Relationship between Rotor Wake Structures and Performance Characteristics over a
Range of Low-Reynolds Number Conditions
THESIS
Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in
the Graduate School of The Ohio State University
By
Mark Louis Sutkowy Jr.
Graduate Program in Aeronautical and Astronautical Engineering
The Ohio State University
2018
Thesis Committee:
Dr. James W. Gregory, Advisor
Dr. Jeffrey P. Bons
Dr. Matthew H. McCrink
Copyrighted by
Mark Louis Sutkowy Jr.
2018
Abstract
Small-scale rotors exhibit degraded aerodynamic efficiency, which has been linked to non-ideal losses within their wake. Many small unmanned aircraft systems
(UAS) are powered by such rotors, and are currently at the forefront of aerospace research for a multitude of innovative applications. As such, a comprehensive understanding of their operational capabilities is critical for implementation in the field.
A great deal of attention has been given to characterize the performance of large- scale rotor models. However, similar studies for small-scale, low Reynolds number (Re) applications has received relatively little attention. This work seeks to gather insight into the behavior of the rotor wake structures as a function of Re, relate this to performance capabilities and the corresponding far-field acoustic signature. Two-component particle image velocimetry (PIV), performance, and acoustic measurements were performed using three small-scale, NACA 0012 rotors operated over a range of low-Reynolds number conditions. Rotor geometry and operational speed (Ω) were varied to obtain the desired Re variation.
Spanwise PIV has demonstrated an absence of tip vortex formation as the operational thrust coefficient (CT) is increased, suggesting the presence of outboard tip stalling. Phase-locked, chordwise PIV has confirmed this hypothesis, showing the development of flow separation and a highly turbulent downstream wake. Thrust and ii torque measurements show degraded rotor performance at the onset of these conditions, especially at low Re.
A vortex identification scheme was used to locate downstream tip vortices and characterize their size, swirl velocity, and aperiodic wandering behavior for different operational conditions. When observed at constant wake age, the wandering motion of the vortices behaved independently of vortex Reynolds number (Rev) scaling. The normalized standard deviation of the tip vortex wander was found to match well with historically observed trends, but a difference in magnitude suggests aperiodicity depends strongly on blade number instead.
Chordwise PIV measurements revealed the wake characteristics at moderate collective angles (θ) produce periodic counter-rotating structures which are attributed to laminar boundary layer vortex-shedding. Acoustic measurements at similar operating conditions show significant broad-band, high frequency peaks. The broad-band peaks were found to correlate well with a physical quantification of the shedding phenomenon.
iii Dedication
FOR MY FAMILY. THEIR SELFLESSNESS MADE HIGHER EDUCATION
POSSIBLE.
FOR MY FRIENDS, WHO ARE STILL HERE WHILE MY ATTENTION WAS
ELSEWHERE.
FOR AVA, THE BRIGHTEST LIGHT AT THE END OF THE TUNNEL.
MOST OF ALL, FOR MY LORD AND SAVIOR, JESUS CHRIST. HE IS THE ONE
WHO CREATED THE QUESTIONS FOR WHICH SCIENTISTS SEEK ANSWERS,
AND HE IS THE ONE WHO ENABLES US TO ANSWER THEM.
IN HIS WISDOM, I CAME TO SEE THAT HOURS SPENT IN A LAB ARE MINUTE
COMPARED TO ETERNITY IN PARADISE.
iv Acknowledgments
In my experience, I have come to learn that progress in scientific research demands resilience and curiosity. For fueling these qualities, I express gratitude to my adviser, Dr. Gregory, and mentor, Dr. McCrink. Their own accomplishments and passion for aerospace have shown that work can be an enjoyable and gratifying experience. Their guidance has been instrumental in my development as a researcher and engineer.
I am grateful to my colleagues for sharing their expertise and unique skill sets which were critical to the completion of this work. I am appreciative of Achal Singhal for his efforts in developing my fluency with experimental techniques. His patience is truly remarkable. Anshuman Pandey and Braxton Harter played critical roles in the acquisition, discussion, and interpretation of the data. Collaboration with Ryan Thorpe and Wenbo Zhu was also valuable during course of my graduate work. Josh Gueth and
Ken Fout are appreciated for sharing their machining expertise which enabled the design, fabrication, and assembly of my experiments.
This research was partially funded by the Government under Agreement No.
W911W6-17-2-0002. The U.S. Government is authorized to reproduce and distribute reprints for Governmnet purposes notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not
v be interpreted as representing the official policies, either expressed or implied, of the
Aviation Development Directorate or the U.S. Government.
I am also grateful to colleagues with the Georgia Institute of Technology Vertical
Lift Research Center of Excellence, including Narayanan Komerath, Ganesh
Rajagopalan, Gloria Yamauchi, Oliver Wong, and Tom Thompson for helpful discussions on this work.
vi Vita
2013...... Co-op Employee, Passport 20 Systems, GE
Aviation, Cincinnati, Ohio
2015...... Intern, Manufacturing Engineer, Ford Motor
Company, Cleveland, Ohio
2015...... B.S. Mechanical Engineering, University of
Dayton, Dayton, Ohio
2016...... Research Intern, Inlets and Nozzles Division,
NASA Glenn Research Center, Cleveland,
Ohio
2016 to present ...... Graduate Research Assistant, Mechanical
and Aerospace Engineering, Aerospace
Research Center, The Ohio State University,
Columbus, Ohio
vii Publications
Conference Publications
1. Sutkowy, M., Pandey, A., McCrink, M., and Gregory, J., “Rotor Wake Structure Development in Low Reynolds Number Conditions,” AIAA SciTech Forum, Orlando, Florida, January 8, 2018, DOI: 10.2514/6.2018-1830
2. Sutkowy, M., Harter, B., McCrink, M., and Gregory, J., “Impact of Wake Structure Characteristics on Small-Scale Rotor Performance over a Range of Reynolds Numbers,” American Helicopter Society 74th Annual Forum, Phoenix, Arizona, May 14, 2018, SKU #:74-2018-0135
3. Wang, Z., Pandey, A., Sutkowy, M., Harter, B., McCrink, M., Gregory, J., and Zhuang, M., “A Comprehensive Approach to Study Aerodynamics and Aerocoustics around Small Multicopter Unmanned Aerial Systems,” AIAA SciTech Forum, Orlando, Florida, January 8, 2018, DOI: 10.2514/6.2018-0268
4. Pandey, A., Sutkowy, M., McCrink, M., and Gregory, J., “Aerodynamic Characterization of a Quad-Rotor Helicopter,” AIAA SciTech Forum, Orlando, Florida, January 8, 2018, DOI: 10.2514/6.2018-1526
Fields of Study
Major Field: Aeronautical and Astronautical Engineering
viii
Table of Contents
Abstract ...... ii
Dedication ...... iv
Acknowledgments...... v
Vita ...... vii
List of Tables ...... xii
List of Figures ...... xiii
Nomenclature ...... xx
Chapter 1. Introduction ...... 1
Chapter 2. Background ...... 5
2.1 Rotor Performance ...... 6
2.2 Rotor Wake Features...... 11
2.3 Acoustic Signature ...... 17
Chapter 3. Description of the Experiment ...... 18
3.1 Rotor Characteristics ...... 18
3.2 Test Stand Design ...... 20
3.3 Performance Measurements ...... 21
ix 3.3.1 Thrust ...... 21
3.3.2 Torque ...... 23
3.4 Flowfield Measurements ...... 27
3.4.1 Enclosure Design ...... 27
3.4.2 Wake Recirculation Study ...... 28
3.4.3 Laser Optics ...... 35
3.4.4 Imaging Configurations ...... 36
3.4.5 Phase-Lock Triggering...... 38
3.4.6 Time Resolved Measurement Configuration ...... 40
3.4.7 Seeding ...... 40
3.4.8 PIV Data Processing ...... 40
3.5 Acoustic Measurements ...... 42
Chapter 4. Results and Discussion ...... 45
4.1 Performance Measurements ...... 45
4.1.1 Thrust Measurements ...... 45
4.1.2 Torque Measurements ...... 52
4.1.3 Performance Uncertainty Measurements ...... 57
4.2 Flowfield Measurements ...... 62
4.2.1 Tip Stalling...... 62
x 4.2.2 Chordwise PIV ...... 64
4.2.3 Spanwise PIV ...... 71
4.2.4 Time Resolved Measurements ...... 75
4.3 Tip Vortex Aperiodicity ...... 81
4.4 Laminar Boundary Layer Shedding ...... 86
4.4.1 Wake Structure Development ...... 86
4.4.2 Acoustic Measurements ...... 86
Chapter 5. Conclusions ...... 90
Bibliography ...... 93
Appendix A. Derivation of Vortex Reynolds Number Approximation ...... 97
Appendix B. Chordwise PIV Measurements ...... 100
xi List of Tables
Table 1. Geometric rotor characteristics...... 19
Table 2. Test conditions - vortex Reynolds number...... 21
Table 3. Test conditions - chord Reynolds number...... 22
Table 4. Profile power estimates for all Re conditions...... 26
Table 5. Time resolved measurement conditions...... 40
Table 6. Operational conditions used for rotor wake flow visualization...... 64
Table 7.Chordwise PIV test conditions...... 64
Table 8. Vortex shedding frequency comparison...... 88
xii List of Figures
Figure 1. Relative hovering efficiency for various size vehicles.4 ...... 2
Figure 2. Effect of airfoil efficiency and induced losses on aerodynamic efficiency.4 ...... 3
Figure 3. (a) Illustration of rotor wake structures and (b) DES results for a rotor wake.8-9 6
Figure 4. Rotor wake control volume...... 7
Figure 5. Rotor wake geometry and corresponding pressure changes...... 9
Figure 6. BEMT computation of radial variation in (a) non-dimensional inflow and (b) thrust per unit span for an untwisted and twisted low-Re rotor...... 12
Figure 7. Tip vortex and vortex sheet development for an (a) untwisted and (b) twisted rotor blade.7 ...... 13
Figure 8. Tip vortex and vortex sheet development for an untwisted blade with (a) CT/σ =
7 0.053 and (b) CT/σ = 0.080...... 13
Figure 9. (a) Variation in turbulent viscosity with vortex Reynolds number and (b) turbulent regions of a transitional vortex.15 ...... 15
Figure 10. Vortex pairing illustrations from (a) a two-bladed rotor experiment12 and (b) a single-bladed water tunnel dye visualization.19 ...... 15
Figure 11. 3-D Printed rotor blades...... 19
Figure 12. Rotor head and rotor blade interface...... 20
Figure 13. Rotor test stand...... 20 xiii Figure 14. Torque sensor calibration curves at 2500 and 3500 RPM...... 24
Figure 15. Drag polars derived from XFLR 5 using a NACA 0012 airfoil...... 25
Figure 16. Mechanical and aerodynamic power measurements for (a) Re = 47,500 and (b)
Re = 89,400...... 27
Figure 17. Laboratory PIV setup...... 28
Figure 18. Rotor wake recirculation...... 29
Figure 19. Thrust measurements from the small and large rotors over a range of rotational speeds within the PIV enclosure and in an open environment...... 30
Figure 20. Torque measurements from the small and large rotors over a range of rotational speeds within the PIV enclosure and in an open environment...... 30
Figure 21. RMS of thrust measurements for the large and small rotors in an open and enclosed operating environment...... 31
Figure 22. Thrust measurements from the S-1000 rotor over a range of rotational speeds within various enclosure sizes...... 32
Figure 23. Torque measurements from the S-1000 rotor over a range of rotational speeds within various enclosure sizes...... 33
Figure 24. Thrust fluctuations of the DJI S-1000 for an (a) open environment and enclosure sizes of (b) DW/DR = 5, (c) DW/DR = 2.5, and (d) DW/DR = 1.5...... 34
Figure 25. RMS of thrust measurements for the S-1000 rotor operating in various enclosure sizes...... 35
Figure 26. Laser optics configuration...... 36
Figure 27. Spanwise PIV configuration...... 37
xiv Figure 28. Spanwise PIV recorded phase increments...... 37
Figure 29. Chordwise PIV configuration...... 38
Figure 30. PIV system schematic...... 39
Figure 31. Tip vortex velocity components...... 42
Figure 32. The OSU anechoic chamber...... 44
Figure 33. (a) Thrust and (b) blade loading measurements for the small sized rotor at varying rotational speeds and collective angles...... 46
Figure 34. (a) Thrust and (b) blade loading measurements for the intermediate sized rotor at varying rotational speeds and collective angles...... 47
Figure 35. (a) Thrust and (b) blade loading measurements for the large sized rotor at varying rotational speeds and collective angles...... 47
Figure 36. Standard deviation of (a) thrust and (b) blade loading for the small size rotor.
