Development, Control and Recovery of Leading- and
Trailing-Edge Vortices in Tandem-Airfoil Configurations
Vom Fachbereich Maschinenbau an der Technischen Universit¨at Darmstadt
zur
Erlangung des Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte
D i s s e r t a t i o n
vorgelegt von
David E. Rival, M.Sc.
geb. in Atlanta, USA
Berichterstatter: Prof. Dr.-Ing. C. Tropea Mitberichterstatter: Prof. Dr.-Ing. R. Radespiel Tag der Einreichung: 10.8.2009
Tag der m¨undlichen Pr¨ufung: 14.10.2009 Darmstadt 2009
Hiermit versichere ich, die vorliegende Doktorarbeit unter der Betreuung von Prof. Dr.-Ing. C. Tropea nur mit den angegebenen Hilfsmitteln selbst¨andig angefertigt zu haben.
Abstract
Particle Image Velocimetry (PIV) has been used to uncover the two distinct aerody-namic mechanisms associated with cruise and hover conditions in tandem-airfoil con-figurations, i.e. dragonfly flight. The studies performed here have been carried out at transitional Reynolds numbers and reduced frequencies corresponding to dragonfly flight, which in turn are relevant to the development of Micro Air Vehicles (MAVs). The associated vortex dynamics of these strongly separated flows have been examined using vortex-tracking, pressure-integration and control-volume methods. To a lesser extent, direct force measurements and URANS-based numerical simulations have been used to support the findings. At first the formation and manipulation of leading-edge vortices (LEVs) and trailing-edge vortices (TEVs) for a single airfoil have been examined using non-sinusoidal kinematics and the theory of optimal vortex formation. Results show that the development of the LEV can be modified without the need for a span-wise flow, contrary to popular belief. In a similar fashion, the TEV can be reduced or even completely mitigated through the superposition of a quick-pitch motion near the bottom of the stroke. Subsequently, the studies of tandem configurations in cruise conditions revealed that the vortex-formation process for the forefoil is strongly influenced by the placement of a hindfoil in its wake, be it static or moving. In strong agreement with observations made on live dragonflies, airfoil configurations with specific out-of-phase kinematics of approximately ψ = 60◦
were found to generate net thrust. The mecha-nism for this thrust production was identified as the formation of a leading-edge suction bubble on the hindfoil induced by the passing of the forefoil LEV. Similarly, the passing forefoil TEV was found to induce a vortex on the hindfoil, which in turn helped reduce power consumption on the upstroke. Finally, time-resolved PIV investigations into tan-dem hovering identified a beneficial interaction at ψ = 90◦
in which the forefoil TEV interacted with the development of the hindfoil LEV. This aerodynamic mechanism in turn adjusted the strength and positioning of the hindfoil TEV, thus affecting thrust and power consumption. This result agrees with observations in nature and analogous force measurements for three-dimensional flapping wings, suggesting again that a span-wise flow is not necessarily required for efficient vortex control but rather a result of the evolutionary restrictions in nature towards root-flapping flight.
Kurzfassung
Die aerodynamischen Vorg¨ange, wie sie beim Reise- und Schwebeflug von Libellen vor-kommen, wurden mit zwei Profilen in einer Tandemanordnung erzeugt und mit Hilfe von Particle Image Velocimety (PIV) erfasst. Hierbei wurden Reynolds Zahlen im Transitionsbereich und reduzierte Frequenzen untersucht, die den Fl¨ugelschlag von Li-bellen abbilden und somit f¨ur die Entwicklung von Micro Air Vehicles (MAVs) relevant sind. Die Dynamik der Wirbelstrukturen, die sich in den stark abgel¨osten Str¨omungen ¨
uber den Profilen bilden, wurde durch die Berechnung der Trajektorien der Wirbelzen-tren, der Druckintegrale und der Kontrollvolumenmethode analysiert. Weiterhin wurden Kraftmessungen und numerische Simulationen (URANS) durchgef¨uhrt, um die Ergeb-nisse abzusichern. Als erstes wurden Entstehung und Beeinflussung der Vorderkanten-und Hinterkantenwirbel an einem einzelnen Profil untersucht, indem eine nicht-sinus-f¨ormige Kinematik und das Konzept der optimalen Wirbelbildung angewendet wurden. Entgegen der g¨angigen Meinung, zeigen die Ergebnisse, dass der Vorderkantenwirbel ohne das Vorhandensein einer Str¨omungskomponente in Querrichtung beeinflusst werden kann. Ebenso kann der Hinterkantenwirbel durch die ¨Uberlagerung einer schnellen Nick-bewegung im Tiefpunkt der HubNick-bewegung reduziert oder sogar komplett getilgt werden. Die darauf aufbauende Analyse der Str¨omung, die sich in der Tandemanordnung ergibt, zeigt f¨ur den Reiseflug, dass die Wirbelbildung am vordern Profil stark durch die Position des hinteren Profils beeinflusst wird. Dies gilt sowohl f¨ur den Fall, in dem das hintere Profil statisch ist, als auch, wenn es ebenfalls bewegt wird. Messungen bei verschiedenen Phasenwinkeln zwischen den Profilen haben gezeigt, dass bei einem Phasenwinkel von ψ = 60◦
in der Summe ein Schub entsteht, was in ¨Ubereinstimmung mit Beobachtungen aus dem Libellenflug ist. Der f¨ur den Schub verantwortliche Mechanismus, konnte als eine Tiefdruck-Blase am hinteren Profil, die durch den vor¨uberziehenden Vorderkantenwirbel des vorderen Profils verursacht wird, identifiziert werden. In gleicher Weise entsteht durch das Vor¨uberziehen des Hinterkantenwirbels des vorderen Profils ein induzierter Wirbel am hinteren Profil, der den Energieverbrauch innerhalb der Aufw¨artsbewegung verringert. Abschlieend wurde die Str¨omung, die sich f¨ur die Tandemanordnung im Schwebeflug ergibt, mit Hilfe von zeitaufgel¨osten PIV-Messungen erfasst. Hierbei ergab sich, in ¨Ubereinstimmung mit den Erkenntnissen von Libellen, ein Phasenwinkel von ψ = 90◦
, bei dem der Hinterkantenwirbel des vorderen Profils eine positive Wirkung auf die Entstehung des Vorderkantenwirbels des hinteren Profils hat. Dieser Mechanis-mus bestimmt die St¨arke und Position des Hinterkantenwirbels am hinteren Profil und beeinflusst dadurch die Schuberzeugung und den Energieverbrauch. Diese Ergebnisse stimmen sowohl mit den Beobachtungen in der Natur, als auch mit Kraftmessungen an dreidimensionalen Schlagfl¨ugelmodellen ¨uberein. Dies legt die Vermutung nahe, dass die Str¨omungskomponente in der Querrichtung des Fl¨ugels nicht f¨ur eine effektive Kon-trolle der Wirbelstrukturen notwendig ist, sondern aus den Einschr¨ankungen bei der evolution¨aren Entwicklung von Schlagfl¨ugel herr¨uhrt.
Acknowledgements
This research was supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) within the national priority program entitled Nature-Inspired Fluid Mechan-ics (SPP1207, Tr 194/40).
First and foremost I would like to acknowledge the support and guidance of my advisor Prof. Dr.-Ing. Cam Tropea. Even before arriving in Germany in January 2006 he had already begun inspiring me through our regular correspondence. Over the years, Professor Tropea entrusted me with a great deal of responsibility, which in turn helped develop my confidence and hone my research skills. I have grown quite a bit under his guidance over the last four years.
I would also like to thank Prof. Dr.-Ing. Rolf Radespiel from the Technische Uni-versit¨at Braunschweig for refereeing this thesis and for his constructive input over the course of the NATO Research and Technology Organization program entitled: AVT-149 Task Group on Unsteady Low Reynolds Number Aerodynamics for Micro Air Vehicles. I am also indebted to the chair of this task group, Dr. Michael Ol from the Air Force Research Laboratories at Wright-Patterson AFB, for his feedback over the course of this project.
I am very thankful for having had the opportunity to work with a group of talented students during the course of my stay at the institute. The following students con-tributed greatly to the research program: Julien Dumond (´Ecole Centrale de Lyon), Kai Aschmoneit, Martin Sharman (U. of Glasgow), Tim Prangemeier, Roland Manejev, Dirk Sch¨onweitz and Guillermo Hass (in conjunction with the U. of Toronto). Without their hard work and creativity, many of the results in this thesis would not have materialized. I would also like to extend my thanks to Marc Wolf and Jake (a.k.a. Riley) Monsour for their help with the experiments.
