ABSTRACT
Title of dissertation: ROTOR HOVER PERFORMANCE
AND SYSTEM DESIGN
OF AN EFFICIENT COAXIAL ROTARY WING
MICRO AIR VEHICLE
Felipe Bohorquez, Doctor of Philosophy, 2007
Dissertation directed by: Professor Darryll Pines
Department of Aerospace Engineering
Rotary-wing Micro air vehicles (MAVs) due to their unique hovering and low-
speed flight capabilities are specially suited for missions that require operation in
constrained spaces. Size restrictions force MAVs to operate in a low Reynolds num-
ber aerodynamic regime where viscous effects are dominant. This results in poor
aerodynamic performance of conventional airfoils and rotor configurations. This dis-
sertation explores the design issues that affect the hover performance of small-scale
rotors and the implementation of a working coaxial MAV prototype.
A computerized hover test stand was used for the systematic testing of single
and coaxial small-scale rotors. Thin circular arcs were chosen for blade manufac-
turing because of their good aerodynamic characteristics at low Reynolds numbers,
and simplified parameterization. Influence of airfoil geometry on single rotor hover
performance was studied on untwisted rectangular blades. Non rectangular blades
were used to study coupled airfoil and blade parameters. Tip tapered geometries
were manufactured by removing material from baseline rectangular blades producing
a coupling between blade planform, twist distribution, and spanwise airfoil shape.
Performance gains were obtained by introducing large negative twist angles over
short radial distances at the blade tips. A parametric study of the blade geometries
resulted in maximum figures of merit of 0.65.
Coaxial rotor performance at torque equilibrium was explored for different
trims and operating conditions. It was found that the upper rotor was marginally
affected by the lower one at spacings larger than 35% of the rotor radius, and that
it produced about 60% of the total thrust. Experiments showed that power loading
was maximized when higher collectives were used at the lower rotor, resulting in
sizable differences in rotational speed between rotors.
The CFD solver INS2d was used for a two-dimensional parametric aerody-
namic study of circular arc airfoils. Lift, drag, and moment coefficients were ex-
plored over a range of Reynolds numbers. Validation with wind-tunnel data showed
that lift predictions were satisfactory; however, drag was under-predicted at low an-
gles of attack. The CFD database was integrated to a BEMT rotor model through
a parameterization that coupled blade planform with twist distribution and airfoil
shape. Thrust and maximum FM predictions were satisfactory for rectangular and
non-rectangular blades with maximum cambers of 6% and below. The BEMT model
was extended to the coaxial rotor case, producing good thrust and power predictions
with errors within 5% of the experimental measurements. The approach validated
the use of analytical and numerical tools commonly used in full-scale analysis, and
proved to be a versatile system design tool.
A fully functional coaxial MAV was developed based on the aerodynamic stud-
ies performed. Transmission, rotors, and swashplate were designed from scratch.
Batteries, motors, and electronics were carefully selected off-the-shelf components.
The prototype has been used as a testing platform for control systems and algo-
rithms.
ROTOR HOVER PERFORMANCE
AND SYSTEM DESIGN
OF AN EFFICENT COAXIAL ROTARY WING
MICRO AIR VEHICLE
by
Felipe Bohorquez
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2007
Advisory Committee:
Professor Darryll Pines, Chair/Advisor
Professor Amr Baz
Professor Inderjit Chopra
Professor Norman Wereley
Professor James Baeder
c? Copyright by
Felipe Bohorquez Artunduaga
2007
ACKNOWLEDGMENTS
Several years ago I showed up at Dr. Pines office looking for an opportunity. I
was just a guy from a distant place with lots of enthusiasm that had missed all the
application deadlines. Somehow, he saw my potential and gave me the chance to be
a grad student at the Rotorcraft Center. I owe my gratitude to him for believing
in me, for his support, and for his guidance through the course of this research. I
would also like to thank my committee, especially Dr. Chopra, for their advice and
support.
I would like to acknowledge the invaluable help of Jayant Sirohi, he intro-
duced me to the lab, and taught me the fundamental skills that were essential for
most of the experimental aspects of this work. We shared endless hours tinkering
with experimental setups and MAV prototypes, while having fun with our peers at
the Smart Structures lab. My colleagues Paul Samuel, Jason Pereira, Joe Conroy,
Anubav Datta, Atul Jayasimha, and Ron Couch not only helped me on various
aspect of my research, but also enriched in many ways my graduate student life.
Some of the students of the CFD lab were incredibly patient and helpful an-
swering thoroughly the many questions I had. Karthik Duraisamy, Eric Shroeder,
Vinod Lakshminarayan, Falcon Rankins, and Brandon Bush, thanks for all your
help.
I would also like to acknowledge some of the staff members, Becky, Julia,
Debora and specially Pat Baker, who guided me through a maze of deadlines and
paper work.
Grad student life would have been less enjoyable without my lunch buddies and
good friends Natalia Perez, Andres Garay, Ashish Purekar, Carlos Malpica, Sandra
Ugrina, and Maria Ribera. We will always cherish the memories of the culinary
delicacies of suburban Maryland.
ii
I owe my deepest gratitude to my family, my mother, father, and sister, and
to Maria Francisca, Gonzalo, and Claudia. They are the foundation of who I am
and have always believed in me.
Finally and most importantly, I would like to thank my wife Patricia for being
the sweetest, most inspiring woman one can imagine. She calmly dealt with the
stress of two Ph.Ds, hers and mine. Her constant love and support kept me going.
Without her this journey would have been impossible.
iii
TABLE OF CONTENTS
List of Tables vii
List of Figures viii
1 Introduction 1
1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation and Problem Statement . . . . . . . . . . . . . . . . . . . 2
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Experimental Low Re Airfoil Studies . . . . . . . . . . . . . . 6
1.3.2 Vehicle Design and Development . . . . . . . . . . . . . . . . 9
1.3.3 Coaxial Helicopter Aerodynamics . . . . . . . . . . . . . . . . 11
1.3.4 Computational Studies . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Current Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.3 Organization of Dissertation . . . . . . . . . . . . . . . . . . . 17
2 Single Rotor Tests 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 System Calibration . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.3 Procedure for Data Acquisition . . . . . . . . . . . . . . . . . 24
2.2.4 Experimental Error Analysis . . . . . . . . . . . . . . . . . . . 26
2.3 First Generation Rotors - Hover Performance . . . . . . . . . . . . . 29
2.3.1 First Generation Rotor Configuration . . . . . . . . . . . . . . 29
2.3.2 Experimental Results - First Generation Rotors. . . . . . . . . 32
2.3.3 Effect of Number of Blades on Rotor Performance . . . . . . . 45
2.4 Third Generation Rotors - Flat Plate Research . . . . . . . . . . . . . 48
iv
2.4.1 Zero Lift Drag Measurements . . . . . . . . . . . . . . . . . . 50
2.4.2 HoverPerformanceofRotorswithRectangularFlatPlateBlades 58
2.5 Third Generation Rotors - Cambered Rectangular Blades . . . . . . . 61
2.5.1 Hover Performance - Effect of Maximum Camber . . . . . . . 61
2.5.2 Hover Performance - Effect of Maximum Camber Chordwise
Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5.3 Hover Performance - Effect of Rotor Solidity . . . . . . . . . . 68
2.6 Third Generation Rotors - Non-Rectangular Blades . . . . . . . . . . 73
2.6.1 Blade Tip Shapes - Full-scale vs. MAV-Scale . . . . . . . . . . 73
2.6.2 Blade Planform Effects on Hover Performance of Rotors with
Flat Plate Blades . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.6.3 Blade Planform Effects on Hover Performance of Rotors with
cambered blades . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.6.4 Maximum FM and Maximum PL Envelope . . . . . . . . . . . 92
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3 Blade Element Momentum Theory
and
Average sectional Airfoil Characteristics 99
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.2 Blade Element Momentum Theory . . . . . . . . . . . . . . . . . . . 100
3.2.1 Derivation of Equations . . . . . . . . . . . . . . . . . . . . . 100
3.2.2 BEMT and Sectional Airfoil Characteristics . . . . . . . . . . 104
3.3 Inverse Method to Obtain 2D-Airfoil Characteristics from Rotor Tests 105
3.3.1 Iteration algorithm . . . . . . . . . . . . . . . . . . . . . . . . 107
3.4 BEMT Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.4.1 Inverse BEMT Results . . . . . . . . . . . . . . . . . . . . . . 108
3.4.2 Induced and Profile Power, FM revisited . . . . . . . . . . . . 112
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
v
4 Sectional Airfoil Characteristics from CFD 117
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2 Geometrical Airfoil Parameterization . . . . . . . . . . . . . . . . . . 118
4.3 CFD Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.3.1 Flow Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.3.2 Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 123
4.3.4 Flow Field Assumptions . . . . . . . . . . . . . . . . . . . . . 124
4.4 Two-Dimensional Validation . . . . . . . . . . . . . . . . . . . . . . . 125
4.5 CFD Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.5.1 Parametric Study on 4% Circular Arc . . . . . . . . . . . . . . 129
4.5.2 Elliptical Leading Edge Database Results . . . . . . . . . . . . 131
4.5.3 Sharp Leading Edge Database Results . . . . . . . . . . . . . 139
4.5.4 Sharp vs. Elliptical Leading Edges . . . . . . . . . . . . . . . 148
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5 Hybrid BEMT-CFD Method for Rotor Analysis and Design 154
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.2 Design Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.3 Geometric Blade Parameterization . . . . . . . . . . . . . . . . . . . 157
5.4 Airfoil Database Interpolation . . . . . . . . . . . . . . . . . . . . . . 161
5.4.1 Interpolation Validation . . . . . . . . . . . . . . . . . . . . . 161
5.5 Approach Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.5.1 Rectangular Blades - Camber Effects . . . . . . . . . . . . . . 164
5.5.2 Rectangular Blades - Rotational Speed Effects . . . . . . . . . 169
5.5.3 Rectangular Blades - Leading Edge Effects . . . . . . . . . . . 170
5.5.4 Non-Rectangular Blades . . . . . . . . . . . . . . . . . . . . . 176
5.6 Blade Planform Optimization - Case Study - . . . . . . . . . . . . . . 180
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
vi
6 Coaxial Rotor Performance and Vehicle Design 186
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.3 Coaxial Tests Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.4 Coaxial Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.4.1 Rotor Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.4.2 Rotor Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.4.3 Rotor Load Sharing . . . . . . . . . . . . . . . . . . . . . . . . 195
6.5 Maximization of Power Loading in a Coaxial Rotor . . . . . . . . . . 196
6.6 Modeling of a Coaxial Rotor . . . . . . . . . . . . . . . . . . . . . . . 201
6.6.1 Blade Element Momentum Theory for a Coaxial Rotor . . . . 202
6.7 Vehicle Design and Integration . . . . . . . . . . . . . . . . . . . . . . 209
6.7.1 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . 210
6.7.2 Concept Selection . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.7.3 Vehicle Configuration . . . . . . . . . . . . . . . . . . . . . . . 213
6.7.4 Vehicle Control . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.7.5 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
6.7.6 Powertrain - Rotor Matching . . . . . . . . . . . . . . . . . . 221
6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7 Concluding Remarks 229
7.1 Summary of Contribution and Results . . . . . . . . . . . . . . . . . 229
7.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 235
A Low Reynolds Number Flow Physics
and Basic Rotor Aerodynamics 238
A.1 Flow Regimes, Reynolds and Mach Numbers . . . . . . . . . . . . . . 238
A.2 Boundary layer Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 239
A.3 Lift Generation and Boundary Layers . . . . . . . . . . . . . . . . . . 243
A.4 Low Reynolds Numbers Sub-Regimes . . . . . . . . . . . . . . . . . . 243
A.5 Laminar Separation Bubble . . . . . . . . . . . . . . . . . . . . . . . 246
vii
A.6 Reynolds Number Effects on Aerodynamic Airfoil Characteristics . . 249
A.6.1 Thin Cambered Airfoil Characteristics . . . . . . . . . . . . . 250
A.7 Fundamental Rotor Aerodynamics . . . . . . . . . . . . . . . . . . . . 256
A.7.1 Momentum Theory for a Hovering Rotor . . . . . . . . . . . . 256
A.7.2 Thrust and Power Coefficients . . . . . . . . . . . . . . . . . . 258
A.7.3 Rotor Hover Efficiency Metrics . . . . . . . . . . . . . . . . . 259
A.7.4 Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . 259
A.7.5 Power Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Bibliography 263
viii
LIST OF TABLES
2.1 First generation rotor geometric characteristics . . . . . . . . . . . . . 30
2.2 Second generation rotor geometric characteristics . . . . . . . . . . . 48
2.3 Third generation rotor geometric characteristics . . . . . . . . . . . . 49
2.4 Test Matrix for rectangular blades with flat plates, four LE shapes
and three chord-thickness configurations . . . . . . . . . . . . . . . . 49
2.5 Test matrix for rectangular blades with circular arc airfoils with var-
ious cambers and solidities . . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Test matrix for rectangular blades with various max camber locations,
chord = 1.5 cm, ?=0.0909 . . . . . . . . . . . . . . . . . . . . . . . . 67
2.7 Flat plate tests matrix for three different tip shapes, and elliptical
leading edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.8 Tests matrix for five different tip shapes with elliptical leading edges
and equal solidity. Linear taper starts at 0.85R . . . . . . . . . . . . 79
2.9 Tests matrix for 6% and 9% camber blades with various linear taper
ratios starting at various chordwise locations . . . . . . . . . . . . . . 80
4.1 Test matrix for CFD calculations, combinations of camber and TR. . 118
6.1 Test matrix for coaxial tests using 6% camber rectangular blades. . . 190
6.2 Third generation rotor geometric characteristics . . . . . . . . . . . . 190
6.3 Coaxial vehicle weight breakdown. . . . . . . . . . . . . . . . . . . . . 216
ix
LIST OF FIGURES
1.1 Relative performance of various micro air vehicles. . . . . . . . . . . . 5
1.2 Re vs. Mass for animal and Man-made vehicles from data in Ref [19]. 6
2.1 First generation hover test stand . . . . . . . . . . . . . . . . . . . . 21
2.2 Second generation hover test stand . . . . . . . . . . . . . . . . . . . 21
2.3 Hover stand experimental setup sketch. . . . . . . . . . . . . . . . . . 23
2.4 Sample screen of data acquisition software developed after a single
rotor test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Thrust and torque calibration curves . . . . . . . . . . . . . . . . . . 25
2.6 Proof of concept design from Ref. [75] . . . . . . . . . . . . . . . . . 30
2.7 Mold used for first generation baseline blades. . . . . . . . . . . . . . 31
2.8 First generation 8% camber 5% TR airfoil. . . . . . . . . . . . . . . . 31
2.9 Rotor, blades and transmission used in the testing of the first gener-
ation rotor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.10 Collective vs. thrust first generation rotor with 8% camber circular
arcsat various rotational speeds. . . . . . . . . . . . . . . . . . . . . . 33
2.11 Collective vs. torque first generation rotor with 8% camber circular
arcs at various rotational speeds. . . . . . . . . . . . . . . . . . . . . 34
2.12 Collective vs. CT first generation rotor with 8% camber circular arcs
at various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . 34
2.13 Collective vs. CP first generation rotor with 8% camber circular arcs
at various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . 35
2.14 CT vs. FM first generation rotor with 8% camber circular arcs at
various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.15 DL vs. FM first generation rotor with 8% camber circular arcs at
various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.16 Collective vs. thrust of first generation rotor with NACA 0012 airfoil
at various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . 37
x
2.17 Collective vs. torque for first generation rotor with NACA 0012 airfoil
at various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . 37
2.18 Collective vs. thrust of first generation rotor with flat plate airfoil at
various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.19 Collective vs. torque for first generation rotor with flat plate airfoil
at various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . 38
2.20 Collective vs. CT of first generation rotor with NACA 0012 airfoil at
various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.21 Collective vs. CP for first generation rotor with NACA 0012 airfoil
at various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . 39
2.22 Collective vs. CT of first generation rotor with flat plate airfoil at
various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.23 Collective vs. CP for first generation rotor with flat plate airfoil at
various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.24 Comparison of CT vs. CP for 8% camber plates, flat plates and NACA
0012 airfoils at 4500 RPM i.e Re=27,000 using the first generation
rotors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.25 Comparison of CT vs. FM for 8% camber plates, flat plates and
NACA 0012 airfoils at 4500 RPM i.e Re=27,000 using the first gen-
eration rotors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.26 Comparison of DL vs. PL of rotors with 8% camber plates, flat plates
and NACA 0012 airfoils at 4500 RPM i.e Re=27,000 using the first
generation rotors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.27 Comparison of CT vs. CP of rotors with twisted and untwisted 8%
camber plates, at 4500 RPM i.e tip Re=27,000 using the first gener-
ation rotors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.28 Comparison of CT vs. FM of rotors with twisted and untwisted 8%
camber plates, at 4500 RPM i.e tip Re=27,000 using the first gener-
ation rotors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
xi
2.29 Comparison of DL vs. PL of rotors with twisted and untwisted 8%
camber plates at 4500 RPM i.e Re=27,000 using the first generation
rotors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.30 Comparison of CT vs. CP of two and three-bladed rotors with 8%
camber and flat plates at 3500 RPM, ?=0.111 . . . . . . . . . . . . . 46
2.31 Comparison of CT vs. FM of two and three-bladed rotors with 8%
camber and flat plates at 3500 RPM, ?=0.111 . . . . . . . . . . . . . 47
2.32 Comparison of DL vs. PL of two and three-bladed rotors with 8%
camber and flat plates at 3500 RPM, ?=0.111 . . . . . . . . . . . . . 47
2.33 Leading edge shapes tested with flat plate airfoils . . . . . . . . . . . 50
2.34 Re vs. equivalent zero lift drag coefficient Cd0 for rotor blades with
blunt leading edges and various TR and chords . . . . . . . . . . . . 52
2.35 Re vs. equivalent zero lift drag coefficient CD0 for rotor blades with
elliptical leading edges and various TR and chords . . . . . . . . . . . 53
2.36 Re vs. equivalent zero lift drag coefficient CD0 for rotor blades with
symmetric sharp leading edges and various TR and chords . . . . . . 54
2.37 Re vs. equivalent zero lift drag coefficient Cd0 for blades with various
LE shapes with TR=2.2% and chord=2.25 cm . . . . . . . . . . . . . 55
2.38 Re vs. equivalent zero lift drag coefficient Cd0 for blades with various
LE shapes with TR=3.3% and chord=2.25 cm . . . . . . . . . . . . . 55
2.39 Re vs. equivalent zero lift drag coefficient Cd0 for blades with various
LE shapes with TR=3.3% and chord=1.5 cm . . . . . . . . . . . . . . 56
2.40 Pressure distribution at the round LE of a flat plate at zero angle of
attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.41 Flow around the LE of a flat plate at low angles of attack . . . . . . . 58
2.42 CT vs. FM at 2500 RPM for rotors with flat plate blades and various
LE shapes, chord=2.25cm, TR=2.2%, tip Re = 43,700 . . . . . . . . 59
2.43 DL vs. PL at 2500 RPM for rotors with flat plate blades, and various
LE shapes, chord=2.25cm, TR=2.2%, tip Re = 43,700 . . . . . . . . 59
xii
2.44 CT vs. CP at 2500 RPM for rotors with flat plate blades, and various
LE shapes, chord=2.25cm, TR=2.2%, tip Re = 43,700 . . . . . . . . 60
2.45 CT vs. FM at 2500 RPM for rotors with circular arc airfoils of various
cambers and elliptical LE, chord=2.25cm, TR=2.2%, tip Re = 43,700 62
2.46 CT vs. CP at 2500 RPM for rotors with circular arc airfoils of various
cambers and elliptical LE, chord=2.25cm and TR=2.2%, tip Re =
43,700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.47 CT vs. CP at 2500 RPM for rotors with circular arc airfoils of various
cambers and elliptical LE, chord=2.25cm and TR=2.2%, tip Re =
33,400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.48 CT vs. CP for rotors with circular arc airfoils of various cambers and
elliptical LE, chord=2.25cm and TR=2.2%, various tip speeds . . . . 65
2.49 Hover performance of rotor with 9% camber rectangular blades at
various RPM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
(a) CT vs. FM for various RPM, 12deg to 18deg collective . . . . . 66
(b) DL vs. PL various RPM - envelope for max DL . . . . . . . . . 66
2.50 CT vs. FM for rotors with 6 % camber airfoils, max camber at dif-
ferent chordwise locations, chord=1.5cm, TR=3.3%, tip Re= 35,000 . 67
2.51 CT vs. CP for rotors with 6 % camber airfoils, max camber at different
chordwise locations. chord=1.5cm, TR=3.3%, tip Re= 35,000 . . . . 68
2.52 FM of two rotor with 6% camber blades and different solidities at
2500 RPM as function of CT and CT/?. . . . . . . . . . . . . . . . . 71
(a) CT vs. FM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
(b) CT/? vs. FM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.53 Hover performance of rotors with 9% camber blades at 2500 RPM. . 71
(a) CT/? vs. FM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
(b) DL vs. PL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.54 DL vs. PL for rotors with 6% and 9% camber blades at different
rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
(a) DL vs. PL for 6% camber rotors . . . . . . . . . . . . . . . . . 72
xiii
(b) DL vs. PL for 9% camber rotors . . . . . . . . . . . . . . . . . 72
2.55 CT vs. FM for rotors with rectangular 6% and 9% camber blades
with elliptical and positive sharp leading edges at 2500 RPM, chord
=2.25 cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
(a) CT vs. FM for 6% camber rotors . . . . . . . . . . . . . . . . . 73
(b) CT vs. FM for 9% camber rotors . . . . . . . . . . . . . . . . . 73
2.56 Geometric effects of symmetric and asymmetric 2:1 linear taper on
blade with baseline 9% circular arc airfoil. . . . . . . . . . . . . . . . 75
(a) 2:1 taper ratio symmetric . . . . . . . . . . . . . . . . . . . . . 75
(b) 2:1 taper ratio asymmetric . . . . . . . . . . . . . . . . . . . . . 75
2.57 Blade tip shapes used with flat plate blades, (a) elliptical asymmetric,
(b) elliptical, (c) 2:1 asymmetric linear. For geometries taper starts
at 0.8R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.58 Effect of symmetric and asymmetric elliptical taper starting @ 0.8R
on flat plate blades with elliptical leading edges. . . . . . . . . . . . . 78
(a) CT/? vs. FM base chord=1.5 cm . . . . . . . . . . . . . . . . . 78
(b) CT vs. CP base chord=1.5 cm . . . . . . . . . . . . . . . . . . . 78
2.59 Effect of 2:1 taper ratio starting @ 0.8R on flat plate blades with
elliptical leading edges. . . . . . . . . . . . . . . . . . . . . . . . . . 79
(a) CT/? vs. FM base chord=2.25 cm . . . . . . . . . . . . . . . . 79
(b) CT vs. CP base chord=2.25 cm . . . . . . . . . . . . . . . . . . 79
2.60 Blade tip shapes used with 9%, (a) elliptical asymmetric, (b) ellip-
tical, (c) 1.66:1 asymmetric linear, (d) 1.66:1 symmetric linear. For
geometries taper starts at 0.85R. . . . . . . . . . . . . . . . . . . . . 80
2.61 Tip shapes tested with 6% and 9% camber airfoils with 2.25 cm chord 81
(a) Linear taper with various ratios starting at 0.8R . . . . . . . . . 81
(b) Linear 2:1 taper starting at 0.6R . . . . . . . . . . . . . . . . . 81
2.62 Effect of symmetric taper distribution on blades with baseline 9%
camber airfoils and elliptical leading edges at 3000 RPM. . . . . . . . 83
(a) CT/? vs. FM base chord=1.6 cm . . . . . . . . . . . . . . . . . 83
xiv
(b) CT vs. CP base chord=1.6 cm . . . . . . . . . . . . . . . . . . 83
2.63 Effect of asymmetric taper distribution on blades with baseline 9%
camber airfoils and elliptical leading edges at 3000 RPM. . . . . . . . 83
(a) CT/? vs. FM base chord=1.6 cm . . . . . . . . . . . . . . . . . 83
(b) CT/? vs. CP base chord=1.6 cm . . . . . . . . . . . . . . . . . 83
2.64 Tip shape geometry effects on twist distribution, airfoil camber, thick-
ness ratio and Reynolds number for elliptical and linear tapers on
rotors with equal solidity . . . . . . . . . . . . . . . . . . . . . . . . . 85
(a) Twist along asymmetric tapered tips . . . . . . . . . . . . . . . 85
(b) Airfoil camber along tapered tips . . . . . . . . . . . . . . . . . 85
(c) Thickness ratio along tapered tips . . . . . . . . . . . . . . . . . 85
(d) Re along tapered tips at 3000 RPM . . . . . . . . . . . . . . . . 85
2.65 Tip shape geometry effects on twist distribution and airfoil camber
on blades with baseline 6% and 9% camber airfoils and various linear
asymmetric taper ratios. . . . . . . . . . . . . . . . . . . . . . . . . . 87
(a) Camber along asymmetric tapered tips . . . . . . . . . . . . . . 87
(b) Twist along asymmetric tapered tips . . . . . . . . . . . . . . . 87
2.66 CT/? vs. FM for rotors with baseline 6% camber airfoils with el-
liptical leading edges and various tip taper ratios at two rotational
speeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
(a) CT/? vs. FM at 2500 RPM . . . . . . . . . . . . . . . . . . . . 89
(b) CT/? vs. FM at 3000 RPM . . . . . . . . . . . . . . . . . . . . 89
2.67 CT/? vs. FM for rotors with baseline 9% camber airfoils with el-
liptical leading edges and various tip taper ratios at two rotational
speeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
(a) CT/? vs. FM at 2500 RPM . . . . . . . . . . . . . . . . . . . . 90
(b) CT/? vs. FM at 3000 RPM . . . . . . . . . . . . . . . . . . . . 90
2.68 CT/? vs. FM for rotors with rectangular, symmetric and asymmetric
2:1 taper ratio blades with elliptical leading edges at 3000 RPM. . . . 91
xv
2.69 CT/? vs. FM for rotors with asymmetric 2:1 taper ratio starting
at 0.8R and 0.6R and symmetric 2:1 taper ratio starting at 0.6R.
Baseline 9% camber blades with elliptical leading edges at 3000 RPM. 91
2.70 CT/? vs. FM for rotors with baseline 6% and 9% camber airfoils,
effect of leading edge on tapered blades. . . . . . . . . . . . . . . . . 93
(a) CT/? vs. FM 6% camber base airfoil at 3000 RPM . . . . . . . 93
(b) CT/? vs. FM 9% camber base airfoil at 3000 RPM . . . . . . . 93
2.71 Maximum PL envelopes for rotors with various blades configurations 96
(a) Envelope of max PL from T vs. PL plots at various rotational
speeds for rotor with 2:1 tapered blades @ 0.8R, 9% baseline
camber and elliptical LE . . . . . . . . . . . . . . . . . . . . . . 96
(b) Comparison of maximum PL envelopes over a thrust range of
50g to 100g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.1 Discretization of rotor disk used for BEMT analysis. . . . . . . . . . . 100
3.2 Algorithm used to obtain average airfoil characteristics from rotor
tests using BEMT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.3 Average lift and drag coefficients vs. angle of attack from Inverse
BEMT using experimental data of first generation rotors at 4500
RPM for flat plates, 8% camber circular arcs, and NACA 0012 airfoils.109
(a) Lift coefficient vs. ? . . . . . . . . . . . . . . . . . . . . . . . . 109
(b) Drag coefficient vs. ? . . . . . . . . . . . . . . . . . . . . . . . . 109
3.4 BEMT results for the rotor with 417a airfoils: AoA span distribution,
CT vs. FM, collective vs. CP, and collective vs. CT. . . . . . . . . . . 110
3.5 Lift and drag sectional characteristics of curved plate with 6% max
camber at 40% chord and 3.33% TR from BEMT compared to wind
tunnel tests of Go417a from Ref. [1] . . . . . . . . . . . . . . . . . . . 111
(a) Cl vs. ? from BEMT . . . . . . . . . . . . . . . . . . . . . . . . 111
(b) Cd vs. ? from BEMT . . . . . . . . . . . . . . . . . . . . . . . 111
3.6 Experimental and BEMT calculated lift to drag ratio of Go417a airfoil.111
xvi
3.7 Lift to drag ratio for the airfoils of the first generation rotor from
inverse BEMT results. . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.8 Experimental and BEMT CT vs. FM for the first generation rotors
with twisted and untwisted 8% camber blades. . . . . . . . . . . . . . 114
(a) Experimental CT vs. FM . . . . . . . . . . . . . . . . . . . . . . 114
(b) BEMT prediction CT vs. FM . . . . . . . . . . . . . . . . . . . 114
3.9 CT vs. CP from BEMT results for the rotors twisted and untwisted
8% camber blades. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.10 BEMT calculated induced power factor ? for the first generation ro-
tors and third gen rotor with Go417a airfoil. . . . . . . . . . . . . . . 115
4.1 Geometrical proportions of circular arc airfoils with elliptical and
sharp leading edges. 4% camber, 1.93 TR shown. . . . . . . . . . . . 120
4.2 Detail of elliptical and sharp leading edges. . . . . . . . . . . . . . . . 120
4.3 Family of circular arcs with elliptical LE, cambers ranging from 0%
to 9% and TR of 2.5% and 3.75%. . . . . . . . . . . . . . . . . . . . . 121
4.4 Sample C-type grid used in the CFD calculations. . . . . . . . . . . . 123
4.5 Sample grid around curved plate airfoil showing clustering at leading
and trailing edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.6 Lift, drag, moment, and lift to drag ratio validation of TURNS,
INS2d, and Fluent with experimental data at Re=60,000 from Ref.