...... 48
Figure 37. Standard deviation of (a) thrust and (b) blade loading for the intermediate sized rotor...... 48
Figure 38. Standard deviation of (a) thrust and (b) blade loading for the large sized rotor.
...... 49
Figure 39. FFT of raw thrust measurements of the small rotor at 2500 RPM...... 50
Figure 40. Raw and filtered thrust data for the small rotor at Ω = 2500 RPM and θ =
3.5o...... 51
Figure 41. Figure of merit for varying Reynolds number conditions...... 53
Figure 42. Induced torque for varying Reynolds number conditions...... 54
xv Figure 43. Comparison between FM measurements and a theoretical approximation. .... 56
Figure 44. Comparison between induced torque measurements and a theoretical approximation...... 56
Figure 45. Variation in thrust measurements taken with the small rotor at 3500 RPM. .. 58
Figure 46. Thrust uncertainty measurements for the small rotor operating at 3500 RPM over a range of θ...... 59
Figure 47. Blade loading uncertainty measurements for the small rotor operating at 3500
RPM over a range of θ...... 59
Figure 48. Figure of merit uncertainty measurements for the small rotor operating at 3500
RPM over a range of blade loading conditions...... 61
Figure 49. Induced torque uncertainty measurements for the small rotor operating at 3500
RPM over a range of blade loading conditions...... 61
Figure 50. Instantaneous vorticity contours using the large blades at 3500 RPM for (a) CT
= 0.001 (b) CT = 0.005 (c) CT = 0.0075 (d) CT = 0.01...... 62
Figure 51. Ensemble average of chordwise vorticity at Rec = 106,700 and CT = 0.007. .. 66
Figure 52. Ensemble average of chordwise vorticity at Rec = 106,700 and CT = 0.008. .. 66
Figure 53. Ensemble average of chordwise vorticity at Rec = 106,700 and CT = 0.009. .. 67
Figure 54. Ensemble average of velocity magnitude at Rec = 96,500 and CT = 0.002. .... 68
Figure 55. Ensemble average of velocity magnitude at Rec = 106,700 and CT = 0.002. .. 68
Figure 56. Ensemble average of velocity magnitude at Rec = 96,500 and CT = 0.008. .... 69
Figure 57. Ensemble average of velocity magnitude at Rec = 106,700 and CT = 0.008. .. 70
xvi Figure 58. Phase average vorticity contours for (a) Rev = 17,100 (b) Rev = 32,600 (c) Rev
= 41,900...... 72
o o Figure 59. Phased averaged vorticity of at Rev = 22,000 for (a) ψ = 1 , (b) ψ = 104 , and
(c) ψ = 155o...... 74
Figure 60. Bi-modal wake geometry at Rev = 22,000...... 74
Figure 61. Spanwise time resolved images for Rev = 27,450 and CT = 0.0059 at (a) t = 8.1 ms (b) t = 12.6 ms (c) t = 18.4 ms (d) t = 21.0 ms (e) t = 23.1 ms and (f) t = 26.8 ms. .... 76
Figure 62. Spanwise time resolved images for Rev = 27,450 and CT = 0.004...... 79
Figure 63. Spanwise time resolved images for Rev = 27,450 and CT = 0.003...... 79
Figure 64. Spanwise time resolved images for Rev = 27,450 and CT = 0.007...... 80
Figure 65. Rotation angle of tip vortex loci...... 82
Figure 66. Tip vortex position variation for (a) Rev = 27,450 (b) Rev = 32,600 (c) Rev =
35,280 (d) Rev = 41,900...... 82
Figure 67. Normalized standard deviation of tip vortex wandering for the component normal to the slipstream boundary...... 84
Figure 68. Normalized standard deviation of tip vortex wandering for the component tangential to the slipstream boundary...... 84
Figure 69. Normal and tangential standard deviation comparison between a two and four bladed rotors (Mula) of similar geometric and operational characteristics.36 ...... 85
Figure 70. Instantaneous chordwise PIV image at Rec = 106,700 and CT = 0.002...... 86
Figure 71. Rotor acoustic measurements for varying Rec at 3500 RPM...... 89
Figure 72. Rotor acoustic measurements for varying Rec at 4500 RPM...... 89
xvii Figure 73. Chordwise ensemble average of velocity magnitude at Rec = 57,500 for a CT of
(a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009.
...... 100
Figure 74. Chordwise ensemble average of vorticity at Rec = 57,500 for a CT of (a) 0.001
(b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009...... 102
Figure 75. Chordwise ensemble average of velocity magnitude at Rec = 96,500 for a CT of
(a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009.
...... 104
Figure 76. Chordwise ensemble average of vorticity at Rec = 96,500 for a CT of (a) 0.001
(b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009...... 106
Figure 77. Chordwise ensemble average of velocity magnitude at Rec = 98,400 for a CT of
(a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009.
...... 108
Figure 78. Chordwise ensemble average of vorticity at Rec = 98,400 for a CT of (a) 0.001
(b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009...... 110
Figure 79. Chordwise ensemble average of velocity magnitude at Rec = 106,700 for a CT of (a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009.
...... 112
Figure 80. Chordwise ensemble average of vorticity at Rec = 106,700 for a CT of (a)
0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009. ... 114
xviii Figure 81. Chordwise ensemble average of velocity magnitude at Rec = 115,000 for a CT of (a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009.
...... 116
Figure 82. Chordwise ensemble average of vorticity at Rec = 115,000 for a CT of (a)
0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009. ... 118
xix Nomenclature
Latin Symbols
Symbol Description Units
A Rotor disk area m2
AR Aspect ratio - c Chord length m
C Profile drag coefficient - do
Cl Two-dimensional section lift coefficient -
Cp Power coefficient -
Cp, actual Actual power coefficient -
Cp, ideal Ideal power coefficient -
C Power coefficient at zero-lift - po
CQ Torque coefficient -
CT Thrust coefficient -
DR Rotor diameter m
Dw Distance between PIV enclosure walls m dCT Differential thrust coefficient - dr Differential non-dimensional blade radius -
xx dS Rotor wake differential control volume area m2 f Frequency Hz fa Rotor wake acoustic frequency kHz fs Spanwise vortex shedding frequency kHz
FM Figure of merit - k Proportionality constant -
L’ Lift per unit span N/m
L/D Lift-to-drag ratio - m Rotor wake mass flow rate kg/s n Inflow profile factor -
N Blade number -
P Air pressure Pa
P Power required for rotor operation W p0 Total air pressure Pa
Pa Aerodynamic power W
Pe Servo electrical power W
Pi Induced power required for rotor operation W
Pm Mechanical or total power W
P Mechanical power at zero-lift W m L 0
Po Profile power required for rotor operation W
Ps Mechanical system power W
Qm Mechanical torque Nm
xxi r Non-dimensional rotor radius -
R Rotor radius m
Rhub Rotor hub radius m
Re Reynolds number -
Rec Chord Reynolds number -
Rev Vortex Reynolds number -
SPL Sound pressure level dB
T Thrust N
U Local rotor tangential velocity m/s
UT Thrust uncertainty N
V Velocity magnitude m/s
V Axial wake velocity m/s vi Rotor induced velocity m/s w Vena contracta axial wake velocity m/s
Δx Shed vortex spacing m z Axial wake distance m
Greek Symbols
α Angle of attack deg
2 Гb Blade bound circulation m /s
2 Гv Vortex circulation m /s
ζ Wake age deg xxii η Efficiency factor -
θ Collective angle deg
κ Induced power factor -
λ Non-dimensional rotor inflow -
λtip Local tip non-dimensional rotor inflow -
ν Kinematic viscosity m2/s
ρ Air density kg/m3
σ Rotor solidity -
σi Standard deviation -
ψ Azimuthal position deg
Ω Rotational speed RPM
xxiii Chapter 1. Introduction
Innovative applications for small unmanned aerial vehicles (UAV) are currently on the rise. The potential scope for these UAVs includes rapid package delivery, military reconnaissance work, disaster zone mapping, and weather forecasting.1-3 Their reliability to successfully execute such tasks can be improved by further optimizing design parameters. In the case of multi-rotor vehicles, there are a large number of variables which dictate capability, and a fundamental understanding of small-scale rotor performance is necessary to influence the design process.
The hovering efficiency of a rotor can be characterized in terms of the power loading (i.e. the weight of a vehicle relative to the power required to keep it airborne).
Fig. 1 from Ramasamy et al. 4 shows there are multiple factors which dictate an optimal hovering efficiency. Particularly, a low effective disk loading (DL) and high figure of merit (FM) are desirable design characteristics for any rotorcraft. In this case, the DL refers to the relative thrust production to rotor disk area, while the FM is defined as the ratio of ideal to actual power required to hover.
1
Figure 1. Relative hovering efficiency for various size vehicles.4
Fig. 1 shows how microrotor vehicles operate with a low FM and low DL relative to their full-scale counterparts. That is, small-scale vehicles tend to exhibit lower aerodynamic efficiency despite the benefits of lower disk loading, which is the case for rotors that power small UAV rotorcraft. Another way to characterize such vehicles is by the Reynolds number (Re) of the flow in which they operate. Bohorquez5 and Otsuka et al.6 have shown that the FM, and therefore aerodynamic efficiency, decreases significantly with Re. For a rotor in hover, the primary factors that adversely affect FM are the profile drag coefficient ( C ) and induced power factor (κ). Respectively, these d0 components are dictated by the lift-to-drag (L/D) of the local airfoil section (airfoil efficiency) and non-ideal losses induced by the change in momentum across the rotor disk. Fig. 2 from Ramasamy et al.4 shows the former plotted against the latter. The trend 2 indicates that the ultimate design goal of an optimal FM cannot be achieved by simply improving airfoil efficiency. Rather, the induced power factor associated with these low-
Re vehicles must be driven down if there is any hope of matching or exceeding the aerodynamic efficiency of large-scale rotorcraft.
Figure 2. Effect of airfoil efficiency and induced losses on aerodynamic efficiency.4
In the case of a rotor, the induced power factor is dictated by a myriad of complex flow phenomena associated with the highly three-dimensional entrained wake. Namely, the developmental dependencies of axially convecting tip vortex filaments on the operational Reynolds number is a current focus in this study. Potential trends associated with tip vortex pairing and wake sheet interactions would provide a physical rational for losses in the wake. Vortex filament size, convection speed, and aperiodic positioning as the slipstream contracts at old wake ages will impact instability growth, and could also be
3 Reynolds dependent. Such questions will be explored as potential explanations for the diminished aerodynamic efficiency of small rotors. Due to the small size of these flowfields, common measurement techniques become quite difficult to employ, and as a result there is a great deal that is not understood about the physics at play. This work seeks to explore these flow phenomena over a range of Re conditions, and relate these findings to the corresponding performance of canonical rotor geometries.
4 Chapter 2. Background
Multi-rotor vehicles have recently gained a great deal of popularity, and are one of the primary platforms for new applications. Although their compact designs make them a versatile option, it is difficult to study the aerodynamics of simultaneously operating rotors. In part, this is due to complications associated with wake interactions from the various rotors. Conventional measurement techniques such as PIV also become difficult due to lack of resolution induced by the small length scales inherent to structures of interest. To simplify this complex problem, it is helpful to first isolate a given rotor, and develop a fundamental understanding of its aerodynamic behavior in a hovering flight condition.
During operation, rotors generate an aerodynamically complex wake which contracts as it propagates. Characteristics of the wake structures are of primary importance as they provide physical insight that allows us to explain trends associated with the rotor performance and acoustic signature. The development of helical vortex filaments dominate the rotor wake. Milluzzo et al.7 has shown that these filaments interact with weaker counter-rotating structures contained in the spanwise shear layer which is shed from the trailing edge of the rotor blade. Fig. 3a, from Gray8, is a theoretical illustration of these typical wake features. Chaderjian9 provides a more physically accurate representation from a detached eddy simulation (DES) of a rotor, 5 shown in Fig. 3b. Here, the helical tip vortex system generated by the three-bladed rotor is shown in pink, and smaller-helical root vortices are generated near the axis of rotation.
In this representation, the wake sheets are shown in blue at a single phase over successively older wake ages. By studying the behavior of these structures over a range of Re conditions, a great deal can be learned about the fundamental aerodynamics of small-scale rotorcraft.