The success of the numerous experimental campaigns in the wind tunnel was in large part due to support from Ilona Kaufhold and her team of machinists who were always prepared to help out in some form or another. From the IT side, Dipl.-Ing. Matthias Quade always showed enormous patience with me, irrespective of the problem or its source. From an administrative standpoint, Silke Wallner and Stephanie Lath at the Lichtwiese provided a great deal of support throughout the course of my stay here in Germany.
The list of colleagues who provided advice over the years is a long one and includes: Dr. Tatjana Hubel, Dr.-Ing. Sven Grundmann, Dr.-Ing. Markus Gnirss, Dr.-Ing. Feras Batarseh, Dipl.-Ing. Jochen Kriegseis and Dipl.-Ing. Benjamin Lambie. A particular thanks goes out to Dipl.-Ing. Matthias Kinzel for his support and friendship. Be it a discussion on Particle Image Velocimetry or a spot at the gym, he was always there to lend a helping hand.
Finally, I would like to acknowledge the encouragement I have received over the years from all my friends and family (wherever they might be hiding), especially from my mom, Rob, Katia, Claudia and Johnny.
Contents
Abstract I Kurzfassung II Acknowledgements III 1 Introduction 1 1.1 Overview . . . 11.2 Inspiration from Nature . . . 1
1.3 Motivation from Engineering . . . 1
1.4 Objectives . . . 2
2 Background 3 2.1 Characterization of Dragonfly Flight . . . 3
2.1.1 Cruise versus Hover . . . 4
2.2 Separated Flow . . . 4 2.2.1 Dynamic Stall . . . 5 2.2.2 Stall Regimes . . . 6 2.2.3 Leading-Edge Vortex . . . 7 2.2.4 Trailing-Edge Vortex . . . 8 2.2.5 Reattachment . . . 9 2.3 Vortex Dynamics . . . 9
2.3.1 Optimal Vortex Formation . . . 9
2.3.2 Vortex Convection . . . 10
2.3.3 Blade-Vortex Interaction . . . 10
2.4 Energy Recovery . . . 11
2.4.1 Thrust Production . . . 11
2.4.2 Power Reduction . . . 12
3 Experimental Setup and Methods 14 3.1 Experimental Rig . . . 14 3.2 Profiles . . . 15 3.2.1 SD7003 Profile . . . 15 3.2.2 Flat-Plate Profile . . . 16 3.3 Wind Tunnel . . . 16 3.3.1 Flow Quality . . . 17 3.3.2 Assumption of Two-Dimensionality . . . 18 3.4 Hover Chamber . . . 19 3.5 Force Measurements . . . 20 3.6 Smoke Visualizations . . . 21
3.7 Particle Image Velocimetry (PIV) . . . 21
3.7.1 Standard PIV System . . . 21
3.7.2 Time-Resolved PIV System . . . 22
3.7.3 Analysis and Uncertainty . . . 23 IV
4 Numerical Setup and Validation 27
4.1 Governing Equations . . . 27
4.1.1 Closure . . . 27
4.1.2 Low Reynolds Number Correction . . . 28
4.1.3 Transition Model . . . 29
4.1.4 Detached Eddy Simulations (DES) . . . 29
4.2 Static Simulations . . . 30
4.2.1 Discretization . . . 30
4.2.2 Mesh . . . 31
4.2.3 Validation . . . 32
4.3 Dynamic Simulations in Wind Tunnel . . . 35
4.3.1 Discretization . . . 35
4.3.2 Mesh . . . 36
4.3.3 Validation . . . 38
4.4 Summary . . . 39
5 Dynamic Stall and its Associated Wake 41 5.1 Parameter Space . . . 41
5.2 Results . . . 41
5.2.1 Effect of Reduced Frequency . . . 41
5.2.2 Pure-Pitch versus Pure-Plunge Decomposition . . . 42
5.2.3 Combined Motion . . . 46
5.3 Summary . . . 49
6 Formation of Leading-Edge Vortices 50 6.1 Parameter Space . . . 50
6.2 Circulatory and Non-Circulatory Decomposition . . . 51
6.3 Results . . . 54
6.3.1 Reference (Sinusoidal) Case . . . 54
6.3.2 Effect of Kinematics on LEV Formation . . . 57
6.3.3 Effect of Kinematics on Wake Formation . . . 58
6.3.4 Vortex Growth and Pinch-Off . . . 58
6.3.5 Vortex Convection . . . 60
6.4 Summary . . . 61
7 Manipulation of Trailing-Edge Vortices 63 7.1 Parameter Space . . . 63
7.2 Results . . . 65
7.2.1 Vorticity . . . 65
7.2.2 Circulation . . . 67
7.2.3 Statistical Analysis of TEV . . . 68
7.2.4 Reattachment . . . 70
7.3 Summary . . . 71
8 Schmidt Propeller 72 8.1 Parameter Space . . . 72
8.2 Results . . . 72
8.2.1 Trajectories and Convective Velocities . . . 72
8.2.2 Vortex Circulation . . . 73
8.2.3 Vortex-Induced Separation . . . 74 V
8.2.4 Control-Volume Analysis . . . 75
8.2.5 Pressure Integration . . . 76
8.3 Summary . . . 77
9 Tandem Configuration in Cruise Conditions 79 9.1 Parameter Space . . . 79
9.2 Hindfoil with Pure-Plunge . . . 79
9.2.1 Vortex Trajectories and Convective Velocities . . . 79
9.2.2 Vortex-Induced Separation . . . 81
9.2.3 LEV Circulation . . . 81
9.2.4 Control-Volume Analysis . . . 81
9.2.5 Pressure Integration . . . 82
9.3 Hindfoil with Pitch/Plunge . . . 83
9.3.1 Vortex Trajectories and Convective Velocities . . . 83
9.3.2 Vortex-Induced Separation . . . 84 9.3.3 LEV Circulation . . . 84 9.3.4 Control-Volume Analysis . . . 85 9.3.5 Pressure Integration . . . 86 9.4 Insight from CFD . . . 86 9.4.1 Mean Forces . . . 86 9.4.2 Time-Resolved Forces . . . 87 9.4.3 Power Consumption . . . 89 9.5 Summary . . . 90
10 Tandem Configuration in Hover 91 10.1 Parameter Space . . . 91 10.2 Time-Resolved Forces . . . 91 10.2.1 Complete Run . . . 92 10.2.2 Representative Cycle . . . 93 10.3 Mean Forces . . . 94 10.3.1 Combined . . . 94 10.3.2 Forefoil . . . 94 10.3.3 Hindfoil . . . 95 10.4 Power Consumption . . . 96 10.4.1 Time-Resolved Power . . . 96 10.4.2 Mean Power . . . 97 10.5 Vortex Interaction . . . 98
10.5.1 Effects on Thrust Force . . . 98
10.5.2 Effects on Normal Force . . . 99
10.5.3 Behavior of TEV . . . 99 10.6 Summary . . . 103 11 Conclusions 104 Bibligraphy 106 Appendix A 112 Appendix B 116 VI
List of Figures
2.1 Topology of LEV on dragonfly forewings, from Thomas et al (2004). . . 3 2.2 Kinematics of dragonfly flight demonstrating the tilting of the body axis
(body attitude) to the horizontal, from Azuma and Watanabe (1988). . 4 2.3 Schematic of the dynamic-stall process, from Leishman (2006). . . 5 2.4 Schematic characterizing light and deep stall, from McCroskey (1982). . . 7 2.5 Creation of thrust on a plunging airfoil, where A is the lift force and S is
the resulting horizontal component; taken from Schmidt (1965). . . 11 2.6 Arrangement and kinematics of airfoils used for the wave-propeller
exper-iments; taken from Schmidt (1965). . . 12 3.1 Experimental setup in wind tunnel with flow direction from right-to-left:
(a) CCD cameras, (b) beam expander, (c) PIV image frames, (d) wall-spanning carbon-fibre profile, (e) embedded piezo-electric force sensors, (f) test-section, (g) laser head, (h) linear motors with linkage system and (i) base structure. . . 14 3.2 One of two SD7003 carbon-fibre profiles mounted on experimental rig;
note piezo-electric force transducers situated just below profile. Also note rear linear motors for each profile rotate to allow for extra degree-of-freedom in kinematics. . . 15 3.3 Plexiglass plates in tandem arrangement; note minimal shadows emitted
from leading and trailing edges allowed for excellent optical access during PIV measurements. . . 16 3.4 Schematic of wind tunnel; note intake (A-A) and test section (C-C)
cross-sections. . . 17 3.5 Turbulence intensities in wind-tunnel test section for the two relevant
chord-based Reynolds numbers used in the wind-tunnel studies; note boundaries represent test-section walls and dashed lines the motion limits of the moving airfoil as well as the PIV measurement plane. . . 17 3.6 Initial tests of SD7003 profile in tunnel at Re = 60000 under static
con-ditions: (a) static lift measurements in tunnel show little variation due to vertical positioning, i.e. blockage effects are small; (b) oil-flow visualiza-tion shows clear posivisualiza-tioning of separavisualiza-tion (s) and reattachment (r) lines in agreement with previous experimental measurements made by Ol et al (2005). . . 18 3.7 PIV measurements taken on x/c = 0.4 plane normal to the freestream
direction in Eiffel-type wind tunnel at TU Braunschweig depicting span-wise flow at bottom of stroke for the pure-plunge reference case; note data taken from Bansmer and Radespiel (2008). . . 19 3.8 The experimental rig enclosed in the hover chamber shown with laserhead