[10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.7 Effect of Re in lift, drag, moment coefficients and L/D ratio on a 4%
camber 1.93% TR circualr arc. . . . . . . . . . . . . . . . . . . . . . . 130
4.8 Change of maximum lift-to drag ratio with Reynolds number for the
4% camber 1.93%TR circular arc. . . . . . . . . . . . . . . . . . . . . 130
4.9 Effect of TR in lift, drag, moment coefficients and L/D ratio on a
4%camber circular arc at Re=60,000 . . . . . . . . . . . . . . . . . . 132
4.10 Cl and Cd vs. ? for circular arcs with elliptical LE and two TR at
Re=60,000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
xvii
(a) Lift coefficient 2.5% TR . . . . . . . . . . . . . . . . . . . . . . 134
(b) Drag coefficient 2.5% TR . . . . . . . . . . . . . . . . . . . . . 134
(c) Lift coefficient 3.75% TR . . . . . . . . . . . . . . . . . . . . . . 134
(d) Drag coefficient 3.75% TR . . . . . . . . . . . . . . . . . . . . . 134
4.11 L/D vs. ? and L/D vs. Cl for circular arcs with elliptical LE and
two TR at Re=60,000. . . . . . . . . . . . . . . . . . . . . . . . . . . 135
(a) L/D vs. ? for 2.5% TR circ arcs. . . . . . . . . . . . . . . . . . 135
(b) L/D vs. Cl for 2.5% TR circ arcs. . . . . . . . . . . . . . . . . . 135
(c) L/D vs. ? for 3.75% TR circ arcs. . . . . . . . . . . . . . . . . 135
(d) L/D vs. Cl for 3.75% TR circ arcs. . . . . . . . . . . . . . . . . 135
4.12 Effect of TR in lift, drag, and moment coefficients and L/D ratio on
6% and 9% camber circular arcs with 2.5% TR and 3.75% TR at
Re=60,000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.13 Contours of L/D vs. ? at Re=60,000 for airfoils with elliptical LE
and two TR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
(a) 2.5% TR airfoils at Re=60k . . . . . . . . . . . . . . . . . . . . 137
(b) 3.75% TR airfoils at Re=60k . . . . . . . . . . . . . . . . . . . 137
4.14 Lift and Drag coefficients vs. ? for airfoils with elliptical LE with
2.5% TR, and 3.75% TR. All cambers at Re=15,000. . . . . . . . . . 138
(a) Lift coefficient at Re=15k, TR=2.5% . . . . . . . . . . . . . . . 138
(b) Drag coefficient at Re=15k, TR=2.5% . . . . . . . . . . . . . . 138
(c) Lift coefficient at Re=15k, TR=3.75% . . . . . . . . . . . . . . 138
(d) Drag coefficient at Re=15k, TR=3.75% . . . . . . . . . . . . . . 138
4.15 Effect of Re in lift, drag, moment coefficient and L/D on a 6% camber
2.5% TR circular arc with elliptical LE. . . . . . . . . . . . . . . . . . 140
4.16 Contour plots of L/D as function of airfoil camber and ? for airfoils
with elliptical LE and 2.5% TR at Re = 15k, 30k, 45k, 60k. . . . . . 141
4.17 Contour plots of L/D as function of airfoil camber and ? for airfoils
with elliptical LE and 3.75% TR at Re = 15k, 30k, 45k, 60k. . . . . . 142
xviii
4.18 Cl and Cd vs. ? for circular arc airfoils with sharp LE and TR= 2.5%
at Re=60,000 and Re=15,000. . . . . . . . . . . . . . . . . . . . . . . 144
(a) Cl vs. ? for 2.5% TR circ arcs sharp LE, Re=60k. . . . . . . . . 144
(b) Cd vs. ? for 2.5% TR circ arcs, sharp LE, Re=60k. . . . . . . . 144
(c) Cl vs. ? for 2.5% TR circ arcs, sharp LE, Re=15k. . . . . . . . 144
(d) Cd vs. ? for 2.5% TR circ arcs, sharp LE, Re=15k. . . . . . . . 144
4.19 Contour plots of L/D as function of airfoil camber and ? for airfoils
with sharp LE and 2.5% TR at Re = 15k, 30k, 45k, 60k. . . . . . . . 145
4.20 Contour plots of L/D as function of airfoil camber and ? for airfoils
with sharp LE and 3.75% TR at Re = 15k, 30k, 45k, 60k. . . . . . . 146
4.21 Effect of TR in lift, drag, and moment coefficients and L/D ratio
on 6% and 9% camber circular arcs with sharp LE having 2.5% and
3.75% TR at Re=60,000. . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.22 Effect of leading edge shape in lift, drag, and moment coefficients
and L/D ratio on 6% and 9% circular arc airfoils with 2.5% TR at
Re=60,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.23 Effect of leading edge shape in lift, drag, and moment coefficients
and L/D ratio on 6% and 9% circular arc airfoils with 2.5% TR at
Re=15,000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.1 Single rotor design algorithm . . . . . . . . . . . . . . . . . . . . . . . 156
5.2 Radial parameters for rectangular blade with uniform 6% camber airfoil.159
5.3 Radial parameters for non-rectangular blade with baseline 6% camber
airfoil at root and 9% camber airfoil at tip. . . . . . . . . . . . . . . . 160
5.4 Comparison of lift drag and L/D results from direct CFD and inter-
polated results for a 4% camber, 1.93% TR at various Re. . . . . . . 163
5.5 Predicted CT vs. FM and CT vs CP for rotor with rectangular blades
and various airfoil cambers at 2500 RPM. . . . . . . . . . . . . . . . 164
5.6 Predicted and experimental CT vs. FM for 0%, 3%, 6% and 9%
camber airfoils with elliptical leading edges at 2500 RPM. . . . . . . 166
xix
5.7 Predicted and experimental CT vs. FM and CT vs CP for rotor with
3% camber airfoils, elliptical leading edges at 2500 RPM. . . . . . . . 167
5.8 Predicted and experimental CT vs. FM and CT vs CP for rotor with
6% camber airfoils, elliptical leading edges at 2500 RPM. . . . . . . . 168
5.9 Predicted and experimental CT vs. FM and CT vs CP for rotor with
9% camber airfoils, elliptical leading edges at 2500 RPM. . . . . . . . 168
5.10 Induced angles of attack(a), and Induced and profile power compo-
nents (b), at various collectives of a 6% camber rectangular blades. . 169
5.11 CT vs. FM, CT vs CP, and PL vs. T for rotor with 6% camber
airfoils, elliptical leading edges at three different RPMs. . . . . . . . . 171
5.12 CT vs. CP considering Re change along span, and assuming constant
Re of section at 75% span. Rotors with rectangular blades and 6%
camber airfoils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.13 Percentile difference in CT, CP, and FM, when Re change along
span is modeled, and when constant Re of section at 75% span is
used. Rotors with rectangular blades and 6% camber airfoils . . . . . 173
5.14 CT vs. FM and CT vs.CP for rotors with 6% camber, sharp and
elliptical leading edges, 2500 RPM, Tip Re?43000. . . . . . . . . . . 174
5.15 Comparison of sectional aerodynamic characteristics of 6% camber
circular arcs with elliptical and sharp leading edges at Re=30,000. . . 175
5.16 Linear taper with various ratios starting at 0.8R . . . . . . . . . . . . 176
5.17 Experimental and predicted CT vs. FM at 3000 RPM for rotors
with various taper ratios on blades with baseline 6% camber airfoils,
elliptical leading edges. . . . . . . . . . . . . . . . . . . . . . . . . . . 178
(a) Experimental CT vs. FM . . . . . . . . . . . . . . . . . . . . . 178
(b) Predicted CT vs. FM . . . . . . . . . . . . . . . . . . . . . . . 178
5.18 Experimental and predicted CT vs. FM for rotors with rectangular
and 2:1 tapered blades, baseline camber 6% . . . . . . . . . . . . . . 179
5.19 Blade planform parameters of sample optimization case. . . . . . . . 180
xx
5.20 Variation of maximum FM with taper start location and taper offset
for a 2:1 taper ratio on blades with 6% baseline camber airfoil. . . . . 181
(a) FM vs. taper start radial location . . . . . . . . . . . . . . . . . 181
(b) FM vs. taper offset . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.21 Variation of maximum FM with taper start location for a 2:1 taper
ratio on blades with 0% 3% and 4.5% baseline camber airfoil. . . . . . 182
5.22 Blade planform defined by 20 control points and Akima splines. . . . 183
6.1 Coaxial experimental setups a) single coax transmission b) indepen-
dent single rotor transmissions. . . . . . . . . . . . . . . . . . . . . . 189
6.2 Average coaxial thrust at 6 different rotor separations vs. upper rotor
rotational speed. Error bars cover 3 standard deviation, calculated
using all data points at each spacing. . . . . . . . . . . . . . . . . . . 191
6.3 Variation of coaxial thrust with rotor spacing for rectangular 6% cam-
ber blades. upper and lower collectives set at 16deg. . . . . . . . . . . 192
(a) Coax thrust vs. rotor spacing at 2000 RPM . . . . . . . . . . . 192
(b) Coax thrust vs. rotor spacing at 2500 RPM . . . . . . . . . . . 192
6.4 Experimental thrust and torque vs. RPM of single rotor with rectan-
gular blades and 6% camber airfoil with at 16 deg collective. Equa-
tions of best quadratic fit shown. . . . . . . . . . . . . . . . . . . . . 194
(a) Trust vs. RPM . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
(b) Torque vs RPM . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.5 Thrust vs. Upper rotor rotational speed. Total, upper and lower
rotor contributions shown. 6% camber airfoils, rectangular blades. . . 195
6.6 Upper rotor percentile contribution to total thrust. Collectives set at
16 deg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.7 Power loading vs. coaxial thrust and percentile rotational speed dif-
ference vs. coaxial thrust, for the collective settings of Table 6.1 . . . 199
(a) Power loading vs. coaxial thrust. . . . . . . . . . . . . . . . . . 199
(b) Percentile rotational speed difference vs. coaxial thrust. . . . . . 199
xxi
6.8 Power loading vs. thrust for coaxial and tandem systems . . . . . . . 200
6.9 Flow model of a coaxial rotor operating in the fully developed wake
of the upper rotor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.10 Contour plot of normalized wake velocities with respect to tip speed
below single rotor plane. Collective at 16 deg, 2500 RPM. . . . . . . 204
6.11 Radial position vs. normalized downwash velocities at two vertical
locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.12 Thrust vs. rotational speed for upper and lower rotors. Experimental
and BEMT results compared. . . . . . . . . . . . . . . . . . . . . . . 206
6.13 Percentileerrorvsupperrotorrotationalspeedfor: lowerrotorthrust,lower
rotor rotational speed, and coaxial thrust. . . . . . . . . . . . . . . . 207
6.14 BEMT predictions for spanwise variation in inflow and thrust for
upper and lower rotors. . . . . . . . . . . . . . . . . . . . . . . . . . . 208
(a) Upper rotor - spanwise variation in inflow . . . . . . . . . . . . 208
(b) Lower rotor - spanwise variation in inflow . . . . . . . . . . . . 208
(c) Upper rotor - spanwise variation in thrust. . . . . . . . . . . . . 208
(d) Lower rotor - spanwise variation in thrust. . . . . . . . . . . . . 208
(e) Upper rotor - spanwise variation in torque. . . . . . . . . . . . . 208
(f) Lower rotor - spanwise variation in torque. . . . . . . . . . . . . 208
6.15 Current generation coaxial vehicle transmission design. . . . . . . . . 213
6.16 Coaxial MAV prototype. . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.17 Exploded and assembled views of the custom designed swashplate. . . 217
(a) Exploded view of the swashplate . . . . . . . . . . . . . . . . . 217
(b) Assembled swashplate . . . . . . . . . . . . . . . . . . . . . . . 217
6.18 Spider system fir changing cyclic pitch from Ref. [88]. . . . . . . . . . 218
6.19 Swashplate prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . 220
6.20 MTOW vs. FM and MTOW vs. ? for various rotor diameters. . . . . 223
(a) MTOW vs. FM @ ? = 1. . . . . . . . . . . . . . . . . . . . . . 223
(b) MTOW vs. ? @ FM=0.6. . . . . . . . . . . . . . . . . . . . . . 223
6.21 Motor characterization test stand showing Eddie-current break. . . . 224
xxii
6.22 Efficiency vs. shaft power of controller-motors-transmission system
at various rotational speeds. . . . . . . . . . . . . . . . . . . . . . . . 226
(a) System efficiency at 7.4V. . . . . . . . . . . . . . . . . . . . . . 226
(b) System efficiency at 11.1V. . . . . . . . . . . . . . . . . . . . . . 226
A.1 Reynolds Vs. Mach number for various types of vehicles . . . . . . . 239
A.2 Effect on boundary layer of favorable and adverse pressure gradients. 242
A.3 Velocity distribution for three different types or airfoils at 3 deg aoa. 244
A.4 Sketch of laminar separation bubble. . . . . . . . . . . . . . . . . . . 246
A.5 Schematic of hysteresis in lift coefficient with variations in a) angle
of attack, and b) Reynolds number from data in (Ref. [1]) . . . . . . 247
A.6 Long and short bubble effect on pressure distribution (Ref. [71]) . . . 248
A.7 Airfoils studied by Schmitz in Ref. [1] . . . . . . . . . . . . . . . . . . 249
A.8 Drag polars for the airfoils studied by Schmitz from data in Ref. [1] . 251
A.9 Reynolds number vs. maximum lift coefficient for the airfoils studied
by Schmitz from data in Ref. [1] . . . . . . . . . . . . . . . . . . . . . 252
A.10 Reynolds number vs. minimum drag coefficient for the airfoils studied
by Schmitz from data in Ref. [1] . . . . . . . . . . . . . . . . . . . . . 252
A.11 Effective AoA vs. lift coefficient at different Reynolds numbers for
airfoils studied by Schmitz,(Eff. aoa=0 at Cl = 0), from data in Ref. [1].253
A.12 Angle of attack vs. lift coefficient for rectangular wings of AR 6 at
two different turbulence levels and Re=20,700 Ref. [17] . . . . . . . . 254
A.13 Angle of attack vs. lift coefficient for AR=6 rectangular wings at
Re=20,700, Ref [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
A.14 Flow model used by momentum theory for a single rotor in axial flight.257
A.15 Disk Loading vs. Power Loading for various Figures of Merit at sea
level (? = 1.225kg/m3). . . . . . . . . . . . . . . . . . . . . . . . . . . 262
xxiii
Chapter 1
Introduction
1.1 Historical Background
At the end of the 19th Century, dozens of people around the world were trying
to develop the first powered heavier-than-air machine that could finally let Man
experience the timeless dream of flying. On December 17 1903, the Wright brothers
achieved that dream, making history at Kitty Hawk North Carolina by successfully
flying their airplane for twelve seconds. However, the foundations of this challenging
enterprisewerelayedoutmorethanacenturyearlierbythesometimescalled?Father
of Aviation?, Sir George Cayley. Cayley was the first researcher that used a scientific
approach to the problem of flying. As is common in the history of science, he
observed nature and tried to learn from it. He studied the cross section of bird
wings and, using a rudimentary whirling arm apparatus, identified the positive effect
of camber on lift generation. His 1804 experiments can be considered the first
aerodynamic airfoil study in history.
The German civil engineer Otto Lilienthal also made important contributions
to the development of the modern airplane. He was the first researcher to make
a serious systematic effort to develop pilot controlled gliders. He built and tested
eighteen different models of his gliders, achieving relative success and inspiring the
Wright brothers to take on the invention of the airplane.
Cayley, Lilienthal, and specially the Wright brothers experimented with small-
scale models that were used to extrapolate the characteristics of their larger scale
prototypes. In a consistent way, they all converged to optimal airfoil shapes similar
to those of the larger birds. The factor known as Reynolds number was a new
concept, unknown to the aviation pioneers. Inadvertently, they became the first
1
low Reynolds number aerodynamic researchers. In fact, the Wright brothers? wind
tunnel was used to test wing sections at Reynolds numbers close to 100,000, as
opposed to the 1.8 million chord-based Re that their flying prototype had during
cruise.
The historic Kitty Hawk flight was the starting point of a race for faster,
larger, and more powerful vehicles that still continues today. Low Reynolds number
fluid mechanics research was a secondary problem that did not have a motivation.
Only until the early 1990s, technology was mature enough in fields as diverse as
electronics, energy storage, and energy conversion to seriously consider practical
uses for small flying machines. Technology made a full circle; scientists were once
again revisiting the results and contributions of the aeronautic pioneers. This time,
however, with the idea of developing smaller and slower vehicles.
1.2 Motivation and Problem Statement
The concept of Micro Air Vehicles (MAVs) was first introduced in 1992 in a DARPA
workshop titled ?Future Technology - Driven Revolutions in Military Operations?.
The idea gained popularity, and a series of feasibility studies were performed in the
following years by the Lincoln Laboratories, and the U.S Naval Research Laboratory.
The positive outcome of the preliminary studies, led to a DARPA sponsored 35
million dollars program.
The program goal was to develop inexpensive flying robots with no dimension
exceeding 6 inches (15.24 cm), an endurance of at least an hour, and a weight of
no more than 100 grams. This is an order of magnitude smaller than any previ-
ously developed system. The final product was intended to be used in a series of
military and civilian mission, such as over-the-hill reconnaissance, indoor/outdoor
covert imaging, biological or chemical agent detection, traffic monitoring, and urban
intelligence gathering. This meant that practical MAVs had to be capable of real-
time imagery and/or sensing with a high degree of autonomy. By the end of phase
II of the program a series of prototypes with different configurations were developed
by the participants. These included the fixed-wing Lockheed-Sanders ?Microstar?,
2
the CalTech-Aerovironment-UCLA ornithopter ?Microbat?, the Lutronix Corpora-
tion micro helicopter ?Kolibri?, MicroCraft?s Small Lift Augmented Ducted Fan
?ISTAR? and Aerovironment?s ?Black Widow? flying wing.
Figure 1.1 shows the 2003 performance in terms of weight and endurance of
a series of small-scale vehicles including the DARPA sponsored designs. The figure
clearly shows that the objective set is far from being reached. Fixed wing vehicles
meet size and weight constraints, but are lacking in endurance. For rotary and
flapping wing vehicles, even with larger and heavier designs, endurance times are
shorter than for the fixed wing configurations. After the termination of the program,
development continued for the Black Widow and the ISTAR, mainly in the areas
of autonomous navigation and control. Since then, only marginal improvements in
flight performance have been achieved.
Fixed wing vehicles are well suited for outdoor reconnaissance missions that do
not require maneuvering in tightly constrained spaces. However, for missions around
or within buildings hovering vehicles have a clear advantage over fixed wings config-
urations. A stealthy hovering MAV can fly indoors without being detected and can
potentially have the ability to ?perch and stare?, providing tactical reconnaissance
and surveillance for extended periods of time with low risk of being detected. Two
different MAV configuration are capable of hovering: flapping and rotary wing vehi-
cles. Up to date, the only hovering flapping wing vehicle built is the MENTOR from
SRI International, shown in Figure 1.1. It can achieve stable hover and forward
flight but its complex kinematics and loud operating conditions - due to the wing
flapping - hindered any further development. On the other hand, rotary wing vehi-
cles can adapt some of the technology used in full-scale vehicles and require simpler
mechanics. Aerodynamic noise is not an issue, and if electric power is used, overall
noise levels can be kept at a minimum. These characteristics make the rotary wing
configurations specially attractive for MAV applications.
The main difficulty in achieving a better performance with rotary wing vehicle
comes from the large hover power requirements. Hover is an intrinsically high-power
flight state with considerably larger energy requirements than cruise for fixed wing
3
vehicles; this fact is independent of scale. If hover extends for a significant fraction
of the mission duration, hover efficiency becomes a key vehicle characteristic that
must be carefully addressed.
The size constraints of MAVs, forces wings and rotors to work at Reynolds
numbers much closer to the ones faced by small birds and insects than to previously
developed man-made systems. Figure 1.2 shows Reynolds number vs. Mass for
a wide range of animals and aircraft. MAVs lie within the shaded region at the
lower left corner of the graph, bounded by Re between 2,000 and 100,000. At these
Reynolds numbers, viscous effects in the flow are dominant over the inertial ones,
boundary layers are thick and undergo several complex phenomena. Separation,
transition, andreattachmentcanalloccurwithinashortchordwisedistance, forming
laminar separation bubbles that have a strong adverse effect on the lifting surface
characteristics.
Considering that hover efficiency depends largely on airfoil characteristics,
sub-scale rotors are expected to have a lower hovering efficiency than full-scale ones.
Before the current investigation was performed, hover efficiency of small-scale rotors
below Reynolds numbers of 100,000 was uncharted territory. The only available
experimentaldatacouldbeobtainedfrommodeltestingintendedforfull-scaledesign
at Re of one million and above.
In the current investigation, initial hover rotor testing showed that small-scale
rotors with airfoils and blade geometries similar to those found in full-scale vehicles
had maximum Figures of Merit (FM), of the order of 0.35. This is a very low value
considering that full-scale helicopters can reach maximum FMs in the vicinity of
0.8. In order to improve these numbers it is necessary not only to consider two
dimensional airfoil characteristics but the overall rotor design. Blade parameters
such as taper twist and collective have to be adapted to the radically different
flow conditions. Some aspects of these fundamental problems are addressed in the
following chapters of the dissertation.
4
Figure 1.1: Relative performance of various micro air vehicles.
5
0.0001
0.001
1000
0.1
10
100,000
10,000,000
1000 10,000 100,000
1,000,00010,000,000100,000,000
Mass
, Kg
Re
Norway Maple
Emperor Moth
Dragonfly
Swift
Common Tem
Kestrel
Peregrine Falcon
Herring Gull
Stork
Albatross
Gossamer Albatross
Sailplane
Ultralight
Hang Glider
Boeing 737
Airbus 310
Boeing 747
Space 
Shuttle
MAVs
Figure 1.2: Re vs. Mass for animal and Man-made vehicles from data in Ref [19].
1.3 Literature Review
Two main areas of research are relevant for the present investigation, low Reynolds
number airfoil aerodynamics, and small-scale single and coaxial hover rotor modeling
and optimization. Airfoil aerodynamics have been studied experimentally on wind
tunnels since the dawn of the 20th Century, and computationally for no more than 20
years. However, only during the last decade computational models have approached
the problem of low Reynolds number flows. The following literature review presents
the main contributions on the different areas of interests.
1.3.1 Experimental Low Re Airfoil Studies
One of the most influential publications in the field was published by Schmitz Ref. [1]
in 1941. The report ?Aerodynamics of the Model Airplane?, was awarded the Lud-
wing Prandtl prize and has a foreword praising the interesting results by Prandtl
himself. Schmitz performed his research in a wind tunnel with turbulence levels
similar to those found in free flight (0.1%). Under those flow conditions, he was
6
able to observe for the first time the hysteresis loops in the lift and drag of airfoils
at Reynolds numbers between 40,000 and 160,000. He also identified the reduction
in the critical Reynolds number due to the added turbulence produced by placing a
wire upstream of the leading edge.
Bruining published in 1979 lift and drag wind tunnel measurements on two-
dimensional circular arc steel plates Ref. [3]. The research studied the effects of
a turbulence-inducing spar mounted at different positions along the blade chord.
He tested at Reynolds numbers of 60,000, 100,000 and 200,000. This relatively
unknown work is relevant to the current research because of the similarities in the
airfoils investigated.
In 1980 a comprehensive airfoil catalog was published by Althaus Ref. [4],
the measurements were made in a low turbulence wind tunnel at the University of
Stuttgart in West Germany. Moment coefficients were not measured, but lift and
drag coefficients of 30 airfoils at Reynolds numbers between 40,000 and 250,000 are
presented.
Carmichaelpublishedin1981his?LowReynoldnumberairfoilsurvey?(Ref.[5]).
This document compiles the state of the art in low Reynolds number flow physics
at the time, while giving a historical perspective of the evolution of the field. A
detailed bibliography of airfoil, wind tunnel, and free flight testing is presented, as
well as a discussion and additional references about the use of boundary layer trips.
Miley Ref. [2] published in 1982 a well known airfoil catalog intended to be
used in the design of small wind turbines. Problems associated with presenting
airfoil data at large angles of attack were addressed as well as airfoil behavior and
the effects of Reynolds number, surface roughness, and turbulence.
MuellerfromtheUniversityofNotreDamehasbeenworkingwithlowReynolds
number flows since the early eighties; he has performed a wide range of wind tun-
nels tests, measuring aerodynamic coefficients and documenting flow characteristics.
Mueller performed fundamental research at low Reynolds number on flow visual-
ization (Ref. [6]), boundary layers and laminar separation bubbles (Ref. [7, 8, 9])
and two and three dimensional aerodynamic characteristics of wings (Ref. [10, 11]).
7
Mueller?s research is specially relevant since in some of his papers he tested circular
arc airfoils, similar to those used in the current work.
Selig, from UIUC, has been working over the last 20 years on systematic wind
tunnel testing of more than 200 airfoils at Reynolds numbers between 40,000 and
500,000. This set of data is especially valuable since all tests were performed at the
same facilities with the same methodology, making quantitative comparison within
the set very accurate. Results have been compiled in a series of 3 volumes (Refs.
[12, 13]). Selig has also worked on inverse design methodologies and optimization
of airfoils at low Reynolds numbers (Ref. [14, 15]).
Laitone from Berkeley, tried to explain why at Reynolds numbers below 70,000
thin curved plates with sharp leading edges are aerodynamically superior to all other
known round-nosed airfoils (Refs. [16, 17, 18]). His research can be considered a
continuation of the seminal work of Schmitz. Laitone performed wind tunnel mea-
surements of forces and moment, as well as pressure distribution tests and some flow
visualization. His main contribution was showing that for certain airfoils aerody-
namic lift is not determined by the Kutta-Joukowsky trailing edge condition.
An excellent review paper that covers the main issues associated with the aero-
dynamics of small-scale flight was written by Lowson (Ref. [19]). He discusses the
flow characteristics that make low Reynolds number testing very sensitive to experi-
mental conditions, leading to a large variation in the results at different experimental
facilities.
A few researchers have done experimental work in the ultra-low Reynolds
number range of 10,000 and below. This range is so distant from regimes faced
by traditional man-made machines, that experimental biologists, and not engineers,
have been the ones producing a significant share of the results. Most of their work
has been on insect flapping flight; however, some steady-state studies relevant for
this dissertation have been performed.
Sunada for example (Refs. [20, 21, 22]), studied the hydrodynamic charac-
teristics of wings at Reynolds number of 4,000. His parametric studies gave results
consistent with trends observed at higher Re in air. Okamoto (Ref. [23]) performed
8
a similar study in a wind tunnel. He used steady state measurement as a baseline
for unsteady wing characterization.
One of the most recognized scientist that has contributed to the understanding
of unsteady low Reynolds number flows is Ellington, a professor of Animal Mechanics
at the Department of Zoology in Cambridge University. He identified the existence
of leading edge vortices on insect wings, and showed in Ref. [24] how they work as
an aerodynamic unsteady mechanism that enhances lift production. A fraction of
his results correspond to wind tunnel and rotor tests used as steady data baseline
Ref. [25].
Dickinson from Caltech has also contributed to the understanding of insect
flight. His research covers several areas including aerodynamics, flight energetics,
control of flight maneuvers, and visual flight control. Using a Dynamically scaled
model of a flapping insect known as the ?Robofly? he identified in Refs. [26, 27] the
three main unsteady aerodynamic mechanisms for enhanced lift production: delayed
stall, wake capture and rotational circulation. Even though his results are more
relevant to flapping flight researchers, they are important for the understanding of
the limitations of conventional fixed and rotary wing vehicles.
1.3.2 Vehicle Design and Development
Design and system integration of vehicles is also an aspect of MAVs that has been
explored by the scientific community. In this section the most representative study
cases that have been published are presented.
In early 2000, Samuel and Sirohi, at the time graduate students of the Alfred
Gessow Rotorcraft Center at the University of Maryland, presented in Ref. [28]
a preliminary design study of a micro coaxial rotorcraft later known as ?MICOR?.
This study was the starting point for the research presented in this dissertation. The
main design issues and challenges faced when implementing a rotary wing MAV were
identified in that paper.
AnotherdesignreportonhoveringMAVscomesfromAlliedAerospaceRef.[29].
The paper covers the basic configuration, control system, ground and flight-test re-
9
sults of the ducted-fan iSTAR, one of the DARPA funded vehicles shown in Figure
1.1. Aerodynamic performance or rotor design is not addressed, however valuable
information about power requirements and control strategies is provided.
The Mesicopter is a multi-rotor rotary-wing vehicle, considerably smaller than
an MAV, having a target weight between 3 and 15 grams. Its aerodynamic design
and subsystem configuration are presented in Ref. [31]. Several prototypes have
been built and evaluated, however successful flight testing has not been reported.
Lockheed Martin and Aerovironment also produced design reports on the ?Mi-
croSTAR? and ?Black Widow? fixed wing MAVs (Refs. [32, 30]). An explanation of
the design methodology and a detailed description of each subsystem are presented is
the papers. Aerovironment?s custom electronics, shown in its infancy, would evolve
over the years to make the Black Widow the most successful autonomous fixed wing
MAV.
Flapping wing vehicles are considerably more complex mechanically and aero-
dynamically than rotary and fixed wing designs. Over the last few years funda-
mental aerodynamic research has increased the understanding of flapping flight,
however few technical publications address the actual design of such vehicles. Wing
design, manufacturing processes and wing performance for the Caltech ?Microbat?
are presented in Ref. [33]. The only successful hovering flapping vehicle up to the
publication date of this document is the MENTOR, developed by SRI. The tech-
nology used was proprietary and only fragments of design information are available
from Ref. [34].
One of the most interesting publications that explores a variety of hovering
MAV configurations is Ref. [35]. The paper provides pictures, characteristics and
some aerodynamic performance measurements of a quad-rotor, a tandem twin fuse-
lage tilt-rotor, a coaxial helicopter and a morphing fixed/rotary-wing vehicle called
the ?mini-morpher?. All configurations were built and flight tested, making this a
particularly useful reference.
A vehicle known as the ?GIANT? was developed at the University of Maryland
over the course of the research presented in this dissertation. Dr. Sirohi and Dr.
10
Tishchenko, designed and implemented a ducted fan, single rotor vehicle that uses
a set of vanes in the downwash to counter act the motor torque. The vehicle was
designed to have high hover endurance and passive stability. The paper in Ref. [36]
describes the vehicle configuration and provides experimental data of ducted and
isolated rotor performance, as well as the effect of the anti-torque vanes on power
consumption.
A paper that makes use of published data to compare flapping, rotating and
fixed wing vehicle?s performance was published by Woods et al. Ref. [37]. Energy
and Power requirements for the three configurations were calculated, and an opti-
mization procedure was used to evaluate the best choice of vehicle for a series of
missions. The paper provides first order equations and airfoil coefficients, useful to
make comparisons and rough power estimates for a variety of missions. A similar
paper by Baxter (Ref. [38]), uses classical flight mechanics equations with simple
corrections to introduce Reynolds number effects to calculate power requirements
of fixed and flapping wing vehicles.
1.3.3 Coaxial Helicopter Aerodynamics
When designing a helicopter, counter-acting the rotor torque is a key issue. The
concept of a coaxial helicopter, with two rotors spinning in opposite directions in-
stead of a tail rotor, has been present since the early stages of vertical flight Ref. [39].
Considering that one of the outcomes of the current research was the development of
a coaxial MAV, the most relevant publications that treat the subject are presented.
A NASA report that gives a broad perspective of the aerodynamic issues
and state of the art of coaxial helicopters technology was published by Coleman in
Ref. [40]. His survey summarizes the main publications on the topic from American,
Russian, Japanese, British and German sources. Experimental data and analysis
that address rotor separation distance, load sharing between the rotors, wake struc-
ture, solidity effects and the unique characteristics of the coaxial configuration are
presented.
Few experimental measurements of coaxial rotor performance are available in
11
the literature, these include the full-scale tests performed by Harrington (Ref. [41])
and Dingledein (Ref. [42]) in the Langley full-scale wind tunnel, and the data pre-
sented by Nagashima in Ref. [43]. Reference [44] is a recent publication that ex-
perimentally explores the maximization of the thrust-torque ratio on a small-scale
coaxial rotor. The results presented are not specific for low Reynolds numbers and
suggest an interesting result that is discussed in chapter 6.
Modeling of coaxial rotors has been undertaken by various authors such as
Saito and Azuma (Ref. [45], Andrew (Ref. [46]) and Zimmer (Ref. [47]). Various
approaches that generally combine momentum theory, blade element theory, vortex,
and lifting line models have been implemented in an attempt to calculate the power
requirements of coaxial rotors. Generally a good predictive capability has been
achieved. However, most of the algorithms require empirical models of the wake
geometry. Bagai in Ref. [48] uses the Free Vortex Method (FVM) to study the
tandem, tilt-rotor and coaxial configurations. The FVM approach makes use of
considerable more computational resources, but has the advantage of not requiring
a prescribe wake geometry.
1.3.4 Computational Studies
Wind tunnel low Reynolds number experimentation requires expensive infrastruc-
ture and a careful and thorough methodology that guarantees the quality of the
results. These difficulties and the limited experimental data available, make numer-
ical tools for aerodynamic design extremely useful. Airfoil synthesis and selection,
as well as vehicle development can be accelerated through the use of Computational
Fluid Dynamics (CFD) as long as proper validation of the numerical results is done.
The following are some relevant publications that cover the development and use of
CFD tools in the low Reynolds number regime.
The two-dimensional incompressible Navier-Stokes solver INS2d developed by
Rogers (Refs. [50, 51]) was used by Kunz in Refs. [52, 53] to analyze and op-
timize the performance of airfoils at Reynolds numbers below 10,000. In Kunz?s
work INS2d was used mostly for single grid steady calculations assuming fully lam-
12
inar flow. Limited experimental validation of the computed airfoil properties was
made. Rotor performance calculations were done using two dimensional airfoil char-
acteristics and a blade element model. Rotor experimental results largely disagree
with the predictions. Lack of blade structural stiffness and airfoil manufacturing
imperfections were blamed for the discrepancies.
At Reynolds numbers above 10,000 the flow is not necessarily fully laminar,
and turbulence models should be used. The occurrence of laminar separation bub-
bles in the range of interest adds another level of complexity to the problem.
Singh and Kellogg (Ref. [55, 56]) used XFOIL, a two dimensional panel method
code (Ref. [57]), to perform parametric studies on thin cambered airfoils in the
Reynolds number range of 60,000 to 150,000. The quality of the predictions is
acceptable, showing how this computationally inexpensive approach can give first
order estimates of airfoil coefficients, useful for preliminary design considerations.
Ramamurti implemented a finite element flow solver based on unstructured
grids to study the characteristics of MAVs. In Ref. [58], inviscid and viscous flow
simulationsoftwofixedwingMAVSarepresented. ThesimulationsusetheBaldwin-
Lomax turbulence model for the viscous cases. Aerodynamic characteristics of ve-
hicle?s finite wings are studied and the effects of the fuselage and propellers on wing
performance were investigated.
Shum in Ref. [59] developed a computational model to investigate laminar
separation bubbles. His model predicts separation, transition, and reattachment
location on two-dimensional airfoils. Drag and bubble size predictions on an Ep-
pler 387 airfoil were compared with experiments showing a good correlation in the
Reynolds number range of 200,000 to 1,000,000.
Bohorquez et al. in Ref. [60] investigated experimentally and computationally
the rotors of the first generation MICOR MAV, shown in Ref. [62]. The compu-
tational aspect of the research was performed using OVERFLOW-D (Ref. [63]), a
ReynoldsAveragedCompressibleNavier-StokessolverdevelopedbytheArmy/NASA
Rotorcraft Division. Overflow-D was used in Ref. [61] for the computational analy-
sis of a martian prototype experiment. The martian vehicle prototype was designed
13
to work at unique flow characteristics, combining low Reynolds numbers and high
Mach numbers of the order of Re=60,000 and M=0.65 at the tip. In Bohorquez
work, for cases in which turbulence is considered, the one-equation Baldwin-Barth
turbulence model was used for closure. Results showed that rotor drag is well pre-
dicted; however, lift was largely over predicted leading to rotor performance values
higher than the ones measured experimentally. Surface flow visualization performed
on the rotor blades showed that the CFD results differ qualitatively from the exper-
iments which show large areas of separated flow on the upper surface of the blades.
This work was one of of the first attempts that compared experiments with CFD
three dimensional predictions of MAV rotors.
The CFD group at the Rotorcraft Center of the University of Maryland has
been working for more than a decade in the Transonic Unsteady Rotor Navier-Stokes
solver - TURNS - . This compressible code was originally developed to solve flows
at high Reynold and Mach numbers typical of full-scale helicopters. Part of the
doctoral research of Gupta (Ref. [64, 65]) consisted in implementing a low Mach
number pre-conditioner that allows now the use of TURNS at much lower Reynolds
and Mach numbers. Schroeder (Ref. [66, 67]) validated the use of TURNS with
the pre-conditioner for airfoil and rotor predictions at Reynolds numbers typical
of MAVs. In his research, Schroeder performed two dimensional calculations for a
variety of airfoils, he also did a three-dimensional case study of an MAV rotor. All
cases were compared with experimental results from Refs. [10, 13, 69]. The work
of Gupta and Schroeder played an important role in the computational aspects of
this dissertation, considering that TURNS was used to obtain some of the results
presented in chapter 4.
Lakshminarayan et al. also from the University of Maryland?s Alfred Gessow
Rotorcraft Center continued the work of Schroeder and published in Ref. [68] a
computational study of an MAV rotor using an updated version of TURNS. Inviscid
phenomena such as thrust, wake structure and tip vortex evolution were accurately
predicted; however, drag and power were largely under-predicted when compared to
the experimental data of Ref. [69].
14
1.4 Current Research
1.4.1 Objectives
The research presented in this dissertation has a total of five objectives.