(a) (b)
Figure 3. (a) Illustration of rotor wake structures and (b) DES results for a rotor
wake.8-9
2.1 Rotor Performance
A first order accurate approximation of rotor performance (i.e. thrust production and power consumption) can be made from the conservation of momentum and energy applied over the control volume of a hovering rotor. To do so, assumptions of a one dimensional, quasi-steady, incompressible and inviscid rotor wake must be assumed.10
6 Thrust can be computed according to Eq. 1 as the force equal and opposite to that imparted by the rotor on the fluid particles in its wake.
T VdSV VdSVmw . (1) 0
Here, the vector equation reflects the net time rate of change of momentum across the wake control surface as defined in Fig. 4. If a quiescent upstream flow and uniform increase in total pressure across the rotor disk are assumed, thrust is reduced to the product of the downstream mass flow and velocity.
Figure 4. Rotor wake control volume.
Similarly, Eq. 2 simplifies energy conservation, and relates the hover power required by the rotor to a net increase in kinetic energy per unit time imparted on the wake. 7 22 PTv 1 VdSV 1 VdSV 1 mw2 . (2) i 2 0 2 2
Relating Eqs. 1-2, Eq. 3 shows that the velocity in the vena contracta (w), a region of the rotor wake illustrated in Fig. 5, is nominally double the induced velocity (vi) through the rotor disk.
vw 1 (3) i 2
By continuity, the increasing flow speed with axial position in the wake will cause the cross-sectional area to contract as shown in Fig. 5. Since the rotor is adding energy to the flow field, there will be a net increase in total pressure across the disk, and a continuous rise in dynamic pressure. The accelerating flow below the wake will correspond to a decrease in static pressure as fluid particles reach the vena contracta. These trends are also shown in Fig. 5, where the pressure changes correspond to that typical for a small- scale quadcopter rotor.
Eqs. 1-2 show that rotor performance largely depends on the wake velocity. That is, for a fixed mass flow rate (푚̇ ), thrust and power scale with w and the square of w, respectively. However, the apparent simplicity of this relationship stems from the assumptions made in the equations above. The presence of a vortex system in the wake introduces complex interactions and viscous dissipation that will reduce the convection speed and adversely affect rotor performance. This is of primary concern for small-scale rotors because the impact of these non-ideal phenomena are exacerbated.
8
Figure 5. Rotor wake geometry and corresponding pressure changes.
The conventional criteria used to quantify the impact of undesirable wake characteristics on rotor performance is the figure of merit, which is derived by first considering the relationship between thrust and power. The non-dimensional form of thrust (CT) is defined as
T C , (4) T AR22 where ρ, A, Ω, and R are air density, rotor disk area, rotational speed, and disk radius, respectively. The power coefficient (CP) is defined as
9 P Tv C32 C i T , (5) P ARAR3 3 3 3 2 relating CP to CT. Since this relationship proceeds directly from the ideal assumptions made in Eqs. 1-2, an adjustment must be made in an effort to quantify non-ideal wake conditions. The FM does this by normalizing CP by a corrected power coefficient as
32 CT C FM p, ideal 2 . (6) C C32 C p, actual T d0 2 8
Here, κ is the induced power factor, σ is blade solidity, and C is the section profile drag d0 coefficient. The first term in the denominator is the corrected induced power requirement which accounts for non-ideal wake losses, while the second is profile power which accounts for the cost of friction and pressure drag over the airfoil. Since corrected or actual power required to drive the rotor will always be greater than the ideal value, FM will always be a value less than unity. Historically, research has shown that FM is significantly low for small-scale rotors relative to full-scale.5,6 These finding are linked to the fact that both κ and are typically larger for a low-Re rotor, which ultimately drives up the induced and profile power terms.10
Specifically, Ramasamy et al.4 have shown that, despite the low disk loading associated with micro-air vehicles (MAV), the corresponding FM, is significantly lower.
In this case, their measured FM values suggest κ and 퐶푑0 values of 1.75 and 0.03, respectively. In comparison, a typical induced power factor and section drag coefficient for full scale vehicles is 1.15 and 0.01. Recently, technological capabilities of computers
10 and cameras have enabled more robust methods to qualitatively and quantitatively observe features in the wake of a rotor. These new techniques have allowed for a more in depth analysis of small-scale wake structures, providing insight into their development and interactions.
2.2 Rotor Wake Features
Directly linked to rotor performance are characteristics associated with the three- dimensional wake which are dominated by tip vortices and influenced by spanwise vortex sheets. 11,12-13 It is conventional to study the temporal development of wake structures by characterizing them at successive wake ages (ζ). The wake age of given structure is synonymous to the total azimuthal position (ψ) traversed by the rotor since it initially shed that structure.
Structure development in the wake depends heavily on a number of variables including rotor blade geometry and operational condition. Therefore, when studying wake dependency on Reynolds number, care must be taken to isolate as many other variables as possible to ensure changes in wake characteristics are solely due to Re variation. Specifically, many have studied the behavior of tip vortex and wake sheet development as a function of blade loading (CT/σ) and blade twist.
Milluzzo et al.7 performed phase-resolved PIV and demonstrated that, relative to an untwisted blade, a twisted profile results in a more uniform spanwise inflow profile and linear increase of the local thrust production. In the case of a rotor, a typical blade twist follows a linear or cubic decrease in geometric pitch from the largest angle at the root to a lowest angle at the tip. Since the local tangential velocity increases with span, a
11 corresponding decrease geometric pitch enables uniform spanwise lift. Conversely, untwisted blades are heavily tip loaded, inducing a linearly increasing inflow, with a parabolic lift increase from root to tip. Using blade element momentum theory (BEMT), these characteristics are highlighted for both blade types at a Reynolds number estimated for an arbitrary quadcopter rotor.
(a) (b)
Figure 6. BEMT computation of radial variation in (a) non-dimensional inflow and
(b) thrust per unit span for an untwisted and twisted low-Re rotor.
Milluzzo et al. showed that this varying inflow profile impacts the convection rate of both tip vortices and wake sheets, as indicated by their flow visualization in Fig. 7. From the smoke flow, it is evident that higher outboard inflow inherent to the untwisted blades forces the tip vortex system to convect more rapidly than the twisted case. Conversely, the relatively greater inflow at mid-span of the twisted blade forces the vortex sheets to convect at a faster rate than those of the untwisted blades.
12 (a) (b)
Figure 7. Tip vortex and vortex sheet development for an (a) untwisted and (b)
twisted rotor blade.7
They also showed the dependency of convection speed on the operational blade loading coefficient (CT/σ). Fig. 8 shows how an increase in CT/σ, achieved by collective angle adjustment, will induce an overall inflow rise, thus convecting wake features more rapidly.
(a) (b)
Figure 8. Tip vortex and vortex sheet development for an untwisted blade with (a)
7 CT/σ = 0.053 and (b) CT/σ = 0.080.
Ramasamy et al.11 show how the behavior of wake structures can directly impact the rotor inflow profile. The degree to which this occurs is partially dictated by the
13 operating thrust coefficient and blade twist. The influence on rotor inflow can promote reduced convection speed of structures in the wake, enabling interaction between aging structures and the following blade passage. This phenomenon, known as blade vortex interaction (BVI), has been shown to induce unsteady blade loading and increase noise levels.14
Bhagwat and Leishman13 have observed characteristics of tip vortex development, and shown there is a logarithmic tip vortex core growth and corresponding velocity decay as the vortices convect below the rotor. Ramasamy et al.4 observed similar characteristics for MAV and found a strong correlation between vortex Reynolds number (Rev) and growth characteristics. Conventionally, Rev is defined as the circulation present in the rotor tip vortex system, normalized by the local kinematic viscosity. There has been great difficulty in modeling these characteristics for small-scale vehicles because the Rev of the vortex system typically exists in a transitional regime.15 Fig. 9a shows variation in the turbulent eddy viscosity with Rev, and Fig. 9b illustrates regions of varying turbulence associated with transitional vortices. In this case, common vortex models such as Lamb-
Oseen16 and Squire17, which respectively assume fully laminar and turbulent vortices, do not accurately predict the radially varying eddy viscosity term. Iversen’s18 core growth model improved upon these by modeling the non-linear turbulence growth in the transitional regime from experimental correlations. More recently, Ramasamy and
Leishman15 have worked to develop a model for transitional tip vortices which numerically computes turbulence as a function of core radius.
14 (a) (b)
Figure 9. (a) Variation in turbulent viscosity with vortex Reynolds number and (b)
turbulent regions of a transitional vortex.15
Another phenomenon that has been studied in great detail involves the temporal development and inherent instabilities of the wake geometry. Rapid growth of these wake instabilities, along with mutual and induced dynamics of the tip vortex filaments results in condition known as vortex pairing. In this case, interaction between two vortex filaments causes them to rotate about a common centroid as both convect axially in the wake. Figs. 10a-b illustrate this pairing phenomenon.
(a) (b)
Figure 10. Vortex pairing illustrations from (a) a two-bladed rotor experiment12 and
(b) a single-bladed water tunnel dye visualization.19
15 Tangler et al.20 and Caradonna et al.12 have experimentally observed this phenomenon for a two-bladed rotor while Quaranta et al.19 have done so by perturbing the wake of a single-bladed rotor. Their work has shown that the divergence of these long-wave instabilities which prompt vortex pairing is quite sensitive to seemingly insignificant sources of asymmetry. These include blade mistracking and minor rotor recirculation.
Blade number, thrust production, and potentially Reynolds number are also factors which govern the onset of pairing. Bhagwat and Leishman21 have developed a free vortex wake model and demonstrated that the presence of pairing from a numerical perspective is caused by truncation and round-off rather than physical disturbances noted in experimental tests.
Although a great deal of attention has been given to study of the rotor wake vortex system, insight into airfoil efficiency can also be obtained from phase-locked chordwise PIV. Benedict et al.22 were one of the few to conduct chordwise PIV measurements on a small-scale rotor. They discovered the presence of a large laminar separation bubble on a conventional NACA 0012 microrotor and a relatively small separation region over a thin cambered foil. Their findings suggest the lift-to-drag ratio
(L/D) can be significantly reduced in low-Re applications by using a thin cambered plate airfoil.
In addition to experimental work, recent advancements in the computational domain have enabled high fidelity modeling of these three-dimensional rotor wakes.
Yoon et al.23 implemented RANS/LES modeling, and demonstrated the loss in efficiency of a DJI Phantom 3 quadcopter as the rotor wake structures interact during forward flight.
16 Zhou et al.24 studied wake interactions between two closely spaced DJI Phantom rotors, and found that decreasing rotor spacing (i.e. blade tip to tip) from 1D to 0.05D, induced thrust fluctuations up to 2.5 times.
2.3 Acoustic Signature
The growing popularity of small-scale rotorcraft not only demands attention to aerodynamic efficiency, but also the inherent acoustic signature. Historically, many have observed the acoustic frequency spectrum produced by commercially available rotors, pointing out likely sources of noise. In the aforementioned work, Zhou et al.24 also integrated over the acoustic spectrum from 20-20,000 Hz, and concluded that the same difference in rotor spacing induced a 3dB increase in noise production. Intaratep et al.25 measured the acoustics of a model DJI 9443 rotor, and found significant broadband noise above 15 kHz which was attributed to laminar boundary layer vortex shedding. Similar types of airfoil self-noise have been studied, such as turbulent-boundary layer trailing edge (TE) noise, separation stall noise, and noise induced by shedding over a blunt TE.26-
27 However, there is a lack of quantitative chordwise measurements to physically capture such phenomena in the case of a rotor.
17 Chapter 3. Description of the Experiment
Three sets of two-bladed rotors were designed for conducting PIV, thrust, and torque measurements. They were geometrically designed to capture a specific Reynolds number range while isolating other aerodynamically dependent parameters. The blades were 3D-printed using a multijet fusion plastic material, and finished with a polished matte black color to reduce laser reflections during PIV testing. The blades were designed with two hollow canals running along the span. Carbon fiber rods were potted in these canals for added strength and rigidity. The spanwise filament layers resulting from the printing process yielded a step height of less than 10% of the blade chord length. A
NACA 0012 airfoil was used for each blade set with a constant-chord rectangular planform. This airfoil was selected since its relatively simple geometry made fabrication at various radii and chord lengths manageable. Additionally, performance of the NACA
0012 has been well characterized in high Reynolds applications. As such, this airfoil provided a good starting point for exploring deviations in performance and flow physics at low-Re conditions. The blades were untwisted with a constant aspect ratio of 5.0. Their dimensions are shown in Table 1, and the blades are illustrated in Fig. 11.