located at left and high-speed camera at right. . . 20 3.9 Plots of dimensionless vorticity for different ensemble sizes at t/T = 0.333
for the pure-plunge reference case. . . 22
3.10 Dimensionless vorticity plot for the reference case at t/T = 0.5 showing the windows used to calculate the circulation of the TEV (left) and LEV (right); note no positive vorticity in TEV window and no negative vorticity in LEV window due to threshold value of zero. . . 23 3.11 Schematics of: (a) both single and combined control volumes
superim-posed on top of vorticity field, and (b) pressure integration over hindfoil for pure-plunge 8◦
(h/c = −0.5, 0◦
) case; note integration follows path A to B to C. . . 25 3.12 Comparison of lift forces extracted from control-volume analysis with
direct-force measurements; note direct force measurements for pure-plunge 8◦
(single) case taken from Ol et al (2009). . . 25 4.1 Cross-section of DES mesh with z = c span discretized with 32 cells. . . . 31 4.2 Validation of force coefficients between experiment and CFD; note all
results at Re = 60000 unless otherwise stated. . . 32 4.3 Comparison of separation (xs), transition (xt) and reattachment (xr)
points between: PIV (top), standard SST k -ω (middle) and SST k -ω (γ-Reθ) model (bottom) at α = 8◦ and Re = 60000; note PIV measurements
taken from Nerger et al (2003). . . 33 4.4 Variation of lift coefficient at α = 20◦
and Re = 60000 unless otherwise stated; note f = 17Hz in experiment is the system’s natural frequency. . 34 4.5 Iso-surfaces portraying the three-dimensional structures in the flow at
α = 20◦
and Re = 60000, where iso-contours of vorticity are set at ωzc/U∞
= ±5. . . 34 4.6 Dimensionless vorticity in the y-z plane at several chord-wise locations
from the leading edge (α = 20◦
and Re = 60000). . . 35 4.7 Numerical domain representative of wind-tunnel test section with forefoil
quarter-chord situated at x/c = 7.25 downstream of inlet; zones a) and d) contain pure block-structured cells whereas zones b) and c) contain dynamic layering and sliding zones. . . 36 4.8 Middle of numerical domain, zones b) and c), with two circular-structured
cores encompassing the airfoils; note airfoils and wind-tunnel walls (top and bottom) with fully-resolved boundary layers. . . 37 4.9 Effect of grid refinement on pressure distribution for single, static airfoil
at α = 8◦
; note three levels correspond to meshes with 10000 cells (), 28000 cells ( ) and 95000 cells (∇). . . 37 4.10 Validation of lift and drag for the pure-plunge reference case; note
refer-ence data taken from experiment, see Ol et al (2009). . . 38 4.11 Validation of lift and drag for pitch/plunge reference case; note reference
data taken from experiment, see Ol et al (2009). . . 39 5.1 Comparison of the wake formation during the downstroke for pure-plunging
motions at two reduced frequencies with αo = 8◦; note at the higher
re-duced frequency a single yet more distinct vortex pair known as a mush-room structure is shed into the wake. . . 42 5.2 Direct lift and moment coefficients for three pure-plunge cases (αo = 5◦, 8◦
and 10◦
) and one equivalent pure-pitch case (αo = 8 ◦
) at k = 0.1; note black circle represents top of stroke. . . 43 5.3 Trajectories of LEV and TEV cores for the single-airfoil reference cases
at k = 0.25. . . 43 VIII
5.4 Comparison of equivalent pure-plunge and pure-pitch kinematics using smoke visualization with a mean angle of attack of αo = 8◦; note sequences
start at t/T = 0 (top of stroke) and follow in constant steps of t/T = 1/16. 44 5.5 Convective velocities of LEV and TEV cores for the single airfoil reference
cases at k = 0.25. . . 45 5.6 Plots of dimensionless vorticity for the pure-plunge reference case (a) and
the pure-pitch case (b) at k = 0.25. . . 45 5.7 Direct lift and moment coefficients for three effective angle-of-attack
vari-ations and a mean angle of attack of αo = 5 ◦
; note black circle represents top of stroke. . . 47 5.8 Direct lift and moment coefficients for three effective angle-of-attack
vari-ations and a mean angle of attack of αo = 10 ◦
; note black circle represents top of stroke. . . 48 5.9 Maximum lift and moment coefficients taken from Figs. 5.7 and 5.8. . . . 49 6.1 Effective angle-of-attack distribution for the four test cases as a function
of (a) period and (b) dimensionless time; note symbols represent 12 mea-surement phases over the period. . . 51 6.2 Contributions of the non-circulatory (added mass) and circulatory
com-ponents to the total lift of the sinusoidal reference case based on classical theory developed by Theodorsen (1935). . . 52 6.3 SD7003 carbon-fibre profile and dummy cylinder shown for comparison;
note cylinder identical to profile in mass (±0.5g) and spanwise distribution. 53 6.4 Comparison of peak theoretical and measured added-mass contributions;
theoretical values derived from Theodorson (1935). . . 53 6.5 Visualization depicting strong leading- and trailing-edge separation at
bot-tom of stroke during a dynamic-tare measurement. . . 54 6.6 Plots of dimensionless vorticity for the reference case, a sinusoidal plunging
motion with a mean angle of attack of αo = 8◦; note masking under airfoil
necessary due to reflections and shadow effects. . . 56 6.7 Plots of dimensionless vorticity during the downstroke (t/T = 0.25) for (a)
sinusoidal, (b) asymmetric, (c) peak-shifted and (d) fast-sinusoidal cases; note t/T = 0.333 used for fast-sinusoidal case to compare equivalent LEV development time. . . 57 6.8 Plots of dimensionless vorticity at top of stroke (t/T = 0) for (a)
asym-metric and (b) fast-sinusoidal cases, demonstrating the importance of the history effects just before the LEV formation process begins. . . 57 6.9 Plots of dimensionless vorticity around bottom of stroke (t/T = 0.5) for
(a) sinusoidal, (b) asymmetric, (c) peak-shifted and (d) fast-sinusoidal cases; note t/T = 0.667 used for fast-sinusoidal case to compare equivalent convection times of LEV and TEV in wake. . . 58 6.10 Development of LEV and TEV circulation as a function of (a) period and
(b) dimensionless time; note solid and hollow symbols represent LEVs and TEVs, respectively. . . 59 6.11 Formation time as a function of (a) period and (b) dimensionless time;
note solid and hollow symbols represent LEVs and TEVs, respectively. . 60 6.12 Trajectories of LEV and TEV cores for the four test cases. . . 61 6.13 Convection velocities of LEV and TEV cores for the four test cases. . . . 61
7.1 Individual components of the quick-pitch motion: effective angle of attack (αef f) for the sinusoidal-plunge and quick-pitch cases, geometric angle
(αgeo) of attack for the quick-pitch case. . . 63
7.2 Effective angle-of-attack distribution for the various quick-pitch cases; note symbols represent the 12 measurement phases over the period. . . . 64 7.3 Plots of dimensionless vorticity showing the TEV formation one phase
before bottom dead center (t/T = 0.417) and at bottom dead center (t/T = 0.5) for the reference (a)-(b), pitch A (c)-(d) and quick-pitch C (e)-(f) cases. . . 66 7.4 Plots of dimensionless vorticity showing the LEV formation at t/T = 0.333
for the reference (a), quick-pitch A (b) and quick-pitch C (c) cases. . . . 67 7.5 Development of LEV (solid) and TEV (hollow) circulation as a function
of period (a), and a close-up of the TEV circulation growth (b). . . 68 7.6 Histograms of dimensionless circulation for the reference case (a) and the
quick-pitch C case (b); mean and standard deviations for both cases shown above for reference. . . 69 7.7 Contour plots of dimensionless stream-wise velocity (u/U∞) for the phases