As reported in the literature experimentation at low Reynolds numbers re-
quires not only the appropriate facilities but also the correct methodology. This
is true for wind tunnel testing and for hover rotor testing. the first objective of
the current research is to develop and implement a test stand and experimental
procedure that can be used to characterize small-scale single and coaxial rotors in
hover.
The second objective is the experimental identification and characterization
of an airfoil or family of airfoils suited for the implementation of efficient MAV
rotors. Structural and manufacturing constraints need to be considered in order to
apply the findings on a working prototype. Rotor design, and the effect of blade
and airfoil parameters on hover performance are also explored in order to determine
basic small-scale rotor design guidelines.
A third objective involves the development of a test stand and methodology
to study coaxial rotors. Basic coaxial rotor parameters such as rotor spacing and
collective settings as well as the effect of lower rotor design on hover performance
are explored at a zero torque condition. This part of the investigation is aimed at
finding the rotor parameters that guarantee the highest efficiency of a given coaxial
rotor system.
The fourth objective is to implement and validate analytical and numerical
models that can be efficiently used for the design and eventual optimization of MAV
rotor systems. Tools widely used in the analysis of full-scale rotors such as Blade
Element Momentum Theory (BEMT) and two-dimensional CFD airfoil calculations,
are combined and validated with the rotor experiments used to accomplish the
previous two research objectives.
A fifth and final objective of this dissertation is the implementation of a fully
functional coaxial rotary-wing MAV that can be used as platform for the evaluation
15
of sensors and the development of control systems. In order to accomplish this, not
only rotor design and aerodynamic issues are considered. Powertrain design, vehicle
configuration, stability and control strategies, energy storage and overall system
integration issues have to be addressed.
1.4.2 Contributions
The key contributions of this research include:
1. Developed a hover stand for the testing of small-scale single and coaxial rotors.
Hardware and software were customized to achieve excellent levels of accuracy,
minimizing testing time and assuring repeatability of the results.
2. Performed systematic single rotor testing varying airfoil and blade parameters
over a range of rotational speeds and collective pitch settings. Rotors with
circular arc airfoils were evaluated on rectangular and tapered blades.
3. Circular arc airfoils were validated as a good choice for the implementation of
rotary wing MAVs.
4. Identified key rotor parameters that affect hover performance of single small-
scale rotors.
5. Performed a CFD parametric study of circular arc airfoils at Reynolds number
ranging from 15,000 to 60,000.
6. Implemented and validated a hybrid Blade Element Momentum Theory / 2D-
CFD algorithm to calculate hover rotor performance at low Reynolds numbers.
7. Developed a blade parameterization that facilitates blade geometry optimiza-
tion within a well defined space.
8. Experimentally studied aerodynamic performance of MAV coaxial rotors at
zero torque condition exploring the effects of main operating parameters.
9. Extended BEMT/CFD algorithm to use in the coaxial rotor configuration.
16
10. Designed and manufactured a swashplate ideal for small-scale single and coax-
ial rotary-wing vehicles.
11. Designed and manufactured working MAV coaxial prototypes stable enough
to be easily flown without a control system. Vehicles are capable of perch
and stare, indoor and limited outdoor flying. Motor characterization and
subsystem integration were issues addressed.
1.4.3 Organization of Dissertation
This dissertation consists of seven chapters that show the logical progression fol-
lowed through the course of the project. Chapter 1 introduces the topic giving
a historical overview and showing the state of the art. A detailed literature re-
view of the relevant topics is given. Chapter 2 describes the rotor test stand and
methodology used in this investigation. The chapter presents the results of single
rotor tests, where the effect of airfoil and rotor parameters in hover performance are
investigated. The geometrical couplings between blade planform and airfoil shape
used in this investigation are introduced illustrating with experimental results the
performance benefits that can be obtained. Chapter 3 explains the modeling tools
and analysis techniques used to model the rotors and to gain insight into the is-
sues affecting their hover performance. Blade element momentum theory (BEMT)
numerical implementation is presented, and a methodology to obtain airfoil char-
acteristic from rotor tests is explained. Chapter 4 presents the methodology and
results of CFD calculations of two dimensional airfoil characteristics at low Reynolds
numbers. Results of a parametric study of aerodynamic characteristics of circular
arc airfoils are presented. These are used to generate a database that covers a vari-
ety of Reynolds numbers, cambers, thickness ratios and angles of attack. In Chapter
5 the implementation of a hybrid method that uses BEMT and the database gen-
erated in Chapter 4 to calculate the performance of single rotors over a wide range
of operating conditions. Blade parameterization is explained and the approach to
perform a blade geometry optimization is presented. Chapter 6 describes the coax-
17
ial rotor testing methodology, and explores the effect on hover performance of key
coaxial rotor parameters. BEMT is extended to the coaxial case and predictions
are validated with experimental results. The last part of the chapter deals with
the development of a practical coaxial MAV prototype. Motor characterization and
general system integration is discussed. Special attention is placed on the concep-
tual design of a simple swashplate ideal for rotary-wing MAVs. Finally, Chapter 7
concludes and gives recommendation for the continuation of this research.
18
Chapter 2
Single Rotor Tests
2.1 Introduction
This chapter explains and describes the methodology and experimental setup used
for evaluating and characterizing small-scale rotors in hover. Experimental results
for single rotors are presented, exploring the effect of operational and geometric
parameters on hover rotor performance.
2.2 Experimental Setup
In order to experimentally measure the performance of small-scale rotors, a hover
tests stand was developed. Based on the performance metrics and the definitions of
the coefficients presented in sections A.7.2 and A.7.3, it was clear that thrust and
power are needed to establish rotor performance. Hence, three physical quantities
need to be simultaneously measured by the test stand: thrust, torque and rotational
speed. Through the course of the investigation two generations of test stands were
developed. The first generation, shown in Figure 2.1 measured thrust and torque
using two axial load cells. Single or coaxial rotor systems could be mounted on
a transmission supported by a stem. The whole system could rotate freely, and
moment was transmitted to a load cell by a 1 inch arm. Thrust was measured by
the second load cell placed directly under the platforms shaft. The rotor system was
mounted inverted such that the airflow traveled from bottom to top, avoiding any
possible influence of in-ground effect on the measurements. The rotational speed of
each rotor was determined using a Hall-effect sensor that was excited by an array of
magnets attached to the rotating frame of the setup. Additionally, the voltage and
19
current supplied to the motors were also measured to obtain the electrical power
consumption. This first generation test stand was very sensitive to vibrations and
imbalances of the rotor system. Hinge alignment and moment arm/load cell contact
were a source of experimental error. Similarly the axial bearings supporting the
main thrust shaft, introduced small amounts of friction, resulting in hysteresis loops
in the thrust measurements when thrust changed direction.
The second generation test stand, shown in Figure 2.2, was designed to correct
the flaws of the first setup. In this design, no hinges or bearings were used. Instead,
torque was directly measured by a Transducer Techniques RTS-5 reaction torque
sensor with a maximum capacity of 5 in-oz, and thrust was measured by a model
31 Sensotec load cell with 1000 g maximum capacity. Both sensors are rated at
1.5mV/V output.
The thrust cell is mounted directly on top of the torque sensor inside an alu-
minum housing. Thrust and torque are decoupled by using two horizontally mounted
steel membranes that cannot transfer axial loads. Measurement do not present a
hysteretical behavior regardless of the changes in thrust and torque, allowing for
both positive and negative measurements. Sensitivity to vibrations produced by the
rotor system was also greatly improved.
2.2.1 Data Acquisition System
The data from the test stand was collected using an electronic data acquisition
system. A National Instruments DAQcard-6062E and a SC-2345 carrier box with
NI-SCC modules were used for signal conditioning and data collection. A custom
code specially tailored for the tests performed was developed in MATLAB and run
on a laptop PC. The user friendly GUI interface is shown in Figure 2.4. The code
was designed to simultaneously acquire up to eight channels while sending up to
two analog control signals. Sampling rate in all tests was 1000 samples per second
for each channel. For single rotor tests, five input channels and one output channel
were used. For the coaxial tests, explained in detail in the Chapter 6, two additional
input channel and a second output channel were required. Figure 2.3 shows a sketch
20
Hinge
Figure 2.1: First generation hover test stand
Torque sensor
Steel 
membranes
and
thrust load cell
Inside 
Transmission
Inverted Rotor
Mounting stem
Figure 2.2: Second generation hover test stand
21
of the experimental setup.
A PI (Proportional - Integral) controller was programmed and embedded in
the code, allowing to control the rotors? rotational speed with a feedback signal
from any channel. The system was setup such that the desired rotational speed
specified by the user was attained using the output from the Hall effect sensors
as feedback. The PI controller output signals can be send to a power booster
circuit or to a custom programmed micro-controller board, depending on the type
of motors used. For conventional dc motors the current of the output control signal
just needs to be amplified. However, for brushless motors, a low power pulse-
width modulated signal fed to a speed controller varied the power and rotational
speed of the motors. The micro-controller changes the duty cycle of the pulse-width
modulated signals proportionally with the dc control voltages sent from the DAQ
system. DC motors can also be used with speed controllers, so if required this setup
is capable of interfacing with any motor or device compatible with standard RC
pulse width modulated protocols.
2.2.2 System Calibration
The thrust and torque sensors were calibrated while mounted on the tests stand to
avoid any changes in the calibration factors produced by pre-loading of the struc-
ture. An excitation voltage of 10V was provided, and maximum thrust and torque
calibration values were 200 g and 0.125 N.m (1.37 in.oz) respectively. As observed
in the calibration plots of Figure 2.5, over the range of interest the behavior of both
sensors is very linear, having each an R2 of practically one.
The limits of the torque and thrust values that can be measured by the balance
are determined by the maximum rating of the sensors. However, it must be taken
into account that the thrust cell is preloaded with the stem/transmission structure
and by unavoidable compression forces present after the balance assembly. This
reduces the thrust range to a maximum of ? 500 g when using a 1kg load cell.
The torque sensor is not preloaded in any way; however, care must be taken not
to damage the sensor by introducing large dynamic forces in the measurements.
22
Data acquisition
hardware
Micro-controller
 board
Speed Controllers
Power Supply
Test stand
Current and 
V
oltage motor 1
Current and 
V
oltage motor 2
Control signal 1 Control signal 2
Hall ef
fect sensor 1 
Hall ef
fect sensor 2 
Thrust load cell
Torque load cell
MATLAB
Figure 2.3: Hover stand experimental setup sketch.
23
Baseline 
Range
Baseline 
Range
Data 
Range
Figure 2.4: Sample screen of data acquisition software developed after a single rotor
test
Rotational acceleration of the blades need to be kept at reasonable values based on
their mass. When testing large rotors with heavy blades or assemblies with large
inertias, shrouded rotors for example, the rotational speed needs to be ramped to
the desired value to avoid damage to the stand.
2.2.3 Procedure for Data Acquisition
Before starting a new set of experiments, the test stand should always be recali-
brated. Torque calibration is not sensitive to balance assembly procedure or ex-
ternal factors such as ambient temperature; however, it is recommended to check
thrust calibration if transmission weight changes and when ambient temperature
has considerably varied. The test transmission is designed such that the rotor hub
can easily be removed by simply loosening a set-screw. Collective pitch of the blades
is manually set on the unmounted rotors using a custom made protractor that pro-
vides a ?1/4 of degree precision. When performing test that sweep over a range of
collectives, it is recommended to start from the angles that will produce the lowest
levels of thrust, and gradually increase the collective. This eliminates the danger
of accidentally overloading the stand. Every time the collective pitch is changed
24
y = -0.0219x
R
2
 = 1
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 50 100 150 200 250
Mass, gr
Signal voltage, V
y = 311.68x
R
2
 = 0.9995
-5
-4
-3
-2
-1
0
1
2
3
4
5
-0.015 -0.01 -0.005 0 0.005 0.01 0.015
Signal voltage, V
Torque, N.m
Figure 2.5: Thrust and torque calibration curves
25
the rotor balance and blade tracking needs to be checked, a standard RC propeller
balancer was used for this purpose. After a collective pitch has been set, tests at
all rotational speeds should be done sequentially. This minimizes the error in blade
pitch setting between tests.
The data acquisition code performs each test in two steps. The first step, where
no data is acquired, consists in finding the control signal that produces the desired
rotational speed. This process takes between ten to fifteen seconds depending on
the controller?s constants and rotor settings. The control voltage found in step one
is stored in memory and the rotor is stopped. In step two, for the single rotor
tests, 10 seconds of data are captured. After the first 1.5 seconds, the control signal
is ramped up to its desired value and kept unchanged for the rests of the data
acquisition process. Since there is always a small drift in the voltage output of the
sensors, an absolute voltage reference for a set of test should not be used. The
small drifting values that accumulate over time can introduce significant errors over
just a few minutes. This is why, the first second of data is used as baseline for the
thrust and torque measurements. The output voltages are found by subtracting the
average value of the first second from the average value of the two last seconds of
acquired data. The buffer time between baseline and data sample is necessary to
avoid any transient effects that might affect the measurements
2.2.4 Experimental Error Analysis
The main sources of error in the experiments are the standard deviations of the
rotational speed and the mean voltages from the thrust and torque sensors. Typical
values of the standard deviations of torque and thrust are about 1% of the mean
values. Rotational speed error is related to the finite number of magnets that excite
the Hall effect sensor. In the current tests four magnets were used, this means that
for any rotational speed measured there is an uncertainty of a quarter revolution
per sample. By counting the number of times a magnet passes over the Hall effect
sensor in one second, there is an error of 0.25?60 = 15 RPM. If a sample of four
seconds is considered, the uncertainty in the rotational speed is cut in half, being
26
3.75 RPM or 0.0131 rad.s?1.
Inordertocalculatetheuncertaintyofthemeasurements, theKline-McClintock
(Ref. [74]) method for error propagation was used. For a given quantity R that de-
pends on a series of independent variables x1,x2,x3 the uncertainty of R (?R) as
function of the uncertainties of the independent variables (?1,?2,?n) is given by:
?R =
bracketleftBiggparenleftbigg
?R
?x1 ?1
parenrightbigg2
+
parenleftbigg?R
?x2 ?2
parenrightbigg2
+
parenleftbigg?R
?xn ?n
parenrightbigg2bracketrightBigg
(2.1)
The uncertainty in thrust and power coefficients, figure of merit and power loading
was calculated considering the errors in the measurements of thrust torque and
rotational speed. Starting from the previously defined thrust and power coefficients
CT = T?A?2R2 (2.2)
First the partial derivatives of the above equation are found
?CT
?T =
CT
T ? 1
?CT
?? = (CT ? ?)
2 ? ?2
?3 =
?2CT
?
Applying Kline-McClintock equation (cf. 2.1)
?CT =
radicalBiggparenleftbigg
CT
T ? ?T
parenrightbigg2
+
parenleftbigg?2C
T
? ? ??
parenrightbigg2
Finally
?CT
CT =
radicalBiggparenleftbigg
?T
T
parenrightbigg2
+ 4
parenleftbigg??
?
parenrightbigg2
(2.3)
Similarly for the power coefficient
CP = P?A?3R3 = Q??A?3R3 = Q?A?2R3 (2.4)
First the partial derivatives of the above equation are found
27
?CP
?Q =
CP
Q ? 1
?CP
?? = (CP ? ?)
2 ? ?2
?3 =
?2CP
?
Applying Kline-McClintock equation (cf. 2.1)
?CP
CP =
radicalBiggparenleftbigg
?Q
Q
parenrightbigg2
+ 4
parenleftbigg??
?
parenrightbigg2
(2.5)
The figure of merit equation in terms of the measured quantities is:
FM = C
3
2
T?
2CP =
T 32
Q??2?A (2.6)
Now partial derivatives are calculated
?FM
?T =
FM
T 32 ?
3
2T
1
2 =
3
2 ?
FM
T
?FM
?Q = (FM ?Q)?
?1
Q2 =
?FM
Q
?FM
?? =
?FM
?
Applying Kline-McClintock equation (2.1)
?FM
FM =
radicalBigg
9
4
parenleftbigg?T
T
parenrightbigg2
+
parenleftbigg?Q
Q
parenrightbigg2
+
parenleftbigg??
?
parenrightbigg2
(2.7)
Finally for the power loading:
PL = CT?RC
P
= TQ? (2.8)
The partial derivatives are:
?PL
?T =
PL
T
?PL
?Q =
?PL
Q
?PL
?? =
?PL
?
28
Applying Kline-McClintock equation (cf. 2.1)
?PL
PL =
radicalBiggparenleftbigg
?T
T
parenrightbigg2
+
parenleftbigg?Q
Q
parenrightbigg2
+
parenleftbigg??
?
parenrightbigg2
(2.9)
The error analysis showed that the average uncertainties for thrust and power
coefficients at collectives between 8 and 18 deg are 1%. For the figure of merit and
power loading the error propagates and the uncertainties are higher being 2% and
1.5% respectively. The previous results do not consider the error that can be in-
troduced through the collective pitch settings. As mentioned before the instrument
used to set the blade pitch has an accuracy of a quarter of a degree; however, the
pitch angle difference between each blade is very small and can be considered negli-
gible. This is why when comparing tests it is advised to look at the curves defined
by the measurements, since the error in collective pitch will move the measured
values along the curves without affecting its shape.
2.3 First Generation Rotors - Hover Performance
As explained in the Introduction this research was largely motivated by the devel-
opment of a coaxial MAV. An initial proof of concept design, show in Figure 2.6 was
conceived and implemented by Samuel and Sirohi (cf. Ref [75]). The performance
of this first generation design was experimentally studied. This section presents the
experimental results obtained.
2.3.1 First Generation Rotor Configuration
When the first generation rotors were conceived back in 1999 there was no existing
vehicle or experimental data that could give hints on how to design an efficient
small-scale rotor. Experience with full-scale vehicles was used in the choice of airfoil,
solidity and number of blades. Each blade had a chord of 1 cm and a length of 7 cm.
Three blades were attached to an aluminum hub trough pins designed to facilitate
the setting of the collective pitch. No lead-lag or flapping hinges were used and the
coning angle was set at zero degrees. The rotor diameter with blades, hub and pins
29
Figure 2.6: Proof of concept design from Ref. [75]
was 17.2 cm, resulting in a solidity of 0.111. Table 2.1 summarizes the geometric
characteristics of this first generation rotor.
The airfoil chosen as baseline was an 8% camber circular arc with rounded
leading and trailing edges as shown in Figure 2.8. The blades consisted of three
layers of graphite/epoxy weave prepreg with a layup of (+45,0,+45). The symmet-
ric composite layup gave the blades excellent bending and torsional stiffness while
keeping weight at a minimum, each blade mass being just ? 0.2 g. To manufacture
Radius 86 (mm)
Root cut-out 15% Radius
No of blades 3
Blade planform Rectangular
Blade chord 1o (mm)
solidity 0.111
Table 2.1: First generation rotor geometric characteristics
30
Figure 2.7: Mold used for first generation baseline blades.
Figure 2.8: First generation 8% camber 5% TR airfoil.
the blades, a mold shown in Figure 2.7 composed of a top concave surface, a bottom
convex surface, and an edge dam was machined from aluminum. The total thickness
of the composite material was 0.5 mm resulting in blades with a thickness ratio of
5%.
Four different rotors with the same diameter, solidity and number of blades
were tested. The baseline 8% camber blades were compared with untwisted NACA
0012 and flat plates with blunt edges having a thickness ratio of 3.7%. The effect of
twist was explored by testing 8% camber blades having a linear twist of -10 degrees.
The rotor, blades and transmission used in these tests are shown in Figure 2.9. Each
rotor was tested at collectives ranging from 0 deg up to 18 deg in steps of 3 deg.
Rotational speeds for each rotor were set from 2500 RPM up to 5000 RPM in steps
of 500 RPM. Having a uniform chord of 1 cm, the tip Reynolds number varied from
15,000 up to 30,000 for the different rotational speeds.
31
Figure 2.9: Rotor, blades and transmission used in the testing of the first generation
rotor.
2.3.2 Experimental Results - First Generation Rotors.
Usually for full-scale helicopters, a specific rotational speed is used to study the
performance of a given rotor. However, this is not the case in the current investi-
gation. As opposed to traditional full-scale analysis, a range of rotational speeds
is always considered for each rotor configuration. This makes difficult to grasp the
relative thrust and power magnitudes when coefficients are used. Collective pitch vs.
thrust and torque measurement for the 8% camber untwisted blades are presented
in Figures 2.10 and 2.11. This is not a common way of showing the results since the
quantities plotted have units; however, the figures are useful to have an idea of the
magnitude of thrust and torque levels covered through the tests. It can be observed
how the measurements define smooth curves for each rotational speed, and how the
thrust and torque levels achieved span over two orders of magnitude. When this
same information is plotted using thrust and power coefficients the resultant curves
are now very close to each other. These relatively small variations in coefficients
translate into clear differences in the figure of merit plots shown in Figure 2.14.
32
0
20406080
100120
0
3
6
9
1
21
51
8
 (deg)
T, (g)
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.10: Collective vs. thrust first generation rotor with 8% camber circular
arcsat various rotational speeds.
The shape of the figure of merit curves is similar to the ones observed in full-scale
helicopters, varying from practically zero at low collectives up to a maximum value
before reducing in magnitude.
Gessow in his classic book (Ref. [76]) explains the shape of the CT vs. FM
curve as follows. Recalling equation A.23, where the total power required by a
hovering rotor is composed of a thrust dependent induced component and a constant
profile component, it is expected to have a large denominator at low thrust levels,
which results in low FM values. As CT increases the relative importance of the
constant profile term is reduced resulting in a higher figure of merit. The FM curve
gradually reduces its slope as the magnitude of the profile power becomes a smaller
part of the total power. Eventually the FM drops as a consequence of blade stall,
where lift is lost and drag is increased.
The maximum FM reached by the 8% camber blades is achieved for all rota-
tional speeds at a collective between 15 and 18 deg, a smaller collective pitch step
was required to determine this angle more accurately. Maximum FM values vary
from 0.5 up to 0.54 as tip speed increases, and are relatively low considering that
modern full-scale helicopters can have maximum FM of the order of 0.8.
33
0
0.005
0.01
0.015
0.02
0
3
6
9
12
15
18
(deg)
Q, (N.m)
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.11: Collective vs. torque first generation rotor with 8% camber circular
arcs at various rotational speeds.
0
0.005
0.01
0.015
0.02
0.025
0
3
6
9
1
21
51
8
 (deg)
C
T
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.12: Collective vs. CT first generation rotor with 8% camber circular arcs
at various rotational speeds.
34
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0369
1
2
1
5
1
8
(deg)
C
P
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.13: Collective vs. CP first generation rotor with 8% camber circular arcs
at various rotational speeds.
0
0.10.20.30.40.50.6
0
0.005
0.01
0.015
0.02
0.025
C
T 
FM
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.14: CT vs. FM first generation rotor with 8% camber circular arcs at
various rotational speeds.
35
0
0.10.20.30.40.50.6
0
1
02
03
04
05
0
DL, (N/m
2
)
FM
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.15: DL vs. FM first generation rotor with 8% camber circular arcs at
various rotational speeds.
It is very important to clarify that since the CT vs. FM curves of Figure 2.14
are obtained at different rotational speeds, a given CT corresponds to different disk
loadings (DL). As mentioned in section A.7.4, this results in an invalid comparison
since at higher DL the contribution of induced power to the total power is larger,
changing the ideal power required for each rotor. To further clarify this, FM is
plotted vs. DL for the 8% camber blades in Figure 2.15.
Effect of airfoil shape
Now that some basic characteristics and performance of a baseline rotor have been
established the effect of some airfoil and blade parameters can be explored. Two
additional sets of rectangular untwisted blades were tested, NACA 0012 and flat
plates. Airfoil shape had a dramatic effect on rotor performance in terms of thrust
capabilities and hover efficiency. Thrust and torque experimental results for the
NACA 0012 and flat plate blades as a function of the collective pitch are show in
Figures 2.16, 2.17, 2.18 and 2.19. This same information is shown in coefficient form
in Figures 2.20, 2.21, 2.22 and 2.23.
36
0
1020304050607080
0
3
6
9
12
15
18
, (deg)
T, (g)
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.16: Collective vs. thrust of first generation rotor with NACA 0012 airfoil
at various rotational speeds.
0
0.0020.0040.0060.008
0.01
0.0120.0140.016
0
3
6
9
12
15
18
, (deg)
Q, (N.m)
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.17: Collective vs. torque for first generation rotor with NACA 0012 airfoil
at various rotational speeds.
37
0
1020304050607080
0
3
6
9
12
15
18
, (deg)
T, (g)
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.18: Collective vs. thrust of first generation rotor with flat plate airfoil at
various rotational speeds.
0
0.0020.0040.0060.008
0.01
0.0120.0140.0160.018
0.02
0
3
6
9
12
15
18
, (deg)
Q, (N.m)
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.19: Collective vs. torque for first generation rotor with flat plate airfoil at
various rotational speeds.
38
0
0.0020.0040.0060.008
0.01
0.0120.014
0
3
6
9
12
15
18
 (deg)
C
T
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.20: Collective vs. CT of first generation rotor with NACA 0012 airfoil at
various rotational speeds.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0
3
6
9
1
21
51
8
(deg)
C
P
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.21: Collective vs. CP for first generation rotor with NACA 0012 airfoil at
various rotational speeds.
39
0
0.0020.0040.0060.008
0.01
0.0120.014
0369
1
2
1
5
1
8
 (deg)
C
T
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.22: Collective vs. CT of first generation rotor with flat plate airfoil at
various rotational speeds.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0369
1
2
1
5
1
8
(deg)
C
P
3000 rpm2500 rpm 3500 rpm 4000 rpm 4500 rpm 5000 rpm
Figure 2.23: Collective vs. CP for first generation rotor with flat plate airfoil at
various rotational speeds.
40
In order to compare the performance of the various rotors, a rotational speed
of 4500 RPM was chosen. This corresponds to a tip Reynolds number of ? 27,000.
Thrust vs. power coefficients are plotted in Figure 2.24. It can be observed how
flat plates and NACA blades have very similar thrust levels while the 8% cambered
plates achieve the largest thrust coefficients at every collective pitch. In fact, the
maximum CT of the 8% camber blades at 18 deg collective is about 50% larger than
the ones achieved by the other two rotors. In terms of power, the flat plates require
higher CP than the NACA blades for the whole CT range, and at collectives below
9 deg, the cambered plate blades require more power than the other two rotors.
Considering now figure of merit, Figure 2.25 shows that above thrust coefficients of
0.01, the cambered blades provide the highest hover efficiency, reaching a maximum
FM of 0.54. The NACA 0012 rotor follows with a maximum FM of 0.37 being
substantially higher than the rotor with flat plates which has a maximum FM of
only 0.3. When using power loading as a performance metric results have to be
interpreted in a different way. Figure 2.26 shows that an overall maximum PL
of 0.147 N/W is achieved with the NACA 0012 rotor, being just a few percent
higher than the maximum PL achieved with the 8% camber blades. However, this
maximum occurs at the low disk loading of 12.6 N/m2 which corresponds to a thrust
of 29.7 grams. Even though the 8% camber rotor has a lower maximum PL, it can
sustain acceptable PL levels for a wide range of disk loadings. The NACA rotor
would be a better choice for the specific case of a vehicle that has to operates at
very low thrust levels with a fixed solidity and at that particular tip speed. In section
2.6.4 it is explained how if blade geometry and tip speed are considered as design
parameters, the rotor?s operational and geometric characteristics can be matched
with the thrust requirements allowing to use the configuration that achieves the
highest hover rotor performance.
Effect of blade twist
A second rotor having 8% camber blades and a linear negative twist of -10 deg was
tested. As explained in section A.7.4 the induced power component of a hovering
41
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0
0.005
0.01
0.015
0.02
C
T 
C
P
8% camber Flat plates NACA 0012
Figure 2.24: Comparison of CT vs. CP for 8% camber plates, flat plates and NACA
0012 airfoils at 4500 RPM i.e Re=27,000 using the first generation rotors.
0
0.10.20.30.40.50.6
0
0.005
0.01
0.015
0.02
C
T 
FM
8% camber Flat plates NACA 0012
Figure 2.25: Comparison of CT vs. FM for 8% camber plates, flat plates and NACA
0012 airfoils at 4500 RPM i.e Re=27,000 using the first generation rotors.
42
0
0.020.040.060.08
0.1
0.120.140.16
0
5
10
15
20
25
30
35
40
45
DL, (N/m
2
) 
PL, (N/W)
8% camber Flat plates NACA 0012
Figure 2.26: Comparison of DL vs. PL of rotors with 8% camber plates, flat plates
and NACA 0012 airfoils at 4500 RPM i.e Re=27,000 using the first generation rotors.
rotor is at a minimum when having uniform inflow. By introducing negative twist
in the blades the inflow at the root is increased while reducing it at the tips. This
produces a more uniform inflow distribution, thus reducing the induced losses of the
rotor. There is always a tradeoff between hover and forward flight performance. Full-
scale helicopters use linear twists that range from -10 deg down to -20 deg depending
on the mission they are designed for. In general forward flight design requirements
are the ones that limit the use of twist in the rotor blades. For the twisted blades the
local blade pitch at 75% of the span is used as reference. The experimental results
that compare the twisted and untwisted rotor blade performance are presented in
Figures 2.27, 2.28 and 2.29.
The results were surprising considering that practically no change in the per-
formance was discernible. Thrust and power measurements for most collectives were
very close to each other, except at the higher values were the stalling characteristics
of the rotor were enhanced, allowing for higher thrust values. The maximum figure
of merit measured for the rotor with twisted blades is slightly lower than for the un-
twisted rotor; however, it seems that because of the large discretization of collective
43
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
0
0.005
0.01
0.015
0.02
0.025
C
T 
C
P
8% camber untwisted 8% camber twisted
Figure 2.27: Comparison of CT vs. CP of rotors with twisted and untwisted 8%
camber plates, at 4500 RPM i.e tip Re=27,000 using the first generation rotors.
0
0.10.20.30.40.50.6
0
0.005
0.01
0.015
0.02
0.025
C
T 
FM
8% camber untwisted 8% camber twisted
Figure 2.28: Comparison of CT vs. FM of rotors with twisted and untwisted 8%
camber plates, at 4500 RPM i.e tip Re=27,000 using the first generation rotors.
44
0
0.020.040.060.08
0.1
0.120.140.16
0
5
10
15
20
25
30
35
40
45
DL, (N/m
2
) 
PL, (N/W)
8% camber untwisted 8% camber twisted
Figure 2.29: Comparison of DL vs. PL of rotors with twisted and untwisted 8%
camber plates at 4500 RPM i.e Re=27,000 using the first generation rotors.
settings (steps of 3 deg), the maximum value was missed. Nevertheless considering
the observed trends, it is not expected to achieve a substantial increase in maximum
FM.
2.3.3 Effect of Number of Blades on Rotor Performance
Tip losses have a large impact on hover rotor performance, experience with full-scale
helicopters has shown that by increasing the number of blades tip losses are reduced.
This is why three-bladed rotors were used in the first generation prototype. At
MAV scale, blade weight considerations are not important since each blade accounts
only for ? 0.2% of the total vehicle mass. A more important design driver is the
mechanical complexity of the hub and vehicle?s subsystems. A larger number of
blades adds complexity to the rotor assembly and increases trimming and balancing
times, butmoreimportantlythe implementationof aswashplate can be substantially
simplified if only two blades are used. Before these tests were performed, it was
not clear if the aerodynamic benefits of a larger number of blades outweighed the
mechanical disadvantages. This is why it was essential to measure experimentally
45
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0
0.005
0.01
0.015
0.02
C
T
C
P
8% camber 2 bladed 8%camber 3 bladed Flat plate 2 bladed flat plate 3 bladed
T
ip Re= 20,800
T
ip Re= 31,200
Figure 2.30: Comparison of CT vs. CP of two and three-bladed rotors with 8%
camber and flat plates at 3500 RPM, ?=0.111
the aerodynamic effects of the number of blades on hover efficiency at MAV scale.
Two-bladed rotors with rectangular planform and a solidity equal to that of
the rotors of the first generation were assembled. Two different airfoils were tested,
flat plates and 8% camber plates. The chord of the two-bladed rotors was set as
1.5 cm and the root cut-out was left unchanged being 15% of the rotor radius. The
comparison tests covered the same collectives and rotational speeds than the first
generation tests. Experimental results showed that for a given rotational speed and
collective, the 8% camber two bladed rotor was able to produce higher thrust than
the three bladed rotor. In other words power requirements for the two-bladed rotor
were reduced, improving the aerodynamic efficiency of the system. These results
are presented in Figures 2.30, 2.31 and 2.32 for a rotational speed of 3500 RPM.
Established helicopter theory predicts a reduction in the tip losses when in-
creasing the number of blades assuming all other parameters constant. However,
at MAV scale keeping some of these parameters unchanged is a difficult task due
to manufacturing and scaling issues. This is the case for airfoil shape and tip
Reynolds number. When the wider blades were manufactured, even though the
46
0
0.10.20.30.40.50.6
0
0.005
0.01
0.015
0.02
C
T
FM
8% camber 2 bladed 8%camber 3 bladed Flat plate 2 bladed Flat plate 3 bladed
T
ip Re= 20,800
T
ip Re= 31,200
Figure 2.31: Comparison of CT vs. FM of two and three-bladed rotors with 8%
camber and flat plates at 3500 RPM, ?=0.111
0
0.05
0.1
0.15
0.2
0.25
0
5
10
15
20
25
DL, N/m
2
PL, (N/W)
8% camber 2 bladed 8% camber 3 bladed Flat plate 2 bladed Flat plate 3 bladed
T
ip Re= 20,800
T
ip Re= 31,200
Figure 2.32: Comparison of DL vs. PL of two and three-bladed rotors with 8%
camber and flat plates at 3500 RPM, ?=0.111
47
Radius 86 (mm)
Root cut-out 15% Radius
No of blades 2
Blade planform Rectangular
Blade chord 15 (mm)
solidity 0.111
Table 2.2: Second generation rotor geometric characteristics
camber matched exactly the one of the first generation blades, thickness could not
be changed proportionally to the increase in chord. This produced a difference in
the airfoil thickness ratio. The first generation blades had an airfoil with 5% thick-
ness ratio. By increasing the chord 50%, from 1 cm to 1.5 cm, the thickness ratio
was reduced to 3.33%. Tip speed in the two sets of tests was kept constant, this
produced an increase of 50% in the chord-based Reynolds number of the two-bladed
rotors. Thinner airfoils and higher Reynolds numbers may be factors that benefit
the airfoil?s aerodynamic performance at the MAV scale. These two factors may
explain the better performance of the two-bladed rotors. Reynolds number and
thickness ratio effects on thin-plate airfoils are studied experimentally in section 2.4
and computationally in Chapter 4.