18 Table 1. Geometric rotor characteristics.
Chord Length [cm] Radius [cm] Solidity 1.91 13.96 0.058 2.77 16.96 0.068 2.95 18.19 0.072
Here, the rotor solidity (σ) accounts for the root cutout dictated by the rotor head assembly, and is defined as,
Nc() R R hub , (7) R2 where N is the blade number, R is the assembly radius, and Rhub is the radius of the root cutout.
Figure 11. 3-D Printed rotor blades.
The rotor head mounting point used a clevis design to constrain the rotor blades during operation. A schematic of the design is shown in Fig. 12. For the purposes of this study, the rotor blades were mounted in an inverted fashion, such that the wake convected
19 vertically upward, and thrust was produced downward. This was done to minimize wake interference from the test stand hardware.
Figure 12. Rotor head and rotor blade interface.
A custom rotor test stand was designed and manufactured with an articulated rotor head assembly to enable servo driven collective pitch control. A solid model of the stand is shown in Fig. 13, along with all key features of the design.
Figure 13. Rotor test stand.
20 The rotor head was coupled to a 5 mm 316 stainless steel rotary shaft which was pinned to a 0.75 in. 1045 carbon steel rod. The rod was 36 in. in length, and was belt driven by a
Baldor AC servo motor using timing pulleys of varying diameter. A gearing ratio of
2.66:1 was used to increase the rotor’s capable range of rotational speeds. The stand was coupled to a 50 lb. Omega LCCA-50 load cell directly beneath the rotational axis of the rotor shaft. The entire assembly was designed as a counterbalance, such that the load cell effectively saw negligible loading while the rotor was not in operation. Therefore, while the rotors produced thrust during operation, the full output range from the load cell could be utilized. The support shaft of the test stand was designed with a low profile to minimize inflow blockage to the rotor. To dynamically calibrate the power output from the motor, an HBM T22 2-Nm torque transducer was mounted and coupled in-line with the bottom of the rotor shaft.
3.3.1 Thrust
Thrust measurement were made over a series of six test conditions which were set to span the range of Reynolds numbers shown in Tables 2-3.
Table 2. Test conditions - vortex Reynolds number.
Solidity Ω [RPM] 0.058 0.068 0.072
Rev 3500 17,100 27,450 32,600 4500 22,000 35,280 41,900
21 Table 3. Test conditions - chord Reynolds number.
Solidity Ω [RPM] 0.058 0.068 0.072
Rec 3500 49,100 78,900 93,750 4500 63,150 83,460 120,500
The chord Reynolds number (Rec) was defined at 75% span as
0.75Rc Re , (8) c where Ω is rotational speed, R is the rotor radius, c is the blade chord length, and ν is the kinematic viscosity.
The function used to approximate vortex Reynolds number was derived using blade element theory, momentum theory, and an assumed inflow profile.10 The underlying assumption is that the bound circulation over the blade can be related to that contained in the shed tip vortices, such that
b CT k , (9) Rc Rc where σ is the solidity of the rotor, CT is the thrust coefficient, and k is a proportionality constant. For an untwisted blade, a value of 3 is assumed for k as suggested by Martin
28 and Leishman. Rev is calculated using
v 3(Rc ) CT Re , (10) for a known blade loading (CT/σ), rotor speed (Ω), blade radius (R), and chord length (c).
A full derivation of Rev is shown in Appendix A. 22 Thrust measurements were made with each of the three blade sets at 3500 and
4500 RPM, over a range of operational blade loading values. Spanwise flow visualization was conducted at a constant CT/σ and AR in an effort to isolate the effect of Reynolds number on wake structure behavior. Since aspect ratio AR and CT/σ are held constant, a different CT was required for each blade set. The capability to dynamically adjust the rotor collective angle during operation enabled fine incrimination of CT until the desired condition was achieved.
Prior to PIV data acquisition, a steady rotation speed (Ω) was attained, and voltage readings were acquired and averaged over 20,000 samples. The thrust was calculated from the load cell calibration function, and CT calculated using Eq. 4. This process was executed four times, and an average CT was obtained. The collective angle of the rotor blades was incrementally adjusted until the subsequent CT measurement converged on the set value.
3.3.2 Torque
Torque measurements were acquired for each of the rotor blades during operation at 2500 and 3500 RPM. The physical design of the test stand prohibited direct integration of a torque transducer for conventional in situ measurements. Instead, electrical power measurements (Pe) were obtained from the servo motor and calibrated to account for mechanical losses. This was done by manually applying a range of arbitrary torque values on the transducer at a given rotational speed, and recording concurrent mechanical torque
(Qm) and Pe values. This allowed for computation of an efficiency factor (η) as,
PQem (11) 23 The slope of the best fit line in Fig. 14 indicates an efficiency factor of approximately
0.0075 and 0.0053 for the 2500 and 3500 RPM cases, respectively.
Figure 14. Torque sensor calibration curves at 2500 and 3500 RPM.
The mechanical torque measurements allowed for computation of mechanical power (Pm) using
PQmm. (12)
In this case, Pm describes the aerodynamic power requirement of the rotor (Pa) and the power required to drive all other mechanical components of the system (Ps). It will be assumed that the aerodynamic power is composed of both induced rotor power (Pi) and profile rotor power (Po). Therefore, mechanical power can be written as
PPPPPPm a s i o s . (13)
For the purpose of evaluating parameters such as rotor FM, it is necessary to isolate the aerodynamic power requirement from Pm. Therefore, the system power will be quantified 24 and subtracted. This is done by first acquired mechanical power measurements at a unique operational condition in which the induced power requirement is negligible. Such a condition occurs at a low collective angle, where the lift production is reduced to zero and a minimum power requirement is observed. Ps can then be evaluated as
PPP (14) s m L0 o where P is the mechanical power at zero-lift. Determination of the profile power m L0 requirement was then estimated using XFLR 5, an open source GUI which utilizes the panel method. A NACA 0012 airfoil was assumed at the appropriate Re conditions to generate drag polars used for the approximation of the zero-lift drag coefficient (C ). d0
These polars are shown in Fig. 15 for all Re conditions.
Figure 15. Drag polars derived from XFLR 5 using a NACA 0012 airfoil.
25 The zero-lift drag coefficients extracted from the panel method analysis were then used to quantify the profile power using
3 3C 3 3 P C AR do AR . (15) opo 8
The resulting Po and Ps values for each Re condition is documented in Table 4.
Table 4. Profile power estimates for all Re conditions.
Rec 34,000 47,500 57,600 63,900 80,700 89.400
R [m] 13.96 13.96 16.96 18.19 16.96 18.19
Ω [RPM] 2500 3500 2500 2500 3500 3500
C 0.024 0.021 0.020 0.019 0.018 0.017 d0
Po 0.637 1.53 1.65 2.27 4.08 5.57
Ps 26.45 42.84 26.30 26.28 42.25 40.91
An example of the system power bias offset subtracted from the mechanical measurements for the 47,500 and 89,400 Re cases is shown in Figs. 16a-b. The resulting aerodynamic power measurements were then used to compute FM using Eqs. 5-6. Since
CCQP , (16) the power coefficient from Eq. 5 was also used to determine the torque coefficient (CQ).
26 (a) (b)
Figure 16. Mechanical and aerodynamic power measurements for (a) Re = 47,500
and (b) Re = 89,400.
3.4.1 Enclosure Design
Fig. 17 is an image of the laboratory setup used to conduct all PIV measurements.
The cubical enclosure is constructed from 80/20 t-slotted extrusion and covered with rigid polypropylene sheeting. It measures 1.96 m W x 1.96 m L x 1.88 m H, which is approximately 2.5-3 rotor diameters in all directions from hub to boundary. Two orthogonal sides of the enclosure have been fitted with clear acrylic windows (45 cm. x
60 cm.) to permit laser sheet entry and a clear camera field of view. A false floor was installed such that the only portion of the test stand exposed to the test volume was the shaft support. This was done to improve rotor inflow uniformity.
27
Figure 17. Laboratory PIV setup.
3.4.2 Wake Recirculation Study
Recirculation effects on a rotor are often observed when it is operated in close proximity to solid boundaries. Physically, it can be described as a scenario in which the complex wake structures impinge on nearby ground and side planes causing the flow to recirculate back toward the tip path plane as shown in Fig. 18. This event induces an increase in rotor inflow. If the rotational speed is maintained, the total pressure increase across the rotor will drop and consequently cause a loss in overall thrust production. The unsteady nature of wake recirculation has also been shown to induce fluctuations in thrust, and is therefore desirable to mitigate as much as possible.29
28
Figure 18. Rotor wake recirculation.
A study was conducted to address the potential for wake recirculation effects during operation within a confined space. The custom small, intermediate, and large sized rotors were tested in addition to a commercially available DJI S-1000 rotor. Each were operated over a range of speeds in variably sized enclosures to observe the effects of recirculation on the mean and fluctuating component of thrust. In this case, all blades were driven with a DJI S-1000 brushless motor which was coupled to an ATI Gamma SI-
130-10 six axis load cell. In order to set the custom rotor blades at a finite collective angle, 3-D printed adapters were designed to attach the blades to the brushless motors.
This enabled operation at a constant CT of 0.002. Figs. 19-20 show the mean thrust and torque from the small and large rotors over a range of speeds while in an open and enclosed environment. The PIV enclosure produced a house length to rotor diameter ratio of 7 and 5.4, respectively. 29
Figure 19. Thrust measurements from the small and large rotors over a range of
rotational speeds within the PIV enclosure and in an open environment.
Figure 20. Torque measurements from the small and large rotors over a range of
rotational speeds within the PIV enclosure and in an open environment. 30 There is little variation in the mean component of thrust and torque caused by the presence of the enclosure. As expected, there is a slight decrease in thrust production from the large rotor during operation at higher speeds in the PIV enclosure of at most 0.1
N. The fluctuations in thrust were also observed in both open and enclosed conditions.
Fig. 21 shows the RMS of the thrust measurements for the large and small rotors over a range of rotational speeds in both open and enclosed environments. At this relatively low operational thrust coefficient, the rotor wake velocity is low relative to typical operating
CT values of approximately 0.006. Therefore, it is expected that recirculation will have a lower effect on the performance measurements. The blades show no conclusive relationship between enclosure size and thrust fluctuations, but the RMS thrust values do increase with speed for both blade sizes.
Figure 21. RMS of thrust measurements for the large and small rotors in an open
and enclosed operating environment.
31 Figs. 22-23 show mean thrust and torque measurements from the DJI S-1000 over a range of enclosure sizes. The two smaller enclosure sizes were achieved by covering the rotor assembly with varying sized cardboard boxes. In this case, the nominal thrust production tends to decrease with enclosure size, especially at higher rotational speeds. It is suspected that the greater drop in performance due to enclosure size for the S-1000, relative to the custom blades, is the fact that the nominal operational CT is significantly higher. That is, the greater fixed collective angle of the S-1000 induces a greater pressure jump across the tip path plane, and when the inflow is obstructed by recirculation the thrust deficit is more evident. Despite this fact, there is still minimal variation between the thrust in an open environment and that in the PIV enclosure. The percent error between these measurements is approximately 8.5% at 4000 RPM and 7% at 2500 RPM.
Figure 22. Thrust measurements from the S-1000 rotor over a range of rotational
speeds within various enclosure sizes.
32
Figure 23. Torque measurements from the S-1000 rotor over a range of rotational
speeds within various enclosure sizes.
Thrust fluctuations were also observed for the S-1000 rotor in various enclosure sizes. Contrary to the minimal difference in fluctuations of the custom blades, Figs. 24a-d clearly show how the onset of recirculation depends on enclosure size. Fig. 24b shows how after the initial transition to operational speed near the 10 second time stamp, a following period of approximately 4 seconds occurs in which thrust fluctuations appear nearly identical to those in the open environment. Following this period, there is a significant jump in the fluctuations which indicates that the rotor wake has started to recirculate back through the disk. The period between startup and recirculation is significantly shortened for DW/DR = 2.5 in Fig. 24c. This is because a smaller enclosure is used and the period of time required for the flow to circulate back to the tip path plane is significantly shortened. Finally, at DW/DR = 1.5, fluctuations begin to occur much sooner,
33 and the magnitude is also significantly greater. In this case, the period of undisturbed inflow evident in Fig. 24b has vanished entirely because recirculation initiates almost immediately.
(a) (b)
(c) (d)
Figure 24. Thrust fluctuations of the DJI S-1000 for an (a) open environment and
enclosure sizes of (b) DW/DR = 5, (c) DW/DR = 2.5, and (d) DW/DR = 1.5.