t/T = 0.333, t/T = 0.417, t/T = 0.5 and t/T = 0.667; reference case (left column), quick-pitch A (center column) and quick-pitch C (right column). 70 8.1 Measured trajectories and convective velocities of LEV and TEV cores for
the various single-airfoil and tandem-airfoil configurations; note h/c refers to vertical orientation of hindfoil. . . 73 8.2 Development of LEV and TEV circulation as a function of period for
the various single-airfoil and tandem-airfoil configurations; note solid and hollow symbols represent LEVs and TEVs, respectively. . . 74 8.3 Comparison of blade-vortex interaction between unloaded and loaded cases;
(a) h/c = −0.5 and αH = 0 ◦
, and (b) h/c = −0.5 and αH = 8 ◦
. . . 74 8.4 Comparison of blade-vortex interaction between unloaded and loaded cases;
(a) h/c = −0.25 and αH = 0◦, and (b) h/c = −0.25 and αH = 8◦. . . 75
8.5 Variation of lift (a) and drag (b) using control-volume analysis; note sym-bols represent measurement phases over the period. . . 76 8.6 Pressure distributions over hindfoil surface for the various tandem
config-urations demonstrating blade-vortex interaction. . . 77 9.1 Trajectories of LEV and TEV cores for hindfoil with pure-plunge motion. 80 9.2 Plunging hindfoil with ψ = 30◦
; LEV and TEV passing above the hindfoil. 80 9.3 Convective velocity of LEV and TEV cores for the cases with the hindfoil
performing pure-plunge. . . 80 9.4 Forefoil and hindfoil performing pure-plunge motion with ψ = 0◦
; devel-opment and convection of induced vortex over the hindfoil surface clearly seen as TEV passes by. . . 81 9.5 Development of LEV circulation (left) and formation time (right) as a
function of period for the various tandem-airfoil configurations. . . 81 9.6 Lift and drag variation over cycle using the standard control-volume
anal-ysis; note t/T referenced to the forefoil position. . . 82 9.7 Pressure distributions over hindfoil suction surface at stroke reversal. . . 82 9.8 Trajectories of LEV and TEV cores for the pitch-plunge cases. . . 83 9.9 Hindfoil performing pitch/plunge with ψ = 120◦
; plots showing the TEV passing below the hindfoil. . . 83
9.10 Convective velocity of LEV and TEV cores for the pitch-plunge cases. . . 83 9.11 Hindfoil with pitch/plunge with ψ = 0◦
(top), ψ = 30◦
(middle), ψ = 60◦
(bottom); note vortex-induced separation much stronger when compared to tandem pure-plunge cases. . . 84 9.12 Development of LEV circulation (left) and formation time (right) as a
function of period for the various tandem-airfoil configurations. . . 85 9.13 Lift and drag variation over cycle using the standard control-volume
anal-ysis; note t/T referenced to the forefoil position. . . 85 9.14 Pressure distributions over hindfoil suction surface at stroke reversal. . . 86 9.15 Time-averaged lift and drag for both tandem arrangements from both the
standard control-volume analysis (experiment) and from CFD. . . 86 9.16 Lift and drag variation for tandem arrangement with pitch/plunge hindfoil
at ψ = 60◦
; observe drop in lift and drag at approximate time step t/T = 0.583, note hindfoil curve shifted in time for comparison. . . 87 9.17 Lift and drag variation for tandem arrangement with pitch/plunge hindfoil
ψ = 90◦
; observe drop in lift and drag at approximate time step t/T = 0.667, note hindfoil curve shifted in time for comparison. . . 88 9.18 Development of leading-edge suction on forefoil for ψ = 60◦
(top) and ψ = 90◦
(bottom): experiment (left) and CFD (right), where time is referenced to forefoil motion. Note leading-edge suction in experiment not visible due to shadow. . . 88 9.19 Variation in power for tandem arrangements with pitch/plunge hindfoil
(ψ = 60◦
and ψ = 90◦
) where energy extraction is observed on upstroke; note hindfoil curve shifted in time for comparison. . . 89 9.20 Development of normal suction on forefoil for ψ = 60◦
(top) and ψ = 90◦
(bottom) responsible for power reduction: experiment (left) and CFD (right), where time is referenced to forefoil motion. . . 89 10.1 Ensemble-averaged normal force (a) and thrust force (b) for the forefoil
during an experimental run. . . 92 10.2 Ensemble-averaged normal force (a) and thrust force (b) for the hindfoil
during an experimental run; note plots of ψ = 90◦
for the hindfoil are shifted in time in a manner that the hindfoil is always at TDC at t/T = 0. 92 10.3 Ensemble-averaged normal force (a) and thrust force (b) for forefoil during
representative cycle. . . 93 10.4 Ensemble-averaged normal force (a) and thrust force (b) for hindfoil
dur-ing representative cycle; note plots of ψ = 90◦
for the hindfoil are shifted in time in a manner that the hindfoil is always at TDC at t/T = 0. . . . 93 10.5 Influence of the phase angle (ψ) on combined/averaged (fore- and hindfoil)
normal force (a) and thrust force (b). . . 94 10.6 Influence of the phase angle (ψ) on mean normal force (a) and thrust force
(b) for forefoil. . . 95 10.7 Influence of the phase angle (ψ) on mean normal force (a) and thrust force
(b) for hindfoil. . . 95 10.8 Nondimensional time-resolved power for the fourth cycle; note plots for
the hindfoil are shifted in time in a manner that the hindfoil is always at TDC at t/T = 0. . . 96 10.9 Nondimensional mean power as function of phase angle (ψ). . . 97 10.10Trajectories and nondimensional circulation of downstroke TEV from
sin-gle airfoil and tandem configuration with ψ = 90◦
. . . 99 XI
10.11Tandem configuration with ψ = 90◦
(left) and single airfoil (right) during downstroke; note time referenced to hindfoil position for tandem case. . . 100 10.12Tandem configuration with ψ = 90◦
(left) and single airfoil (right) around bottom dead center; note time referenced to hindfoil position for tandem case. . . 101 10.13Tandem configuration with ψ = 90◦
(left) and single airfoil (right) during upstroke; note time referenced to hindfoil position for tandem case. . . . 102 11.1 Validation of pure-plunge reference case downstroke: experiment (left)
and CFD (right). . . 112 11.2 Validation of pure-plunge reference case upstroke: experiment (left) and
CFD (right). . . 113 11.3 Validation of pitch/plunge reference case downstroke: experiment (left)
and CFD (right). . . 114 11.4 Validation of pitch/plunge reference case upstroke: experiment (left) and
CFD (right). . . 115 11.5 Validation of downstroke for ψ = 60◦
tandem case with pitch/plunge hindfoil (time referenced to forefoil): experiment (left) and CFD (right). . 116 11.6 Validation of upstroke for ψ = 60◦
tandem case with pitch/plunge hindfoil (time referenced to forefoil): experiment (left) and CFD (right). . . 117 11.7 Validation of downstroke for ψ = 90◦
tandem case with pitch/plunge hindfoil (time referenced to forefoil): experiment (left) and CFD (right). . 118 11.8 Validation of upstroke for ψ = 90◦
tandem case with pitch/plunge hindfoil (time referenced to forefoil): experiment (left) and CFD (right). . . 119
List of Tables
3.1 Validation of control-volume analysis technique for lift and drag forces; note large discrepancy in pure-plunge 8◦
(single) case since direct-force measurements taken from different facility, see Ol et al (2009). . . 26 4.1 Comparison of laminar separation bubbles between PIV and CFD data
at α = 8◦
and Re = 60000. . . 32 5.1 Description of combined pitching and plunging cases tested. . . 41 6.1 Characteristics of various test cases. . . 50 7.1 Characteristic timing of kinematics and the crossing of the static-stall
angle for the various quick-pitch cases. . . 65 7.2 Dimensionless circulation of the TEV at various points in the cycle. . . . 69 8.1 Comparison of lift and drag coefficients for the various cases. . . 76
Symbol Unit Description
A m2 vortex integration window
c m chord length
Cd - sectional drag coefficient, 2d/ρU∞2c
Cl - sectional lift coefficient, 2l/ρU∞2c
Cm - sectional moment coefficient, 2m/ρU∞2c 2