Since substantial performance gains were achieved with the two bladed rotor, it
can be considered as a second generation design, Table 2.2 summarizes its geometric
characteristics. In fact, this two bladed rotor was used in the implementation of a
second generation coaxial vehicle that used a swashplate for lateral control.
2.4 Third Generation Rotors - Flat Plate Research
Based on the performance requirements of the MAV prototypes that were developed
in parallel to this research, it was decided to increase the rotor diameter by 30%
from 17.2 cm (6.77 in) to 22.4 cm (8.81 in). Rotors with this new diameter were
used in the rest of the experimental results of this dissertation. This configuration
48
Radius 112 (mm)
Root cut-out 13.5% Radius
No of blades 2
Blade planform various
Blade chord various
solidity various
Table 2.3: Third generation rotor geometric characteristics
chord=1.5cm chord=2.25cm % chord=2.25cm
LE shape thickness=0,05cm thickness=0,08cm thickness=0,05cm
TR=3.3 TR=3.3 TR=2.2
1) ? ? ?
2) ? ? ?
3) ? ? ?
4) ? ? ?
Table 2.4: Test Matrix for rectangular blades with flat plates, four LE shapes and
three chord-thickness configurations
is considered the third generation, and its main characteristics are summarized in
Table 2.3
The performance improvements observed in the second generation rotor pre-
sented in Section 2.3.3 can be attributed to the higher Reynolds numbers and the
smaller blade thickness ratios product of a larger blade chord. In order to quan-
tify the aerodynamic effects of various geometric and operating parameters on thin
plates, a set of experiments using flat plates were performed.
Two different chords and plate thickness were combined to obtain three dif-
ferent sets of blades. Two of those had an equal thickness ratio of 3.3%, and a third
one had a smaller thickness ratio of 2.2%. Additionally the three sets of blades
were tested with four different leading edge shapes shown in Figure 2.33. Table 2.4
summarizes the blade configurations tested.
49
1) Blunt
5) Sharp 
    negative camber
4) Sharp 
    positive camber
3) Sharp 
    symmetric
2) Elliptical
Figure 2.33: Leading edge shapes tested with flat plate airfoils
2.4.1 Zero Lift Drag Measurements
All the blade configurations of Table 2.4 were tested over a range of rotational
speeds such that the tip Reynolds numbers of the rotors varied from 21,000 up
to 71,000. The zero-lift torque was measured and used to calculate the equivalent
zero-lift drag coefficient of the airfoil shapes. Transmission losses and hub drag
were measured without blades, and were two orders of magnitude smaller than the
torque produced by aerodynamic drag, ruling out any significant contribution to
the measurements. In order to relate power coefficient and profile drag coefficient,
a simple blade element analysis was carried out. The profile drag coefficient is a
function of Reynolds number and Mach number, which vary along the blade span.
Profile power is obtained by integrating the sectional drag force along the blade
span as follows.
P0 = ?Nb
integraldisplay R
Rc
Dydy. (2.10)
Where ? is the rotational speed, Nb is the number of blades, Rc is the root cutout
and D is the drag force per unit span at a distance y from the rotational axis. The
drag force is given by
50
D = 12?(?y)2cCd. (2.11)
Where c is the blade chord. Assuming an average drag coefficient Cd0 it can be
shown that
CP0 = 18?Cd0 hence Cd0 = 8CP0? . (2.12)
For full-scale helicopters, the zero lift drag coefficient Cd0 is generally used
as an average drag coefficient over the operational range of angles of attack of
the rotor. This approach is used to calculate the profile drag contributions in the
FM calculation (cf. Eq.A.23). As it is shown in Chapter 3, this is not a good
methodology at MAV scale. However, at zero lift, an average value of Cd0 can be
used to calculate profile power. Using equations 2.10 and 2.12 with the Blasius
expression for laminar drag coefficient (cf. Eq. A.7), it was found that the blade
section at 75% of the rotor span has the average drag coefficient of a rectangular
blade.
At each rotational speed the zero-lift profile drag coefficient for the different
blades was calculated using Equation 2.12. Results were compared with the laminar
and turbulent theoretical predictions for a flat plate using the Reynolds number
that corresponds to the blade section at 75% span. Since the Cd0 values obtained
experimentally correspond to flat plate with two surfaces, theoretical predictions
from equations A.7 and A.9 were multiplied by two in the figures that show the
experimental results.
The effect of thickness ratio on blunt, elliptical, and symmetrically sharp lead-
ing edges is shown on Figures 2.34, 2.35 and 2.36. For the blunt leading edge plates
(Figure 2.34), it can be observed how Reynolds number has a minimal effect on the
drag coefficients. The three curves exhibit a slight negative slope as Re increased,
much smaller than for the other two leading edge shapes. As expected an increases
in TR produces a large increase in CD0 of the order of 40%. These results suggest
that the boundary layer is separating in extensive regions of the blades. Since Cd0
51
y = -0.0004Ln(x) + 0.0463
y = -0.0017Ln(x) + 0.0594
y = -0.0002Ln(x) + 0.0318
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
20000
30000
40000
50000
60000
70000
Tip Re
equivalent Cdo
Laminar theory Turbulent theory Blunt LE, TR = 2.2 Blunt LE, TR=3.3, large chord Blunt LE TR =3.3 small cord
Figure 2.34: Re vs. equivalent zero lift drag coefficient Cd0 for rotor blades with
blunt leading edges and various TR and chords
values barely change with Re, the extent of those regions is not being affected by
the increased tip speed.
For the plates with elliptical leading edges (Fig. 2.35),Reynold number has a
substantial effect on drag coefficient. Cd0 values are largely reduced as tip speed
increases. For the lowest TR, which as in the previous case has the lowest Cd0,
a change of the order of 30% was observed. The larger thickness ratios not only
increased Cd0 values, favorable Reynolds number effects were also reduced. This is
quantified by the coefficient of the logarithmic curve fits. The curve that corresponds
to the lower thickness ratio has a coefficient with larger absolute value multiplying
the logarithm. The strong dependency of Cd0 with Re suggest that the extent of the
regions of laminar, turbulent and separated flow is being modified by the changing
flow conditions. Since separated flow regions produce the largest drag forces, these
seem to be reducing in size as tip Re increases.
For the symmetrically sharp leading edge case, the blades with lower TR have
drag coefficients comparable to the theoretical fully turbulent prediction, having a
large dependency with Re. As observed for the other leading edge geometries, an
52
y = -0.0033Ln(x) + 0.0639
y = -0.0044Ln(x) + 0.0754
y = -0.0065Ln(x) + 0.0922
0.01
0.015
0.02
0.025
0.03
20000
30000
40000
50000
60000
70000
Tip Re
equivalent Cdo
Laminar theory Turbulent theory Elliptical LE, TR = 2.2 Elliptical LE, TR=3.3, large chord Elliptical LE TR =3.3 small cord
Figure 2.35: Re vs. equivalent zero lift drag coefficient CD0 for rotor blades with
elliptical leading edges and various TR and chords
increase in TR increases the drag coefficient values. In fact, Cd0 measurements of the
3.3% TR blades are very similar for the elliptical and symmetrically sharp leading
edge geometries. The sharp symmetric leading edge shape is minimizing the drag
generation, which can be linked to the behavior of the boundary layer. Lower drag
levels suggest larger laminar flow regions and/or smaller separated flow zones.
The idea behind having two blade geometries with different chords but equal
thickness ratios is to verify that the non-dimensionalization with TR and Reynolds
number is not being affected by unaccounted parameters. Surface finishing, geomet-
ric manufacturing accuracy, blade bending stiffness, all can introduce higher order
effects that can make the two geometries non-identical. The results obtained are
satisfactory in that respect, even though some differences in the measurements were
observed.
By comparing the experimental results of blades having the same TR and
chord, the leading edge shape effects can be isolated. In Figures 2.37, 2.38 and
2.39, all the leading edge geometries of Figure 2.33 are compared. For the three
TR and chord combinations the highest drag coefficients correspond to the blunt
53
y = -0.006Ln(x) + 0.0901
y = -0.0017Ln(x) + 0.0443
y = -0.0067Ln(x) + 0.0898
0.01
0.015
0.02
0.025
0.03
0.035
20000
30000
40000
50000
60000
70000
Tip Re
equivalent Cdo
Laminar theory Turbulent theory Sharp sym LE, TR = 2.2 Sharp sym LE, TR=3.3, large chord Sharp sym LE TR =3.3 small cord
Figure 2.36: Re vs. equivalent zero lift drag coefficient CD0 for rotor blades with
symmetric sharp leading edges and various TR and chords
leading edge airfoils. Similarly the lowest drag values were always achieved with
symmetrically sharp leading edges. Asymmetrically sharp leading edges produced
lower drag than the elliptical ones in two of the TR-chord combinations. Only for
the blades with the shorter 1.5 cm chord and a TR of 3.3% elliptical blades had
slightly lower Cd0 values.
In ideal conditions an infinitely thin and smooth plate parallel to the flow de-
velops a laminar boundary layer on both surfaces. Transition of the boundary layer
occurs after a given length at a critical local Reynolds number (Rec). The critical
Re value may vary depending on surface roughness and free flow turbulence levels.
Experiments have shown (cf. Ref. [1]) that critical Reynolds numbers for flat plates
are usually between 500,000 and 1 million. Since the Reynolds numbers covered
in the current tests are much lower than Rec for flat plates, flow transition and/or
separation are being produced by factors other than surface friction. Previous wind
tunnel investigations on flat plates (Ref. [1]) have shown that when rounded leading
edges were used, only turbulent drag measurements were obtained. Since the test
plates have a finite thickness, the flow undergoes a local acceleration from the stag-
54
0.01
0.015
0.02
0.025
0.03
20000
30000
40000
50000
60000
70000
Tip Re
equivalent Cdo
Turbulent theory Laminar theory Elliptical (2) Sharp positive camber (4) Sharp Symmetric (3) Blunt (1)
Figure 2.37: Re vs. equivalent zero lift drag coefficient Cd0 for blades with various
LE shapes with TR=2.2% and chord=2.25 cm
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
20000
30000
40000
50000
60000
70000
Tip Re
equivalent Cdo
Turbulent theory
Laminar theory
Blunt (1)
Sharp positive camber (4)
Sharp symmetric (3)
Elliptical (2)
Figure 2.38: Re vs. equivalent zero lift drag coefficient Cd0 for blades with various
LE shapes with TR=3.3% and chord=2.25 cm
55
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
20000
30000
40000
50000
60000
70000
Tip Re
equivalent Cdo
Turbulent theory Laminar theory Blunt (1) Elliptical (2) Sharp positive camber (4) Sharp symmetric (3)
Figure 2.39: Re vs. equivalent zero lift drag coefficient Cd0 for blades with various
LE shapes with TR=3.3% and chord=1.5 cm
nation point to the upper and lower surfaces along the leading edge curvature as
shown in Figure 2.40. The adverse pressure gradients faced by the boundary layer
can induce early transition, and if large enough, they can lead to separation as it
seems to be the case for the blades with blunt leading edges. The pressure gradients
generated are reduced by tapering the leading edge, gradually increasing the plate?s
thickness forming a sharp wedge similar to shape 3) of Figure 2.33. This delays
transition and avoids massive separation at the leading edge.
The previous discussion supports the experimental results obtained with the
different leading edge shapes and thickness ratios. The higher drag coefficients were
measured with the configurations that are expected to produce the largest adverse
pressure gradients. However, there is another factor that helps explain the large
drag coefficients measured. So far it was assumed that the free flow and the plate
were perfectly aligned. However, in practice a perfect alignment is hard to achieve,
specially in a rotor environment. Even a small deviation in angle of attack may
have an effect on the location of the stagnation point, possibly resulting in strong
pressure gradients. This makes difficult to achieve laminar or fully attached flow in
56
1
0
Cp
Pressure contours
Figure 2.40: Pressure distribution at the round LE of a flat plate at zero angle of
attack
the experiments performed.
The experimental results presented show that thickness ratio and leading edge
shape play an important role in the drag characteristic of flat plate blades. As
thickness ratio is increased, drag coefficients are adversely affected independently of
leading edge geometry. Nevertheless, leading edge shape has a large influence in the
drag characteristics. The minimum values measured correspond to a symmetrically
sharp leading edge with a TR of 2.2%, and are very close to the fully turbulent theo-
retical prediction (cf. Fig. 2.36). It is important to keep in mind that the turbulent
skin friction models are not exact analytical solutions, but just approximations that
are valid for a specific Reynolds number range. The model used in the plots is the
one from Eq. A.9, and it is generally used down to a Re of about 100,000. Below
that value, laminar flow is usually assumed. Since the theoretical turbulent predic-
tions are not reliable, based on the experimental result presented in this section it
can be concluded that the flow in the rotor blades cannot be assumed laminar in
spite of the low Reynolds numbers faced by the rotor blades. Most likely the flow
is transitional, and is separating at different regions of the blades surface. These
experiments do not provide the information necessary to identify regions of laminar,
turbulent or separated flow, but clearly show that even when no lift is produced and
57
Separation 
Bubble
Separation
Transition
Stagnation Point
Turbulent 
Boundary 
Layer
Figure 2.41: Flow around the LE of a flat plate at low angles of attack
when the adverse pressure gradients along the chord are kept at a minimum, the
flow is separating producing large amounts of drag. It is possible that root and tip
effects or an axial flow produced by centrifugal forces are influencing the boundary
layer, increasing the values of the equivalent drag coefficients calculated.
2.4.2 Hover Performance of Rotors with Rectangular Flat Plate Blades
The rotor configuration that consistently had the lowest zero lift drag coefficients
(chord = 2.25 cm , TR=2.2%) was tested for a range of collective pitch angles
and rotational speeds with each of the leading edge blade shapes of Figure 2.33.
The objective of the experiments is to experimentally determine if the reductions
in drag measured at the zero lift condition translates into hover performance gains
when thrust is produced by the rotor. Collective pitch was varied from 8 deg to 18
deg in steps of two degrees, and rotational speeds ranged from 1500 to 2500 RPM
in steps of 500 RPM. Figures 2.42, 2.43 and 2.44 show a representative set of results
at 2500 RPM, this corresponds to a tip Reynolds number of 33,400.
From the previous plots it can be observed that leading edge shape has a
strong influence in hover performance. Maximum FM values were increased from
0.35 for the blades with blunt leading edges up to 0.42 with the sharp positive
camber leading edges, this is a 20% improvement. The rotors with sharp symmetric
58
0.25
0.3
0.35
0.4
0.45
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
C
T
FM
Blunt (1) Elliptical (2) Sharp symmetric (3) Sharp positive camber (4)
Figure 2.42: CT vs. FM at 2500 RPM for rotors with flat plate blades and various
LE shapes, chord=2.25cm, TR=2.2%, tip Re = 43,700
0
0.05
0.1
0.15
0.2
0.25
51
0
1
5
2
0
DL (N/m
2
)
PL, (N/W)
Blunt (1) Elliptical (2) Sharp symmetric (3) Sharp positive camber (4)
Figure 2.43: DL vs. PL at 2500 RPM for rotors with flat plate blades, and various
LE shapes, chord=2.25cm, TR=2.2%, tip Re = 43,700
59
0
0.0010.0020.0030.004
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
C
T
C
P
Blunt (1) Elliptical (2) Symmetric sharp (3) Sharp positive camber (4)
Figure 2.44: CT vs. CP at 2500 RPM for rotors with flat plate blades, and various
LE shapes, chord=2.25cm, TR=2.2%, tip Re = 43,700
and elliptical leading edge blades had a similar performance, with a maximum FM
of about 0.36, being slightly superior for the sharp symmetric geometry. The blades
with the sharp asymmetric leading edge (LE geometry (4)) not only achieved the
highest FM values but had improved stall characteristics. As opposed to the other
configurations that stalled at collectives below 16 deg, the rotor with asymmetric
sharp leading edges reached a FM plateau at 12 deg collective that continued until
18 deg collective.
There is a link between the zero-lift drag values measured and the hover per-
formance of the rotors. Blunt leading edges which tend to produce flow separation,
achieved the lowest FMs, while the rotors with sharp symmetrical leading edges,
that had the lowest zero-lift drag values, only achieved intermediate FM levels. Fig-
ure 2.44 can be used to discern were the hover performance gains originate. The
best performing rotor not only is requiring less power for a given thrust, but is able
to produce more thrust at any given collective than the other configurations. This
result can be linked to a good behavior of the boundary layer, separation regions
seem to be smaller int this case due to the leading edge effects on the boundary
layer. These experiments cannot provide insight of the boundary layer behavior;
60
however, they have been useful to recognize the importance of leading edge shape
in flat plate airfoils and served to identify a specific leading edge geometry that
maximized rotor performance.
2.5 Third Generation Rotors - Cambered Rectangular Blades
As was observed in the experimental results of the first generation rotors (cf. 2.3.2)
camber has a strong effect on the hover performance of rotors that use thin plate
airfoils. In the previous sections only results for flat plates and 8% circular arc
camber blades were presented. Here the effect of camber magnitude and maximum
camber location are explored.
2.5.1 Hover Performance - Effect of Maximum Camber
Based on the geometric characteristics of the third generation rotors (two-bladed,
22.4 cm in diameter), three additional maximum camber magnitudes on circular arc
airfoils were experimentally studied. Two different blade chords were used such that
Reynolds number and thickness ratio effects were considered. Table 2.5 presents the
test matrix used in this section.
Max camber c=1.5cm TR=3.3% ? = 0.0909 c=2.25cm TR=2,2% ? = 0.1279
0% ? ?
3% ?
6% ? ?
9% ? ?
Table 2.5: Test matrix for rectangular blades with circular arc airfoils with various
cambers and solidities
For the rotor configurations of Table 2.5, collective pitch angles were varied
from 0 deg to 18 deg in steps of 2 deg, and rotational speeds were varied from 2000
RPM to 3000 RPM in steps of 500 RPM. Figures 2.45, 2.47 and 2.46 show the
61
0
0.10.20.30.40.50.6
0
0.005
0.01
0.015
0.02
0.025
0.03
C
T
FM
Flat plates 3% Camber 6% Camber 9% Camber
Figure 2.45: CT vs. FM at 2500 RPM for rotors with circular arc airfoils of various
cambers and elliptical LE, chord=2.25cm, TR=2.2%, tip Re = 43,700
results for the rotors with a chord of 2.25 cm (? = 0.1279) at a rotational speed of
2500 RPM (tip Re ? 43,700)
From Figure 2.45 it can be observed that as camber increases, the maximum
FM that the rotors achieve also increases. As expected the poorest performance in
terms of FM was obtained with the flat plates, that had a maximum FM of 0.36.
With the 3% camber blades maximum FM jumped to 0.51, achieving a 41% increase
with respect to the rotor with flat plate blades. A more moderate but still significant
improvement was achieved with the 6% camber blades that reached a maximum FM
of 0.57 being just 1 percentage point below the maximum of 0.58 measured with
the 9% camber blades. Maximum FM occurred at different CT and collective pitch
for all rotors. With increasing camber, maximum FM moved to gradually higher
collectives and CT values. Figure 2.46 shows the CT vs. CP plots for the set of
rotors considered. As in the CT vs. FM plot (cf. 2.45), it can be observed that the
rotor?s maximum thrust increases at the expense of additional power. The envelope
for minimum power for any given thrust is delimited by the dotted line in Figure
2.46. Each rotor configuration defines a section of the boundary. If a infinite number
62
0
0.0010.0020.0030.0040.0050.006
0
0.005
0.01
0.015
0.02
0.025
0.03
C
T
C
P
Flat plates 3% Camber 6% Camber 9% Camber
Boundary of
 minimum  power 
Figure 2.46: CT vs. CP at 2500 RPM for rotors with circular arc airfoils of various
cambers and elliptical LE, chord=2.25cm and TR=2.2%, tip Re = 43,700
0
0.05
0.1
0.15
0.2
0.25
0.3
0
5
10
15
20
25
30
DL (N/m
2
)
PL (N/W)
Flat plates 3% Camber 6% Camber 9% Camber
Figure 2.47: CT vs. CP at 2500 RPM for rotors with circular arc airfoils of various
cambers and elliptical LE, chord=2.25cm and TR=2.2%, tip Re = 33,400
63
of cambers were used, each point of the dotted line would correspond to a single
camber magnitude.
Experimental results are presented in terms of power loading in Figure 2.47.
This figures shows that even though maximum figure of merit is higher with the
9% camber blades, depending on the disk loading at which the rotor is operating,
0% 3%, 6% and 9% camber airfoils can be an optimum choice. This is essentially
the same observation that was performed in the CT vs. CP plot when a line for
minimum power for a given thrust was defined.
However, since the tests were performed with rotors having similar geometric
characteristics and the comparisons are performed at a fixed rotational speed of
2500 RPM, care must be taken not to draw an erroneous conclusion. Figure 2.48
has an additional curve that corresponds to the PL for the rotor with 9% camber
blades at a lower rotational speed of 1666 RPM. It can be observed that in the
lower disk loading range (3 to 13 N/m2) the rotor with 9% camber blades easily
outperforms the other configurations. What these results are saying is that for a
given disk loading or thrust coefficient, rotational speed and rotor solidity should be
adjusted such that the required thrust is produced with the airfoil that can achieve
the highest FM.
The performance of the rotor with 9% camber blades over a range of rotational
speeds is shown in Figure 2.49. It is important to note that for all rotational speeds
maximum FM occurs at the same collective, in this particular case somewhere in the
vicinity of 16 deg (cf. Figure 2.49(a)). On the other hand, maximum PL for each
rotational speed corresponds to a collective of about of 9 deg (cf. Figure 2.49(b)).
The envelope for maximum PL, shown as a dotted line, corresponds to a curve
defined by the power loading at maximum FM for the range of rotational speeds
considered. This means that in order to minimize the power requirements of a rotor
for a given thrust, collective pitch should be set such that the rotor is working at
its maximum FM. Thrust is then controlled by varying the rotational speed. This
is a radical difference with respect to full-scale helicopters where rotational speed
is fixed and collective is used to control thrust. Having a fixed collective not only
64
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
5
10
15
20
25
30
DL (N/m
2
)
PL (N/W)
Flat plates 2500 RPM 3% Camber 2500 RPM 6% Camber 2500 RPM 9% camber 2500 RPM 9% Camber 1666 RPM
Figure 2.48: CT vs. CP for rotors with circular arc airfoils of various cambers and
elliptical LE, chord=2.25cm and TR=2.2%, various tip speeds
gives performance benefits, but also simplifies the design and implementation of a
working prototype as discussed in Chapter 6.
2.5.2 Hover Performance - Effect of Maximum Camber Chordwise Location
The effect of maximum camber location was investigated by manufacturing and
testing blades very similar to the Go 417a thin cambered airfoil, which was studied
by Schmitz in his wind tunnel experiments (Ref. [1]). The Go 417a, shown in figure
A.7, is a thin cambered airfoil with 5.8% maximum camber at 40% of the chord
and 2.9% TR. The airfoil manufactured for the rotor tests has a nominal maximum
camber of 6% at 0.4c with a chord of 1.5cm and a TR of 3.33%. Both leading and
trailing edges were rounded. By inverting the airfoil, a camber location at 60% of
the chord was achieved. Table 2.6 shows the test matrix for this set of experiments.
Collective pitch was varied form 0 deg to 18 deg in steps of 2 deg, except for the rotor
with aft camber that reached only 16 deg. The measurements presented correspond
to a rotational speed of 3000 RPM (tip Re ? 35,000).
Figure 2.50 shows that maximum camber chordwise location has a marked
65
0.480.500.520.540.560.580.60
0.018
0.02
0.022
0.024
0.026
0.028
C
T
FM
9% Camber 1333 rpm 9% Camber 1666 rpm 9% Camber 2000 rpm 9% Camber 2500 rpm 9% Camber 3000 rpm
(a) CT vs. FM for various RPM, 12deg to 18deg collective
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
5
10
15
20
25
30
35
40
45
DL (N/m
2
)
PL (N/W)
9% Camber 1333 RPM 9% Camber 1666 RPM 9% Camber 2000 RPM 9% Camber 2500 RPM 9% Camber 3000 RPM
Envelope of maximum PL
PL for max FM
(b) DL vs. PL various RPM - envelope for max DL
Figure 2.49: Hover performance of rotor with 9% camber rectangular blades at
various RPM.
66
Max camber Max camber @ 0.4c Max camber @ 0.5c Max camber @ 0.6c
6% ? ? ?
Table 2.6: Test matrix for rectangular blades with various max camber locations,
chord = 1.5 cm, ?=0.0909
0
0.10.20.30.40.50.6
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
C
T
FM
6% Max camber @ 0.4c 6% Max camber @ 0.5c 6% Max camber @ 0.6c
Figure 2.50: CT vs. FM for rotors with 6 % camber airfoils, max camber at different
chordwise locations, chord=1.5cm, TR=3.3%, tip Re= 35,000
effect on hover rotor performance. The blades with the maximum camber at 40%
chord achieved a maximum FM of 0.57, almost 10% higher than the rotors with
circular arc airfoils. On the other hand, performance was degraded down to a
maximum FM of 0.45 when the camber was shifted aft of the midchord. Based on
the results of Figure 2.51 it seems that the forward camber enhanced the lift to
drag ratio mostly by increasing lift. For each collective, similar power levels were
achieved with the circular arc and forward camber airfoil; however, thrust levels
with the circular arc were lower. The rotor with the aft camber airfoil required for
each collective considerably more power than the other two cases and reached the
highest thrust levels.
The results presented only consider the maximum camber location disregard-
67
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0
0.005
0.01
0.015
0.02
C
T
C
P
6% Max camber @ 0.4c 6% Max camber @ 0.5c 6% Max camber @ 0.6c
Figure 2.51: CT vs. CP for rotors with 6 % camber airfoils, max camber at different
chordwise locations. chord=1.5cm, TR=3.3%, tip Re= 35,000
ing the camber distribution. Different camber lines will affect the velocities along
the surface and the adverse pressure gradients faced by the boundary layer, altering
the sectional aerodynamic characteristics. Maximum camber magnitude and loca-
tion are not enough to determine the sectional characteristics of a uniform thickness
curved plate airfoil; however, the experimental results show that there is plenty
of room to improve on the performance of circular arc airfoils by optimizing this
geometric parameter.
2.5.3 Hover Performance - Effect of Rotor Solidity
Rotor solidity is one of the most important parameters chosen in the rotor design
process. In this section, the effect of rotor solidity using circular arc airfoils is
explored by comparing the performance of the two pairs of rotors with 6% and 9%
camber airfoils of Table 2.5. For each airfoil, rotors having solidities of ?1 = 0.0909
and ?2 = 0.1279 were tested. The differences in solidity corresponds to an increase
in chord of 50%, from 1.5 cm to 2.25 cm. Collective pitch angles ranged from 0 deg
to 18 deg in steps of 2 deg, rotational speeds ranged from 2000 RPM up to 3500
68
RPM in steps of 500 RPM.
When there is a difference in solidity between rotors, the blade loading coef-
ficient defined CT/? is generally used as abscissa when comparing figures of merit.
By doing this, the effects of operating conditions are minimized and the rotors can
be judged on their design alone. The blade loading coefficient is defined as
CT
? =
T
?A(?R2)
parenleftbigg A
Ab
parenrightbigg
= T?A
b(?R)2
. (2.13)
Where Ab is total blade area. Fixing the rotational speed at 2500 RPM, the hover
performance of rotors with similar airfoils but different solidities were compared.
The results plotted on Figures 2.52 and 2.53 show that for the 6% and 9% camber
blades, the maximum figures of merit and range of CTs achieved are considerably
higher for the rotors with larger solidities.
For a fixed rotational speed, as it is usually the case in full-scale helicopters,
the rotor operating at the highest CT/? for a fixed DL or CT generally has a higher
FM, assuming that airfoil characteristics and induced losses are not affected by the
chord change and that the rotors are operating at collectives well below stall. The
better hovering efficiency of lower solidity rotors is a result of the reduced wetted
blade area that minimizes viscous losses. This case is illustrated in Figure 2.52 by
points A and B. On the CT vs FM plot, A and B are vertically aligned; however,
when plotted vs. CT/?, point A is at a higher blade loading coefficient.
At MAV scale there are a series of factors that question the use of that design
rule. Figure 2.52 shows that the rotor with higher solidity is the best choice at CTs
above 0.01 since FM values for this rotor are higher in that range. As shown by
points C and D, blade loading coefficients are lower for the more efficient rotor.
The difference in maximum FM values between the two rotors can be explained
by the degraded airfoil performance of the lower solidity rotor that is operating at
a lower tip Re. Additionally, manufacturing constraints become relevant. Blades
with larger chords can be manufactured with airfoils having smaller thickness ratios,
improving sectional aerodynamic characteristics and simultaneously maximizing the
chord-based Reynolds number. It seems that the reduction in net blade area of the
69
rotor does not compensate for the increased viscous loses resulting from the lower
Reynolds numbers and the increase in airfoil thickness ratio. This is a similar
explanation as the one given in Section 2.3.3 where the effects of number of blades
were explored.
Rotational speed
There is another difference with respect to full-scale rotors that affects the design
consideration and the choice of solidity for an MAV rotor. As previously men-
tioned rotational speed is usually fixed for full-scale vehicles, and thrust differences
produced by different solidities are usually compensated with a collective pitch ad-
justment. At MAV scale, rotational speed can be varied over a wide range, adding
another dimension to the problem. For example, an MAV rotor designer may have
to determine what is the best choice between a rotor with low solidity and higher ro-
tational speed or a rotor with high solidity and low rotational speed. Experimental
results of Figure 2.54 show that for both the 6% and 9% camber blades, rotors with
wider chords achieved higher performance levels at a lower rotational speed than
the lower solidity rotors over a given range of disk loadings. The rotors with wider
blades, in spite of the lower rotational speeds, have a higher tip Reynolds number
than the lower solidity ones. Since the thickness ratio of the airfoils could not be
kept constant, it is not possible to know in what measure the better performance
was a result of the higher Reynolds numbers or the lower airfoil thickness ratios.
Anyhow, by having blades with larger chord, two of the parameters that affect rotor
performance were improved.
It is important to understand that the results obtained define some trends that
apply to the test space explored, for radically different configurations, airfoil shapes
or operating conditions, the effect of the factors considered needs to be reevaluated.
Leading edge shape effects on cambered blades
As discussed in sections 2.4.1 and 2.4.2, leading edge shapes had a strong effect on
the zero-lift drag and hover performance of rotors with flat plate blades. The next
70
0
0.10.20.30.40.50.6
0
0.005
0.01
0.015
0.02
C
T
FM
6% camber, 
 = 0.0909
6% camber, 
 = 0.1279
A
B
C
D
(a) CT vs. FM
0
0.10.20.30.40.50.6
0
0.05
0.1
0.15
0.2
C
T
/
FM
6% camber, 6% camber, 
 = 0.0909  = 0.1279
A
B
C
D
(b) CT/? vs. FM
Figure 2.52: FM of two rotor with 6% camber blades and different solidities at 2500
RPM as function of CT and CT/?.
0
0.10.20.30.40.50.6
0
0.05
0.1
0.15
0.2
0.25
C
T
/
FM
9 % camber,   = 0.0909 9 % camber,   = 0.1279
(a) CT/? vs. FM
0
0.05
0.1
0.15
0.2
0.25
0
5
10
15
20
25
30
DL, (N/m
2
)
PL, (N/W)
9 % camber,   = 0.0909 9 % camber,   = 0.1279
(b) DL vs. PL
Figure 2.53: Hover performance of rotors with 9% camber blades at 2500 RPM.
71
0
0.05
0.1
0.15
0.2
0.25
0
5
10
15
20
25
30
35
DL (N/m
2
 )
PL, (N/W)
6% camber,   
1
 3500 RPM
6% camber,   
2
 3000 RPM
Tip Re = 40,000 
Tip Re = 52,000 
(a) DL vs. PL for 6% camber rotors
0
0.05
0.1
0.15
0.2
0.25
0
5
10
15
20
25
30
35
DL (N/m
2
 )
PL, (N/W)
9 % camber,   
1
 3000 RPM
9 % camber,   
2
 2500 RPM
Tip Re = 35,000 
Tip Re = 43,700 
(b) DL vs. PL for 9% camber rotors
Figure 2.54: DL vs. PL for rotors with 6% and 9% camber blades at different
rotational speeds.
logical step is to investigate if the performance of rotors having cambered blades
can be further improved by the choice of leading edge. In order to do this, the
rotors that achieved the highest FM with elliptical leading edges were also tested
with leading edge (4) of Figure 2.33. That particular leading edge, called ?sharp
positive camber?, had the best results on flat plate blades (cf. Fig 2.42), producing
the highest figures of merit.
Figure 2.55 compares the performance of the 6% and 9% camber rotor with
elliptical and sharp positive camber leading edges at a rotational speed of 2500
RPM. For the 6% camber blades, maximum FM was increased from 0.57 to 0.6,
while simultaneously reaching higher thrust coefficients. For the 9% camber blades
leading edge had a more subtle effect, maximum figure of merit did not have a
substantial change; however, stalling characteristics were clearly enhanced. As with
the flat plates, 6% and 9% camber rotors showed improved stalling characteristics
when using sharp leading edge shapes. At the higher collectives, the rotors reached a
plateau were FM suffered only minor chages. For the 6% camber blades the plateau
extended from 12 deg to 18 deg and for the 9% camber blades from 14 deg to 18
deg.