The RMS of the thrust measurements from the S1000 in each enclosure size is shown in Fig. . As expected, there is a direct relationship between the RMS values and
34 rotational speed. A decrease in enclosure size also results in an RMS increase. Also, the rate at which the fluctuations increase with rotational speed grows significantly as enclosure size decreases. Despite the small differences noted between the open environment and the PIV enclosure, laboratory spatial constraints limited the enclosure size. Based on the observations made here, it is expected that even a slightly larger enclosure size would likely delay the onset of recirculation, but would still exist in the flow. As such, the existing PIV enclosure was accepted for conduction of rotor wake flow visualization.
Figure 25. RMS of thrust measurements for the S-1000 rotor operating in various
enclosure sizes.
3.4.3 Laser Optics
In this PIV configuration, a New Wave 532-nm wavelength Nd:YAG laser was used to produce the laser sheet for particle illumination. It has the capability to generate 35 dual laser pulses at 50 mJ. Three 90-degree turning mirrors were used to orient the beam at the correct height. One cylindrical lens was then used to generate a laser sheet, and one
1000 mm focal length spherical lens was used to reduce the sheet width to approximately
2 mm at the region of interest. The optical configuration was mounted on a traverse to easily adjust the laser sheet position for chordwise measurements at various spanwise locations. A schematic of the optical configuration is shown in Fig. 26.
Figure 26. Laser optics configuration.
3.4.4 Imaging Configurations
Phase-resolved, two-component PIV was taken with the laser sheet aligned in the spanwise and chordwise directions. The spanwise measurements enabled rotor wake visualization, and provided insight into development of the vortex system and spanwise wake sheet interactions. In this configuration, two 16-bit LaVision Imager sCMOS cameras were implemented with 85 mm focal length lenses. The dual camera configuration enabled a larger field of view, with each camera spanning approximately
36 one rotor radius. An illustration of the rotor position relative to the cameras and laser is shown in Fig. 27.
Figure 27. Spanwise PIV configuration.
In the spanwise laser configuration, images were taken with the rotor positioned in 26° increments relative to the position parallel to the laser plane. This enabled wake structure visualization as they aged and convected further beneath the rotor plane. Fig. 28 shows the phase increments used for these experiments.
Figure 28. Spanwise PIV recorded phase increments.
37 Imaging in the chordwise plane showed flow characteristics over the suction side of the blade and downstream vortex shedding. These features are of interest when assessing the aerodynamic behavior of the airfoil section. For these measurements, a single sCMOS camera was implemented with a 75-300 mm zoom lens. In this case, the region of interest around the blade’s small chord length was best resolved with the lens set to a 300 mm zoom. A schematic of the chordwise configuration is shown in Fig. 29.
Figure 29. Chordwise PIV configuration.
To select a time differential between successive PIV image pairs, an optimization study was performed. A time difference of 30 μs was found to provide adequate particle correlation with minimal noise.
3.4.5 Phase-Lock Triggering
A twice-per-rev optical encoder using an external green laser was mounted directly above and orthogonal to the rotor plane, to track rotor speed. A phototransistor, mounted directly below the rotor received the encoder laser to detect blade passage. A schematic of the PIV system is shown in Fig. 30.
38 One hundred phase-locked images were taken for each test case, as minimal statistical variation was observed beyond this number of samples. This was accomplished by extracting the rotor signal from the optical encoder, generating a designated phase offset using an automated LabVIEW program, and providing the PIV timing unit with the delayed signal. The timing unit was used to synchronize and trigger the PIV camera and laser. The sCMOS cameras have a minimum 10 ms response time which required an additional offset of this magnitude to be subtracted from the delayed signal. All components within the enclosure were painted matte black to minimize laser reflection and image saturation.
Figure 30. PIV system schematic.
39 3.4.6 Time Resolved Measurement Configuration
Spanwise time resolved measurements were also conducted to gain more insight into wake structure evolution. For these tests, a Phantom v1210 high speed camera was synchronized to a Photonics DM Series, dual cavity, 532-nm wavelength Nd:YAG laser.
In this case, a high speed programmable timing unit (PTU) was implemented and data was acquired using the same facility. Details of the time resolved measurements are shown in Table 5.
Table 5. Time resolved measurement conditions.
Parameter Units Value Frame Rate kHz 12.34 Images acquired - 2000 (~9.5 rotor revolutions)
Blade Radius cm 12.7 Rotational Speed RPM 3500
CT Range - 0.003 – 0.007
3.4.7 Seeding
An atomizer was used to generate olive oil seeding particles which were injected into the test enclosure via plastic tubing. The seeding entry point was near the rotor inflow, and a constant flow rate was maintained for the duration of a given test. No mid-test adjustments in seeding were required as the density of the particles remained relatively consistent during data acquisition.
3.4.8 PIV Data Processing
PIV vector calculation was performed using DaVis 8.4.0. Multi-pass, sequential cross-correlation processing parameters were selected. A first-pass window size of 48 x 40 48 pixels was used, followed by three passes using a 24 x 24 pixel window with 50% overlap. For each test case, all images were exported to MATLAB and individually processed.
For each test case, all spanwise images were individually processed for vortex identification using the delta criterion as described by Epps.30 The centers were located by taking the mean of all boundary points surrounding each vortex in a given image. To fine-tune the vortex center position, several vertical and horizontal velocity profiles were taken, and the profile with the largest peak swirl velocity was assumed to be that which crosses the centerpoint. The derivative of the maximum horizontal and vertical profiles were taken, and the position corresponding to the greatest magnitude provided the X and
Y-coordinates of the tip vortex center.
Tip vortex sizes and induced velocity profiles for a given phase were computed by finding the average quantities among one hundred instantaneous images at a given phase. To obtain these averages several corrections were made to the instantaneous images. First, tip vortex aperiodicity was accounted for by aligning the center of each core to a uniform location prior to averaging. This location corresponded to the vortex center found from the phase averaged image. Second, the induced velocity profiles were corrected by subtracting out the axial convection speed from the total vertical velocity.
This was achieved by computing a time average of the entire velocity field, and subtracting the local velocity at the respective core location from the entire velocity cross section through the core. Since the induced velocity is theoretically zero at this location,
41 the difference of the operation is the induced velocity. An example of the velocity components is shown in Fig. 31.
Figure 31. Tip vortex velocity components.
Successive instantaneous images from the chordwise PIV measurements suggest less than one degree of azimuthal walking. This is attributed to expected small fluctuations in latency of the Arduino when generating the signal offset. Consequently, it was required that the instantaneous images were aligned prior to computing an ensemble average for each data set. The alignment was done by converting the raw PIV images to a binary form and adjusting a threshold to extract the location of the airfoil TE. The corresponding location in the vector matrices were then shifted to a uniform location.
Acoustic experimentation was conducted in the OSU anechoic chamber, shown in
Fig. 32. Internal dimensions of the facility (wedge-tip to wedge-tip) measure 5.14 m W x 42 4.48 m L x 2.53 m H. Far-field acoustic pressure was recorded at a polar angle of ninety degrees relative to the rotor tip path plane. The microphone was approximately 1.82 m from the rotor hub, and was oriented with the microphone tip in that direction. Wright31 showed that, after neglecting the hydrodynamic effect, the sound pressure level is approximately the same on the inflow and wake side of the rotor. However, to avoid interference from pressure waves generated by the wake, the microphone was positioned on the inflow side of the rotor.
All acoustic measurements were made by operating the rotors independent of the custom test stand due to lack of mobility. To replicate the rotor collective angles corresponding to an operational CT of 0.002, the same 3-D printed adapters were used, as described in the recirculation study. This allowed for use of the custom rotor blades independent of the custom test stand. A DJI S1000 motor was used to drive the blades, and was powered by a 22V DC power supply in conjunction with an electric speed controller (ESC). The rotational speed was dictated by a prescribed duty cycle generated by an Arduino UNO pulse width modulation (PWM) output. Rotor frequency was monitored and modulated using a Melexis US5881 Unipolar Hall-effect switch in a feedback loop to the Arduino.
Acoustic far-field measurements were acquired using a Bruel & Kjær 4939-A-011 microphone. The signal was band-pass filtered from 20 Hz to 25 kHz using a Bruel &
Kjær Nexus 2690 signal-conditioning amplifier, and recorded using a National
Instruments PXI-6133 A/D board and LabView software. The microphones were calibrated using a Bruel & Kjær 94 dB, 1 kHz sine wave generator. Voltage signals were
43 collected at 200 kHz with 65,536 data points per block, resulting in a frequency resolution of 1.526 Hz. One hundred blocks were recorded for each case resulting in roughly 60 s of data, allowing for adequate convergence of statistics. Data was processed in MATLAB using a discrete Fourier transform (DFT) with a fast Fourier transform
(FFT) algorithm. The pressure power spectrum was converted into sound pressure level
(SPL) in decibels (dB), using
p SPL 20log10 , (17) p0 where p0 = 20 μPa. Background (quiescent) noise amplitudes were acquired, but were not subtracted from the data. This background noise is shown in the acoustic data plots as the lowest SPL curve. Validation of the chamber demonstrated a cutoff frequency of 160 Hz for the design, so SPL amplitudes of frequencies below this cutoff frequency may not be accurate.32
Figure 32. The OSU anechoic chamber.
44 Chapter 4. Results and Discussion
4.1.1 Thrust Measurements
The motivation for thrust measurements in this work was twofold. First, thrust measurements were necessary to ensure PIV data acquisition was acquired at a desired blade loading condition. This was achieved by taking dynamic thrust measurements immediately prior to PIV. Subsequent micro-adjustments to blade collective angle were made to lock into the desired CT/σ. Secondly, thrust measurements from the rotor were made to understand their performance characteristics as a function of both collective angle and Reynolds number.
Fig. 33a shows the variation in thrust as a function of Ω and θ for the smallest blade set. As expected, thrust tends to increase with rotational speed and collective angle. The increase in thrust with θ is linear at lower collective angles (<10o). However, as θ is increased further, the gain in thrust begins to asymptote. Fig. 33b shows the corresponding increase in blade loading with collective angle, but little variation with speed. This result has been documented by others, and occurs because the computation of thrust coefficient contains a simultaneously increasing thrust in the numerator and rotational speed in the denominator.33 The mathematical relationship between the two
45 results in a condition in which blade loading scales almost independently from speed, especially higher Ω.
(a) (b)
Figure 33. (a) Thrust and (b) blade loading measurements for the small sized rotor
at varying rotational speeds and collective angles.
Figs. 34-35a show similar measurements for the medium and large sized blades.
As blade size increases, thrust production at a given collective angle and rotational speed also increases. This is simply because the larger blade size generates a greater bound circulation across the span, as defined by the Kutta-Joukowski theorem in Eq. A.2.
Despite the variation in thrust with blade size, Figs. 34-35b show that blade loading remains relatively constant. This is because, the increase in rotor thrust is also accompanied by an increase in solidity. The two increasing parameters nearly cancel in the computation of blade loading.
46 (a) (b)
Figure 34. (a) Thrust and (b) blade loading measurements for the intermediate sized
rotor at varying rotational speeds and collective angles.
(a) (b)
Figure 35. (a) Thrust and (b) blade loading measurements for the large sized rotor
at varying rotational speeds and collective angles.
The s-beam load cell used to acquire the thrust measurements is rigidly mounted, directly beneath the test stand structure. As such, any vibrations in the test stand hardware propagated to the load cell during operation. Although the configuration was capable of
47 generating repeatable mean thrust measurements for a given condition, the inherent vibrations also induced a fluctuating component. Figs. 36-38 represent the standard deviation of the thrust and blade loading for all three blade sets.
(a) (b)
Figure 36. Standard deviation of (a) thrust and (b) blade loading for the small size
rotor.
(a) (b)
Figure 37. Standard deviation of (a) thrust and (b) blade loading for the
intermediate sized rotor.
48
(a) (b)
Figure 38. Standard deviation of (a) thrust and (b) blade loading for the large sized
rotor.
The standard deviation of each data set was computed by first filtering out high frequency fluctuations in the raw data which can be attributed to vibrations, rather than physical variations in thrust. A third-order, low-pass butterworth filter was designed, and the applied cutoff frequency was selected based on a fast Fourier transform (FFT) of the raw data. Specifically, the cutoff was selected as the lowest peak frequency corresponding to a physical phenomenon. Fig. 39 shows the FFT of the raw thrust data from the small blade operating at 2500 RPM.
49
Figure 39. FFT of raw thrust measurements of the small rotor at 2500 RPM.
