uc=9.3964(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-24

Tap21-25

Figure C.1.10 Band-filtered Rpp correlations (RCDB Case 2)

153

0.25

Uc/U

∞

1250-1450Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=7.7445(m/s)
0

1

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
1450-1650Hz

0.25

Rpp
Rpp

Rpp

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]
1550-1750Hz

0.25

0

-0.5

-0.5
Uc=7.9735(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

d) -1 0
0

U =8.007(m/s)
c
0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

∞

1650-1850Hz

1

1750-1950Hz

1
0.5
Rpp

0.5
Rpp

b) 0

0.5

0

0

0

-0.5

-0.5

e) -1

Uc=7.8107(m/s)

-1

1

0.5

c) -1

0

-0.5

-0.5

a)-1

1350-1550Hz

1

Uc=9.4241(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-21

Tap21-22

0.25

f) -1

Tap21-23

Uc=10.2175(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-24

Tap21-25

Figure C.1.11 Band-filtered Rpp correlations (RCDB Case 2)

154

0.25

Uc/U

∞

1850-2050Hz

1

0.5
Rpp

Rpp

0.5
0

Uc=10.4809(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

2150-2350Hz

0.5
Rpp

Rpp

Uc=10.8436(m/s)

1

0.5
0

-0.5

0

-0.5

Uc=12.0262(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25 d)

-1

Uc=12.7282(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

0.25

2250-2450Hz

1
0.5
Rpp

c)

b) -1

2050-2250Hz

1

-1

0

-0.5

-0.5

a)-1

1950-2150Hz

1

0

-0.5

e)

Tap21-21

-1

Uc=11.3696(m/s)
0

0.05 0.1 0.15 0.2
Time Delay [t*u ∞/c]

Tap21-22

Tap21-23

0.25

Tap21-24

Tap21-25

Figure C.1.12 Band-filtered Rpp correlations (RCDB Case 2)

155

Uc/U

∞

C.2 Coherence-Phase

RCDB: Case 1 (Tap1-Tap2)
Coherence

1
0.5
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.1 Coherence and Phase (RCDB Case1: Tap1-Tap2)

RCDB: Case 1 (Tap2-Tap3)
Coherence

0.4
0.2
0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.2 Coherence and Phase (RCDB Case1: Tap2-Tap3)

156

RCDB: Case 1 (Tap3-Tap5)
Coherence

0.2
0.1
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.3 Coherence and Phase (RCDB Case1: Tap3-Tap5)

RCDB: Case 1 (Tap5-Tap6)
Coherence

0.4
0.2
0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.4 Coherence and Phase (RCDB Case1: Tap5-Tap6)

157

RCDB: Case 1 (Tap6-Tap7)
Coherence

0.4
0.2
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.5 Coherence and Phase (RCDB Case1: Tap6-Tap7)

RCDB: Case 1 (Tap7-Tap9)
Coherence

1
0.5
0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.6 Coherence and Phase (RCDB Case1: Tap7-Tap9)

158

RCDB: Case 1 (Tap9-Tap11)
Coherence

0.5

0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.7 Coherence and Phase (RCDB Case1: Tap9-Tap11)

RCDB: Case 1 (Tap11-Tap21)
Coherence

0.5

0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.8 Coherence and Phase (RCDB Case1: Tap11-Tap21)

159

RCDB: Case 1 (Tap21-Tap22)
Coherence

1
0.5
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.9 Coherence and Phase (RCDB Case1: Tap21-Tap22)

RCDB: Case 1 (Tap22-Tap23)
Coherence

1
0.5
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.10 Coherence and Phase (RCDB Case1: Tap22-Tap23)

160

RCDB: Case 1 (Tap23-Tap24)
Coherence

1
0.5
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.11 Coherence and Phase (RCDB Case1: Tap23-Tap24)

RCDB: Case 1 (Tap24-Tap25)
Coherence

1
0.5
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.12 Coherence and Phase (RCDB Case1: Tap24-Tap25)

161

RCDB: Case 1 (Tap25-Tap25A)
Coherence

0.5

0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.13 Coherence and Phase (RCDB Case1: Tap25-Tap25A)

RCDB: Case 1 (Tap25-Tap25B)
Coherence

1
0.5
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.14 Coherence and Phase (RCDB Case1: Tap25-Tap25B)

162

RCDB: Case 1 (Tap25-Tap25C)
Coherence

0.4
0.2
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

0

Phase delay [Rad]

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]

5
0
-5

Figure C.2.15 Coherence and Phase (RCDB Case1: Tap25-Tap25C)

163

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164

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