72
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
C
T
FM
6% Camber,  elliptical LE (2) 6% Camber, sharp pos. camber LE (4)
20 deg collective 
(a) CT vs. FM for 6% camber rotors
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
C
T
FM
9% Camber, elliptical LE (2) 9% Camber, sharp pos. camber LE (4)
20 deg collective 
(b) CT vs. FM for 9% camber rotors
Figure 2.55: CT vs. FM for rotors with rectangular 6% and 9% camber blades with
elliptical and positive sharp leading edges at 2500 RPM, chord =2.25 cm
2.6 Third Generation Rotors - Non-Rectangular Blades
Up to this point, rotor hover performance has been studied using rectangular blades.
In this section the effect of blade planform shape on flat and cambered blades is
presented.
2.6.1 Blade Tip Shapes - Full-scale vs. MAV-Scale
In full-scale helicopters blade tips play an important role in the aerodynamic perfor-
mance of the rotor since they face the highest dynamic pressures and highest Mach
numbers. Advanced full-scale helicopter blade tips make use of sweep, taper or a
combination of two. The positive effects in performance originate from the reduc-
tion of the Mach number normal to the leading edge and the change of tip vortex
characteristics. Blade tip shape effects in full-scale helicopters are not completely
understood and are a topic of ongoing research. MAV rotor operating conditions
differ considerably from larger size vehicles (cf. Fig. A.1): while for full-scale he-
licopters a typical hover tip Mach number is about 0.6, for the third generation
MAV rotors at 3000 RPM Mach number is only M=0.102. The lower tip speeds
and smaller pressure gradients between upper and lower surfaces of the blades rad-
ically change the design drivers of the blade tips. Since compressibility effects are
73
negligible and rotary-wing MAV vehicles are designed to spend a large fraction of
their mission duration in hover, blade tips main design driver is to reduce the power
requirement by promoting a more uniform inflow while keeping profile loses at a
minimum.
In general, when blades are tapered the only geometric rotor parameter that
changes is the solidity. The blade area is reduced along the span without changes
in the airfoil or twist distribution. For the tests performed in this investigation,
rectangular blades with a uniform airfoil shape along the span were modified by
simply removing some material from the tips. By doing this, four rotor and airfoil
parameters can be simultaneously affected:
? Solidity
? Twist distribution
? Airfoil camber
? Airfoil thickness ratio
If the baseline rectangular blade has a flat plate airfoil, only solidity and
airfoil thickness ratio will be modified. However, if the airfoil is cambered all the
four geometric parameters listed above are modified as function of the resultant
blade shape. The use of this methodology with baselines blades having circular
arcs is specially convenient considering that when cutting a circular arc in two,
the resultant segments are circular arcs too. This fact, as will be shown Chapter
5, facilitates the parameterization of the airfoil geometry and implementation of a
table lookup algorithm.
Figure 2.56 illustrates how the use of linear taper at the tip of a blade with
baseline 9% camber circular arc changes the parameters previously mentioned. The
use of asymmetric taper with respect to the midchord along the blade span with the
offset toward the leading edge introduces negative twist (cf. Fig. 2.56(b)). If the
taper is symmetric, no twist is introduced (cf. Fig. 2.56(a)). Obviously, if positive
twist is required, the taper would be offset towards the trailing edge.
74
A-A
B-B
C-C
4.5% camber 5% TR
9% camber 2.5% TR
6.75% camber 3.75% TR
A A
B B
C C
leading
edge
(a) 2:1 taper ratio symmetric
A-A
B-B
C-C
4.5% camber 5% TR
9% camber 2.5% TR
6.75% camber 3.75% TR
-10 deg
-5 deg
0 deg
A A
B B
C C
leading
edge
(b) 2:1 taper ratio asymmetric
Figure 2.56: Geometric effects of symmetric and asymmetric 2:1 linear taper on
blade with baseline 9% circular arc airfoil.
75
As explained in Section A.7.4, blade and airfoil parameters determine in large
part the power required by a hovering rotor. Taper and twist are generally used to
optimize the rotor performance by reducing the induced and profile power require-
ments. Taper reduces the local solidity of the blades and consequently the inflow.
With the right combination of twist and taper, a more uniform inflow and higher lift
to drag ratios can be achieved along the blade span. In the following sections, the
experimental results for rotors having various representative tip shapes on blades
with flat and cambered airfoils are presented.
2.6.2 Blade Planform Effects on Hover Performance of Rotors with Flat Plate
Blades
The effect of different planform shapes on rotors with flat plate blades with elliptical
leading edges is studied in this section. In order to isolate the effects of taper alone,
the airfoil shape needs to be kept unchanged over the blade span. In larger scale
vehicles this can be easily achieved by proportionally reducing the airfoil thickness as
the chord is reduced. However, because of manufacturing constraints, the thickness
of the blades used in this investigation is fixed to a single value along the blade
length. This leads to an increase in thickness ratio proportional to the taper, even
though airfoil camber is unmodified and equal to zero.
Three different blade geometries were tested and compared with the results for
rectangular blades; these are shown in Figure 2.57. Blade planform was modified by
simply cutting material from tips and carefully reshaping the leading edge. Ideally,
these tests should have been performed using a single baseline rotor with rectangular
blades. However, two baseline rotors with different chord were used to compare the
effect of the linear and nonlinear tapers. Nevertheless, this should not affect the
conclusions drawn. Table 2.7 shows the test matrix used.
Figure 2.58 shows the effect on performance of the elliptical tip shapes. As
expected there is no significant difference between the symmetric and asymmetric
shapes. This is not surprising considering that both geometries have the exact same
solidity and thickness ratio distribution. However, with respect to the rectangular
76
(a) (b) (c)
2:1
Figure 2.57: Blade tip shapes used with flat plate blades, (a) elliptical asymmetric,
(b) elliptical, (c) 2:1 asymmetric linear. For geometries taper starts at 0.8R.
Planform Baseline chord=1.5cm Baseline chord=2.25cm
Rectangular ? (? = 0.0909) ? (? = 0.1279)
Elliptical sym ? (? = 0.0722)
Elliptical non-sym ? (? = 0.0722)
2:1 lin taper @ 0.8R ? (? = 0.1222)
Table 2.7: Flat plate tests matrix for three different tip shapes, and elliptical leading
edge
77
blades, a clear reduction in rotor efficiency was observed. This drop in FM is mostly
a result of loss of thrust at the blade tips due to the reduced blade area.
Comparing the performance of the rotors with rectangular and linearly tapered
tips leads to the opposite result. Figure 2.59 shows an increase in maximum FM
from 0.35 to 0.37 as a result of the linear taper. The difference between the two tip
geometries, other than the solidity, are the solidity and thickness ratio distributions.
The linearly tapered blades have a larger blade area closer to the tip, which results
in a higher thrust weighted solidity than the elliptical blades. These issues are
explored in more detail in the following section using cambered blades.
From the results presented in this section, it can be concluded that when
using flat plate airfoils, blade planform only matters in the extent it affects solidity
distribution. As long as the thrust weighted solidity is kept constant, the effects on
rotor performance will be minimal. Nevertheless, blade planform can be modified to
reduce rotor power consumption as was observed with the linearly tapered blades.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
C
FM
Elliptical non symmetric Elliptical Symmetric Rectangular 
T
/
(a) CT/? vs. FM base chord=1.5 cm
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
C
C
Elliptical non symmetric Elliptical Symmetric Rectangular
T
P
(b) CT vs. CP base chord=1.5 cm
Figure 2.58: Effect of symmetric and asymmetric elliptical taper starting @ 0.8R on
flat plate blades with elliptical leading edges.
2.6.3 Blade Planform Effects on Hover Performance of Rotors with cambered
blades
This section explores the combined effects of different blade tip shapes on three
different baseline rotors. The first set used a rotors with a 9% camber airfoil and
78
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
FM
Flat plate 2:1 taper @ 0.8R Flat plate rectangular
C
T
/
(a) CT/? vs. FM base chord=2.25 cm
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
C P
Flat plate 2:1 taper @ 0.8R Flat plate rectangular
C
T
/
(b) CT vs. CP base chord=2.25 cm
Figure 2.59: Effect of 2:1 taper ratio starting @ 0.8R on flat plate blades with
elliptical leading edges.
Tip shape 9% camber Solidity
Rect. baseline ? ? = 0.0.0909
Elliptical sym ? ? = 0.0882
Elliptical non-sym ? ? = 0.0882
lin taper 1.66:1, sym ? ? = 0.0882
lin taper 1.66:1, non-sym ? ? = 0.0882
Table 2.8: Tests matrix for five different tip shapes with elliptical leading edges and
equal solidity. Linear taper starts at 0.85R
a chord of 1.6 cm (? = 0.909), and was used to evaluate five different tip shapes.
The other two sets focused on the study of 6% and 9% circular arc airfoils using as
baseline the higher solidity rotor (? = 0.1279). Tables 2.8 and 2.9 summarize the
tests performed.
Effect of solidity distribution
The blade shapes of Figure 2.60 have a equal solidity of 0.0882 but different solidity
distributions. Tips a) and b) are nonlinear and shapes c) and d) are linear tapers. In
terms of thrust weighted solidity (?e), geometries a) and b) have a ?e of 0.0832, and
geometries c and d have a ?e of 0.0836. The thrust weighted solidity is a concept
similar to the mean chord used for fixed wing analysis. It is used in helicopter
79
Taper start Tip shape 6% camber 9% camber Solidity
Rect. baseline ? ? ? = 0.1279
Taper @ Lin taper 1.33:1 asym ? ? ? = 0.1247
0.8R Lin taper 1.66:1 asym ? ? ? = 0.1228
Lin taper 2:1 asym ? ? ? = 0.1215
Lin taper 2:1 sym ? ? ? = 0.1215
Taper @ Lin taper 2:1 asym ? ? = 0.1151
0.6R Lin taper 2:1 asym ? ? = 0.1151
Table 2.9: Tests matrix for 6% and 9% camber blades with various linear taper
ratios starting at various chordwise locations
a) b) d)c)
1.66:11.66:1
Figure 2.60: Blade tip shapes used with 9%, (a) elliptical asymmetric, (b) elliptical,
(c) 1.66:1 asymmetric linear, (d) 1.66:1 symmetric linear. For geometries taper
starts at 0.85R.
80
a) b) d)c)
1.33:1
1.66:1
2:1 2:1
(a) Linear taper with various ratios starting at 0.8R
a)
b)
2:1 2:1
(b) Linear 2:1 taper starting at 0.6R
Figure 2.61: Tip shapes tested with 6% and 9% camber airfoils with 2.25 cm chord
81
rotors to account for the effect of the planform, weighting the influence of the tips
more heavily than the inner rotor regions. The equivalent thrust weighted solidity
is defined as
?e = 3
integraldisplay 1
0
?(r)r2 dr. (2.14)
Where ? is the local solidity and r is the radial station. The difference in
thrust weighted solidities between the linear and nonlinear taper distributions is
small; however, its effects were clearly measured. Figure 2.62 shows the CT vs. FM
and CT vs. CP plots for the symmetric linear and elliptical tapers of Figure 2.60(b
and d). In both cases maximum FM increased with respect to the rotor with the
baseline rectangular blades, being higher for the symmetric elliptical tip shape. The
performance gains are a product of the lower power requirements of the rotor over
the thrust range covered. However, maximum thrust coefficients were reduced, as
it occurred with the flat plate blades, because of the smaller blade area.
The thrust weighted solidity definition makes use of an important assump-
tion. As for the mean-wing concept in fixed wings, lift coefficient is assumed con-
stant along the span and equal for all the rotors compared. This means that the
definition breaks down if rotors with radically different planforms or airfoils are
compared. This is what is happening in this case. For the rotors considered there
are airfoil differences produced by the various planform shapes. This is why thrust
weighted solidities are not used in this dissertation to compare the various rotor
configurations.
Figure 2.63 shows the results for the asymmetric tip configurations. In this case
the performance gains are higher than with the symmetric blade geometries. The
maximum figure of merit achieved with the asymmetric linear taper is 0.585, being
11.5% higher than with the rectangular baseline blades. The use of the asymmetric
elliptical tip also improved FM. Maximum FM reached a value of 0.570 at a slightly
lower CT. In both cases, the asymmetric geometries were superior to the symmetric
ones, trading off higher efficiency for lower thrust levels.
The effects of blade tip geometry on twist distribution, airfoil camber, thick-
82
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.05
0.1
0.15
0.2
C
T 
/
FM
Rectangular Elliptical sym Linear sym 
(a) CT/? vs. FM base chord=1.6 cm
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0
0.005
0.01
0.015
0.02
C
T
C
P
Rectangular Elliptical sym Linear sym 
(b) CT vs. CP base chord=1.6 cm
Figure 2.62: Effect of symmetric taper distribution on blades with baseline 9%
camber airfoils and elliptical leading edges at 3000 RPM.
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.05
0.1
0.15
0.2
FM
Elliptical non sym Rectangular Linear non sym
C
T
/
(a) CT/? vs. FM base chord=1.6 cm
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0
0.005
0.01
0.015
0.02
C
T
C
P
Elliptical non sym Rectangular Linear non sym
(b) CT/? vs. CP base chord=1.6 cm
Figure 2.63: Effect of asymmetric taper distribution on blades with baseline 9%
camber airfoils and elliptical leading edges at 3000 RPM.
ness ratio and Reynold number, are plotted in Figure 2.64. The elliptical tip shape
gradually reduces the blade chord to the point where it becomes zero at the tip
of the blade. This has negative consequences on the airfoil parameters that deter-
mine the aerodynamic sectional characteristics. As can be observed in Figure 2.64,
thickness ratio becomes very large over the outer 5% of the rotor, while the airfoil
camber rapidly approaches zero. Similarly, the chord Reynolds number is reduced
to extremely low values. The linear tapered blades undergo similar airfoil geomet-
ric changes. However, because of the finite chord length, thickness ratio, Reynolds
83
number and camber are kept at reasonable values even at the blade tip.
Now that the airfoil differences between the elliptical and linearly tapered
tips have been identified, the results of Figure 2.62 can be better understood. Both
symmetric elliptical and linearly tapered blades outperform the rectangular baseline.
However, the linearly tapered blades suddenly stalled at collectives above 14 deg,
while the elliptical ones had a more gradual stall that allowed the rotor to reach
higher efficiency levels. It seems that the outer portions of the blades that have the
smaller cambers and face relatively low Reynold numbers are stalling at the higher
collectives producing large amounts of drag. Since these regions of separated flow
are smaller in the elliptical blades, less drag is produced.
On the other hand, when asymmetric taper is used, negative twist angles are
introduced. Figure 2.64(b) shows how the tip negative twist delays the stall of the
outer blade regions. The final result is that the asymmetric linearly tapered blades
with a finite blade tip chord and negative twist outperformed the elliptical blades
(cf. Fig. 2.63). The performance gains are not only a result of improved airfoil char-
acteristics and operating conditions. The negative twist and reduced blade tip area
have also a strong influence in the rotor inflow, helping in the reduction of induced
power. In order to have a quantitative interpretation of the experimental results a
model that considers the varying airfoil characteristics and the inflow changes needs
to be used. Chapter 5 addresses that issue in detail. For now, the experimental
results are presented and used to identify the trends and relevant parameters that
affect rotor performance.
Now that the elliptical and linear taper blade tips have been compared, the effects
of geometric variation of linear tapers on rotor performance are explored.
Effect of taper ratio, baseline camber and taper start location
In this segment of the chapter, the experimental results of the test matrix of Table
2.9 are presented. For the tip geometries with camber starting at 0.8R the taper
ratio is gradually increased using two rectangular baselines blades with 6% and 9%
camber. Figure 2.65 shows the camber and twist distributions along the tapered
84
-14-12-10
-8-6-4-2
0
0.80
0.85
0.90
0.95
1.00
Twist angle (deg)
Elliptical Linear 1.66:1 ratio
Non-dimensional radial location, r
(a) Twist along asymmetric tapered tips
0123456789
0.8
0.85
0.9
0.95
1
Non-dimensional radial location, r
Airfoil camber %
Elliptical Linear 1.66:1 ratio
(b) Airfoil camber along tapered tips
0123456789
10
0.8
0.85
0.9
0.95
1
TR %
Elliptical Linear 1.66:1 ratio
Non-dimensional radial location, r
(c) Thickness ratio along tapered tips
0
5000
100001500020000250003000035000
0.8
0.85
0.9
0.95
1
Re
Elliptical Linear 1.66:1 ratio
Non-dimensional radial location, r
(d) Re along tapered tips at 3000 RPM
Figure 2.64: Tip shape geometry effects on twist distribution, airfoil camber, thick-
ness ratio and Reynolds number for elliptical and linear tapers on rotors with equal
solidity
85
portion of the blade for the the various taper ratios. The Figure shows the very large
negative twist angles that are achieved over such a short radial distance. For the
6% camber blades with a taper ratio of 2:1 a negative twist of ? -6 deg is obtained,
while with the baseline 9% camber, the 2:1 taper ratio produces a negative twist
of -10 deg. The effect in hover performance of the different tip configurations was
studied in experiments where collective pitch angles were varied from 0 deg to 18
deg in steps of 2 deg, and rotational speeds were varied from 2000 RPM up to 3000
RPM in steps of 500 RPM. Some representative results at 2500 and 3000 RPM are
presented in Figures 2.66 and 2.67 where FM is plotted vs. blade loading coefficient
for the different tip geometries and the two baseline cambers.
Results show that the blade tip geometric modification had a strong and clear
beneficial effect on rotor performance at all rotational speeds. For all cases, the
asymmetric taper increased the maximum FM with respect to the baseline case.
For the blades with 6% baseline airfoil, the 2:1 and 1.66:1 taper ratios had similar
performance levels reaching maximum FM of ? 0.62 at 3000 RPM and ? 0.60 at
2000 RPM. This is an improvement over the baseline of ? 7% at 3000 RPM and ?
5% at 2500 RPM. The 2:1 taper ratio provided slightly higher FM values of about
1% over the entire blade loading coefficient range. The use of the 1.33:1 taper ratio
produced a substantial reduction in thrust, however, maximum FM increased only
by 2.5% with respect to the baseline.
The blade tip modifications on the blades with a baseline 9% camber are
shown on Figure 2.67. Large performance improvements were also observed in these
set of tests. At 3000 RPM maximum figures of merit of ? 0.64 were achieved
with the 1.66:1 taper ratio. The rotor with 2:1 taper performed almost as well,
reaching a maximum FM of 0.63. These improvements correspond to a increase
of approximately 8.5% and 7% in maximum FM. As tip speed is reduced, the gap
in performance between the rotors with rectangular and tapered blades tends to
be reduced. However, it still is significant even at the lower rotational speeds. As
opposed to the tests with the 6% camber baseline airfoil, which showed sign of
stall at collectives between 14 deg and 16 deg, the tapered blades with 9% baseline
86
0123456789
10
0.80
0.85
0.90
0.95
1.00
Airfoil camber %
Linear 1.33:1, 9%
Linear 1.66:1, 9%
Linear 2:1, 9%
Linear 2:1, 6%
Linear 1.66:1, 6%
Linear 1.33:1, 6%
Non-dimensional radial location, r
(a) Camber along asymmetric tapered tips
-12-10
-8-6-4-2
0
0.80
0.85
0.90
0.95
1.00
Twist angle, (deg)
Linear 1.33:1, 9%
Linear 1.66:1, 9%
Linear 2:1, 9%
Linear 2:1, 6%
Linear 1.66:1, 6%
Linear 1.33:1, 6%
Non-dimensional radial location, r
(b) Twist along asymmetric tapered tips
Figure 2.65: Tip shape geometry effects on twist distribution and airfoil camber on
blades with baseline 6% and 9% camber airfoils and various linear asymmetric taper
ratios.
87
camber stalled at higher collectives closer to 18 deg.
In the rotors with different taper ratios, the changes in camber and twist
along the span have been coupled. In order to isolate the effect of twist, rotors with
blades having symmetric taper are now compared to blades with asymmetric taper.
A representative result for the 2:1 taper ratio blades with 6% baseline camber is
shown in Figure 2.68. The results show that the lack of negative twist in the tapered
blades practically eliminates any advantages of the planform modification. With the
symmetric tips, the maximum FM dropped ? 5.5% from 0.62 to 0.585 with respect
to the asymmetric taper. This illustrates how the right combination of both airfoil
and rotor parameters are required to obtain a gain in performance.
Taper start location was the final parameter studied in this set of tests. The
taper radial starting point was moved from 0.8R to 0.6R, and a 2:1 taper ratio was
used on the baseline 9% camber blades. By doing this, the rotor solidity was reduced
to ? = 0.1151. The initial and final values of the camber and twist distributions
of Figure 2.65 do not change, but the starting taper radial location is now 0.6R.
The experimental result are presented in Figure 2.69. In terms of maximum FM,
practically no change was obtained by moving the taper start inwards. However,
since the solidities are different, and the maximum FM occurred at the same CT/?,
lower thrust levels were achieved. Higher collectives could be reached (20 deg)
before rotor stall occurred. But, even at the highest collectives, the maximum blade
loading coefficient reached were below the ones achieved by the higher solidity rotor.
The result obtained with the symmetric taper starting at 0.6R are consistent with
the previous results, where the advantages from the planform modifications are
minimized when no negative twist is introduced.
Effect of leading edge shape on tapered blades
The effect of leading edge shape has been explored in the zero-lift drag generation
of flat plates and in the hover performance of flat and cambered rectangular blades.
It was found that the zero-lift drag coefficients and the maximum figure of merit
improved with respect to a baseline elliptical leading edge. In this section, the best
88
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0
0.05
0.1
0.15
0.2
FM
Rectangular Linear 1.33:1 Linear 1.66:1 Linear 2:1
C 
T
/
(a) CT/? vs. FM at 2500 RPM
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0
0.05
0.1
0.15
0.2
FM
Rectangular Linear 1.33:1 Linear 1.66:1 Linear 2:1
C 
T
/
(b) CT/? vs. FM at 3000 RPM
Figure 2.66: CT/? vs. FM for rotors with baseline 6% camber airfoils with elliptical
leading edges and various tip taper ratios at two rotational speeds.
89
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.05
0.1
0.15
0.2
0.25
FM
Rectangular Linear 1.33:1 Linear 1.66:1 Linear 2:1
C 
T
/
(a) CT/? vs. FM at 2500 RPM
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.05
0.1
0.15
0.2
0.25
FM
Rectangular Linear 1.33:1 Linear 1.66:1 Linear 2:1
C 
T
/
(b) CT/? vs. FM at 3000 RPM
Figure 2.67: CT/? vs. FM for rotors with baseline 9% camber airfoils with elliptical
leading edges and various tip taper ratios at two rotational speeds.
90
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0
0.05
0.1
0.15
0.2
FM
Rectangular Linear sym 2:1 ratio Linear asym 2:1 ratio
C 
T
/
Figure 2.68: CT/? vs. FM for rotors with rectangular, symmetric and asymmetric
2:1 taper ratio blades with elliptical leading edges at 3000 RPM.
0
0.10.20.30.40.50.60.7
0
0.05
0.1
0.15
0.2
0.25
FM
Rectangular Linear asym 2:1 @ 0.8R Linear asym 2:1, @ 0.6R Linear sym 2:1 @ 0.6R
20 deg coll. 
20 deg coll. 
18 deg coll. 
C 
T
/
Figure 2.69: CT/? vs. FM for rotors with asymmetric 2:1 taper ratio starting at
0.8R and 0.6R and symmetric 2:1 taper ratio starting at 0.6R. Baseline 9% camber
blades with elliptical leading edges at 3000 RPM.
91
performing linear tapered blades with elliptical leading edges were modified in order
to evaluate their performance with sharp leading edges (geometry (4) of Figure 2.33).
The experimental results for blades with the 2:1 and 1.66:1 taper ratio and the 6%
and 9% baseline camber airfoil are shown in Figure 2.70. Representative results
at 3000 RPM are compared with rotors having blades with the same planform but
elliptical leading edges and with rectangular baseline blades. On the 6% baseline
camber (cf. Fig. 2.70(a)), the sharpening of the leading edge improved even further
the hover rotor performance of the rotor, reaching a maximum FM of 0.65. This
is a ? 5% increase with respect to the tapered blades with elliptical leading edges
and a ? 13% increase with respect to the rectangular baseline. A more moderate
effect was observed in the blades with 9% baseline camber (cf. Fig. 2.70(a)), where
maximum FM moved from 0.64 to 0.65 with the leading edge modification. However,
with respect to the baseline rotor with rectangular blades an overall maximum FM
increase of about 10% was achieved with the combined effects of planform and
leading edge modifications.
Another important aerodynamic characteristic affected by the leading edge
modifications is the rotor stall behavior. As observed before with the rectangular
blades, the sharp leading edges produced a high efficiency plateau where FM had
minor changes keeping a value close to its maximum for a range of thrust coefficients
and collectives. While the tapered blades with elliptical leading edges showed signs
of stall at 14 deg and 16 deg collective for the 6% and 9% baseline camber respec-
tively, when sharp leading edges were used collectives up to 18 deg and 22 deg were
achieved without observing a substantial drop in FM.
2.6.4 Maximum FM and Maximum PL Envelope
Hover rotor performance has been studied and quantified in terms of FM and in
some cases power loading. Maximum FM has been used as the metric to compare
the efficiency of different rotor and blade configurations. However, the large number
of cases considered forced the experiments to be performed at fixed operational
conditions of tip speed and collective settings. This resulted in measurements that
92
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.05
0.1
0.15
0.2
FM
Rectangular ellipt LE 2:1 asym taper ellipt LE 2:1 asym taper sharp LE
18 deg collective 
C 
T
/
(a) CT/? vs. FM 6% camber base airfoil at 3000 RPM
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.05
0.1
0.15
0.2
0.25
FM
Rectangular ellipt LE 1.66:1 asym taper  ellip LE 1.66:1 asym taper sharp LE
18 deg collective 
22 deg collective 
C 
T
/
(b) CT/? vs. FM 9% camber base airfoil at 3000 RPM
Figure 2.70: CT/? vs. FM for rotors with baseline 6% and 9% camber airfoils, effect
of leading edge on tapered blades.
93
covered a wide range of disk loadings. As explained in Section A.7.4, figure of merit
comparisons should be performed at the same disk loading in order to be valid,
and only for a few cases where solidity or baseline blade camber did not change
this condition was met. In order to determine the best rotor configuration for a
given DL, which is the situation faced by a designer that has a fixed vehicle weight
and rotor diameter, the envelope for maximum power loading has to be considered.
In Section 2.5.1 it was shown how for a given rotor configuration, maximum FM
occurred at the same collective pitch at all rotational speeds. In order to match
thrust requirements, the best power consumption is achieved by varying rotational
speed while keeping the collective pitch for maximum FM fixed. In this way, the
envelope for maximum PL is defined for a given rotor over a range of rotational
speeds. Figure 2.71(a)) shows the various thrust vs. PL curves at several rotational
speed, and the maximum PL envelope for a rotor with 2:1 taper ratio starting at
0.8R and with 9% baseline camber and elliptical leading edges.
A good approach to choose the best rotor for a given application within a pool
of candidates is to compare the envelopes of maximum PL at the DL of interest.
For the MAV design space a reasonable thrust range to consider is between 50g
to 100g. Figure 2.71(b) shows the maximum PL envelopes for some rotors with
rectangular and tapered blades having 6% and 9% baseline camber airfoils. It is
important to clarify that for each rotor configuration, a given thrust is produced at
different rotational speeds. From the figure it can be observed that in general the
rotors with the highest maximum FM at a given rotational speed also achieve the
highest PL values over the entire DL or thrust range. Configurations having similar
maximum FM values at a given tip speed will have comparable power loadings over
a wide range of disk loadings.
The highest power loading of all the configurations tested was achieved by the
tapered blades at 0.8R. Small differences exist between the rotors with baseline 6%
and 9% camber airfoils, however the exact performance values are hard to estimate
accurately considering the reduced number of data points used to obtain every PL
curve.
94
An important conclusion that can be drawn from this plot is that various blade
configurations can produce very similar hover performance levels at the same disk
loading while having different rotational speeds. This is an important point when
designing the transmission of an MAV. As it is shown in Chapter 6, matching the
motor with the transmission can be as important as the aerodynamic hover efficiency
of the rotor, a good transmission choice is a key factor in the overall efficiency of
a vehicle. Being able to operate a rotor at very different rotational speeds without
sacrificing aerodynamic efficiency gives the designer flexibility in the development
of a practical system.
95
0
0.10.20.30.4
0
2
0
4
0
6
0
8
0
100
120
T,(g)
PL ,(N/W)
1333 RPM 1666 RPM 2000 RPM 2500 RPM 3000 RPM
Maximum PL
 envelope
(a) Envelope of max PL from T vs. PL plots at various rotational speeds for
rotor with 2:1 tapered blades @ 0.8R, 9% baseline camber and elliptical LE
0.160.18
0.2
0.220.240.260.28
50
60
70
80
90
100
T, (g)
PL (N/W)
9%, 1.66:1 ratio, sharp LE 6%, 2:1 ratio, sharp LE 9%, 2:1 ratio @0.6R, ellip LE 9%, rect, ellip LE 6%, rect, ellip LE 9%, rect, ellip, small chord 6%, rect, ellip, small chord
(b) Comparison of maximum PL envelopes over a thrust range of 50g to 100g.
Figure 2.71: Maximum PL envelopes for rotors with various blades configurations
96
2.7 Summary
The first part of this Chapter presented the hardware, methods an approaches used
to experimentally study the hover performance of single small-scale rotors. The
hover stand used for data acquisition was described in detail, explaining its capabil-
ities, calibration and accuracy limitations. The second part presented the results of
tests designed to explore a series of geometric airfoil, blade and rotor parameters as
well as operational conditions that determine the rotor?s hover performance. Using
rectangular blades, the effect of the following geometric parameters was studied:
? Number of blades
? Solidity
? Twist
? Airfoil shape (streamlined vs thin plate)
? Airfoil camber
? Airfoil maximum camber location
? Airfoil thickness ratio
? Leading edge shape
? Zero lift drag (using flat plates)
The investigation used mostly circular arc airfoils since they offer a unique
series of advantages in terms of manufacturing ease and parameterization. The
fact that a section of a circular arc is also a circular arc was used to study the
coupled effects of camber reduction, taper and twist along tapered sections of the
blade span. Blade material was removed from the blade tips in order to obtain the
coupling between the previously mentioned parameters. Several taper geometries
were studied on flat plates, and blades with 6% and 9% baseline camber. The
effects on hover performance of taper distribution (linear vs. non linear), taper
97
ratio, taper offset (symmetric vs asymmetric), and taper radial starting location
were also measured.
The systematic investigation of geometric rotor, blade and airfoil parameters
helped identify rotor configurations that achieved large improvements over the ro-
tors used in the first generation coaxial MAV prototype, which had limited thrust
capabilities and poor hover efficiency. Maximum figures of merit of up to 0.65 were
measured on two-bladed rotors with 1.66:1 asymmetrically tapered blades and sharp
leading edges. Experimental results were used to determine that the collective for
maximum FM is practically independent of rotational speed. This has important
implications in vehicles implementation which should control rotational speed rather
than collective pitch for varying thrust levels.
Even though FM is a good performance metric, its use is limited by the fact
that valid FM comparison can only be performed at equal disk loadings. The large
variations in solidity, twist and blade camber of the different configurations tested
produced large differences in the DL at which maximum FM occurred. In order to
have a clear method of comparison that can be used over a range a DL, the boundary
of maximum power loading can be obtained for each rotor considered. The resultant
PL vs. DL curves can be compared to unequivocally determine the most efficient
rotor over a range of DL.
98
Chapter 3
Blade Element Momentum Theory
and
Average sectional Airfoil Characteristics
3.1 Introduction
On Chapter 2, systematic experimentation was used to explore the effects on hover
performance of a series of parameters. Results served to identify the combinations
of airfoil and blade geometries within a well defined test space that had a beneficial
effect on hover efficiency. Qualitative knowledge of the airfoil and rotor physics
was used to converge to configurations with improved performance. However, a
purely experimental approach is not sufficient to have a deeper understanding of
the problem. It is important to implement a model that can capture the relevant
aerodynamic phenomena and the effect of the parameters identified in the experi-
ments.
The first part of the chapter explains the implementation of a mathematical
model using Blade Element Momentum Theory (BEMT). A methodology to obtain
from rotor tests an estimate of the average sectional aerodynamic characteristics
of the airfoils is explained. Other results of the analysis such as the induced and
profile power contributions, the inflow and induced angles of attack distributions
are discussed and used to have a better understanding of the experimental results
of Chapter 2.
99
y
dy
R
Differential 
annulus
Figure 3.1: Discretization of rotor disk used for BEMT analysis.
3.2 Blade Element Momentum Theory
While simple momentum theory can be used as a first approximation to estimate
the efficiency of rotors, a more accurate aerodynamic theory is needed to incorpo-
rate blade geometry, sectional orientation, twist distribution, and airfoil character-
istics. The blade element momentum theory (BEMT) is a method that combines
the momentum and blade element theories and allows one to estimate the inflow
distribution along the blade.