Here, the middle peak is at approximately 15.7 Hz which corresponds to the shaft frequency of the servo motor. Since the gearing ratio between the servo and rotor shaft is
2.66, the rotor shaft frequency should appear at 41.8 Hz. This corresponds to the third peak in the spectrum. For the purpose of filtering, a cutoff of approximately 15 Hz was selected. An example of the raw and filtered data is shown in Fig. 40.
50
Figure 40. Raw and filtered thrust data for the small rotor at Ω = 2500 RPM and
θ = 3.5o.
The trends among each rotor show little variation in the fluctuating component of the measurement as a function of θ. However, there is a notable difference with change in rotational speed. This corresponds with elevated vibrations in the structure when operating in close proximity to a resonance peak. Specifically, it was qualitatively noted that resonance frequencies occurred at 3000 and 4000 RPM. The elevated fluctuations at
3000 RPM shown in Figs. 36-38 confirm this is the case. No data was acquired at 4000
RPM due to the severity of vibrations. The magnitude of the thrust fluctuations remained nearly constant with blade size, while corresponding blade loading fluctuations drop with blade size due to inherently lower blade radius and area used in the computation of CT.
51 4.1.2 Torque Measurements
Fig. 41 shows the figure of merit calculated over a range of chord Reynolds number conditions. Here, the Rec conditions at 57,600 and 63,900 were recorded for the intermediate and large size rotors at 2500 RPM, while the remaining data sets represent all three blade size at 3500 RPM. The corresponding dotted lines are fourth order polynomial fits to the data. In all cases, the blade loading is incremented by performing a sweep of the rotor collective angle. It is expected that Reynolds number should scale directly with FM. However, this only holds true when Rec is scaled geometrically. That is, variation in Rec by rotational speed caused a different trend entirely.
The reason FM trends depend more heavily on rotational speed can be linked directly to the corresponding variation in inflow velocity. The lower inflow during operation at
2500 RPM, especially at lower collective angles, implies that axial convection of the tip vortex system and spanwise wake sheets is much slower. As such, a given rotor will always operate in closer proximity to these structures when it is spinning more slowly.
Operation near complex wake structures exacerbates non-ideal conditions and imparts a greater power requirement from the rotor to maintain operation. Thus, the corresponding figure of merit will experience a greater deficit in such conditions.
Observing the test cases at a constant 3500 RPM, a significant difference is seen between the low-Reynolds number condition relative to the two larger. Specifically, the higher Reynolds numbers exhibit a slightly greater maximum figure of merit, which is achieved at a lower blade loading value. Effectively, this implies the larger blades are more aerodynamically efficient while operating at a lower collective angle. Although it
52 seems the difference in Reynolds number between the two larger blades is not negligible, their performance characteristics appear to be almost identical. This is attributed to the
5 11 diminishing effects of profile drag as Rec approaches 10 . Ramasamy et al. demonstrates comparable figure of merit values, specifically in comparison to the higher Rec cases.
Since their experiments were conducted at chord Reynolds numbers exceeding 150,000, it is evident that figure of merit becomes progressively less dependent on Rec as it is scaled up.
Figure 41. Figure of merit for varying Reynolds number conditions.
Fig. 42 shows the induced torque relative to blade loading for the same Rec conditions. Again, the trends at 2500 RPM can be distinguished from those at 3500 RPM.
The 2500 RPM conditions show a notably higher induced torque value relative to those at
-2 3500 since CQ goes by Ω . However, regardless of the rotational speed, low Re conditions at low θ show a significant induced torque spike relative to the higher Re 53 cases. This is attributed to the low inflow velocities for the low Re conditions, especially at low collective angles. As θ increases, a minimum torque demand accompanied by a peak FM is observed. This aerodynamically optimal operating condition is achieved at higher blade loading for the low Re cases, relative to the two highest. As this condition is approached, the torque requirement for all operational conditions at 3500 RPM coalesces to a single curve. This suggests that, for the studied Rec conditions, the induced drag acting on the rotors tends to dominate as thrust coefficient is increased. At low blade loading values, there is minimal induced drag for all cases due to limited thrust production, but a significant profile drag component inherent to the low Rec condition.
Hence, the elevated torque requirement is a reflection of the degraded aerodynamic efficiency, and confirms that low-Reynolds conditions are dominated by profile drag.
Figure 42. Induced torque for varying Reynolds number conditions.
54 Figs. 43-44 show the measured FM and induced torque data relative to a theoretical approximation of the terms. Figure of merit was estimated from Eq. 6, while
induced torque is computed from the relationship in Eq. 16 and the Cp, actual approximation. In both cases, C and κ must be estimated for the particular operating do condition. This was done by applying a least-squares fit to the experimental data for each
Reynolds number condition. The range of predicted and κ values at high Re conditions is comparable to those suggested by Ramasamy et al.4 The elevated zero-lift drag coefficients predicted for the low Re cases is likely caused by the domination of profile drag for these conditions. The correlations match quite well for both FM and induced torque. However, a significant deviation from theory is observed in the torque comparison at low blade loading. This suggests the potential cause for the discrepancy may potentially be the result of non-ideal losses in the wake which are not accurately modeled by this theoretical approximation.
55
Figure 43. Comparison between FM measurements and a theoretical
approximation.
Figure 44. Comparison between induced torque measurements and a theoretical
approximation.
56 4.1.3 Performance Uncertainty Measurements
In addition to thrust variations induced by vibrational effects, an uncertainty analysis was performed for a single test case. Specifically, a series of ten separate measurements were made using the small rotor over a range of collective angles while operating at 3500
RPM. The uncertainty analysis was performed at these conditions since the induced torque measurements suggest the greatest amount of variability. Therefore, the results proceeding from this analysis will set an upper bound for the expected variability among the remaining test cases. The procedure used for each of the ten measurements included assembly of the rotor blade, setting of the collective angle, transition to operational speed from rest, and a 30 second measurement window during which voltage data were acquired at 25 kHz.
Each of the resulting data sets are shown in Fig. 45. The variation among them remains relatively consistent with collective angle, suggesting there is a unique bias offset associated with each run. This is attributed to small fluctuations in the load cell offset used to compute thrust from the calibration curve.
57
Figure 45. Variation in thrust measurements taken with the small rotor at 3500
RPM.
The uncertainty of thrust and blade loading were computed directly from the standard deviation of the data sets assuming a 95% confidence interval as
2 U T . (18) T N where N is the number of data sets used in the computation. The mean of the data sets and corresponding uncertainty are shown in Figs. 46-47.
58
Figure 46. Thrust uncertainty measurements for the small rotor operating at 3500
RPM over a range of θ.
Figure 47. Blade loading uncertainty measurements for the small rotor operating at
3500 RPM over a range of θ.
59
Figure of merit and induced torque uncertainties were computed from the general uncertainty equation as
2 x Uuxa , (19) a where Ux represents the uncertainty of a given parameter, x. The first term in the summation is the sensitivity coefficient, which quantifies the impact of a given variable on the overall uncertainty of the parameter in question. The second term is the uncertainty of that variable. The governing equations used to compute the sensitivity coefficients are from the definition of FM in Eq. 6, and the definition of induced torque,
C CC 32 C Q PT do . (20) 2 8
In each case, the sources of uncertainty are assumed to come exclusively from CT and
C .The uncertainty associated with these measurements are shown in Figs. 48-49. do
60
Figure 48. Figure of merit uncertainty measurements for the small rotor operating
at 3500 RPM over a range of blade loading conditions.
Figure 49. Induced torque uncertainty measurements for the small rotor operating
at 3500 RPM over a range of blade loading conditions.
61
4.2.1 Tip Stalling
Since blade loading is held constant while seeking Reynolds dependent wake characteristics, it is necessary to determine an appropriate condition at which to operate.
In this case, blade loading is controlled via adjustment of the rotor collective angle. Since the rotor solidity varies among the different blades, a different collective angle is required for each to achieve the appropriate CT offset. To determine an appropriate blade loading at which to study the flowfield, instantaneous PIV images were obtained while executing a sweep of θ.
a) b)
c) d)
Figure 50. Instantaneous vorticity contours using the large blades at 3500 RPM for
(a) CT = 0.001 (b) CT = 0.005 (c) CT = 0.0075 (d) CT = 0.01.
62 Fig. 50 shows representative instantaneous vorticity contours of the rotor wake propagating beneath the large rotor at 3500 RPM. The PIV images were recorded for a constant wake age of 155 degrees over a CT range from 0.001 to 0.01. In this case, the wake age is defined for a given vortex as the angular displacement experienced by the leading edge of the rotor blade since that blade produced the vortex. Results indicate that for low CT, thrust is relatively low, and the resultant weak bound circulation over the blade generates weak tip vortices. As CT is increased between 0.005-0.0075, which corresponds to a θ range of 7-13 degrees, the tip vortices appear more highly concentrated and will allow for a more significant statistical analysis of the vortex characteristics in the wake. As CT is increased beyond this range, the vortices begin to break down entirely, despite the steady increase in thrust. This suggests that, at this higher θ range, local outboard blade tip stall begins to occur. The absence of a pressure gradient over the pressure and suction side of the blade tips caused a significant decrease in lift. The result is weak and unsteady vorticity shedding near the slipstream boundary.
While performance measurements and chordwise flow visualization were conducted over a range of blade loading conditions, the spanwise PIV data sets were acquired with each blade set at a constant CT/σ. Based on the spanwise tip stalling observations, all phase-locked spanwise PIV measurements were conducted at a blade loading of 0.087. This enabled operation for each of the rotors within a CT range which produced well-defined and repeatable vortex systems. The operational parameters used for each of the rotors are summarized in Table 6.
63 Table 6. Operational conditions used for rotor wake flow visualization.
Blade R [cm] σ CT CT/σ
Small 13.96 0.058 0.0050
Medium 16.96 0.068 0.0059 0.087
Large 18.19 0.072 0.0062
4.2.2 Chordwise PIV
Chordwise PIV measurements were acquired over a set of five test cases. Three of the cases correspond to each blade set at a fixed spanwise location equal to 90% of the respective rotor radius. The remaining two cases were conducted with the large rotor at
85% and 95% of the radius. In each test case, images were acquired over a range of collective angles which effectively varied the operational thrust coefficient. Table 7 summarizes the details of all chordwise PIV measurement conditions.
Table 7.Chordwise PIV test conditions.
Measurement Test Case R [cm] σ Rec CT Location [%R] 1 13.96 0.058 90 57,500
2 16.96 0.068 90 96,500
3 85 98,400 0.001-0.009
4 18.19 0.072 90 106,700
5 95 115,000
64 Figs. 51-53 show an ensemble average of spanwise vorticity in the wake of the large rotor blades for varying CT. In this case, the laser sheet was aligned at 90% of the total span or approximately 85% of the total lifting surface of the blades. In Fig. 51, there is a region of concentrated vorticity which spans the majority of the chord-length, and the wake is marked by characteristic regions of counter-rotating vorticity shed at the trailing edge. As θ is increased, counter-rotating flow towards the trailing edge of the airfoil becomes visible in the ensemble average. This suggests that flow separates without reattachment, forming a highly turbulent region which propagates further downstream in the wake. If the collective angle is increased further, the point of separation moves toward the LE, and a wider band of counter-rotating flow develops in the wake. The physics captured in these chordwise measurements directly correlate with the breakdown of tip vortex formation noted in the spanwise measurements. In Fig. 50, the absence of tip vortices becomes evident starting at thrust coefficients beyond 0.0075. Comparatively, the chordwise measurements reveal a coherent flow field at CT = 0.007, followed by a highly turbulent downstream wake for CT ≥ 0.008. These flow field characteristics, which dominate the ensemble average, confirm that the absence of tip vortices can be attributed to blade tip stalling as thrust coefficient exceeds a critical value of approximately 0.0075.
65
Figure 51. Ensemble average of chordwise vorticity at Rec = 106,700 and CT = 0.007.
Figure 52. Ensemble average of chordwise vorticity at Rec = 106,700 and CT = 0.008.
66
Figure 53. Ensemble average of chordwise vorticity at Rec = 106,700 and CT = 0.009.
Figs. 54-57show the velocity magnitude over the suction side of the rotor blade at constant collective angles for varying Reynolds number conditions. Huang and Lin34 demonstrated that for a fixed wing NACA 0012 airfoil, the onset of laminar separation occurs as Rec is increased beyond a critical value. In fact, they suggest that fully attached laminar flow is only attainable for Rec < 20,000 at very low angles of attack. Although the highly three-dimensional flow field inherent to a rotor wake makes a direct quantitative comparison difficult, it is feasible to expect that separation will be present for the Reynolds number range studied here.