Cn - sectional normal coefficient, 2n/ρ ˙h2maxc
Cp - pressure coefficient, 2(p − p∞)/ρU 2 ∞c
CP - sectional power coefficient, 2P/ρU∞3 c
Ct - sectional thrust coefficient, 2t/ρ ˙h2maxc
d N sectional drag force
f Hz motion frequency
fpro Hz quick-pitch pronation frequency
fsup Hz quick-pitch supination frequency
h0 m plunge amplitude
h(t) m plunge position
˙h m/s plunge velocity
˙hmax m/s maximum plunge velocity
k - reduced frequency, πf c/U∞
k m2/s2 turbulent kinetic energy
l N sectional lift force
m Nm sectional aerodynamic moment
p N/m2 pressure
p∞ N/m
2 free-stream pressure
P Nm/s power
Re - Reynolds number, U∞c/ν
Ret - turbulent Reynolds number, k/νω
s m airfoil span
S m airfoil spacing
t s time
tpause s duration of pause
tpro s beginning of pronation
t<stall s crossing of static-stall angle
tsup s end of supination
t N sectional thrust force
T s period T u - turbulence intensity U∞ m/s free-stream velocity u m/s stream-wise velocity v m/s normal velocity w m/s span-wise velocity x m stream-wise direction y m normal direction z m span-wise direction XIV
Greek Letters
Symbol Unit Description
α ◦
angle of attack
αo ◦ mean angle of attack
α1 ◦ pitch amplitude
αef f(t) ◦ effective angle of attack
αgeo(t) ◦ geometric angle of attack
αplunge(t) ◦
effective angle of attack for pure plunge
δij - Kronecker delta ηP - thrust efficiency Γ m2/s circulation ΓT EV m2/s TEV circulation ρ kg/m3 density ν m2/s kinematic viscosity νt m2/s eddy viscosity φ ◦
phase between pitch and plunge
ψ ◦
airfoil phasing
ω 1/s specific dissipation rate
ωz 1/s spanwise vorticity component
Subscripts Symbol Description i, j indices F forefoil H hindfoil Abbreviations Symbol Description
CFD computational fluid dynamics DES detached eddy simulations
LCTM local correlation-based transition model LES large eddy simulations
LEV leading-edge vortex
PIV particle image velocimetry
RANS Reynolds averaged Navier-Stokes SST shear stress transport
TEV trailing-edge vortex
1 Introduction
1.1 Overview
As the title suggests, this thesis concerns itself with the behavior of leading-edge vortices (LEVs) and trailing-edge vortices (TEVs) in the context of tandem-airfoil configurations, i.e. dragonfly flight. A particular focus has been placed on the vortex dynamics, including the concepts of optimal vortex formation and blade-vortex interaction with the goal to better understand the thrust production and power reduction. In the first few chapters an overview of the state-of-the-art as well as descriptions of the experimental and numerical methods are provided. Thereafter in chapters 5, 6 and 7, studies focused on single-airfoil kinematics are performed. In chapters 8 and 9, blade-vortex interaction for a static and moving hindfoil is investigated, respectively. In chapter 10, vortex interaction for a hovering configuration is studied. Finally, in chapter 11, the results from the various studies are compiled together to allow for general conclusions to be made regarding the vortical interactions associated with tandem configurations.
1.2 Inspiration from Nature
Biologists over several decades have examined the specific wing kinematics of dragonflies and have uncovered distinct patterns associated with various flight maneuvers. These studies include those by: Norberg (1975); Alexander (1984); Azuma et al (1985); Azuma and Watanabe (1988); Rueppell (1989); Thomas et al (2004); Wang and Russell (2007). Very little regarding the aerodynamic mechanisms themselves could, however, be ex-tracted from these above-mentioned studies. Although present-day dragonflies fly with Reynolds numbers on the order of Re = O(103), dragonflies from the Carboniferous
and Permian periods, such as Meganeuropsis permiana with its giant wingspan up to 75cm, flew well into the (laminar-to-turbulent) transitional Reynolds-number regime, as reported on in Beckemeyer (2006). Thus dragonfly scaling and their overall efficient and highly-maneuverable flight provide a source of great inspiration to aerodynamicists struggling to uncover the nuances of biological flight.
1.3 Motivation from Engineering
Accompanied with an ever-growing interest in Micro Air Vehicle (MAV) development based on bio-inspired flight, i.e. combined lift and propulsion from flapping wings, lies the challenging goal to scale down such vehicles to lower Reynolds numbers (Re < 50000), as reported on by Pines and Bohorquez (2006). In such low Reynolds-number regimes, one of the major limitations is the separation-prone nature of the laminar boundary layer when faced with an adverse pressure gradient. Such a fundamental limitation on an airfoil’s performance at these Reynolds numbers was investigated on a laminar profile using force measurements and smoke visualization by Mueller and Batill (1982). More recently Ol et al (2005) examined the transition to turbulence and the position of a laminar separation bubble for low angles of incidence and at Re = 60000. Performance
has been found to drop quite dramatically at these Reynolds numbers, where the laminar-to-turbulent transition and therefore reattachment occur very late, if at all.
At these transitional Reynolds numbers relatively little work has been performed to-date to understand the performance of the flapping tandem arrangement inspired by dragonfly flight. Most notable are the studies by Jones and Platzer (1999) and Warkentin and DeLaurier (2007) in which various arrangements including a plung-ing/flapping forewing and stationary hindwing were investigated. Such an arrangement is based on the development of Schmidt (1965) in which the oscillatory wake of the forewing would allow for thrust generation on the hindwing. One immediate benefit of such a tandem arrangement for MAVs is the potential improvement in flight stability where out-of-phase flapping could significantly dampen cyclic lift production as well as inertial fluctuations of individual wings.
Coincidentally, such aerodynamic interactions are today receiving much attention in the context of energy harvesting. Examples of such devices are so-called wingmills, as described in McKinney and DeLaurier (1981), Kinsey and Dumas (2006) and Platzer et al (2009). Other applications include small-scale wind energy production using vertical-axis wind turbines (VAWTs), which can benefit from these vortical interaction effects, as described in Bertenyi et al (2007). The issue of gust response on larger horizontal-axis wind turbines is also of relevance here, see Barlas and van Kuik (2007).
1.4 Objectives
At the onset of this research project, a number of outstanding questions pertaining to vortex development and vortex interactions in tandem configurations were identified. Naturally the objective was to answer as many of these questions as possible, as well as to tie in the results into some general conclusions relating to dragonfly flight. Some of the most pertinent questions addressed in this thesis include:
• How does the vortical wake behind a pure-plunging or pure-pitching airfoil vary with increasing reduced frequency?
• To what extent can dynamic stall increase the loading on a moving airfoil?
• Is a span-wise flow an absolute requirement to stabilize the LEV or can specific two-dimensional kinematics be used to control the formation process?
• Can the strength of the TEV be reduced without influencing the size and strength of the LEV?
• What is the influence of a static or moving hindfoil on the vortex formation process over the forefoil?
• How does the vortical wake of the forefoil influence the flow field and loading on the hindfoil?
• Does the mechanism of vortex interaction vary between cruise and hover condi-tions?
• Which airfoil phasings are beneficial for cruise and hover conditions and what are the aerodynamic mechanisms responsible for these interactions?