3.2.1 Derivation of Equations
The rotor disk area can be discretized into concentric annuli of a area dA = 2piydy,
as show in Figure 3.1. Applying simple one dimensional momentum theory, to a
rotor in axial vertical flight, the thrust produced by each annulus is:
dT = 2?(Vc +vi)vi dA = 4pi?(Vc +vi ydy. (3.1)
Where vi is the induced velocity and Vc is the climb speed. Normalizing the previous
equation, the differential thrust coefficient is
dCT = 2?(Vc +vi)vi dA?piR2 (?R)2 = 4Vc +vi?R
parenleftBig vi
?R
parenrightBig parenleftBig y
R
parenrightBig
d
parenleftBig y
R
parenrightBig
. (3.2)
100
In terms of the total inflow ratio ? and the induced inflow ratio ?i the equation
becomes simply:
dCT = 4??i dr = 4?(???c)rdr. (3.3)
Where ?c in the component of the total inflow ratio product of the climb speed of
the rotor. Using blade element theory (BET) the incremental thrust coefficient of
the annulus is:
dCT = 12 ?Cl r2 dr = ?Cl?2 (?r2 ??r)dr. (3.4)
Now the key step is taken, the differential thrust coefficient expressions from mo-
mentum (Eq. 3.3) and blade element theories (Eq. 3.4)are combined to obtain a
quadratic equation in ?:
?Cl?
2 (?r
2 ??r)dr = 4?(???c)rdr. (3.5)
Factoring ? the canonic form is obtained:
?2 +
parenleftbigg ?C
l?
8 ? ?c,
parenrightbigg
? ? ?Cl?8 ?r = 0. (3.6)
The solution to this equation is:
?(r, ?c) =
radicalBiggparenleftbigg
?Cl?
16 ?
?c
2
parenrightbigg
+ ?Cl?8 ?r ?
parenleftbigg ?C
l?
16 ?
?c
2
parenrightbigg
. (3.7)
In this chapter only a hovering condition is considered, such that ?c = 0. This
simplifies Equation 3.7 to :
?(r) = ?Cl?16
bracketleftBiggradicalBigg
1 + 32?C
l?
?r ? 1
bracketrightBigg
. (3.8)
It is a common practice to solve the BEMT equations numerically by discretizing
the blades into a series of small elements. The rotor inflow ratio at each n element,
?(rn), is given in the numerical implementation of the BEMT in hovering conditions
by:
101
?(rn) = ?Cl?16
bracketleftbiggradicalbigg
1 + 32?C
l?
?(rn)rn ?1
bracketrightbigg
(3.9)
where rn and ?(rn) are the radius and the pitch angle at the midspan of each of
the n elements, respectively. Equation (3.9) allows one to solve for the inflow as a
function of the radius for any given airfoil section and blade geometry. After the
inflow is determined, the incremental thrust of each blade element is obtained using
?CT = ?Cl?2 (?(rn)r2n ??(rn)rn)?r (3.10)
The total thrust is calculated by numerically integrating over the blade. The in-
duced power coefficient, CPi, and profile power coefficients, CP0, are also calculated
numerically using the following equations:
CPi =
integraldisplay r=1
r=0
?dCT (3.11)
CPo = ?2
integraldisplay r=1
r=0
Cd(r)r3 dr (3.12)
The induced power factor is found using Eq. (3.13) after solving Eq. (3.11) to find
the induced power coefficient and numerically integrating Eq. (3.10) along the blade
span to calculate the rotor thrust coefficient CT.
? = CPi
C32T /?2
(3.13)
The preceding equations are modeling the viscous losses through the airfoil?s drag
coefficient, and part of the induced losses by calculating a non-uniform inflow dis-
tribution. However, the induced effects of the finite number of blades have not been
considered. The tip and root losses are a product of the pressure differences between
the upper and lower surfaces of the blade. This results in an increased inflow at
the root and tip, reducing the induced angles of attack in these regions. The inflow
distribution with tip losses is such that the lift produced at the inner and outer-most
sections of each blade have to be zero. Tip loses were included in the present model
using Prandtl?s tip-loss function. Prandtl derived a factor to adjust the inflow as
102
function of the number of blades Nb, and radial position r. The correction factor F
is given by:
F =
parenleftbigg 2
pi
parenrightbigg
cos?1 ef. (3.14)
And f is:
f = Nb2
parenleftbigg 1 ? r
r?
parenrightbigg
. (3.15)
Where the induced inflow angle ? is a function of the inflow and the radial position:
? = ?(r)r . (3.16)
Prandtl?s tip loss factor is included in the model by modifying the momentum equa-
tion of the differential thrust coefficient. Equation 3.3 for a hovering rotor case
(?c = 0) becomes:
dCT = 4F ?2 r. (3.17)
Equating the differential thrust coefficient from BET with Eq. 3.17, a quadratic
equation in ? is obtained again. Following the same procedure as in equations 3.5
to 3.8, an expression for the inflow as function of the radial position is derived:
?(r) = ?Cl?16F
bracketleftBiggradicalBigg
1 + 32F?C
l?
?r ? 1
bracketrightBigg
. (3.18)
However, since F is a function of ?, equation 3.18 needs to be solved iteratively. An
initial values of F equal to 1 is assumed, and the resultant ? is used in Eq. 3.14 to
recalculate F. Only a few iteration are necessary to achieve convergence. Figure of
merit and power loading can be calculated by applying their definitions using the
thrust and power obtained from BEMT. Details of the previous derivations can be
found in Refs. [76], [77], and [78].
103
3.2.2 BEMT and Sectional Airfoil Characteristics
In the BEMT equations presented in the previous section, the airfoil characteristics
were included through the use of the lift coefficient slope (Cl?) and the local drag
coefficient Cd. For full-scale helicopters excellent results can be achieved assuming
an average drag coefficient equal to the zero-lift drag (Cd0) and a constant Cl?.
However, at low Reynolds numbers that approach may not yield good result. As
shown in Figures A.8 and A.11, lift coefficient curves can be highly non-linear,
while drag coefficients usually have a rapid growth with angle of attack. BEMT
calculations at low Re using an ideal Cl? of 2pi, and a Cd equal to Cd0 will produce
very inaccurate thrust and power predictions. One way to overcome this problem is
to implement a table look-up scheme. Lift and drag coefficients for a given airfoil
can be tabulated as function of the angle of attack, and used by a BEMT code that
uses that interpolates the coefficient tables to calculate the induced angles of attack
along the elements of the blade.
The BEMT equations derived use Cl? to calculate Cl and obtain the differential
thrust at each blade station. Since the airfoil tables give a lift coefficient, a Cl? can
be defined for each point of the lift coefficient curve such that Cl?(?) = Cl/?.
This approach works well in most cases, however for cambered airfoil at very low
angles of attack, Cl?(?) becomes very large resulting in numerical problems. To
implement a robust algorithm that can use airfoil tables at all angles of attack, the
inflow equation(Eq. 3.5) was left as a function of Cl. The resultant equation is not
a quadratic equation in ? anymore:
?Cl
2 r
2 dr = 4?F(???c)rdr. (3.19)
Since Cl is a function of the induced angle of attack which is dependent on the in-
flow, a solution for the inflow is found numerically using a table look-up scheme for
Cl. After convergence is achieved, the induced angles of attack along the span are
used to interpolated the drag coefficients from the tabulated data and calculate the
profile power. The use of BEMT at low Reynolds number needs the use sectional
104
airfoil characteristics in order to provide reasonable results. As discussed in the
Appendix, the existing airfoil databases are still limited in size and rarely cover the
Reynolds number range of interest for MAV development. Sectional aerodynamic
characteristics of any airfoil can potentially be obtained from wind tunnel tests.
However, wind tunnel experimentation is time consuming, expensive and requires
vast amounts of infrastructure, making difficult to perform comprehensive paramet-
ric studies. The other possibility is the use of CFD to obtain, for a fraction of the
cost and time, the sectional characteristics to feed the BEMT model. Both exper-
imental and computational choices require expertise, and have specific advantages
and shortcomings.
3.3 Inverse Method to Obtain 2D-Airfoil Characteristics from Rotor Tests
During the initial stages of the present investigation the computational resources to
use CFD were not available, and the use of a wind tunnel was not possible. Never-
theless, the sectional airfoil characteristics of the rotors investigated were required
to advance in the understanding of the problem. With this motivation, a methodol-
ogy to obtain the ?average? two dimensional airfoil characteristics solely from rotor
tests was developed.
The appendix presents wind tunnel tests that show the strong dependency
of airfoil characteristics with Reynolds number. For a rotor, the local chord-based
Reynolds number varies along the span. If only one set of Cl and Cd curves are used
at all the blade sections, these can be considered ?average?, or representative airfoil
characteristics.
Ideally, the rotor blades should be rectangular with a single airfoil shape along
the span. In this way the airfoil characteristic obtained will be ?averaged? in terms
of Reynolds number only. If the blades have various airfoil shapes, the resultant lift
and drag coefficient can be considered as equivalent aerodynamic characteristics.
Figure 3.2 shows a block diagram of the algorithm with the inputs and outputs and
the iterative processes involved. The following section explain in more detail how
the algorithm was implemented.
105
Input data:
wRotor parameters: solidity, # blades,
  twist distribution, collectives
wExperimental CP and CT
wInitial  guess of Cl and Cd vs.    
Calculation of  inflow distribution
and induced angle of attack of 
blade sections
BEMT CT is calculated
BEMT CT = exper. CT BEMT CT = exper. CT
Calculation of  inflow distribution
and induced angle of attack of 
blade sections
BEMT CP is calculated
BEMT CP = exper. CP BEMT CP = exper. CP
Cl vs.    is
 modified
Cd vs.    is
 modified
Output: Cl and Cd  vs. 
Inflow dist, k vs. CT, CPi, CP0 etc...
Figure 3.2: Algorithm used to obtain average airfoil characteristics from rotor tests
using BEMT.
106
3.3.1 Iteration algorithm
The algorithm requires as inputs the experimental thrust and power coefficients
and the physical parameters of the rotor such as collective pitch, taper and twist
distribution, number of blades, solidity etc. An initial guess of the Cl vs. ? and Cd
vs. ? functions is also needed. There are three main iterations in the algorithm.
One to find the Cl vs. ? curve, another to find the Cd vs. ? curve, and a third one
for the inflow calculations.
Lift coefficient iteration: Using the initial guess of the airfoil?s Cl vs. ? function,
the rotor thrust coefficient (CT) is calculated for each collective pitch setting. These
values are compared with the experimental ones, and depending on the sign of the
error the Cl vs. ? curve is modified until there is a good agreement between the
calculated and the experimental values.
Drag coefficient iteration: The drag coefficient iteration has to be performed after
the Cl vs. ? curve has been found. This is required because the power coefficient
has an induced component that will directly depend on the lift characteristics of the
airfoil. Again, an initial guess of the CD vs. ? curve is given. For each collective,
the calculated power coefficients (CP) are compared with the experimental ones.
The Cl vs. ? plot is modified until good agreement between the calculated and the
experimental values is obtained.
Inflow calculation iteration: Every time the inflow distribution is calculated, an
initial guess of the local lift coefficient for each blade element is required. Solving
Eq. (3.19) the inflow is calculated and the induced angle of attack is obtained
(? = ? ? ?r). The induced angle of attack is used as an input in a table look-up
subroutine that interpolates the Cl vs. ? vector. The local Cl value is updated and
the cycle continues until the difference in Cl between two consecutive iterations is
negligible. This inflow calculation occurs in each of the lift coefficient iterations.
By implementing this simple algorithm it is possible to calculate the airfoil charac-
teristics by using just the hover rotor tests. A simple numerical error analysis was
performed to determine the minimum number of elements to use. The blade span
107
was initially divided into 10 elements and the BEMT power and thrust coefficients
were calculated. The number of elements was then increased in steps of 5, until
the difference between two consecutive runs was below 0.5%. A minimum of fifty
elements was required to meet the error criteria.
3.4 BEMT Analysis Results
In this section some sample results of the inverse BEMT algorithm are presented.
Experimental results are used to apply the inverse BEMT algorithm. The calculated
airfoil characteristics and the model results are presented and discussed.
3.4.1 Inverse BEMT Results
The experimental data of the first generation rotors at 4500 RPM, (cf. section 2.3.2)
was used in the inverse BEMT algorithm. The Cl and Cd curves obtained for the
flat plates, NACA 0012, and 8% circular arc airfoils are presented in Figure 3.3. The
tip Reynolds number of the rotor tests at 4500 RPM is approximately 27,000 and
the BEMT results correspond to a representative blade section at 75% span with a
Re of about 20,000.
The experimental results obtained with the third generation rotor using rect-
angular blades and an airfoil very similar to the Go417a were used to tests the
algorithm (cf. 2.5.2). Since the Go417a was tested by Schmitz at a Reynolds num-
ber of 42,000, reliable experimental wind tunnel data was available for validation.
Rotor measurements at positive and negative collectives at a rotational speed
of 3000 RPM were used in the inverse BEMT algorithm. At the operating condition
considered the Re at 75% span was approximately 26,000. Figure 3.4 shows for
collectives between 4 deg and 18 deg the induced angles of attack distributions along
the span, and the comparison between experimental and BEMT predictions for FM,
CT and CP. The plots show that the experimental rotor performance parameters
over the entire collective range are well matched.
The the Cl and Cd curves calculated and the experimental data of Schmitz
108
0
0.20.40.60.8
1
02468
1
0
1
2
, (deg)
Cl
Flat plates NACA 0012 8% camber
(a) Lift coefficient vs. ?
0
0.020.040.060.08
0.1
0.120.140.160.18
0.2
0246
8
1
0
1
2
, (deg)
Cd
8% camber NACA 0012 Flat Plates
(b) Drag coefficient vs. ?
Figure 3.3: Average lift and drag coefficients vs. angle of attack from Inverse BEMT
using experimental data of first generation rotors at 4500 RPM for flat plates, 8%
camber circular arcs, and NACA 0012 airfoils.
are shown in Figure 3.5. The BEMT predictions for this case are much closer to
the wind tunnel measurement. However, the lift coefficient curve slope is shallower
than in Schmitz experiments and has a lower Clmax. The drag coefficient predictions
showed a good qualitative agreement with the experiments in the range of -5 deg
to 8 deg angle of attack, were the shape of the curve was captured. Nevertheless,
drag was overpredicted by about 40% over most of that range. At 7 deg the curves
intersect resulting in lower drag values for the BEMT Cd.
The result obtained have a good qualitative correlation with the Go417a exper-
109
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
Angle of attack vs. radial station
Non dimensional radial station, r
 , (deg)
0 0.005 0.01 0.015 0.02
0.2
0.3
0.4
0.5
0.6
0.7
C
T
 vs. FM 
C
T
FM
Exp
BEMT
0 5 10 15 20
0.5
1
1.5
2
2.5
3
x 10
-3 Collective vs. C
P
0
 , (deg)
C
P
Exp
BEMT
0 5 10 15 20
0
0.005
0.01
0.015
0.02
Collective vs. C
T
 
0
 , (deg)
C
T
Exp
BEMT
Increasing collective 
Figure 3.4: BEMT results for the rotor with 417a airfoils: AoA span distribution,
CT vs. FM, collective vs. CP, and collective vs. CT.
110
-0.6-0.4-0.2
0
0.20.40.60.8
1
1.2
-20
-15
-10
-5
0
5
10
15
20
Exp, Re = 42k BEMT, Re = 26k
, (deg)
(a) Cl vs. ? from BEMT
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-20
-15
-10
-5
0
5
10
15
20
, (deg)
Exp, Schmitz Re = 42k BEMT, Re = 26k
(b) Cd vs. ? from BEMT
Figure 3.5: Lift and drag sectional characteristics of curved plate with 6% max
camber at 40% chord and 3.33% TR from BEMT compared to wind tunnel tests of
Go417a from Ref. [1]
-10
-5
05
101520253035
-15
-10
-5
0
5
10
15
20
BEMT Exp. from Schmitz
, (deg)
Figure 3.6: Experimental and BEMT calculated lift to drag ratio of Go417a airfoil.
imental data, the general curve shape and magnitudes are capture by the model. The
results achieved can be considered satisfactory, specially taking into account the dif-
ferences in Reynolds number, airfoil shape, turbulence levels, surface finishing, and
three-dimensional effects that modify the airfoil aerodynamic characteristics. The
wind tunnel data, should only be considered as a rough estimate of the performance
expected.
111
3.4.2 Induced and Profile Power, FM revisited
The BEMT model implemented provides analytical tools that can help in the under-
standing of the issues influencing the rotor performance. In Chapter 2 the induced
and profile contributions to the total rotor power could not be determined exper-
imentally. However, by using the rotor tests to obtain a set of averaged airfoils
characteristics, the components of the power requirements can be identified.
As derived in section A.7.4, the profile component in the denominator of the
figure of merit equation (cf. Eq A.23) is obtained by assuming a constant drag co-
efficient equal to the zero-lift drag coefficient of the airfoil. For full-scale helicopters
this is an approach that gives good results. For more detailed analysis it is common
to approximate the sectional drag coefficients below stall by using a second order
polynomial:
Cd(?) = Cd0 +d1? +d2?2 (3.20)
Where d1 and d2 are empirically determined coefficients. Typical values for full-scale
helicopter airfoils are Cdo=0.01, d1=0.025, and d2=0.65. For the BEMT results the
drag coefficient at low angles of attack (< 5 deg) can be also be approximated using
Eq. (3.20). For the NACA 0012 blades Cdo is 0.025 d1=0 and d2=1.5, and for the 8%
camber flat plate Cd0 is 0.0525 d1=0 and d2=1.5. This shows that there is not only
a large difference in the value of Cd0, but more importantly, the magnitude of the
quadratic coefficient d2 is considerably larger at the Reynolds numbers faced by the
small-scale blades. Additionally, the collectives at which the maximum FM occur
are much higher than in full-scale rotors. Hence, the induced angle of attacks along
the span cover a wide range; this is shown for the rotors with the Go 417a airfoil in
Figure 3.4. These two facts explain why, assuming a drag coefficient independent
of the angle of attack in Equation A.23 is ill advised at low Re. In order to have
a good estimate of the profile losses, the variation of drag coefficient with angle of
attack needs to be included in the analysis. Since the average drag coefficient for
a given operating condition changes in function of the Reynolds number and airfoil
112
02468
1012
, (deg)
Cl/Cd
8% camber NACA 0012 Flat plate
0
2
4
6
8
10
12
Figure 3.7: Lift to drag ratio for the airfoils of the first generation rotor from inverse
BEMT results.
shape it is difficult to define a simplified approach that does not require the airfoil
characteristics.
The lift to drag ratio plots of the three airfoils used in the first generation
rotor are plotted in Figure 3.7. The highest L/D not surprisingly was achieved
by the 8% camber circular arc, reaching a value of about 12, and was followed by
the NACA 0012 airfoil, that achieved an L/D of 8. As expected the the lowest
lift to drag ratios were obtained with the flat plates, reaching only a maximum of
approximately 5. The calculated maximum lift to drag ratios are much lower than
the values achieved at higher Re, and the curves are relatively flat and uniform.
This means that as long as the airfoils used in MAV rotors have the characteristics
discussed, an optimization of the induced angles of attack by tapering and twisting
the blades will have a reduced effect on the hover efficiency of the rotor.
Using the airfoil characteristics of the 8% camber circular arcs the BEMT
model was used to calculate the profile and induced power contributions to the
total power required by the rotor. Additionally, a prediction of the effect of a 10
deg negative linear twist along the blades was performed. Figure 3.8 shows the
experimental and the numerical results in terms of CT vs. FM.
Very small differences were measured between the twisted and untwisted blades
at collectives below 15 deg. The rotor with untwisted blades achieved its maximum
113
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
C
T
FM
Exp. twisted blades Exp untwisted blades
18 deg coll.
(a) Experimental CT vs. FM
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
C
T
FM
BEMT twisted blades BEMT untwisted blades
18 deg coll.
20 deg coll.
(b) BEMT prediction CT vs. FM
Figure 3.8: Experimental and BEMT CT vs. FM for the first generation rotors with
twisted and untwisted 8% camber blades.
FM at 15 deg collective, while the rotor with twisted blades seems to have its maxi-
mum FM at a collective between 15 and 18 deg. Because of the coarse discretization
in collective it seems that this maximum FM value was missed. The BEMT results
of Figure 3.8(b) show that an improvement in maximum FM of approximately 3.5%
is predicted by the model at a collective of 18 deg. An additional point was ob-
tained by running the code for a collective of 20 deg. This last point matches the
experimental results for 18 deg collective. The difference between the experimental
and BEMT results seems to be a result of experimental error, the first generation
rotor had a lack of stiffness at the root of the blades, which resulted in a variation of
a few degrees in the pitch of the blades at the highest collectives where the bending
and pitching moments are larger. The problem was fixed in the second generation
rotors by using stiffer blade supporting clips.
Figure 3.9 shows the CT vs. CP plots using the model results for the rotors
with twisted and untwisted 8% camber blades. For each rotor, the total CP and
its induced and profile components are shown. In a typical full-scale rotor, usually
30% of the power is consumed by the profile losses and 70% by the induced losses.
However, from Figure 3.9, it can observed how the profile drag accounts for a larger
fraction of the total power. At a CT of 0.0174, where the maximum FM for the
untwisted rotor occurs, the profile losses are 40% of the total power requirement.
114
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.005
0.01
0.015
0.02
C
T
C
P
Twisted blades total CP Twisted blades Induced CP Twisted blades profile CP Untwisted blades total CP Unwisted blades Induced CP Untwisted blades profile CP
Figure 3.9: CT vs. CP from BEMT results for the rotors twisted and untwisted 8%
camber blades.
1
1.021.041.061.08
1.1
1.121.141.161.18
0
0.005
0.01
0.015
0.02
0.025
C
T
 Induced power factor,  k
8 % camber untwisted 8 % camber twisted NACA 0012 Flat plates Go417a
Figure 3.10: BEMT calculated induced power factor ? for the first generation rotors
and third gen rotor with Go417a airfoil.
115
The BEMT calculated induced power factor (?) for the first generation rotors are
shown in Figure 3.10. The values obtained are well below the commonly assumed
full-scale value of 1.15. The use of blade twist resulted in a substantial reduction
of ? from 1.09 to 1.05 at the CT for maximum FM for the rotor with 8% camber
airfoils.
The BEMT results for the 417a airfoil (Fig. 3.6) show an improved lift to
drag ratio with respect to the other airfoils studied, reaching a maximum of 15 at
an angle of attack of 4.6 deg. At the CT for maximum FM, the contribution of the
profile power to the total power was reduced to 35%; however, the induced power
factor increased with respect to the first generation rotors, being 1.14, resulting in
a maximum FM of 0.556.
3.5 Summary
This chapter presented the derivation and numerical implementation of a BEMT
single rotor model. An algorithm used to obtain averaged two-dimensional airfoil
characteristics from rotor tests was explained, giving some sample results of its use.
The results showed that simplifications of the airfoil characteristics by assuming
constant lift coefficient slope or a constant drag coefficient, an approach commonly
used in full-scale rotor analysis, should not be applied in the modeling of small-
scale rotors due to the radical difference in airfoil behavior at the Reynolds numbers
considered.
116
Chapter 4
Sectional Airfoil Characteristics from CFD
4.1 Introduction
In order to accurately predict the performance and to eventually optimize the design
of MAV-scale rotors, it is fundamental to have the means to obtain the sectional
aerodynamic characteristics of the airfoils used in the implementation of the system.
Lift and drag coefficients are key airfoil parameters that need to be provided to the
rotor model for performance calculations. For rotary wing vehicles, the Reynolds
number varies along the blade span, and depends on the local chord and rotational
speed. Thus, for every airfoil considered, lift and drag coefficients for a range of
Reynolds numbers and angles of attack are required. The methodology presented in
the previous section, where the average airfoil characteristics were calculated using
BEMT and experimental hover performance data, is very limited since it can only
provide approximate airfoil characteristics at a Reynolds number with a certain de-
gree of uncertainty. This is why other alternatives to obtain the airfoil characteristics
had to be considered. The use of CFD for the calculation of two-dimensional airfoil
characteristics is an approach that has great advantages over wind-tunnel testing.
Even though experimental validation is required, after a successful algorithm and
methodology have been developed, CFD can produce results much faster and at a
fraction of the cost. This is why CFD was used for the analysis of a family of airfoils
that served for the compilation of a database later used in a design code.
117
Camber
TR% 0% 1.5% 3.0% 4.5% 6.0% 7.5% 9%
2.5 ? ? ? ? ? ? ?
3.75 ? ? ? ? ? ? ?
Table 4.1: Test matrix for CFD calculations, combinations of camber and TR.
4.2 Geometrical Airfoil Parameterization
The experimental aspect of the current research was focused on the study of rotors
using thin cambered circular arcs. Rectangular blades had a uniform airfoil shape
for the entire rotor span. However, by introducing blade planform modifications, the
airfoils? geometrical characteristics were varied in the radial direction. As discussed
in section 2.6.1, rectangular blades were tapered by removing material from the
leading and/or trailing edges. The resultant airfoil shapes were also circular arcs
but with reduced cambers and increased thickness ratios with respect to the original
airfoil.
The test matrix of Table 4.1 covers circular arcs with cambers from 0% up to
9% and two thickness ratios of 2.5% and 3.75%. The camber and thickness ratio
ranges that were encountered in the different blade geometries tested in Chapter
2 are covered by the test matrix. For each combination of camber and thickness
ratio, two airfoil geometries were considered; one with elliptical leading edge and
the other with sharp leading edge. In both cases, the trailing edges were tapered
into sharp wedges symmetric with respect to the camber line. Figure 4.3 show the
entire family of airfoil with elliptical leading edges.
The extent of the leading and trailing edges were defined as function of the
thickness and chord respectively, as shown in Figure 4.1. The proportions for the
airfoil with elliptical leading edge in Figure 4.1, were used by Mueller in Ref. [10]
in the wind tunnel tests of a 4% camber 1.93% TR airfoil. The parameterization
produces airfoils with leading edges that extend for larger chordwise distances as
thickness ratios are increased. This is consistent with the airfoil proportions used in
118
the rotor experiments of Chapter 2. Nevertheless, due to manufacturing limitations,
the shape of the leading and trailing edges of the airfoils used in the rotor tests do not
match exactly the idealized shapes used in the CFD calculations. The manufactured
blades, instead of having the initially intended elliptical leading edge have a rounded
shape that approaches an ellipse and that undergoes small variations along the span.
Similarly, for the blades with sharp leading edges, the extent and angle of the leading
edge wedge is not uniform, varying by small amounts along the span.
Computational considerations play an important role in the way the airfoil ge-
ometries are defined, specially at the leading and trailing edges. Rounded, smooth
leading edges are required for a good resolution of the stagnation point. This mean
that even if the experimental airfoil geometry has a razor sharp leading edge, better
results will be achieved in the CFD calculations by approximating the sharp edge
by a small diameter circle or curve. Figure 4.2 shows a detail of the the leading
edge geometries studied. Experimentally, it has been observed that trailing edge
geometry does not play a determinant role in the lift and drag sectional character-
istics at low Reynolds numbers (cf. Ref [10]). However, most of the grid generation
algorithms require a sharp trailing edge to close the grid and define the wake cut
boundary condition. This is why trailing edge geometries slightly differ from the
experimental ones, which are not perfectly sharp. The geometrical approximations
used are expected to improve the quality of the CFD calculations, allowing the flow
solvers to converge to an accurate solution.
4.3 CFD Methodology
This section explains the grid generation methodology, the flow field assumptions
and boundary conditions used for the calculations of two-dimensional airfoil char-
acteristics with TURNS and two other CFD codes: INS2d and Fluent.
119
c
2.5 t
thickness = t
0.18 c
c
3.5 t
thickness = t
0.18 c
Elliptical LE
Sharp LE 
Figure 4.1: Geometrical proportions of circular arc airfoils with elliptical and sharp
leading edges. 4% camber, 1.93 TR shown.
Figure 4.2: Detail of elliptical and sharp leading edges.
120
TR = 2.5% TR = 3.75%
Figure 4.3: Family of circular arcs with elliptical LE, cambers ranging from 0% to
9% and TR of 2.5% and 3.75%.
4.3.1 Flow Solvers
The literature review of this dissertation references the works of Schroeder and
Guptaasbeingthebasisforthecomputationalpartofthepresentresearch. Schroeder
aswellasGuptaworkedwiththestructured-gridEuler/Navier-StokessolverTURNS
(Ref. [80]). TURNS is a compressible fully implicit solver that includes all the
viscous terms in all directions using the Spalart-Allmaras turbulence model. The
original code was modified by Gupta (Refs. [64, 65]) to include a low-Mach number
preconditioner. Schroeder used the modified code for two-dimensional airfoil and
three-dimensional rotor calculations at low Reynolds numbers (Refs. [66, 67]).
The second code evaluated was INS2d, an incompressible Navier-Stokes solver
developed by Rogers (Refs. [50, 51]). The code solves the incompressible Navier-
Stokes equations in two-dimensions using the method of artificial compressibility.
As for TURNS, an implicit line-relaxation scheme is used. Kunz used INS2d to
perform an airfoil camberline optimization at ultralow Reynolds numbers assuming
fully laminar flow(Ref [53, 52]).
121
The third code evaluated was Fluent, a commercially available flow solver
widely used in the industry. Fluent was configured as a two-dimensional incom-
pressible implicit solver. Fluent, TURNS and INS2d can handle steady and time
accurate calculations. Details of the flow field assumptions are presented in section
4.3.4.
4.3.2 Grid Generation
Grid generation is an essential aspect of all numerical methods that use finite dif-
ferences, finite volumes or finite element to solve partial differential equations. The
quality of the grid not only affects the convergence time, but can also have a large
influence in the quality of the results. For the current research a hyperbolic grid
algorithm that generates structured C-Type grids was used.
Hyperbolic methods are very effective for generation of grids where the wall
boundaries are well defined and the far field boundaries are arbitrary, the grids
generated with this method are orthogonal to the airfoils surface. Computational
cost of two-dimensional grid generation is minimal when compared to the total
computational cost of the flow solution. In the algorithm used, the total number
of points along the surface can be specified, as well as clustering factors to increase
the grid density at the leading and trailing edges. These regions are of interest at
low Re since it is where key boundary layer phenomena are expected to occur.
Initial normal spacing was set to obtain approximately 25 cells in the boundary
layer. Figures 4.4 and 4.5 show respectively a typical C-type grid and the grid around
a curved plate airfoil showing the clustering at the leading and trailing edges. All
the grids used in this investigation had a Jmax of 217 points and a Kmax of 81 points.
The airfoil contour was defined with 160 points leaving 57 points for the definition
of the wake-cut boundary. The outer grid radius was placed at 15 chord lengths. A
simple grid sizing study was performed by increasing the J and K grid dimensions
in three steps: 109 by 41, 217 by 81 and, 435 by 163. The relative errors in lift and
drag coefficients were calculated with respect to the results of an 871 by 327 grid
using a 4% camber circular arc at 60,000 Re. The 217 by 81 grid had a relative error
122
ame001 10Nov 006 Plot3DD
Figure 4.4: Sample C-type grid used in the CFD calculations.
close to 0.5% in lift and drag. Hence, it was chosen since it was a good compromise
between error and computational cost.
4.3.3 Boundary Conditions
The grids used in the INS2D, TURNS, and Fluent calculations were identical. The
three solvers used at the airfoil surface a viscous no-slip wall, with wall-normal vector
pointing in the positive computational direction. At the outer C-grid boundary
TURNS uses a free-stream boundary condition; however INS2D and Fluent divide
the outer boundary in two: a boundary that imposes a velocity field based on free
stream velocity, and an outflow boundary using extrapolated velocity and constant
static pressure. Both INS2D and TURNS also require the specification of a wake-
cut boundary condition, where the boundary points are updated by averaging the
values of its surroundings. The wake cut boundary cannot be specified in Fluent,
instead the wake points need to be defined as interior points.
123
Figure 4.5: Sample grid around curved plate airfoil showing clustering at leading
and trailing edges.
4.3.4 Flow Field Assumptions
Two main flow field assumption were used in the computational aspects of the
current research. First the flow was assumed fully turbulent, and second the flow
was assumed steady.
The experiments performed in Section 2.4.1, showed that zero-lift drag coeffi-
cient of flat plates airfoils over the Reynolds number range of interest were consis-
tently above the theoretical laminar prediction. This means that the flow is either
laminar and undergoes separation, or that it is transitional. Current CFD state of
the art does not appear to be capable to model transition, however three different
approaches are possible: The flow can be assumed either fully laminar, fully turbu-
lent, or a transition chordwise location can be can specified. Of these three options,
assuming a fully turbulent flow is the more acceptable considering the Reynolds
number range of interest (15,000 to 60,000), and the large number of geometries
to evaluate. When turbulent flow is assumed,it is necessary to calculate the tur-
bulent viscosity using a turbulence model. Based on previous investigations (Refs.
124
[66, 67, 68]) it was decided to use the one equation model of Spalart and Allmaras
(Ref. [79]). The Spalart-Allmaras model was designed specifically for aerospace
applications involving wall-bounded flows and has been shown to give good results
for boundary layers subjected to adverse pressure gradients. In its original form, the
Spalart-Allmaras model is effectively a low-Reynolds-number model, requiring the
viscous-affected region of the boundary layer to be properly resolved. An additional
advantage is that its relative simplicity minimizes computational costs, making it
specially suited for a large number of calculations when limited computing power is
available.
The steady assumption greatly reduces computational cost, however it may be
an oversimplification of the problem; specially at the higher angles of attack where
separation is occurring, some wind tunnel experimental investigations (cf. [71]) have
identified periodic vortex shedding in airfoils under similar conditions. However
based on Schroeder?s and Lakshminarayan?s results, the steady state assumption
does not introduce large errors in the calculations.
A third flow characteristics to model is its compressibility. It is reasonable to
assume incompressible flow since the highest tip Mach number reached in rotor tests
is below 0.3. Hence, Incompressible flow was assumed when using the flow solvers
INS2D and Fluent. However, the flow solver TURNS is a compressible code that
solves the energy equations. Only for this case the flow was assumed compressible.