The ensemble average in Fig. 54 shows the horizontal component of velocity is significantly greater than the vertical component. This is because the horizontal component is comparable to the local tangential velocity of the rotor. Also, a momentum deficit near the quarter chord position is evident. In this case, the deficit corresponds to the development of a laminar separation bubble in this region. Figs. 54-55 show that at the same operational thrust coefficient, the chordwise position of the separation bubble
67 moves closer to the leading edge as Rec is increased. These observations are concurrent with Benedict et al.22, who also noted the presence of a laminar separation bubble for a
NACA 0012 micro-rotor operating in even lower Reynolds number conditions (10,000 <
Rec < 50,000).
Figure 54. Ensemble average of velocity magnitude at Rec = 96,500 and CT = 0.002.
Figure 55. Ensemble average of velocity magnitude at Rec = 106,700 and CT = 0.002.
68 Similar flow features are noted over the same Reynolds number range and spanwise location at significantly higher collective angles (CT = 0.008). Fig. 56 shows a momentum deficit indicative of a separation bubble at x/c = 0.20. As Rec is increased
(Fig.57), the bubble shifts further toward the leading edge of the airfoil. In this condition, a momentum deficit also appears closer to the trailing edge, indicating flow separation.
This observation is confirmed by the fact that this operational condition is concurrent with the fully turbulent downstream wake shown by the vorticity contours in Fig. 52.
Figure 56. Ensemble average of velocity magnitude at Rec = 96,500 and CT = 0.008.
69
Figure 57. Ensemble average of velocity magnitude at Rec = 106,700 and CT = 0.008.
Recalling Figs. 33-35, growth in thrust (i.e. the derivative of the thrust curves) tends to decrease as the collective angle is increased. The eventual decrease in thrust growth appears to occur near the same blade loading conditions as the breakdown of the tip vortex system. In other words, the gradual decrease in thrust growth with respect to collective angle is physically manifested by characteristics of the structures within the wake. Specifically, as tip stalling begins to occur lift production is diminished at the outermost spanwise location. Since the inboard portion of the blade has yet to stall, the overall thrust production will continue to grow with collective angle. As θ is increased further, stalling will begin to occur progressively further inboard, and eventually prompt an overall loss in thrust.
This stalling phenomenon is also observable in the FM and torque curves. In Fig. 41 the blade loading value at which the peak FM is achieved typically occurs at a higher value for the lower Reynolds number conditions. In other words, lower Reynolds
70 conditions will operate with peak aerodynamic efficiency at higher collective angles. This is due to the fact that outboard tip stalling initiates at lower collective angles for higher
Reynolds number and higher angles for lower Rec.
Similarly, the minimum induced torque for a given Reynolds number in Fig. 42 also depends on blade loading. In this case, minimum torque achieved by the rotor occurs at higher CT/σ for lower Rec, and vice versa. Since the chordwise PIV indicates separation also occurs according to this trend, it is concluded that the onset of tip stalling will have a significant impact on the torque requirement for the rotor.
4.2.3 Spanwise PIV
Phase averaged vorticity plots are shown in Fig. 58 for a constant wake age at three vortex Reynolds numbers. Fig. 58a shows the flow field characteristics for the smallest rotor at operating at 3500 RPM for an Rev of 17,100. The cases shown in Figs. 58b and
58c were taken with the largest rotor at rotational speeds of 3500 and 4500 RPM, respectively. This produced corresponding Rev values of 32,600 and 41,900. In each case, the spanwise wake sheets shed from the trailing edges exhibit a linear profile as they propagate downstream in the wake. This linear behavior suggests a linearly increasing spanwise inflow distribution, which is characteristic of untwisted rotor blades.7
Comparing the three cases, it is evident that the vorticity strength concentrated in the wake sheets and the tip vortices becomes progressively greater with Rev, which is the result of increasing lift production and bound blade circulation. In the two higher Rev cases, elongation of tip vortices is noted at later wake ages.
71 (a)
(b)
(c)
Figure 58. Phase average vorticity contours for (a) Rev = 17,100 (b) Rev = 32,600 (c)
Rev = 41,900.
72 A more in depth study of the instantaneous images reveals a bi-modal wake geometry exclusive to the 22,000 vortex Reynolds number condition. The small rotor operating at
4500 RPM was used in this case, and Fig. 59 shows phase averaged images as the vortices develop in the wake. The tip vortex at a wake age of 280 degrees, shown in Fig.
59b, appears as a two-lobed structure which appears to divide into two smaller vortices of lesser concentration in Fig. 59c. A more in depth analysis of the instantaneous images suggests this apparent splitting phenomenon is likely an artifact of a bi-modal wake geometry. That is, successive instantaneous images at a constant wake age show intermittent vortex pairing which occurs at a consistent spatial location in roughly 50% of the 100 image set. The other half of this set shows development of the vortex system without the pairing phenomenon in the recorded field of view. An example of each condition is shown in Fig. 60a-b. When these images are averaged together, the two- lobed structure appears resulting from an ensemble of the vortices from each mode. The pairing phenomenon was also observed for the next two larger Rev conditions tested
(27,450, and 32,600). These vortex Reynolds numbers were achieved with the intermediate and large sized rotor while operating at 3500 RPM. In both of these cases, the spatial location of the pairing phenomenon was repeatable relative to the lower Rev case.
73
(a) (b) (c)
o o Figure 59. Phased averaged vorticity of at Rev = 22,000 for (a) ψ = 1 , (b) ψ = 104 ,
and (c) ψ = 155o.
a) b)
Figure 60. Bi-modal wake geometry at Rev = 22,000.
74 4.2.4 Time Resolved Measurements
The apparent vortex splitting phenomenon discussed in the previous section was studied further by conducting time resolved flowfield measurements of a single representative Rev condition of 27,450. The time resolved measurements allow for image acquisition approximately 81 μs apart. When operating at a rotational speed of 3500
RPM, this corresponds to 1.7 degrees per frame or about 211.5 frames per revolution.
Therefore, such measurements will provide great insight into development and interaction of wake structures. Figs. 61a-f show time resolved images for the intermediate sized rotor operating at 3500 RPM over successive closely spaced instances. The time stamps marked on the images indicate the time elapsed since vortex “A” was shed from the rotor.
75 (a)
(b)
(c)
Figure 61. Spanwise time resolved images for Rev = 27,450 and CT = 0.0059 at (a) t =
8.1 ms (b) t = 12.6 ms (c) t = 18.4 ms (d) t = 21.0 ms (e) t = 23.1 ms and (f) t = 26.8 ms. 76 (d)
(e)
(f)
77 In Figs. 61a-b the tip vortices A1, A, B, and C appear to convect in the wake as expected. For this particular operating condition, the divergence of inherent long-wave wake instabilities are observed as vortices B and C begin to revolve about their common centroid. The evolution of this vortex pairing phenomenon between vortices B and C is shown in Figs. 61c-f. As the time progresses, it is observed how the vortices continue to rotate about each other and simultaneously convect and diffuse in the wake.
Caradonna et al.12 and Tangler et al.20 have shown that the onset of this pairing is strongly a function of the thrust production. That is by decreasing thrust, the rotor inflow, effective wake velocity, tip vortex spacing, and vortex circulation will be decreased.
Bhagwat and Leishman21 have shown that the divergence rate of the wake instability tends to decrease with increasing thrust. That is, the wake becomes more stable over time. They also observed that the divergence rate increases with vortex circulation and decreases with vortex spacing. Therefore, it can be concluded that vortex spacing plays a more significant role in the stability of the wake.35 Figs. 62 and 63 show the vortex pairing phenomenon for the same rotor operating at the same speed with CT of 0.004 and
0.003, respectively. In this case, the successive decrease in thrust coefficient implies a decrease in rotor thrust, vortex spacing, and circulation. As expected, the pairing phenomenon occurs closer to the rotor tip path plane, suggesting the presence of a more rapidly diverging long-wave instability.
78
Figure 62. Spanwise time resolved images for Rev = 27,450 and CT = 0.004.
Figure 63. Spanwise time resolved images for Rev = 27,450 and CT = 0.003.
Fig. 64 shows the wake geometry for the same rotor at the same speed with a larger thrust coefficient of 0.007. The increase in thrust coefficient causes greater vortex spacing and greater circulation, but the previously observed pairing event is not evident
79 in the set field of view. This suggests either pairing does not occur prior to vortex dissipation, or that it occurs at a much later wake age than was previously observed in lower thrust conditions.
Figure 64. Spanwise time resolved images for Rev = 27,450 and CT = 0.007.
Although the pairing phenomenon is dramatically impacted by thrust production, more work will have to be conducted to determine whether there is an explicit Re dependency. To explore the potential for vortex Reynolds dependency, it is necessary to isolate the impact of wake convection speed for the various test cases. Effectively this would require a method to systematically control Rev with a minimum impact on the wake velocity. Since the time resolved measurements were observed over a range of CT conditions, the vortex Reynolds number was effectively varied according to Eq. 10. That is, a lower CT results in a lower Rev which corresponds to closer spacing between the pairing event and the tip path plane, relative to high Rev cases. The closer proximity of this event to the tip path plane likely contributes to performance characteristics in Figs. 80 41-42. Specifically, the dramatic increase in torque and significantly lower FM, especially at low collective angles, suggests the wake instability induces a degraded level of performance.
The dynamic characteristics of the tip vortices were also observed over a range of
Rev. The direction of tip vortex wandering in the rotor wake has been shown to depend significantly on wake age. Mula et al.36 has shown that the greatest aperiodic motion of the vortices tends to be in the direction normal to the slipstream boundary. Using a delta criterion vortex identification scheme, the coordinates of each vortex location, obtained from 100 instantaneous phase-locked PIV images, were determined. These vortex locations at a given wake age were then averaged, and a third order polynomial was fit to the average vortex locations to determine the slipstream boundary.
The locus of vortex locations at each wake age was rotated by the angle between direction normal to the slipstream and the horizontal, as illustrated in Fig. 65. This allowed for subsequent statistical analysis of the vortex wander in both normal and tangential directions relative to the slipstream. Fluctuations in the vortex position were then observed over multiple wake ages for several test cases. Fig. 66 shows the collection of vortex locations as they persist in the wake. The shape of the slipstream boundary remains consistent at higher vortex Reynolds number, while the wandering motion appears to be more tangentially oriented in the low Rev case. Additional work will be required to determine whether the wandering motion behaves differently as Rev is decreased further. 81
Figure 65. Rotation angle of tip vortex loci.
a) b)
c) d)
Figure 66. Tip vortex position variation for (a) Rev = 27,450 (b) Rev = 32,600 (c) Rev
= 35,280 (d) Rev = 41,900. 82
The standard deviation (σi) of the tip vortex location, normalized by the blade chord length (c), was observed for vortex motion in the directions normal and tangential to the slipstream boundary. The inherent deficit in aerodynamic efficiency of rotors operating in low Reynolds number conditions suggests a corresponding trend in the behavior of the wake structures dynamics. A study of the magnitude and direction of tip vortex fluctuations was pursued to examine potential growth or changes in these characteristics for varying Rev.
Figs. 67-68 show that vortices tend to fluctuate more significantly in the normal slipstream component than the tangential. In addition to the greater magnitude, growth of the normal fluctuations increases with wake age at a slightly faster rate than the tangential component. Despite anticipated change, comparison of the same statistics among different vortex Reynolds numbers suggests there is no dependency over the studied range.
83
Figure 67. Normalized standard deviation of tip vortex wandering for the
component normal to the slipstream boundary.
Figure 68. Normalized standard deviation of tip vortex wandering for the
component tangential to the slipstream boundary.
84 In a similar analysis, performed by Mula et al.36, data from a NACA 0012 rotor was observed at a vortex Reynolds number of approximately 25,000. The rotor used in
Mula’s experiment was more than three times the diameter of that used in this experiment, but was operated at less than half the rotational speed. The normalized standard deviation of the tip vortex wandering motion from both experiments yields a greater motion in the normal component of the slipstream relative to the tangential, as shown in Fig. 69.
Figure 69. Normal and tangential standard deviation comparison between a two and
four bladed rotors (Mula) of similar geometric and operational characteristics.36
However, the initial growth rate of both components is greater than that at early wake ages found by Mula, and the difference between both is less. This implies that as the vortices convect in the wake, their tendency to wander is not as significant in the direction normal to the slipstream boundary. Also, the overall magnitude of the 85 normalized standard deviation differs by nearly an order of magnitude between the two experiments. This difference could be attributed to more significant vortex wandering caused by a greater tendency for vortex interaction of the four-bladed rotor.