2 Background
2.1 Characterization of Dragonfly Flight
Over the past decades a great deal of analytical, numerical and experimental investi-gations into tandem-wing aerodynamics, almost exclusively for hover conditions, have been performed with the hope of better understanding the complex aerodynamic inter-action associated with dragonfly flight. Among these investigations, Bosch (1978); Lan and Sun (2001); Sun and Lan (2004); Wang and Sun (2005) considered the problem from a two-dimensional standpoint with either the assumption of inviscid or laminar flow. Isogai et al (2004); Maybury and Lehmann (2004); Yamamoto and Isogai (2005); Usherwood and Lehmann (2008); Lehmann (2008, 2009) all investigated the interaction in three dimensions, primarily using force measurements and to a lesser extent particle image velocimetry (PIV). Certain researchers have shown that through wake capture, large increases in performance are attainable for single wings flapping independently, suggesting that the tandem interaction is only secondary in importance. These stud-ies include: Savage et al (1979); Gustafson and Leben (1991); Anderson et al (1998); Birch and Dickinson (2003), in which some go as far as arguing that dragonfly flight is a function of single-wing wake capture and not of vortex interaction between two stag-gered wings. On the other hand, Scharpf and Mueller (1992) showed that even for static tandem configurations at low Reynolds numbers, drag reduction and lift augmentation existed for several stagger and gap configurations.
Thus a very broad range of explanations for the unsteady aerodynamic mechanisms in dragonfly flight exist, where some claim that the performance is linked to the stabi-lization of the span-wise (three-dimensional) flow in the leading-edge vortex (LEV), see Ellington (2006); Lehmann (2009). In contrast Thomas et al (2004) have argued that dragonflies use a quasi two-dimensional dynamic stall mechanism during both cruising and accelerating flight, with a LEV over the airfoil on the downstroke and attached flow on the upstroke, as shown by the topology in Fig. 2.1. Thus by no means does a uni-versal explanation for the unsteady aerodynamic mechanisms associated with dragonfly flight exist at the present time.
Figure 2.1: Topology of LEV on dragonfly forewings, from Thomas et al (2004).
2.1.1 Cruise versus Hover
By studying the kinematics of live dragonflies, Alexander (1984) observed that in-phase flapping was used for high-lift maneuvers such as take-off, whereas out-of-phase flapping was common for steady-state flight conditions where energetics were a consideration. In fact Azuma and Watanabe (1988), among others, recognized that dragonflies modulated not their flapping frequency but their forewing/hindwing phasing (ψ) depending on their flight speed, i.e. transition from hover to cruise conditions. This phasing was found to vary from 60◦
≤ ψ ≤ 90◦
, with lower frequencies observed in hover and higher frequencies found for cruise flight.
In Fig. 2.2 another defining character of dragonfly flight is demonstrated by the tilting of the body axis to the horizontal (body attitude) such that the fore- and hindwing stroke planes approach vertical and horizontal configurations in cruise and hover conditions, respectively. These two extreme flight conditions have been investigated in chapters 9 and 10, which can be considered the bounding limits to the dragonfly’s flight envelope.
Figure 2.2: Kinematics of dragonfly flight demonstrating the tilting of the body axis (body attitude) to the horizontal, from Azuma and Watanabe (1988).
Thus these two bounding flight conditions, cruise and hover, define also the character of the dragonfly’s flow field. In cruise conditions the dragonfly has been observed to combine out-of-phase flapping with intermittent gliding, as reported by Rueppell (1989), where intermittent separation due to dynamic stall on the wings’ suction sides is ex-pected. Here maximum thrust generation approaches the total drag, i.e. maximum forward speed is achieved. In contrast, hover flight is characterized by the continuous development of leading- and trailing-edge vortices in both stroke directions, amounting to permanently-stalled flow conditions, as reported on by Wang and Sun (2005).
2.2 Separated Flow
Both natural and biomimetic flight occur predominantly at low Reynolds numbers (Re < 50000). Mueller and Batill (1982) have shown that at these scales performance is strongly hampered by the separation-prone nature of the laminar boundary layer. Contrary to classical aerodynamics, separation becomes the norm, not the exception. Such a funda-mental limitation to the aerodynamics suggests that lift must be generated through
steady aerodynamic mechanisms such as dynamic stall, as outlined by Ellington (2006). Dynamic stall, as reported on by McCroskey (1982), consists of the formation and de-layed convection of a leading-edge vortex (LEV) over the downstroke, which in turn can be very advantageous to lift production as well as maneuvering. For cruise conditions, as studied in chapters 5 through 9, dynamic stall has been investigated in great detail, first for a single airfoil and then in tandem configurations.
2.2.1 Dynamic Stall
Lift can be augmented well beyond the maximum static lift coefficient by rapidly chang-ing the incidence of an airfoil, as first noted by Kramer (1932). This process is known as dynamic stall and is best understood for purely-pitching airfoil kinematics, for ex-ample as a cause of stall flutter on helicopter rotors, see Leishman (2006). During the dynamic-stall process the viscous effects lag behind the instantaneous angle of attack and the inviscid pressure field. During the rise in lift, the pressure field about the airfoil adjusts to the instantaneous angle of attack at the speed of sound, whereas the viscous effects act at a much slower timescale, as described by Cebeci et al (2005). This delay allows for a huge overshoot in the maximum static lift coefficient, often followed by lift hysteresis with respect to the instantaneous effective angle of attack, as can be seen in Fig. 2.3.
Figure 2.3: Schematic of the dynamic-stall process, from Leishman (2006).
The different stages of the dynamic-stall process for a pitching airfoil are presented in Fig. 2.3. At relatively low mean angles of attack, the first stage is found to be quasi-steady. With increasing angle of attack the lift coefficient rises linearly following an inviscid solution. When approaching the static-stall angle the flow reverses in the boundary layer. However, unlike for the steady case, the flow remains globally attached when crossing the static-stall angle. Stage 2 begins with the formation of a closed separation at the leading edge, in the form of a LEV. At low Mach numbers such as studied here, the LEV can lead to lift values up to 100% higher than the maximum static lift, see Cebeci et al (2005) and Leishman (2006). This is achieved due to low pressure associated with the LEV. The LEV convects over the airfoil surface at approximately
0.333 < uconv U∞
< 0.5, (2.1)
as shown by Beddoes (1976) and Galbraith et al (1986), while growing in strength and increasing lift until it reaches the trailing edge, at which point the separation opens. This marks the beginning of Stage 3 where lift peaks and then drops sharply. Here drag increases due to the fully-separated nature of the flow. After the effective angle of attack drops below the static-stall angle the flow begins to reattach. Once again, the viscous flow lags behind the instantaneous effective angle of attack and reattachment may be delayed substantially. Here the flow typically reattaches from front to back and finally returns to an attached state in Stage 5.
McCroskey (1982) found the key parameters for dynamic stall to be the mean effective angle of attack (α0) and the reduced frequency (k). The mean angle of attack is the
effective incidence about which the motion oscillates. The reduced frequency is a measure of the level of unsteadiness of the flow, as discussed by Cebeci et al (2005). The reduced frequency relates the forcing frequency of the oscillating airfoil to the convective scale of the flow. In this study the reduced frequency is defined in the following manner:
k = πf c U∞
, (2.2)
where f is the pitching/plunging frequency, c the airfoil chord and U∞ the freestream
velocity. Mean angle of attack and reduced frequency are indisputably the dominant parameters of the dynamic-stall process. However, to what extent the specific airfoil kinematics impact the vortex formation and pinch-off processes remains virtually un-known and is the subject of chapters 6 and 7.
2.2.2 Stall Regimes
For a given reduced frequency, dynamic stall can be characterized by the maximum effective angle of attack of the motion, see McCroskey (1982). The first category of dynamic stall is referred to as stall onset, for which there is only limited separation. Stall onset represents the maximum unsteady lift that can be achieved, without incurring significant drag and pitching moment penalties. By increasing the maximum effective angle of attack one moves into the light-stall regime. For light stall, separation is on the order of the airfoil thickness and no distinct vortices are formed. Also noteworthy is that for this regime hysteresis is first encountered. Going beyond the maximum effective angle of attack of light-stall results in the vortex-dominated, deep-stall regime. As can be seen in Fig. 2.3, deep stall incurs significant pitch-down moments and drag penalties, particularly as the LEV pinches off at Stage 3 and the flow becomes fully-separated. Deep stall with the conventional sinusoidal airfoil kinematics generally results in significant
hysteresis loops for the lift coefficient. A schematic from McCroskey (1982) with the classification of light and deep stall is presented in Fig. 2.3.