4.4 Two-Dimensional Validation
Wind tunnel experiments of a 4% camber 1.93% TR circular arc with elliptical
leading edge and the proportions shown in Figure 4.1 were used to validate the CFD
calculation at a Reynold number of 60,000. The experiments performed by Mueller
(Ref. [10]) at the Hessert Center for Aerospace Research of the University of Notre
Dame, are one of the few reliable aerodynamic studies of thin circular plates. The
Reynold number of the experiment is an upper bound of the aerodynamic regime
faced by MAVs. Nevertheless, due to the limited availability of experimental data,
Mueller?s results were used for validation.
125
Figure 4.6 shows the experimental and predicted lift, drag and moment co-
efficients as well as the lift-to-drag ratio obtained with the three codes evaluated.
The lift coefficient curves from TURNS and INS2d have an excellent agreement over
most of the angles of attack explored. However, both codes predict an early stall,
INS2d 2 deg earlier than TURNS. Fluent?s results closely follow TURNS at the
higher angles of attack; however, Fluent did not capture the nonlinearities of the Cl
curve. This produced a considerable over prediction of the lift coefficient values at
angles of attack below 5 deg.
The blades and hubs that used in experimental part of the current investigation
have large torsional stiffness and can safely be assumed rigid in the implementation
of a model. This makes the accuracy of the moment calculations of low relevance.
The computational moment results of TURNS and INS2d are very close to the
experiments at angles of attack below stall. However,at angles above stall large
differences exist.
Drag calculations were very similar for all three codes, being a few percent
higher with Fluent. In all cases, drag was under-predicted at the lower angles of
attack. Qualitatively the shape of the Cd vs, ? curve was captured by TURNS and
INS2d; however, experimental values almost doubled the CFD prediction over the
region of -1 to 4 deg. These differences translate into large errors in the predictions
of the lift-to-drag ratio of the airfoil. While the experiments obtain a maximum
L/D of 23, computational results predict an L/D between 27 and 31. Nevertheless,
TURNS and INS2d accurately predict the angle of attack for maximum lift to drag
ratio. The objective of this computational study is to identify the relative perfor-
mance of the airfoils studied and the effects of various geometrical and operational
parameters on lift and drag. Hence, results are still useful as long as there is a
good qualitative correlation with the experimental results, even when substantial
quantitative differences are present.
The large differences in the computational drag predictions, as noted before
by Schroeder and Lakshminarayan (Refs. [66, 68], seems to be a consequence of the
fully turbulent flow assumption. At low angles of attack the turbulence model is
126
likely to under-predict the length of the separation bubble and possibly the extent
of regions of separated flow at the lower surface of the airfoil, where at low angles
of attack large adverse pressure gradients exist.
Nevertheless, not only the simulations face challenges. As discussed by Mueller
and Barber (Refs. [10, 81]), the lift and drag wind-tunnel measurements can be
affected by the presence of endplates; used when two-dimensional flow wants to
be achieved. The interaction between the endplate?s boundary layer and the flow
around the wing creates a corner flow that can largely increase the two dimensional
drag measured. At Reynolds numbers between 60,000 and 200,000, Ref. [82] shows
an increase in the measured minimum drag coefficient of 18% when endplates were
used. These unresolved experimental issues difficult the validation of codes and
models, slowing the development of CFD algorithms (cf. Ref. [83]).
The comparisons of the performance of the three CFD solvers led to a choice
of a single code to perform the parametric airfoil studies and to generate the airfoil
databases. The failure of Fluent to match qualitatively and quantitatively the ex-
perimental results reduced the choice to INS2d and TURNS. Lift predictions were
more accurate in TURNS at the higher angles of attack, other than that the results
were practically identical. However, factors other than accuracy were considered.
Computational cost and convergence easy were very different between the two codes.
The incompressible nature of INS2D largely reduced the convergence time with re-
spect to TURNS, being typically less than half the time. Additionally at lower
Reynolds numbers (45,000 and below) TURNS was harder to converge, being very
sensitive to the time step used. This why INS2D was chosen to perform the database
calculations presented in this chapter.
4.5 CFD Results
This section presents the effect of various geometrical an operating conditions on
the aerodynamic characteristics of the thin plate airfoils studied.
127
-10 -5 5 10 15
-0.5
0
0.5
1
1.5
 (deg)
C
l
-10 -5 5 10 15
0
0.05
0.1
0.15
0.2
0.25
 (deg)
C
d
-10 -5 5 10 15
-0.2
-0.15
-0.1
-0.05
0
0.05
 (deg)
C
m
TURNS
INS2d
Experiment
-10 -5 5 10 15
-10
0
10
20
30
40
 (deg)
C
l
/C
d
TURNS
FLUENT
INS2d
Experiment
0 0
00
TURNS
FLUENT
INS2d
Experiment
TURNS
FLUENT
INS2d
Experiment
Figure 4.6: Lift, drag, moment, and lift to drag ratio validation of TURNS, INS2d,
and Fluent with experimental data at Re=60,000 from Ref. [10].
128
4.5.1 Parametric Study on 4% Circular Arc
The 4% camber 1.93% TR circular arc airfoil used for the experimental validation
is used to show the effects of Reynolds number and thickness ratio on the lift, drag,
and moment coefficients.
Reynolds Number Effects using INS2d
Four Reynolds numbers were explored from 15,000, up to 60,000 in steps of 15,000.
Figure 4.7 shows the results obtained. As Reynolds number is reduced, the slope
of the linear section of the Cl curve undergoes a small but noticeable reduction.
However, maximum lift coefficients are practically unchanged, except at a Reynolds
number of 15,000, were the lift overshoot that occurred at an angle of attack of 6
deg is not present anymore. Instead, the lift coefficient curve continues to increase
without showing signs of stall.
As expected, the reduction in Reynolds numbers results in a drag coefficient
increase. However, as opposed to the lift, the drag curves keep the same propor-
tions even at the lowest Reynolds numbers. All the curves show a clear change of
slope that consistently occurs between 5 deg and 6 deg. This sudden drag increase
identifies the occurrence of the airfoil stall. The changes in lift and drag coefficients
with Reynolds number translate into a reduction of the lift-to-drag ratios the airfoil
can achieve. Figure 4.8 shows the practically linear evolution of the maximum L/D
with changing Reynolds number. One thing to note is that in spite of the radical
changes of the Cl curve at 15,000 Re, the angle of attack for maximum L/D does
not change, remaining constant at 5 deg.
Thin circular arc airfoils are characterized by relatively large negative pitching
moments, CFD results showed that as Reynolds number is reduced there is a ten-
dency for the moment coefficients to approach zero below stall. This is indicative of
changes in the pressure distribution of the airfoil.
129
-4 -2 0 2 4 6 8 10 12 14
-0.5
0
0.5
1
 (deg)
C
l
Re=15k
Re=30k
Re=45k
Re=60k
-4 -2 0 2 4 6 8 10 12 14
0
0.05
0.1
0.15
0.2
0.25
 (deg)
C
d
-4 -2 0 2 4 6 8 10 12 14
-0.2
-0.15
-0.1
-0.05
0
0.05
 (deg)
C
m
-4 -2 0 2 4 6 8 10 12 14
-10
0
10
20
30
40
 (deg)
C
l
/C
d
Re=15k
Re=30k
Re=45k
Re=60k
Re=15k
Re=30k
Re=45k
Re=60k
Re=15k
Re=30k
Re=45k
Re=60k
Figure 4.7: Effect of Re in lift, drag, moment coefficients and L/D ratio on a 4%
camber 1.93% TR circualr arc.
1520253035
0
20000
40000
60000
80000
Re
L/D max
Figure 4.8: Change of maximum lift-to drag ratio with Reynolds number for the 4%
camber 1.93%TR circular arc.
130
Thickness Ratio Effects
Three thickness ratios were evaluated; starting from the baseline 1.93%, the TR was
increased to 2.5% and 3.75%. These are the same TR used in the generation of the
database. Figure 4.9 shows the results obtained.
As thickness ratio increases, the Cl curves below stall shift slightly to the right
without a substantial change in maximum lift coefficient. Drag coefficients increased
with TR, showing also a slight shifting to the right at angles above stall. The effects
of the curve-shifting and drag increase are clearly observed in the lift-to-drag plot.
Maximum L/D reduced in magnitude and occurred at higher angles of attack with
increasing thickness ratio. Maximum L/D was ? 13% smaller for the 3.75% thick
airfoil with respect to the baseline.
The only aerodynamic positive effect of the TR increase is observed in the Cm
plot. Higher TR reduce significantly the negative pitching moment of the airfoil. A
reduction of ? 13% in the Cm values below stall were obtained with the 3.75% thick
airfoils.
4.5.2 Elliptical Leading Edge Database Results
In this section the results obtained in the generation of the database are presented.
The combined effects of camber, TR and Reynolds number are shown in graphical
format, identifying the trends observed.
Camber Effects
The effects of maximum camber on lift and drag were studied following the test
matrix of Table 4.1. Maximum camber was increased in steps of 1.5% from 0% up
to 9%. Figure 4.10 shows the lift and drag coefficients at Re=60,000 for the 2.5%
and 3.75% thick airfoils.
Results show that as camber increases, the maximum lift coefficients reached
before stall also increase. For the lower camber airfoils, lift slope is positive over
the entire angle of attack range; for these cases, stall is identified in the Cl curve
131
-4 -2 0 2 4 6 8 10 12 14
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
C
l
1.93%TR, Re=60k
2.5%TR, Re=60k
3.75%TR, Re=60k
-4 -2 0 2 4 6 8 10 12 14
0
0.05
0.1
0.15
0.2
0.25
 (deg)
C
d
1.93%TR, Re=60k
2.5%TR, Re=60k
3.75%TR, Re=60k
-4 -2 0 2 4 6 8 10 12 14
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
 (deg)
C
m
1.93%TR, Re=60k
2.5%TR, Re=60k
3.75%TR, Re=60k
-4 -2 0 2 4 6 8 10 12 14
-10
0
10
20
30
40
 (deg)
C
l
/C
d
1.93%TR, Re=60k
2.5%TR, Re=60k
3.75%TR, Re=60k
 (deg)
Figure 4.9: Effect of TR in lift, drag, moment coefficients and L/D ratio on a
4%camber circular arc at Re=60,000
132
by a change in slope. For airfoils having a camber of 3% and higher Cl does not
increase monotonically anymore, and stall is identified by a drop in Cl. As for the
lift coefficient, overall drag coefficient values increase with camber. In all cases, stall
is identified by a clear change in slope, where the drag coefficient start growing at a
higher rate.
In terms of lift-to-drag ratio at a Re of 60,000, the highest values were achieved
by airfoils with cambers between 4.5% and 6%, reaching maximum L/D in the range
of 28 to 31 as observed in Figures 4.11 and 4.13.
Even though similar maximum L/D values can be achieved with a relatively
wide range of cambers, there are two important factors to consider: the shape of
the L/D curve and the Cl at which the maximum occurs. It is desirable to have a
relatively ?flat? L/D vs. ? curve in order to minimize the performance changes with
variations in angle of attack. And it is also desirable to achieve the maximum L/D
at a high lift coefficient. Figures 4.11(b) and 4.11(d) show the L/D ratios for the
2.5% TR and 3.75% TR airfoils at Re=60,000. By plotting L/D vs. Cl it is simpler
to compare the relative performance of the various airfoils. For example the 3%
and 7.5% camber arcs have similar maximum L/D. However, for the 7.5% camber
arc, the maximum occurs at more than twice the Cl than for the 3% camber airfoil.
When using ?high-lift? airfoils lower rotational speed and/or lower solidities can be
used. As shown in Chapter 2 these are factors that greatly affect rotor performance.
In the pool of airfoils studied acceptable levels of L/D and Cl are obtained
with cambers of 4.5% and above. The 9% camber airfoil is a special case. Even
tough its maximum L/D is relatively low (? 23) it has a very wide plateau. This
allows the airfoil to have an almost constant L/D ratio over a wide range of lift
coefficients.
Thickness Ratio Effects
The effect of thickness ratio in the aerodynamic characteristics of a 4% camber
circular arc were discussed in section 4.5.1. In this section the effects of increasing
the TR over the entire airfoil family are presented. Figures 4.10 and 4.11 show
133
vertically aligned the Cl, Cd, L/D vs. ? and L/D vs. Cl for all airfoils with 2.5%
and 3.75% TR at a Re of 60,000.
-4 -2 0 2 4 6 8 10 12 14
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
, (deg)
C
l
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(a) Lift coefficient 2.5% TR
-4 -2 0 2 4 6 8 10 12 14
0
0.05
0.1
0.15
0.2
0.25
, (deg)
C
d
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(b) Drag coefficient 2.5% TR
-4 -2 0 2 4 6 8 10 12 14
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
, (deg)
C
l
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(c) Lift coefficient 3.75% TR
-4 -2 0 2 4 6 8 10 12 14
0
0.05
0.1
0.15
0.2
0.25
, (deg)
C
d
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(d) Drag coefficient 3.75% TR
Figure 4.10: Cl and Cd vs. ? for circular arcs with elliptical LE and two TR at
Re=60,000
Most of the trends observed in section 4.5.1 for the 4% camber airfoil are also
valid for the entire airfoil family. By increasing the TR both lift and drag curves
were shifted to the right; and while maximum lift values were not affected, drag was
generally higher, leading to changes in the L/D curves. The relative magnitude of
the curves did not change with TR, however the extent of the high L/D plateaus
was reduced and overall lower L/D values were achieved. These effects are shown
for the 6% camber and 9% camber airfoils in Figure 4.12, these two airfoils are
of special importance in this investigation since they were used as baseline in the
134
-4 -2 0 2 4 6 8 10 12 14
-20
-10
0
10
20
30
40
, (deg)
C
l
/C
d
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(a) L/D vs. ? for 2.5% TR circ arcs.
-0.5 0 0.5 1 1.5
-20
-10
0
10
20
30
40
C
l
C
l
/C
d
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(b) L/D vs. Cl for 2.5% TR circ arcs.
-4 -2 0 2 4 6 8 10 12 14
-20
-10
0
10
20
30
40
, (deg)
C
l
/C
d
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(c) L/D vs. ? for 3.75% TR circ arcs.
-0.5 0 0.5 1 1.5
-20
-10
0
10
20
30
40
C
l
C
l
/C
d
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(d) L/D vs. Cl for 3.75% TR circ arcs.
Figure 4.11: L/D vs. ? and L/D vs. Cl for circular arcs with elliptical LE and two
TR at Re=60,000.
blades with modified planforms. The results obtained suggest that lower thickness
ratios are preferable based on aerodynamic considerations. However, manufacturing
constraints may force the use of thicker airfoils at the expense reduced performance.
Reynolds Number Effects
Foreachairfoil, CFDcalculationswereperformedatfourdifferentReynoldsnumbers
15000, 30000, 45000, and 60,000. This range covers two of the sub-regimes defined
by Carmichael (Ref. [5] and explained in section A.4. The range from 10,000 to
30,000 is practically unexplored and is expected to have mostly laminar flow with a
tendency to separate. The range of 30,000 to 70,000 has generally transitional flow.
135
-5 0 5 10 15
-0.5
0
0.5
1
1.5
, (deg)
C
l
6%c 2.5%TR-60k
6%c 3.75%TR-60k
9%c 2.5%TR-60k
9%c 3.75%TR-60k
-5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
, (deg)
C
d
6%c 2.5%TR-60k
6%c 3.75%TR-60k
9%c 2.5%TR-60k
9%c 3.75%TR-60k
-5 0 5 10 15
-0.2
-0.15
-0.1
-0.05
0
0.05
, (deg)
C
m
6%c 2.5%TR-60k
6%c 3.75%TR-60k
9%c 2.5%TR-60k
9%c 3.75%TR-60k
-5 0 5 10 15
-10
0
10
20
30
40
, (deg)
C
l
/C
d
6%c 2.5%TR-60k
6%c 3.75%TR-60k
9%c 2.5%TR-60k
9%c 3.75%TR-60k
Figure 4.12: Effect of TR in lift, drag, and moment coefficients and L/D ratio on
6% and 9% camber circular arcs with 2.5% TR and 3.75% TR at Re=60,000
136
-15
-10
-5
0
5
10
15
20
25
-4 -2 0 2 4 6 8 101214
0
1
2
3
4
5
6
7
8
9
, (deg)
camber %
(a) 2.5% TR airfoils at Re=60k
-15
-10
-5
0
5
10
15
20
25
0
1
2
3
4
5
6
7
8
9
, (deg)
camber %
-4 -2 0 2 4 6 8 101214
(b) 3.75% TR airfoils at Re=60k
Figure 4.13: Contours of L/D vs. ? at Re=60,000 for airfoils with elliptical LE and
two TR.
This is the regime were laminar separation bubbles and complex boundary layer
phenomena usually occur. The assumption of fully turbulent flow approximates
the real flow conditions, and hence the Reynolds number effects. Since no reliable
experimental data is available for validation at the lower Reynolds numbers, the
results should be interpreted as a qualitative guideline of the actual airfoil behavior.
Because of the large number of cases studied, a clear and convenient way of
showing the effects of Reynolds number on the airfoils performance is with contour
plots. At each Re, the L/D ratio vs. angle of attack values can be compared for the
different cambers in a single plot. Figure 4.16 shows the results obtained with the
airfoils having 2.5% TR . The color scale used is the same for all four plots such that
a given color or contrast value represents the same L/D. As Reynolds numbers are
reduced, the maximum L/D ratios that the airfoils achieve reduce gradually from
regions of L/D ? 28 at Re of 60,000, to regions of L/D ? 13. However not only the
overall magnitude of the curves changed, as for the sample case of section 4.5.1 the
L/D characteristics curves for all cambers had the tendency to flatten out, reducing
the effects of angle of attack in the L/D ratio.
For the family of airfoils studied the maximum L/D is achieved at the higher
Re at an angle of attack were a sudden increase in Cl occurs. This nonlinearity was
137
-4 -2 0 2 4 6 8 10 12 14
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
, (deg)
C
l
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(a) Lift coefficient at Re=15k, TR=2.5%
-4 -2 0 2 4 6 8 10 12 14
0
0.05
0.1
0.15
0.2
0.25
, (deg)
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
C
d
(b) Drag coefficient at Re=15k, TR=2.5%
-4 -2 0 2 4 6 8 10 12 14
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
, (deg)
C
l
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(c) Lift coefficient at Re=15k, TR=3.75%
-4 -2 0 2 4 6 8 10 12 14
0
0.05
0.1
0.15
0.2
0.25
, (deg)
C
d
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
(d) Drag coefficient at Re=15k, TR=3.75%
Figure 4.14: Lift and Drag coefficients vs. ? for airfoils with elliptical LE with 2.5%
TR, and 3.75% TR. All cambers at Re=15,000.
138
present in the experimental validation case and was accurately captured by INS2d
and TURNS. The lift ?bump? is still present at Reynold numbers as low as 30,000.
However at Re of 15,000 for all airfoil geometries the Cl curves become smoother,
monotonically growing over the entire angle of attack range. Figure 4.14 shows Cl
and Cd vs. ? for all the cambers at Re=15,000.
Figure 4.15 shows the lift, drag, and moment coefficients as well as the L/D
for a 6% camber airfoil at all Reynolds numbers. This plot is representative of how
the Reynolds number affects the lift and drag curves. The behavior illustrated here
is useful to have a better understanding of the L/D contour plots of Figures 4.16
and 4.17. As Reynolds number is reduced, the steep lift slope change that took
place at ?= 5 deg at Re=60,000, moved to ?= 6 deg at Re=45,000 and to ?= 8
deg at Re=30,000. This shifted the angle of attack for maximum L/D by a degree
at Re=45,000. Below Re=45,000 the drag became large enough to negate the effect
of the lift increase, shifting the maximum L/D to a lower ?. The combined effects
of Re on lift and drag gradually moved the angle of attack from 6 deg to 7 deg ,
then to 4 deg, and finally to 5 deg at Re=15,000. Qualitatively, this behavior was
typical of airfoils with cambers of 3% and larger.
A similar behavior was obtained for the airfoils with 3.75% TR that suffered
a reduction in the maximum L/D values as Reynolds numbers decreased. The L/D
results at the different Re for the 3.75% TR airfoils are shown in Figure 4.17. As
for the 2.5% TR airfoils, maximum L/D regions shifted to lower angles of attack at
the lower Re as a consequence of the smoother Cl curve shape. The increase in TR
had moderate adverse consequences on the performance of airfoils at all Reynolds
numbers, lowering by a few percentile points the overall values. However a positive
effect was the widening of the high L/D plateaus at all Re.
4.5.3 Sharp Leading Edge Database Results
In the rotor experiments of chapter 2, it was found that the performance of the
rotors was improved when using airfoils having a sharp leading edge. In this section
the CFD results for the airfoils with sharp leading edges are presented.
139
-5 0 5 10 15
-0.5
0
0.5
1
1.5
, (deg)
C
l
6%c 2.5%TR-15k
6%c 2.5%TR-30k
6%c 2.5%TR-45k
6%c 2.5%TR-60k
-5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
, (deg)
C
d
6%c 2.5%TR-15k
6%c 2.5%TR-30k
6%c 2.5%TR-45k
6%c 2.5%TR-60k
-5 0 5 10 15
-0.2
-0.15
-0.1
-0.05
0
0.05
, (deg)
C
m
6%c 2.5%TR-15k
6%c 2.5%TR-30k
6%c 2.5%TR-45k
6%c 2.5%TR-60k
-5 0 5 10 15
-10
0
10
20
30
40
, (deg)
C
l
/C
d
6%c 2.5%TR-15k
6%c 2.5%TR-30k
6%c 2.5%TR-45k
6%c 2.5%TR-60k
Figure 4.15: Effect of Re in lift, drag, moment coefficient and L/D on a 6% camber
2.5% TR circular arc with elliptical LE.
140
camber %
camber %
camber %
camber %
Re=15,000 Re=30,000
Re=45,000 Re=60,000
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
-15
-10
-5
0
5
10
15
20
25
-15
-10
-5
0
5
10
15
20
25
Figure 4.16: Contour plots of L/D as function of airfoil camber and ? for airfoils
with elliptical LE and 2.5% TR at Re = 15k, 30k, 45k, 60k.
141
-15
-10
-5
0
5
10
15
20
25
camber %
camber %
camber %
camber %
Re=15,000 Re=30,000
Re=45,000 Re=60,000
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
-15
-10
-5
0
5
10
15
20
25
Figure 4.17: Contour plots of L/D as function of airfoil camber and ? for airfoils
with elliptical LE and 3.75% TR at Re = 15k, 30k, 45k, 60k.
142
Figure 4.18 shows the lift and drag coefficient curves vs. ? for all the database
airfoils at Reynolds number of 60,000 and 15,000. The effects of camber did not
change with respect to the elliptical leading edge airfoils:
? As camber increases higher lift coefficients are reached below stall.
? As camber increases also does maximum Cl.
? Higher Cl values result also in higher Cd values.
? Airfoil stall occurs at increasingly high angles of attack as camber increases.
A reduction in the Reynolds number did not affect the relative magnitude
of the plots. However as for the elliptical leading edge case, at the lowest Re lift
coefficient curves did not show signs of stall. Instead Cl curves grew monotonically
over the entire angle of attack range. Stall is identified by the drag coefficient
behavior, which undergoes a changes in slope as the airfoils stall.
In terms of lift to drag ratio the contour plots as function of airfoil camber
and angle of attack are shown in Figure 4.19 for the airfoils with 2.5% TR and in
Figure 4.20 for the airfoils with 3.75% TR. Similar qualitative results were observed
for the sharp and the elliptical LE in terms of L/D:
? A reduction in Re results in lower L/D.
? A reduction in Re gradually lowers the angle of attack for max L/D.
? A reduction in Re extends the high L/D plateau.
The effect of TR in the aerodynamic performance of the sharp leading edge
airfoils was also qualitatively very similar to the results for the elliptical LE airfoils.
Figure 4.21 shows the TR effects on the 6% and 9% camber airfoils. These sample
results show that:
? An increase in TR increases Cd
? An increase in TR tends to reduces L/D values
143
? An increase in TR tends shift the lift and drag curves to the right, resulting
in an effective reduction in angle of attack.
? An increase in TR resulted in changes in the stall characteristics of the airfoils.
Stall was delayed by about 2 degrees for most of the cambers.
-4 -2 0 2 4 6 8 10 12 14
-0.5
0
0.5
1
1.5
, (deg)
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
C
l
(a) Cl vs. ? for 2.5% TR circ arcs sharp LE,
Re=60k.
-4 -2 0 2 4 6 8 10 12 14
0
0.05
0.1
0.15
0.2
0.25
, (deg)
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
C
d
(b) Cd vs. ? for 2.5% TR circ arcs, sharp LE,
Re=60k.
-4 -2 0 2 4 6 8 10 12 14
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
, (deg)
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
C
l
(c) Cl vs. ? for 2.5% TR circ arcs, sharp LE,
Re=15k.
-4 -2 0 2 4 6 8 10 12 14
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0% camber
1.5% camber
3% camber
4.5% camber
6% camber
7.5% camber
9% camber
, (deg)
C
d
(d) Cd vs. ? for 2.5% TR circ arcs, sharp LE,
Re=15k.
Figure 4.18: Cl and Cd vs. ? for circular arc airfoils with sharp LE and TR= 2.5%
at Re=60,000 and Re=15,000.
144
-10
-5
0
5
10
15
20
25
30
35
-10
-5
0
5
10
15
20
25
30
35
camber %
camber %
camber %
camber %
Re=15,000 Re=30,000
Re=45,000 Re=60,000
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
-10
-5
0
5
10
15
20
25
30
35
-10
-5
0
5
10
15
20
25
30
35
Figure 4.19: Contour plots of L/D as function of airfoil camber and ? for airfoils
with sharp LE and 2.5% TR at Re = 15k, 30k, 45k, 60k.
145
-4 -2 0 2 4 6 8 10 12 14
0
1
2
3
4
5
6
7
8
9
-2 0 2 4 6 8 10 12
-2 0 2 4 6 8 10 12 14
-10
-5
0
5
10
15
20
25
30
-10
-5
0
5
10
15
20
25
30
camber %
camber %
camber %
camber %
Re=15,000 Re=30,000
Re=60,000
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
9
8
7
6
5
4
3
2
1
0
-4 -2 0 2 4 6 8 10 12 14
, (deg)
Re=45,000
-10
-5
0
5
10
15
20
25
30
Figure 4.20: Contour plots of L/D as function of airfoil camber and ? for airfoils
with sharp LE and 3.75% TR at Re = 15k, 30k, 45k, 60k.
146
-5 0 5 10 15
-0.5
0
0.5
1
1.5
, (deg)
C
l
6%c 2.5%TR-60k
6%c 3.75%TR-60k
9%c 2.5%TR-60k
9%c 3.75%TR-60k
-5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
, (deg)
C
d
6%c 2.5%TR-60k
6%c 3.75%TR-60k
9%c 2.5%TR-60k
9%c 3.75%TR-60k
-5 0 5 10 15
-0.2
-0.15
-0.1
-0.05
0
0.05
, (deg)
C
m
6%c 2.5%TR-60k
6%c 3.75%TR-60k
9%c 2.5%TR-60k
9%c 3.75%TR-60k
-5 0 10 15
-10
0
10
20
30
40
, (deg)
C
l
/C
d
6%c 2.5%TR-60k
6%c 3.75%TR-60k
9%c 2.5%TR-60k
9%c 3.75%TR-60k
5
Figure 4.21: Effect of TR in lift, drag, and moment coefficients and L/D ratio on
6% and 9% camber circular arcs with sharp LE having 2.5% and 3.75% TR at
Re=60,000.
147
4.5.4 Sharp vs. Elliptical Leading Edges
In this section some representative CFD results of elliptical and sharp leading edge
airfoils are compared. Figure 4.22 shows the lift and drag coefficients for the 6%
and 9% camber airfoils at Re=60,000. From the figure it can observed that:
? Sharp leading edges lead to an increase in maximum Cl.
? Sharp leading edges lead to an increase in Cd.
? Sharp leading edges delay stall.
? Leading edge shape had minimum effect on the moment coefficients.
? Sharp leading edges shift the lift, drag, and moment coefficient curves to the
right, increasing by about 1 degree the angle of attack with respect to the
elliptical LE airfoils.
? Sharp leading edges result in a higher maximum L/D ratio for the 6% camber
airfoil. However, for the 9% camber airfoil overall lower L/D were achieved.
By comparing the performance of the 6% and 9% camber airfoils with sharp en
elliptical leading edges at Re=15,000 the effects of reducing the Reynolds number
can be identified. Figure 4.23 shows that:
? As Reynolds number is reduced the gain in maximum lift coefficient obtained
with the sharp leading edges is not present anymore. Both elliptical and sharp
LE airfoil geometries produce very similar lift coefficient curves.
? The use sharp leading edges results in higher drag coefficients at all Reynolds
numbers. Nevertheless the drag increase is considerably lower at 15,000 than
at 60,000 Re.
? As Reynolds number is reduced the difference in lift to drag ratio between
elliptical and sharp leading edge airfoils is reduced. However, elliptical leading
edges tend to perform better.
148
From the previous comparisons of the computational results it can be con-
cluded that sharp leading edge airfoils can have beneficial effects in the airfoil per-
formance at specific operational points and with only certain airfoil cambers. For
circular arcs with a cambers 6% and below maximum L/D are generally improved
thanks to an additional lift production. However for larger cambers the drag increase
negates the beneficial lift effects resulting in overall lower L/D values.
The lack of experimental two dimension data makes difficult to validate the
trends identified specially the sharp leading edge effects. This is why the obser-
vations made have to be carefully interpreted. The flow assumptions used in the
simulations can have a strong effect on the final results. Hence, even though full con-
vergence could be consistently achieved, it is possible that the steady flow assump-
tion is overlooking some unsteady phenomena that can accentuate the differences
between elliptical and sharp leading edge geometries.
4.6 Summary
This chapter presented the CFD methodology and results used to calculate the two-
dimensional airfoil characteristics of a well defined family of circular arc airfoils. The
CFD solvers TURNS, Fluent and INS2d were evaluated, validating the results with
published wind-tunnel experiments of a circular arc airfoil at a Reynolds number
of 60,000. Validation showed that TURNS produced the best lift coefficient predic-
tions, followed by INS2d which under-predicted lift at the higher angles of attack.
Fluent was unable to capture the non-linearities of the lift measurements. All three
codes under-predicted drag; however, TURNS and INS2d qualitatively captured the
trends in L/D ratio.
Because of the compressible nature of TURNS two issues arose:
? Computational cost was considerably higher with TURNS than with INS2d.
? Convergence with TURNS was difficult at the lower Reynolds numbers
This is why INS2d was chosen to calculate a database of airfoil characteristics that
explored the effect in aerodynamic performance of the following geometric and op-
149
-5 0 5 10 15
-0.5
0
0.5
1
1.5
, (deg)
C
l
6%c 2.5%TR-60k-ellip
6%c 2.5%TR-60k-sharp
9%c 2.5%TR-60k-ellip
9%c 2.5%TR-60k-sharp
-5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
, (deg)
C
d
6%c 2.5%TR-60k-ellip
6%c 2.5%TR-60k-sharp
9%c 2.5%TR-60k-ellip
9%c 2.5%TR-60k-sharp
-5 0 5 10 15
-0.2
-0.15
-0.1
-0.05
0
0.05
, (deg)
C
m
6%c 2.5%TR-60k-ellip
6%c 2.5%TR-60k-sharp
9%c 2.5%TR-60k-ellip
9%c 2.5%TR-60k-sharp
-5 0 5 10 15
-10
0
10
20
30
40
, (deg)
C
l
/C
d
6%c 2.5%TR-60k-ellip
6%c 2.5%TR-60k-sharp
9%c 2.5%TR-60k-ellip
9%c 2.5%TR-60k-sharp
Figure 4.22: Effect of leading edge shape in lift, drag, and moment coefficients and
L/D ratio on 6% and 9% circular arc airfoils with 2.5% TR at Re=60,000.
150
-5 0 5 10 15
-0.5
0
0.5
1
1.5
, (deg)
C
l
-5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
, (deg)
C
d
6%c 2.5%TR-15k-ellip
6%c 2.5%TR-15k-sharp
9%c 2.5%TR-15k-ellip
9%c 2.5%TR-15k-sharp
-5 0 5 10 15
-0.2
-0.15
-0.1
-0.05
0
0.05
, (deg)
C
m
-5 0 5 10 15
-5
0
5
10
15
, (deg)
C
l
/C
d
6%c 2.5%TR-15k-ellip
6%c 2.5%TR-15k-sharp
9%c 2.5%TR-15k-ellip
9%c 2.5%TR-15k-sharp
6%c 2.5%TR-15k-ellip
6%c 2.5%TR-15k-sharp
9%c 2.5%TR-15k-ellip
9%c 2.5%TR-15k-sharp
6%c 2.5%TR-15k-ellip
6%c 2.5%TR-15k-sharp
9%c 2.5%TR-15k-ellip
9%c 2.5%TR-15k-sharp
Figure 4.23: Effect of leading edge shape in lift, drag, and moment coefficients and
L/D ratio on 6% and 9% circular arc airfoils with 2.5% TR at Re=15,000.
151
erational parameters:
Operational parameters
? Angle of attack -4 to 14 deg.
? Reynolds number 15,000 to 60,000.
Geometric parameters
? Camber 0% to 9%.
? Thickness ratio 2.5% 3.75%.
? Leading edge shape elliptical and sharp.
The most relevant results of the parametric study are:
? Camber determines maximum Cl achieved by airfoils.
? Camber determines stall angle of the airfoils.
? Increasing TR has adverse effects in drag, lowering L/D values.
? The highest L/D ratios were achieved by airfoils having cambers between 4.5%
and 6.0%.
? Sharp leading edges increase maximum Cl values and delay stall; however,
only for specific cases improvements in maximum L/D were observed.
? A reduction in the Reynolds number adversely affects L/D ratio as a conse-
quence of a simultaneous reduction in lift slope and an increase in drag.
? Leading edge geometry has a reduced relevance in airfoil performance as
Reynolds number decreases.
? A reduction in Re gradually lowers the angle of attack for max L/D.