4.4.1 Wake Structure Development
Fig. 70 is an instantaneous PIV image showing the downstream wake of the small rotor operating at CT = 0.002. The Karman vortex structures are characteristic features of this flow field, and demonstrate an obvious periodic behavior until the onset of complete turbulent separation for high Rec conditions at high θ. The prominence of these structures, particularly in lower speed laminar conditions, suggest this phenomenon can be attributed to laminar boundary layer vortex shedding.
Figure 70. Instantaneous chordwise PIV image at Rec = 106,700 and CT = 0.002.
4.4.2 Acoustic Measurements
In an effort to better characterize the periodic behavior of this rotor wake, the frequency content of the flow field was observed for all rotor blades at a fixed thrust 86 coefficient of 0.002. Fig. 71 shows the acoustic spectrum for these test cases at 3500
RPM measured from a microphone positioned normal to the rotor tip path plane. The purple plot represents a baseline acoustic recording of the anechoic chamber without the motor running. In each case, the shaft and blade passage frequencies are apparent at 58
Hz and 117 Hz, respectively. Motor noise is also present in the data just below 104 Hz.
Of particular interest are the broad band peaks apparent in the high frequency end of the spectrum for all three test conditions. The largest of the three bands is noted for the highest Rec case around 12 kHz, and a corresponding jump in SPL is evident at 13 kHz for the intermediate Reynolds number condition. Finally, a distinct high frequency band is noted for the low Rec case at approximately 34 kHz. In each case, there is a significant increase in the SPL of about 10 dB. The similar acoustic measurements made by
Intaratep et al.25 on a model DJI Phantom rotor also featured a broad band peak at high frequencies. This acoustic signature proceeding from their measurements was attributed to laminar boundary layer vortex shedding at approximately 14 kHz.
To confirm the source of this high frequency content, the physical chordwise PIV measurements were analyzed to determine the actual shedding frequency of counter- rotating vortices from the TE of the blade. To approximate the frequencies at each Rec condition, a characteristic length was extracted from the instantaneous images as the distance between subsequent shed vortices near the TE. The frequency was then computed as,
f U (21) s x
87 where U is the tangential velocity of the rotor blade relative to the mean velocity in the wake at the local spanwise location of the measured vortices. Table 8 shows the shedding frequencies computed from Eq. 21 relative to the corresponding frequencies from Fig. 70.
The results show good agreement between the physical and acoustic measurements, confirming that the noise source originates from laminar boundary layer shedding.
Acoustic measurements were also generated for the same rotor blades at a rotational speed of 4500 RPM, and these results are shown in Fig. 72. In this case, the high frequency peaks evident at 3500 RPM are still present, but appear slightly attenuated over a wider band of frequencies. The difference is attributed to an enhanced level of wake turbulence expected for the same rotors at higher speed. Subsequent high-speed flowfield measurements would have to be made to confirm the physical characteristics corresponding to this acoustic data.
Table 8. Vortex shedding frequency comparison.
Rec U Δx fs fa
[-] [m/s] [mm] [kHz] [kHz]
47,500 36.6 1.1 33 34
80,700 44.3 3.0 15 13
89,400 47.2 3.4 14 12
88
Figure 71. Rotor acoustic measurements for varying Rec at 3500 RPM.
Figure 72. Rotor acoustic measurements for varying Rec at 4500 RPM. 89 Chapter 5. Conclusions
The prospect of utilizing small-scale rotorcraft for new applications, demands a more comprehensive understanding of their physical limitations. In this work, the degraded aerodynamic efficiency of small-scale rotors was explored. Rotor performance and flowfield measurements were conducted to gain insight into the effect of low-Reynolds number scaling. Thrust, torque, and FM measurements were made at various rotational speeds, using a set of variably sized, custom designed, rotor blades. The NACA 0012 blades were untwisted with a rectangular planform. Spanwise and chordwise 2-
Component PIV were performed to visualize features of the rotor wake structures and observe flow development over the suction side of the airfoil. Time resolved imaging was also conducted to show the evolution and interaction of the vortex system. An Rev range of 17,000-42,000 was captured while maintaining a constant blade AR and CT/σ. Vortex identification was used to study the dynamic characteristics of the tip vortices, quantify vortex core growth, and swirl velocity decay as a function of wake age. The acoustic signature of the rotor was also measured, and correlated to physical flow features.
FM measurements show that the lowest Rec condition operates with the lowest aerodynamic efficiency, while the two higher Reynolds numbers performed significantly better. This is attributed to the decaying profile drag penalty at higher Rec. The low-
Reynolds cases had a notably higher induced torque demand, especially at low collective angles. This is attributed to the close proximity maintained by the wake structures due to
90 the inherent lower inflow velocities of the low Re conditions. It was also observed that the low Re cases tend to achieve a peak FM (and minimum induced torque) at higher collective angles, since the onset of blade tip stalling occurs at high θ for low Re.
Spanwise PIV measurements revealed that tip vortex formation is heavily dependent on the thrust coefficient. At high CT values, a complete lack of tip vortex formation was explained by the absence of an outboard pressure gradient caused by local tip stalling.
Chordwise PIV measurements revealed a separated and highly turbulent wake at high θ, confirming outboard stall at high CT conditions.
Chordwise ensemble averages of the velocity magnitude revealed other aerodynamic characteristics for the NACA 0012 rotors. Specifically, laminar separation tends to occur at low-Reynolds numbers, and is exacerbated as Rec and θ are increased. At low collective angles, a laminar separation bubble was observed on the suction side of the airfoil, and moved toward the LE with increasing Rec. A similar trend was noted at higher collective angles, resulting in transition to a fully turbulent wake beyond a threshold
Reynolds number.
Time resolved spanwise imaging revealed presence of a vortex pairing phenomenon in the rotor wake. The divergence rate of the instability depends significantly on thrust, as the axial location of the pairing was closer to the rotor plane at lower CT conditions. From the phase averaged spanwise measurements, the pairing was only observed in the field of view for the four lower Rev conditions. This suggests the close proximity of pairing to the rotor plane contributes to the elevated torque requirement and low FM at these conditions. As such, inherently low Rev rotors will likely be operated more efficiently by increasing CT and delaying the pairing onset.
91 A statistical analysis of the vortex aperiodicity over a range of Reynolds numbers indicates the standard deviation of the wandering motion remained relatively constant in both the normal and tangential orientations. When compared to experiments performed on a four bladed rotor at similar Rev, a similar statistical analysis yielded much lower standard deviation values. Qualitatively, the pairing of vortices in the wake was a phenomenon observed in lower Rev regimes, but did not impact quantitative fluctuations in position because only one of the paired vortices was tracked at a given wake age.
Instantaneous PIV images indicate the presence of periodic, counter-rotating structures entrained in the wake for sufficiently low collective angles. Acoustic measurements made normal to the rotor wake captured high-frequency broad band peaks characterized by a 10 dB spike at 12, 13, and 34 kHz for the low, intermediate, and high
Rec conditions. By inferring the shedding frequency from the physical PIV data, a direct comparison to the acoustic signature confirms that laminar boundary layer vortex shedding is a physical source of noise for small-scale rotors operating under low-
Reynolds number conditions.
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96 Appendix A. Derivation of Vortex Reynolds Number Approximation
Derivation of the vortex Reynolds number proceeds from both blade element and momentum theory. Leishman10 documents the use of each theory to derive two different relationships for the differential thrust coefficient. That derived from blade element theory is related to the local blade bound circulation, while that from momentum theory is manipulated in terms of an assumed inflow profile and integrated along the blade span.
The two differential forms are then equated, and the assumption is made that the bound circulation around the rotor blade (Γb) can be equated to that of the tip vortex (Γv). This allows Rev to be evaluated explicitly in terms of blade parameters and operating conditions.
Beginning with blade element theory the differential thrust coefficient can be written in terms of the local spanwise lift coefficient as10
rC2 dC l dr . (A.1) T 2
From the Kutta-Joukowski theorem, the local blade circulation is related to the local lift per unit span (L’), and is defined as
LU' b , (A.2) where U is the local tangential velocity of the rotor blade. In terms of the lift coefficient,
Eq. A.2 can be written as,
2 C b , (A.3) l Rrc
97 where R is the blade radius, r is the non-dimensional blade radius, c is the blade chord, and Ω is the rotational speed. Combining Eqs. A.1 and A.3, the differential thrust coefficient is written as
r dC b dr . (A.4) T Rc
Shifting attention to momentum theory, a second relationship for the differential thrust coefficient is considered for a rotor in hover. Here, dCT is represented in terms of the local rotor non-dimensional inflow (λ) as
2 dCT 4 rdr . (A.5)
A general form of the rotor inflow can then be written in terms of the radial location as
n ()rr tip . (A.6)
Combining Eqs. A.5-A.6 , and integrating along the blade span, the thrust coefficient becomes
4 2 C tip . (A.7) T 22n
Combining results from the blade element theory (A.4) and momentum theory (A.5), the bound circulation around the blade can be related to the rotor inflow as
4 22rn Rc tip . (A.8) b
Solving Eq. A.7 for the rotor tip inflow (λtip), and substituting into Eq. A.8, the bound circulation becomes a function of blade loading,
C T 22n Rcr 2n . (A.9) b
98 In this case, we are interested in approximating the circulation in the tip vortex filament.
Therefore, the blade bound circulation will be computed at the most outboard location (r
= 1). Also, we can now employ the assumption that the outboard blade bound circulation is equivalent to vortex circulation (Γb = Γv). Eq. A.9 then simplifies to,
C T 22n Rc . (A.10) v
Recalling Eq. A.6, n is a coefficient which characterizes the rotor inflow profile. In practice, this is evaluated for various conditions as described in Table A. 1.
Table A. 1. Rotor inflow coefficient
n Blade Type Description Inflow profile which is constant with 0 Ideal twist spanwise location Inflow increases linearly from blade root 0.5 Untwisted blade to tip Inflow increase linearly over most of the rotor span with a more rapid increase near 1 Untwisted blade (w/ severe tip loss) the blade tip due to the added induced velocity of the tip vortex
For the purpose of this work, a value of 0.5 was chosen for the inflow coefficient to represent the untwisted rotor blades. By definition, Reynolds number can be defined as the ratio of circulation to kinematic viscosity. Therefore, combining this definition with
Eq. A.10, the vortex Reynolds number is defined as
v 3Rc CT Rev . (A.11)
99 Appendix B. Chordwise PIV Measurements
a) d)
b) e)
c) f)
Figure 73. Chordwise ensemble average of velocity magnitude at Rec = 57,500 for a
CT of (a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i)
0.009. 100 g) h)
i)
101 a) d)
b) e)
c) f)
Figure 74. Chordwise ensemble average of vorticity at Rec = 57,500 for a CT of (a)
0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009.
102 g) h)
i)
103 a) d)
b) e)
c) f)
Figure 75. Chordwise ensemble average of velocity magnitude at Rec = 96,500 for a
CT of (a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i)
0.009.
104 g) h)
i)
105 a) d)
b) e)
c) f)
Figure 76. Chordwise ensemble average of vorticity at Rec = 96,500 for a CT of (a)
0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009.
106 g) h)
i)
107 a) d)
b) e)
c) f)
Figure 77. Chordwise ensemble average of velocity magnitude at Rec = 98,400 for a
CT of (a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i)
0.009.
108
g) h)
i)
109 a) d)
b) e)
c) f)
Figure 78. Chordwise ensemble average of vorticity at Rec = 98,400 for a CT of (a)
0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009.
110
g) h)
i)
111 a) d)
b) e)
c) f)
Figure 79. Chordwise ensemble average of velocity magnitude at Rec = 106,700 for a
CT of (a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i)
0.009.
112 g) h)
i)
113 a) d)
b) e)
c) f)
Figure 80. Chordwise ensemble average of vorticity at Rec = 106,700 for a CT of (a)
0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009.
114 g) h)
i)
115 a) d)
b) e)
c) f)
Figure 81. Chordwise ensemble average of velocity magnitude at Rec = 115,000 for a
CT of (a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i)
0.009.
116 g) h)
i)
117 a) d)
b) e)
c) f)
Figure 82. Chordwise ensemble average of vorticity at Rec = 115,000 for a CT of (a)
0.001 (b) 0.002 (c) 0.003 (d) 0.004 (e) 0.005 (f) 0.006 (g) 0.007 (h) 0.008 (i) 0.009.
118
g) h)
i)
119