Figure 2.4: Schematic characterizing light and deep stall, from McCroskey (1982). To-date most studies on dynamic stall have been restricted to sinusoidal kinematics. In chapters 6 and 7, airfoil kinematics have been chosen to minimize the time spent in the fully-separated state. Ideally the fully-separated flow of Stage 4 would be completely bypassed. Whether specific airfoil kinematics can provide relief from the large hysteresis loops by allowing reattachment to occur earlier and more quickly, has been systematically investigated in this thesis.
2.2.3 Leading-Edge Vortex
Thomas et al (2004) visualized the flow around free-flying dragonflies demonstrating the existence of large, quasi two-dimensional leading-edge vortices (LEVs) over their high aspect ratio forewings. Such LEVs can be characterized by a region of strong rotation with a maximum size on the order of the airfoil chord, in turn fed by a shear layer emanating from the leading edge. The production of LEVs is a common mechanism used to augment lift in biological flight and has been studied in detail for pitching airfoils in the context of rotorcraft. Numerous experimental studies exist, particularly related to helicopter aerodynamics, and have been reviewed in detail by Leishman (2006). Carr et al (1977) measured the lift, drag and moment coefficients for a pitching airfoil undergoing dynamic stall and found large hysteresis loops that characterize the lift coefficient curve. Lift augmentation was present, as were peaks in drag and negative aerodynamic moment. For a harmonically plunging and pitching foil Read et al (2003) found instantaneous lift coefficients with values up to 15 and mean lift values over the cycle as high as 5.5. More abstract investigations into LEV formation have been performed by Panda and Zaman (1994) and Ohmi et al (1990) for pitching and pitching/plunging airfoils, respectively.
2.2.4 Trailing-Edge Vortex
In classical inviscid aerodynamics (potential theory), lift L is related to the bound cir-culation ΓB through the Kutta-Joukowski theorem
L = ρU∞ΓB. (2.3)
An increase of circulation around the airfoil results in an augmentation of lift. In order to fulfill Kelvin’s law
DΓ
Dt = 0, (2.4)
circulation equal and opposite to the clockwise lift-augmenting circulation must also be produced. For gradual changes in lift, this counter-clockwise circulation is continually shed from the trailing edge in the form of a vortex sheet. For rapid changes in lift, however, the vortex sheet may roll-up into a distinct starting vortex or trailing-edge vortex (TEV), should the pressure difference across the trailing-edge be large enough.
The lift on an impulsively started airfoil in an incompressible inviscid flow was first modeled by Wagner (1925). As the airfoil moves away from the starting vortex, the lift coefficient asymptotes toward the quasi-steady lift coefficient. However, lift remains below 80% of the steady-lift value until the airfoil has traveled approximately three chord lengths, which at k = 0.25 (as used in the wind-tunnel measurements) represents a quarter of a cycle. Therefore the TEV has significant lift-diminishing effects for the reduced frequencies studied here. Similarly Kuessner (1936) modeled the airfoil gust response for sharp-edged gusts. In a similar manner to the Wagner function, lift was also found to asymptote to the steady-lift value, but at an even slower rate than that of the Wagner function.
For the strongly separated flows studied here, however, flow visualization by Panda and Zaman (1994) have shown that not only a starting vortex at the trailing edge but also a leading-edge separation in the form of a LEV are produced for harmonic oscillating airfoils with large amplitudes. For this case the TEV is generated through the rolling-up of the trailing-edge shear layer as the LEV convects by. Subsequently the vortex pair has been shown to convect downstream to form the so-called mushroom-wake struc-ture. Similar to a starting vortex described above, a TEV is unfavorable to the overall lift production as it counteracts the airfoil circulation (bound vortex). TEV formation is especially detrimental for cyclic motions that produce a TEV on every downstroke, since the relative spacing of the shed vortices to this bound vortex is small. Contrary to the fast convection of a starting vortex, Panda and Zaman (1994) have shown that the mushroom-wake structure asymptotically approaches the free-stream velocity, thus countering the bound vortex for a relatively long period of time during the cycle. As previously mentioned, dragonflies periodically produce a LEV on their forewings. How-ever, Thomas et al (2004) found that no distinct TEV was formed, but rather a shear layer consisting of many smaller TEVs was produced instead. Replacing a large TEV with a shear layer consisting of smaller vortices is beneficial for lift as described by Katz and Plotkin (2001).
In order to satisfy Kelvin’s law for the case of full separation, the production of both the positive and negative circulation in the flow must be equal to the net lift. Sources of circulation include the bound vortex as well as vortical structures emanating from both leading and trailing edges. During the growth phase of the LEV, counter-rotating vorticity is continually shed from the trailing edge and Kelvin’s law is fulfilled. As the LEV passes over the trailing edge the shear layer rolls up into a distinct vortex. At the same time vorticity is still produced at the leading edge and the flow around the airfoil
develops into a fully-separated state. The lingering flow separation after LEV pinch-off does not increase lift but rather the drag, as discussed in Carr et al (1977), McCroskey (1982) and Leishman (2006).
2.2.5 Reattachment
As described in section 2.2.1, a strong hysteresis in the forces is caused by the delay in reattachment due to the slow viscous time scales. In section 7.2.4 this reattachment process is accelerated through manipulation of the TEV and proves vital to efficient propulsion for flapping flight. Ahmed and Chandrasekhara (1994) studied the reattach-ment process of an oscillating airfoil typical for helicopter applications with a reduced frequency of k = 0.05. They found a slow reattachment process with the flow reattach-ing from front to rear. Separation was maintained until the effective angle of attack fell well below the static-stall angle. Fast flow reattachment not only reduces the size of the force and moment hysteresis, but more specifically the time spent in the disadvanta-geous low-lift/high-drag regions of the cycle. Green and Galbraith (1995) found that flow reattachment is comprised of two overlapping mechanisms. The first is a pitch-rate inde-pendent convection wave, which is followed by the pitch-rate deinde-pendent re-establishment of a boundary layer and fully-attached flow. A more detailed description of the reattach-ment mechanisms can be found in Sheng et al (2007). For the measurereattach-ments performed in this thesis, the temporal and spacial resolution does not allow the two mechanisms to be distinguished from each other.
2.3 Vortex Dynamics
2.3.1 Optimal Vortex Formation
The dimensionless vortex formation time ˆT , often referred to as simply the formation time, is a measure of the state of development of a vortex. In a review paper on this topic, Dabiri (2009) suggests the existence of a level of universality among all unsteady vortical flows through the concept of an optimal vortex formation number, where the vortex stops entraining fluid from the shear layer and pinches off at a value of ˆT ≈ 4. The formation time is defined in its general form in the following manner:
ˆ
T = CΓ
D∆U, (2.5)
where C is a constant factor depending on the physical configuration of the vortex generator, Γ is the instantaneous vortex strength, D the characteristic length scale and ∆U the shear-layer feeding velocity. For the specific case of a two-dimensional plunging airfoil with only one LEV shed per cycle, the vortex generator constant has been set to C = 1 and the limiting length scale of the forming vortex set as D = 2c. The shear-layer feeding velocity was approximated by ∆U = 2πf ho, being the maximum plunge velocity
of the airfoil leading edge over the downstroke. Milano and Gharib (2005) showed that maximum lift on a pitching and plunging plate was generated when ˆT ≈ 4, such that the LEV would pinch off just at the end of the half stroke. A follow-up study using a translating low-aspect-ratio plate with a start-up motion was performed by Ringuette et al (2007) to better understand the impact of the tip vortex in the start-up process. Again the LEV was found to pinch off at around ˆT ≈ 4. This universal behavior in the formation of the LEV is a focus throughout this thesis, with close attention being given
to the influence of the airfoil kinematics and the presence of the hindfoil on this pinch-off process.
Further to this, it has been shown that vorticity can be entrained beyond the forma-tion number ˆT = 4 by varying piston kinematics and nozzle diameter in vortex gener-ators, as shown by Dabiri (2005) and Shusser et al (2006). Although vortex generation on a dynamic airfoil is somewhat more complex than that of a vortex ring generator, it is expected that similar results can be achieved for the dynamic-stall process on an airfoil. Both single and tandem airfoil kinematics may possibly be used to increase the formation number and therefore the maximum circulation of the LEV. These results are presented in sections 6.3.4, 9.2.3 and 9.3.3.
2.3.2 Vortex Convection
Based on the reduced frequency of k = 0.25 used for most of the wind-tunnel tests in this thesis, a particle of fluid moving at U∞ would convect x/c ≈ 12.5 over one full period of
the plunging motion. Panda and Zaman (1994) have shown that the actual convection speed (uconv) of shed vortices varies between
0.6 < uconv U∞
< 0.8 (2.6)
in the vicinity of the airfoil, largely dependent on the position, size and strength of the vortex. These vortices then rapidly accelerate to the free-stream velocity in the immediate wake. Therefore one can infer that a LEV shed at the bottom of the plunge stroke would travel x/c ≈ 6 in the time that the airfoil would require to return back to the top of the stroke. This relatively large spacing between the LEVs suggests that the history effects in the dynamic-stall process are not strongly influenced by the previous cycle’s shed vortices in the wake. Rather the history effects are expected to be more strongly influenced by the slow time scales associated with boundary-layer separation and reattachment present in the dynamic-stall process itself. In other words the state of the boundary-layer at the top of the stroke, just before the LEV formation process begins, should play a much larger role than the LEV shed from the previous cycle. A further discussion on these history effects is treated in Section 6.2. Measurement and comparison of these history effects are then presented in Section 6.3.2.
When considering the influence of vortex convection for the tandem hover configura-tion in chapter 10, the vortex inducconfigura-tion in contrast plays a dominant role on the vortex trajectories. Here the shed vortices linger in the general vicinity for as long as a full pe-riod and therefore influence further vortex development and therefore the instantaneous forces substantially. An overview of such wake-capture effects in hover is given in Birch and Dickinson (2003).
2.3.3 Blade-Vortex Interaction
The interaction between a vortex and an arbitrary body can be characterized by the orientation of the vortex (parallel, streamwise or normal) as well as the proximity at which the vortex passes, i.e. inviscid versus viscous interaction. A large body of analyt-ical, numerical and experimental studies exists in the literature and has been reviewed in great detail by Rockwell (1998). Parallel blade-vortex interaction, a subset of vortex-body interactions examined extensively by Wilder and Telionis (1998), occurs in a wide spectrum of applications and Reynolds numbers. For high Reynolds-number applica-tions ranging from rotorcraft and wind turbines, it is often adequate to use an inviscid
treatment as in Yao and Liu (1998). However, at the other end of the spectrum, when examining the flow of miniaturized flapping drones referred to as Micro Air Vehicles (MAVs), large separations are inevitable, as described in Mueller (2001). For such cases the dynamic pressures are extremely low, i.e. ∆p = O(10Pa), such that the integra-tion of fast-response miniaturized pressure sensors in small wind-tunnel models becomes impractical. Therefore a shift to velocity-field measurements such as PIV is necessary to quantitatively examine such interactions at these low Reynolds numbers. In fact, in stark contrast to pressure measurements, the accuracy of pressure extraction using PIV remains insensitive to flow velocity magnitude, as discussed by van Oudheusden et al (2007). This methodology is treated in section 3.11.
2.4 Energy Recovery
When evaluating the performance of a propulsive system, one usually considers the overall propulsive efficiency defined as:
ηP =
tU∞
P , (2.7)
where t is the thrust, U∞ the velocity of the system and P the power exerted on the
surrounding fluid. It therefore becomes clear that to maximize the propulsive efficiency, one needs to simultaneously address the two independent quantities of thrust and power. These two factors in the context of unsteady tandem configurations will now be discussed, first examining thrust production through leading-edge suction and then power reduction through normal suction.
2.4.1 Thrust Production
Knoller (1909) and Betz (1912) were the first to postulate that a plunging symmetric airfoil can create not only lift but also thrust, generalizing this effect as a consequence of the horizontal component of the lift force when the airfoil is plunged up or down, as shown in Fig 2.5. Katzmayr (1922) subsequently verified the so-called Knoller-Betz effect in his wind-tunnel experiments, where he measured the thrust force exerted on an airfoil in a wavy wake.
Figure 2.5: Creation of thrust on a plunging airfoil, where A is the lift force and S is the resulting horizontal component; taken from Schmidt (1965).
Schmidt (1965) tested a tandem-airfoil configuration in which the wake of the rotating forefoil was used to produce thrust on the static hindfoil. This propulsion system was referred to as a Wellpropeller, translated as wave propeller. Using a hindfoil profile geometry with a maximum thickness-to-chord ratio of t/c = 0.25, the best results were achieved. Compared to conventional airfoils the relatively thick profile was beneficial in creating high thrust forces through the so-called leading-edge suction effect. This
suction bubble, which is developed through a strong flow acceleration at the leading edge, is described in detail for static airfoils by Katz and Plotkin (2001). Although Schmidt’s experiments were performed at reduced frequencies of 1 ≤ k ≤ 4, i.e. at one order of magnitude higher than the experiments in this study, similar trends can be expected. Furthermore, any details regarding this thrust-producing mechanism were not measured for this configuration and are therefore uncovered in chapters 8 and 9.
Figure 2.6: Arrangement and kinematics of airfoils used for the wave-propeller experi-ments; taken from Schmidt (1965).
Warkentin and DeLaurier (2007) tested a tandem-wing ornithopter configuration with reduced frequencies in the range of 0.4 ≤ k ≤ 1. They found that a close spacing of approximately one chord length between the wings and phase angles between ψ = 0◦
and ψ = 50◦
were most beneficial for thrust efficiency. A synergy due to the tandem-wing interactions was obtained such that the propulsive efficiency was at times double that of the single-airfoil reference case, even when taking the larger reference area of the tandem arrangement into consideration. Furthermore it was hoped that asymmetric flapping at phase angles of ψ = 180◦
would produce high thrust efficiencies and simultaneously minimize vertical accelerations for a given MAV design. Unfortunately this phasing presented a substantial drop in performance. Significant leading-edge suction for a static hindwing, based on Schmidt’s wave-propeller configuration, could not be achieved in this experiment. A possible explanation for this is the geometry of the thin membrane wings used in the study with their sharp leading edges, thus not providing the necessary surface area for the suction force. Again no details pertaining to the flow field at the hindwing leading edge were measured.
2.4.2 Power Reduction
The aerodynamic power consumption (P ) for a pitching and plunging airfoil over a given period (T ) is defined as:
P = 1 T
Z
T
(L ˙h + M ˙α)dt, (2.8)
where the moment (M ) term can be neglected due to the relatively low pitch velocities ( ˙α). Subsequently the nondimensional power coefficient can be defined as follows:
CP =
2P ρU∞
3c. (2.9)
In the case of hover, the maximum plunge velocity ( ˙hmax) is used to nondimensionalize
en-lieu of the free-stream velocity (U∞).
Akhtar et al (2007) showed that bluegill sunfish could modify their thrust production but also their side forces through the interaction of the dorsal-fin wake on the tail fin. During a distinct phasing between the two fins, the effective angle of attack could be increased at the tail fin such that a much stronger LEV was established. The stronger
LEV was found to increase the suction on the side of the fin. Similarly the studies of Gopalkrishnan et al (1994) revealed vortex energy capture by a foil in the wake of a D-shaped cylinder, mimicking the behavior of trout in fast-moving water. Here particular modes of vortical interaction were observed such that the vortices from the cylinder interacted with the vortices shed by the foil. For the case of destructive interaction, cylinder vortices were repositioned and weakened by vorticity produced by the foil, thus maximizing propulsive efficiency. As a follow-up to this work, Beal et al (2006) showed that euthanized trout (or an analogous passively-mounted foil) could travel upstream (produce thrust) without any power input. This passive propulsion was possible through simple reorientation of the oncoming vortical energy shed from the upstream cylinder. This latter study is an excellent example of how both thrust production through leading-edge suction and vortical-energy extraction through normal suction are coupled and strongly dependent on vortex-airfoil phasing.
As an extension to this concept of energy harvesting through normal suction, studies by McKinney and DeLaurier (1981), Kinsey and Dumas (2006) and Platzer et al (2009) all have investigated the capacity of a pitching/plunging foil to extract energy from a stream through aerodynamic flutter. The latter study has shown substantial improve-ments when positioning a second moving foil in the wake. Further to this, Zhu and Peng (2009) recently showed that the development of the LEV and its subsequent convec-tion over the foil surface can be used to develop a strong pitch-down moment, which in turn can be used for power extraction. Finally, Dabiri (2007) developed a model based on vortex dynamics showing that the energy extraction mechanism is a function of the oncoming vortex positioning and strength relative to the body. Unfortunately, such a treatise is often too simple for use in the highly-separated flow field associated with dragonfly flight.