? A reduction in Re extends the high L/D plateau of airfoils.
152
The databases generated in this chapter can now be used in rotor design code
that can include varying geometric and operational parameters in the predictions of
rotor hover performance. Chapter 5 explains the approach used for the implemen-
tation of this type of analysis.
153
Chapter 5
Hybrid BEMT-CFD Method for Rotor Analysis
and Design
5.1 Introduction
Blade Element Momentum Theory (BEMT) was introduced in Chapter 3 as a rel-
atively simple and inexpensive but powerful tool for rotor analysis. One of BEMT
main features is the capability to incorporate geometric and operational rotor pa-
rameters such as: blade planform, twist distribution, number of blades, and collec-
tive pitch settings. However, BEMT requires the sectional airfoils characteristics
in order to calculate the nonlinear axis-symmetric inflow. At larger scales linear
aerodynamics can be used to model the airfoil characteristics. However, at MAV
scale Reynolds number effects hinder the use of that approach. This is why ob-
taining a good estimate of the airfoil characteristics is key to achieve satisfactory
results with the model. In this Chapter the coupling of the BEMT algorithm with
the CFD databases calculated in Chapter 4 is explained. The use of this approach
as a design tool requires the definition of a geometric blade parameterization, the
implementation of an interpolation procedure, and a methodology to explore the
test space. These are discussed in the following sections.
5.2 Design Algorithm
Figure 5.1 shows the block diagram of the algorithm implemented for the rotor anal-
ysis. The inputs to the algorithm can be divided into two categories: the fixed basic
rotor configuration and operational parameters, and the parameters that depend
on blade geometry. Basic rotor configuration parameters are: diameter, number
154
of blades, rotational speed, and collective pitch. The parameters that depend on
the blade planform shape include: solidity, twist distribution, spanwise airfoil ge-
ometry and chord-based Re. Based on a blade parameterization that couples the
blade and airfoil shapes along the span, the radial distribution of airfoil geometries,
twist distribution, and local chord-based Reynolds number are calculated. All the
previous geometric and operational parameters are given to the BEMT rotor model.
As explained in Chapter 3, an iterative process finds an inflow distribution that is
consistent with the airfoil characteristics. For each inflow iteration the lift and drag
coefficients need to be obtained at every radial station; this is done by interpolating
the CFD database. After inflow convergence is achieved the output parameters are
calculated for the rotor studied.
By prescribing changes in the parameterized blade planform the combined
effects of airfoil and blade geometries can be explored with a minimum of com-
putational cost. This approach is an alternative to the more sophisticated three-
dimensional CFD and vortex models which capture the physics involved with more
detail. Nevertheless, not necessarily better results will be obtained with those meth-
ods since:
? Similar difficulties and predictive errors faced in the two-dimensional CFD
calculations are also found when studying the three-dimensional cases. How-
ever, computational cost increases exponentially, greatly limiting the use of
this approach as a design tool.
? Vortex methods can be used to model the three dimensional near and far
wake effect to calculate the inflow. However, these methods need also the
two-dimensional airfoil characteristics, which are obtained from wind tunnel
experiments or two-dimensional CFD.
In the next sections the geometric blade-airfoil parameterization and the in-
terpolation implementation is explained in more detail.
155
 
Inflow calculations 
Along the span 
Calculation of tip  
losses  
Calculation of 
profile power 
Interpolation of 
Cd from CFD 
database 
Calculation of 
induced  power 
ind 
Output: 
          Total power required 
          Thrust 
          Torque 
          Inflow distribution 
 
Single rotor BEMT 
model 
, Re, TR
camber  Calculation of  
Induced angles  
of attack 
Interpolation of 
Cl  from CFD 
2D database 
Cl
Blade planform
modification scanning
test-space
Cd
Basic Rotor Parameters:
# of blades
rotor diameter
rotational speed
collective pitch
Initial Blade
 geometry
Calculation of spanwise
 airfoil geometries
following parameterization
Figure 5.1: Single rotor design algorithm
156
5.3 Geometric Blade Parameterization
A rotor that has untwisted rectangular blades with a uniform airfoil along the span
does not have any varying geometric blade or airfoil parameters. This is the simplest
possible case where the only variable along the blade is the chord-based Reynolds
number.
On the other hand, if the blades have a non-rectangular planform with more
than one airfoil along the span, the analysis becomes more complex due to the larger
number airfoil sectional characteristics that the model needs. For a single rotor
configuration operating at a specified design point it is reasonable to perform wind
tunned tests or a detailed CFD studies to obtain the airfoil characteristics. However,
if an open design problem is faced a different approach needs to be taken. This is
the case for MAV rotor design. Rotary-wing MAVs can be used for a wide range
of mission that will cover a broad range of payloads, rotor diameters and rotational
speeds. Hence, it is desired to have a flexible design tool that can adapt to changing
requirements. By coupling the BEMT model with the circular arc database through
a geometric blade/airfoil parameterization, a large design space can be explored.
The following blade planform constraints were used in the implementation of the
model and are mainly based on manufacturing limitations:
? Planform is defined by 8 points shown in figure 5.3. Each point location is
defined in a cartesian plane by a radial and chordwise coordinate: P(r,c).
? Points that define the root edge (points 1 and 8) must be at the same radial
position. Thechordwiselocationofthesetwopointscanbeatanyintermediate
position between CLE and CTE as long as c(1) > 0 and c(8) < 0.
? The points that define the tip edge (points 4 and 5) must be at the same radial
position equal to r=1. As for the points that define the root edge, points 4
and 5 can be at any intermediate chordwise position between CLE and CTE
as long as c(4) > 0 and c(5) < 0.
? The pairs of points 2-3 and 7-6 must have the same chordwise coordinate such
157
that c(2) = c(3) = CLE and c(7) = c(6) = CTE. In this way the line segments
defined by points 2-3 and 7-6 are always parallel to each other and to the
radial axis.
Additionally to the blade planform constraints, the following airfoil constraints were
imposed too:
? Airfoils must be circular arcs.
? Cambers must be in the range of 0% to 9% (range covered in database).
? Camber is defined based on a rectangular baseline blade, were points 1,2,3 and
4 are aligned defining the leading edge, and points 5,6,7 and 8 are also aligned
defining the trailing edge.
? Camber can be defined as constant or varying linearly between the root and
the tip edges - based again on a rectangular blade -. This is equivalent to have
a conical section.
? Material thickness can be defined as uniform or linearly varying along the span.
Chord and material thickness should be defined to have realistic thickness
ratios, in the range of 1.5% to 6%.
These geometrical constraints can results in a quadrilateral or a 5, 6, 7 or 8 side
polygon. The previous blade planform parameterization is consistent with the non-
rectangular geometries tested experimentally in section 2.6. The points are es-
sentially defining the areas to be removed from a baseline rectangular blade. As
explained in section 2.6.1 this results in changes in the solidity, twist, camber, and
thickness ratio distributions of the blades. Figure 5.3 shows the geometric blade
and airfoil parameters along a sample non-rectangular blade. In this case the airfoil
shape of the baseline rectangular blade (dashed lines) was 6% at the root and 9%
at the tip.
158
1(rroot,CLE) 2 3 4(rtip,CLE)
5(rtip,CTE)678(rroot,CTE)
CLE
CTE
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.01
0.02
r
chord (m)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
10
r
camber %
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
r
TR %
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2.5
5
x 10
4
r
Re
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1
0
1
r
twist (deg)
r
Figure 5.2: Radial parameters for rectangular blade with uniform 6% camber airfoil.
159
1(rroot,c1) 2 3 4(rtip,c4)
5(rtip,c5)678(rroot,c2)
CLE
CTE
r
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.01
0.02
r
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
10
r
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
r
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
r
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-10
0
10
r
chord (m)
camber %
TR %
x 10
4
Re
twist (deg)
Figure 5.3: Radial parameters for non-rectangular blade with baseline 6% camber
airfoil at root and 9% camber airfoil at tip.
160
5.4 Airfoil Database Interpolation
The airfoil databases calculated in Chapter 4 covered four parameters: Reynolds
number, camber, thickness ratio, and angle of attack. Hence, for a each blade
section interpolation in these four dimensions has to be performed. In order to
minimize the computational cost of the interpolation the parameter that depend on
blade geometry are interpolated first, obtaining the Cl and Cd curves vs. ?. Finally
during the inflow calculations, the local Cl and Cd curves obtained in the previous
step are interpolated in angle of attack until convergence is achieved.
The successive databaseinterpolation isperformed usingsimplelinearmethods
that allow for extrapolation in case boundaries are exceeded. Linear interpolation
is a good option when very few or many data points are available. In the current
investigation only two thickness ratios and four Reynolds numbers were considered.
Based on the results presented in Chapter 4 it was decided to implement a simple
linear approach in order to avoid overshooting or unwanted oscillations in the predic-
tions. Polynomial and cubic spline methods are likely to face these problems due the
proximity of the points. Other interpolation schemes such as the Akima (Ref. [84])
splines minimize the overshooting or over-fitting problems; however, Akima splines
need a minimum of 5 points to be used. Akima or cubic spline methods could have
been implemented in the angle of attack interpolation; however, since lift and drag
data points are uniformly spaced at one degree intervals the gains in accuracy were
insignificant.
5.4.1 Interpolation Validation
The interpolation methodology was validated by comparing the lift and drag coeffi-
cients obtained from a direct CFD calculation with database interpolated results. A
4% camber, 1.93% TR airfoil with elliptical leading edge was used for this purpose,
it is the same airfoil used in the experimental CFD validation of Chapter 4. The
results obtained are shown in Figure 5.4 where lift, drag, and L/D at various Re
are plotted. For the most part interpolation results show a good agreement with
161
the CFD calculations. At Re = 60,000 maximum percentile differences in L/D ratio
are of the order of 3%. At the other Reynolds numbers (45,000,30,000 and 15,000)
similar errors are obtained over most of the angle of attack range. However, for
negative angles and at an angle of attack of 6 degrees, differences can be as large as
15% (@Re=30k). In general the interpolation errors are a result of an under esti-
mation of the lift coefficient. The highly nonlinear Cl vs. ? curves introduce errors
in the vicinity of steep changes in the curve?s slope. These errors can be minimized
by increasing the number of interpolating points. An increase in the number of
Reynolds numbers and thickness ratios data points would have a noticeable impact
on the predictive capability of the interpolating algorithm.
162
-4 -2 0 2 4 6 8 10 12 14
0
0.05
0.1
0.15
0.2
0.25
, (deg)
Cd
Re=15k direct CFD
Re=60k direct CFD
Interpolated from DB
-4 -2 0 2 4 6 8 10 12 14
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
, (deg)
Cl
Re=15k direct CFD
Interpolated from DB
Re=15k direct CFD
Re=60k direct CFD
Interpolated from DB
-4 -2 0 2 4 6 8 10 12 14
-10
-5
0
5
10
15
20
25
30
35
, (deg)
Cl/Cd
Increasing Re
Figure 5.4: Comparison of lift drag and L/D results from direct CFD and interpo-
lated results for a 4% camber, 1.93% TR at various Re.
163
0 0.005 0.01 0.015 0.02 0.025
0
0.1
0.2
0.3
0.4
0.5
0.6
C
FM
0% camber
1.5% camber
3% camber
4.5 % camber
6% camber
7.5% camber
9% camber
0 0.005 0.01 0.015 0.02 0.025
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
-3
C
P
0% camber
1.5% camber
3% camber
4.5 % camber
6% camber
7.5% camber
9% camber
T
C
T
Figure 5.5: Predicted CT vs. FM and CT vs CP for rotor with rectangular blades
and various airfoil cambers at 2500 RPM.
5.5 Approach Validation
The accuracy and predictive capabilities of the present approach were validated us-
ing some of the experimental results of Chapter 2. First, The effects of basic rotor
parameters such as rotational speed, collective pitch, solidity, and airfoil camber are
validated using rectangular bladed rotors. Second, calculations using non rectangu-
lar planform shapes that include the combined effects of spanwise twist distribution
and airfoil geometry are presented.
5.5.1 Rectangular Blades - Camber Effects
The effect of airfoil camber on third generation rotors (i.e. 2 blades, R=112mm,
chord=22.5 mm) was explored using the BEMT code coupled with the database of
airfoils with elliptical leading edges. The individual results for the different airfoils
are explored further into this section; for now the general trends observed in the
calculations are identified. Figure 5.5 shows the CT vs. FM and CT vs CP plots for
rotors with airfoil having cambers ranging from 0% up to 9% at 2500 RPM.
As it was experimentally confirmed the following trends are present:
? As camber is increased the range of thrust coefficients the rotors can achieve
164
is extended.
? Rotor stall occurs at increasingly higher collectives as airfoil camber is in-
creased.
? Maximum FM occurs at increasingly higher collectives as airfoil camber is
increased.
? Airfoil camber has a strong influence on maximum hover efficiency of the
rotors.
? However, maximum FM values using cambers between 3% and 9% range be-
tween 0.52 and 0.57.
? Maximum FM above 0.5 are achieved with cambers equal or larger than 3%.
? Below stall power coefficient is directly dependent on airfoil camber.
Now that the rotors are modeled as a whole system two main sources of errors
are present: errors linked to the inflow calculations resulting from the assumptions
of BEMT, and errors from the airfoil sectional characteristics calculated in Chap-
ter 4. Since rotational speed and blade geometry are identical for all the rotors of
Figure 5.5, the predicted performance is directly linked to the airfoil characteris-
tics. Hence, any errors in the two-dimensional airfoil characteristics will propagate
affecting the thrust and power calculations. Based on the validation results of the
CFD calculations (section 4.4) maximum lift coefficients and drag coefficients at the
lower angles of attack were under-predicted by the flow solver.
The Analytical and experimental FM vs. CT plots for flat plates, 3%, 6%, and
9% camber circular arcs are shown in Figure 5.6. There is a clear qualitative agree-
ment between experimental and numerical results, however the following differences
consistent with the errors in sectional characteristics are present:
? Experimental maximum FM tends to be larger than predicted.
? Experimental maximum CT tends to be larger than predicted.
165
0 0.005 0.01 0.015 0.02 0.025 0.03
0
0.1
0.2
0.3
0.4
0.5
0.6
0% camber
3% camber
6% camber
9% camber
FM
C
0 0.005 0.01 0.015 0.02 0.025 0.03
0
0.1
0.2
0.3
0.4
0.5
0.6
C
T
FM
0% camber
3% camber
6% camber
9% camber
T
Figure 5.6: Predicted and experimental CT vs. FM for 0%, 3%, 6% and 9% camber
airfoils with elliptical leading edges at 2500 RPM.
? Rotor stall is smoother in the experiments than in the predictions.
The following figures compare in more detail the experimental and predicted per-
formance for the rotors with 3%, 6%, and 9% camber airfoils.
Figure 5.7 compares the experimental and numerical results of the rotor with
3% camber airfoil. There is an excellent agreement in thrust coefficients for collec-
tives of 16 deg and below. The experimental and predicted values differ by about 2%
at a collective of 12 deg. These are very small errors considering that experiments
are not perfect, and that small deviations in pitch might occur due to centrifugal
and aerodynamic forces. In terms of figure of merit, BEMT predicts a maximum
FM ? 5% larger than the experimental one, there is a vertical offset between the
experimental and BEMT curves, resulting from an under-prediction of the power
required.
For the rotor with 6% camber airfoils Figure 5.8 shows the experimental and
predicted results. As for the 3% camber blades there is good agreement in the trust
predictions. The average difference between experimental and numerical thrust
values at collectives of 14 and below is 2.25 grams which is about 3% of the thrust
produced at 14 deg collective. FM is over predicted at the lower collectives, however
as collective is increased the errors are reduced, under-predicting the maximum FM
166
0 0.0020.0040.0060.0080.010.0120.0140.0160.0180.02
0
0.1
0.2
0.3
0.4
0.5
0.6
C
T
FM
BEMT
Experimental
0 0.0020.0040.0060.0080.010.0120.0140.0160.0180.02
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x 10
-3
C
T
C
P
BEMT
Experimental
Figure 5.7: Predicted and experimental CT vs. FM and CT vs CP for rotor with 3%
camber airfoils, elliptical leading edges at 2500 RPM.
by only 2.5%.
The results for the rotor having the 9% camber airfoils are presented in Figure
5.9. In this case the thrust prediction are not as good as for the other rotors. Above
collectives of 8 degrees the BEMT predictions largely under estimate the thrust
produced, resulting in a maximum figures of merit ? 8% below the experimental
values at a thrust coefficient 10% smaller.
In the three cases considered the effects of under-predicting the sectional drag
coefficients are clearly observed in the CT vs CP plots (Figures 5.7, 5.8, 5.9). Using
the 6% camber blades as example, the calculated spanwise induced angles of attack
and the induced and profile power components for the different collectives are plotted
in Figure 5.10. At the lower collectives (0 deg ? ?0 ? 4 deg) where profile power is
dominant or comparable to induced power, experimental CP values are greater than
predicted, this results in higher FMs over the lower collective range. As collective
is increased larger portions of the blades work at angles of attack were sectional
characteristics were accurately calculated by the CFD flow solver producing a good
agreement between experimental and model results. At the collective for maximum
FM, profile power is ? 30% of the total power, and induced angles of attack do not
exceed 6 deg. At the highest collectives (?0 ? 16) large blade sections work in the
167
0 0.005 0.01 0.015 0.02 0.025
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
-3
C
T
C
P
0 0.005 0.01 0.015 0.02 0.025
0
0.1
0.2
0.3
0.4
0.5
0.6
C
T
FM
BEMT
Experimental
BEMT
Experimental
Figure 5.8: Predicted and experimental CT vs. FM and CT vs CP for rotor with 6%
camber airfoils, elliptical leading edges at 2500 RPM.
0 0.005 0.01 0.015 0.02 0.025 0.03
0
0.1
0.2
0.3
0.4
0.5
0.6
C
T
FM
0 0.005 0.01 0.015 0.02 0.025 0.03
1
2
3
4
5
6
7
x 10
-3
C
T
C
P
BEMT
Experimental
BEMT
Experimental
Figure 5.9: Predicted and experimental CT vs. FM and CT vs CP for rotor with 9%
camber airfoils, elliptical leading edges at 2500 RPM.
168
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-2
0
2
4
6
8
10
12
r
Induced AoA, deg
Increasing
 Collective
0 deg
18 deg
0.0020.0040.0060.0080.010.0120.0140.0160.0180.020.022
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
-3
C
T
C
P
BEMT Total
BEMT Induced
BEMT Profile
(a) (b)
Figure 5.10: Induced angles of attack(a), and Induced and profile power components
(b), at various collectives of a 6% camber rectangular blades.
range were airfoil lift coefficients are under-predicted.
From the experimental validation it can be concluded that for rotors having
uniform airfoils with cambers of 6% and smaller, the BEMT-CFD model used pro-
vides reasonable performance estimates, being able to predict the thrust and power
of the rotor within the experimental error at the operating points of interest, i.e in
the vicinity of the maximum FM. Since experimental data for airfoils with interme-
diate camber between 6% and 9% was not obtained, it is not possible to evaluate
the quality of the predictions for those airfoils, however results for the 9% camber
airfoils show that in that range the predictive capabilities of the model degrade.
5.5.2 Rectangular Blades - Rotational Speed Effects
Using the rotor with 6% camber airfoils the rotational speed effects on the model
results are explored. Since BEMT implementation is done using non dimensional
coefficients, the effects of rotational speed on FM, which is also non-dimensional,
can only be included through changes in the airfoil characteristics which depend on
the local chord Reynolds number. Figure 5.11 shows CT vs. FM , CT vs. CP, and
the thrust vs. power loading predictions for the rotor with 6% camber at various ro-
169
tational speeds. The Figures show how as rotational speed increases,hover efficiency
in terms of FM increases, this is consistent with the experimental results of Chapter
2. Nevertheless, since rotational speeds are different for the cases considered, rotors
are working at very different disk loadings, making FM comparisons meaningless.
Power loading is a better performance metric for this case. The bottom plot of
Figure 5.11 shows the experimental and calculated thrust vs. power loading curves.
For all cases thrust was predicted over most of the collective range within the exper-
imental error, except at the highest collectives (?0 ? 16) where variation is larger.
Errors in the power loadings calculations are significant at the lower collectives, but
in the range of interest -close to maximum FM condition- power loading predictions
have errors in the order of 2%-5%.
A common practice used to simplify the analysis of rotors with rectangular
blades is to assume constant sectional airfoil characteristics along the span. If the
blades have a uniform airfoil, a representative blade section at 75% span is generally
used. Using the BEMT model and the database generated in Chapter 4, the validity
of that simplification can be easily asserted. Figure 5.12 compares The CT vs. CP
plots of a rotor with 6% camber airfoils at 2500 RPM when the Re variation a
long the span is modeled and when an equivalent section at 75% span is used. The
differences observed are very small over the range of interest. Figures 5.13 (a) and
(b) show the percentile differences in thrust and power coefficients at the different
collectives. The resultant differences in FM are shown in Figure 5.13 (c). The results
show that differences in FM predictions between 1% and 2% are obtained using the
representative airfoil section approach. For a quick estimate of rotor performance
CFD airfoil characteristics at a single Re could be used to do the calculations without
a significant loss in accuracy. Nevertheless, this method can only be used with rotors
having rectangular blades and a uniform airfoil along the span.
5.5.3 Rectangular Blades - Leading Edge Effects
Experimentally it was found that sharp leading edges improved the rotor perfor-
mance by increasing maximum FM and by improving the rotor stall characteristics.
170
0.0020.0040.0060.0080.010.0120.0140.0160.0180.020.022
0
0.1
0.2
0.3
0.4
0.5
0.6
C
T
FM
2000 rpm
2500rpm
3000rpm
0.0020.0040.0060.0080.010.0120.0140.0160.0180.020.022
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
-3
C
T
C
P
2000 rpm
2500rpm
3000rpm
0 20 40 60 80 100 120 140
0.05
0.1
0.15
0.2
0.25
0.3
T,(g)
PL,(N/W)
BEMT 2000 rpm
BEMT 2500 rpm
BEMT 3000 rpm
Exp 2000 rpm
Exp 2500 rpm
Exp 3000 rpm
(a) (b)
(c)
Figure 5.11: CT vs. FM, CT vs CP, and PL vs. T for rotor with 6% camber airfoils,
elliptical leading edges at three different RPMs.
171
0 0.005 0.01 0.015 0.02 0.025
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x 10
-3
C
T
C
P
2000 RPM var Re
2500 RPM var Re
3000 RPM var Re
2000 RPM const Re
2500 RPM const Re
3000 RPM const Re
Figure 5.12: CT vs. CP considering Re change along span, and assuming constant
Re of section at 75% span. Rotors with rectangular blades and 6% camber airfoils.
Up to this point, the BEMT model used only the database of airfoils with ellipti-
cal leading edges for the calculations. In this section the sharp leading edge airfoil
database is used in a validation case in order to verify if the trends observed exper-
imentally can be captured by the model.
Figure 5.14 shows the CT vs. FM and the CT vs. CP plots of rotors with
6% camber airfoils at 2500 RPM. The figures show a trend that contradicts the
experimental results presented in Chapter 2. The model predicts a reduction in
maximum FM and an earlier rotor stall. The lower performance of the rotors with
sharp leading edges is a result of higher power coefficients at the lower collectives
and lower thrust levels at the higher collectives. Since rotor geometry and operating
conditions are identical for the rotors compared, the performance differences are di-
rectly linked to the airfoil characteristics. In section 4.5.4 the effects of sharpening
the leading edges at a Reynolds number of 60,000 were presented. Under those flow
conditions, beneficial effects in the thrust coefficients were observed, obtaining in
some cases an increase in maximum lift to drag ratio. However, at lower Reynolds
numbers those benefits are no longer present, negatively affecting the airfoil per-
formance. Figure 5.15 shows the sectional characteristics at Re=30,000 for a 6%
172
0 2 4 6 8 10 12 14 16 18
0
2
4
6
8
10
12
collective, deg
C
T
 % difference
2000 rpm
2500 rpm
3000 rpm
0 2 4 6 8 10 12 14 16 18
0
0.5
1
1.5
2
2.5
collective, deg
C
P
 % difference
2000 rpm
2500 rpm
3000 rpm
0 2 4 6 8 10 12 14 16 18
0
2
4
6
8
10
12
14
16
18
20
collective, deg
FM % difference
2000 rpm
2500 rpm
3000 rpm
(a) (b)
(c)
Figure 5.13: Percentile difference in CT, CP, and FM, when Re change along span
is modeled, and when constant Re of section at 75% span is used. Rotors with
rectangular blades and 6% camber airfoils
173
0 0.005 0.01 0.015 0.02 0.025
0
0.1
0.2
0.3
0.4
0.5
0.6
C
T
FM
Elliptical LE
Sharp LE
0 0.005 0.01 0.015 0.02 0.025
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
-3
C
T
C
P
Figure 5.14: CT vs. FM and CT vs.CP for rotors with 6% camber, sharp and
elliptical leading edges, 2500 RPM, Tip Re?43000.
camber airfoils with sharp and elliptical leading edges. These airfoil characteristics
are very close to the operating conditions of the representative 75% span section
that has a Re of 32700 at 2500 RPM. The figure shows that the overshoot in lift
coefficient that occurs at an angle of attack of 8 deg with elliptical leading edges
disappears in the sharp leading edge case. Additionally, there is an increase in drag
coefficient that results in a reduction in lift to drag ratio over the range of 4 deg to
9 deg angle of attack. From these results it can be concluded that the sharp leading
edge database generated in Chapter 4 does not provide reliable airfoil data for the
implementation of the BEMT model. Even though general airfoil behavior at a
Reynolds number of 60,000 showed the expected trends, the database as a whole
seems to have fundamental problems. CFD methodology for that case should be
reviewed, it may be possible that a time accurate solution needs to be implemented
since it is likely that transient phenomena are responsible for the benefits obtained
in the rotor tests. Hence, only the airfoil database with elliptical leading edges is
used in the analysis of non rectangular blades.
174
-5 0 5 10 15
-0.5
0
0.5
1
1.5
, (deg)
C
l
Ellip
Sharp
-5 0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
, (deg)
C
d
-5 0 5 10 15
-0.2
-0.15
-0.1
-0.05
0
0.05
, (deg)
C
m
-5 0 5 10 15
-5
0
5
10
15
20
, (deg)
C
l
/C
d
Ellip
Sharp
Ellip
Sharp
Ellip
Sharp
Figure 5.15: Comparison of sectional aerodynamic characteristics of 6% camber
circular arcs with elliptical and sharp leading edges at Re=30,000.
175
a) b) d)c)
1.33:1
1.66:1
2:1 2:1
Figure 5.16: Linear taper with various ratios starting at 0.8R
5.5.4 Non-Rectangular Blades
In this section the model capability to predict the performance of rotors having non
rectangularbladesisevaluated. Asexplainedinsection5.3bladeplanformiscoupled
with spanwise twist and airfoil camber. Hence, for a single blade configuration the
database needs be interpolated in four dimensions: Reynolds number, thickness
ratio, camber, and angle of attack.
An initial validation with experimental results needs to be performed before
exploring new configurations. Blades with a baseline airfoil camber of 6% with
elliptical leading edges were tested having three taper ratios: 1.33:1, 1.66:1, and
2:1. The taper radial start location was set at 0.8 R and material was removed only
from the trailing edges, producing fully asymmetric tapers as shown in Figure 5.16.
Experimental and model results at a rotational speed of 3000 RPM are presented in
Figure 5.17(a) and (b). The values plotted correspond to collectives that range from
6deg to 18 deg. Figure 5.18 compares in the same figure experimental and predicted
values for a selected case. The model is able to predict the following trends observed
experimentally:
? As taper ratio is increased thrust coefficients are reduced. Accuracy of the
predictions is similar to the ones obtained with rectangular blades.
176
? As taper ratio is increased maximum FM improves.
? The highest Figure of merit of the four configurations plotted is achieved with
a 2:1 taper ratio.
The following differences between predictions and experiments are present:
? Maximum FM occurs experimentally at lower collectives.
? Experimental maximum FM values are higher than predicted. However, errors
are relatively small, on the order of 2%.
The results obtained are encouraging and show that quantitative differences
in performance that are present in the experiments are captured by the model.
Nevertheless, as previously stated, inaccuracies in the airfoil database and some of
BEMT assumptions limit the use of the current approach as a preliminary design
tool. Experimental results may even be used to fine tune the model by introducing
empirical corrections to the database.
177
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0 0.005 0.01 0.015 0.02 0.025
C
FM
Rectangular
Linear 1.33:1
Linear 1.66:1
Linear 2:1
T
(a) Experimental CT vs. FM
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0 0.005 0.01 0.015 0.02 0.025
C
T
FM
BEMT 2:1
BEMT Rectangular
BEMT 1.66:1 
BEMT 1.33:1
(b) Predicted CT vs. FM
Figure 5.17: Experimental and predicted CT vs. FM at 3000 RPM for rotors with
various taper ratios on blades with baseline 6% camber airfoils, elliptical leading
edges.
178
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0 0.005 0.01 0.015 0.02 0.025
C
T
FM
Exp Rect
Exp 2:1
BEMT Rect
BEMT 2:1
Figure 5.18: Experimental and predicted CT vs. FM for rotors with rectangular and
2:1 tapered blades, baseline camber 6%
179
5.6 Blade Planform Optimization - Case Study -
In this section a simple grid search method is used to find the rotor planform with
the highest maximum FM at 3000 RPM. Using a baseline rectangular Blade , airfoil
camber and tip geometry were the only parameters explored. The design space was
bounded by baseline cambers between 0% to 6%, a linear taper ratio that ranged
from 1:1 to 2.33:1, and a taper start location from 0.75R to 0.9R. The taper offset,
defined with respect to the rectangular blade?s radial axis was also varied over its
entire range. Figure 5.19 shows the geometric blade variables described.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
-0.5
Taper start location
Taper 
offset
Non dimensional radial psoistion, r
Normalized chord
0.25
-0.25
Figure 5.19: Blade planform parameters of sample optimization case.
The grid search was automated to find the collective pitch for maximum FM
within 0.25 deg for each blade configuration. Local maximums were then compared
to find the global maximum within the defined space. It was found that the highest
FM equal to 0.6084 was obtained with a baseline 6% camber airfoils with a fully
asymmetric 2:1 taper (only cutting the trailing edge) starting at 84% of the span.
Even though differences between the different configurations are small, clear gradi-
ents are identified. For example Figures 5.20 (a)and (b) show how maximum FM
changes with taper offset and taper radial start location for the optimal configura-
tion found.
Additional trends were identified by plotting the results of the grid search.
180
0.740.760.780.8 0.820.840.860.880.9 0.92
0.604
0.6045
0.605
0.6055
0.606
0.6065
0.607
0.6075
0.608
0.6085
Taper start radial location
maximum FM
(a) FM vs. taper start radial location
-0.5-0.4-0.3-0.2-0.10 0.1 0.2 0.3 0.4 0.5
0.54
0.55
0.56
0.57
0.58
0.59
0.6
0.61
0.62
Taper offset/chord
maximum FM
(b) FM vs. taper offset
Figure 5.20: Variation of maximum FM with taper start location and taper offset
for a 2:1 taper ratio on blades with 6% baseline camber airfoil.
An interesting observation present in the analytical and experimental results of this
research is the dependency of the best taper start location with baseline camber.
As airfoil camber is reduced the optimal planform shape converges to an almost
rectangular blade when flat plates are used. For the blades with 6% camber airfoils
and a 2:1 taper ratio the best performance was achieved with the taper starting
at 0.84 R. This point moved gradually to 0.86R, 0.89R, and 0.97R for blades with
cambers of 4.5%, 3% and 0% respectively. Figure 5.21 shows these results.
The previous case study explored just a subset of the parameters that define
the blade geometry and rotor operating conditions. The design space can be largely
expanded by tapering the blade root in order to introduce positive twist, and by
using a baseline rectangular conical section that has a camber distribution along
the span as explained in section 5.3. Blade parameterization could be taken a step
further by increasing the number of points that define the blade planform. Currently,
the BEMT model implementation uses 80 radial sections along the span. The points
that define the blade?s leading and trailing edges could be splined, producing smooth
blade planforms. An example of such approach is shown in Figure 5.22 were the
chordwise position of 20 control points joined by Akima splines were used to define
the blade?s edge.
181
0.75 0.8 0.85 0.9 0.95 1
0.442
0.444
0.446
0.448
0.45
0.452
0.454
0.456
0.458
Taper start radial location
maximum FM
0.740.760.780.8 0.820.840.860.880.9 0.92
0.578
0.58
0.582
0.584
0.586
0.588
0.59
0.592
0.594
Taper start radial location
maximum FM
0.740.760.780.8 0.820.840.860.880.9 0.92
0.601
0.602
0.603
0.604
0.605
0.606
0.607
0.608
0.609
Taper start radial location
maximum FM
(a) 0 % camber (a) 3 % camber
(a) 4.5 % camber
Figure 5.21: Variation of maximum FM with taper start location for a 2:1 taper
ratio on blades with 0% 3% and 4.5% baseline camber airfoil.
182
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
-0.5
Normalized chord
0.25
-0.25
Non dimensional radial location, r
Control Points
Splined Planform
Figure 5.22: Blade planform defined by 20 control points and Akima splines.
However, model limitations and assumptions have to be taken into account.
First of all the airfoil database needs to be refined by improving the CFD solution
or by empirically correcting the values. Even though for the validation cases good
qualitative and quantitative results were obtained, these represent only a fraction of
the potential design space. Second, the blade geometries considered never diverged
from the traditional shapes used in full-scale helicopters. Radically different geome-
tries may have a strong effect on tip losses, possibly introducing large errors in the
calculation of the inflow and producing misleading results. This is why the current
research was limited to simple cases.
In spite of its limitations the model implemented is a valuable design tool.
Reasonable estimates of power and thrust can be obtained for a wide range of rotor
geometries and operating conditions in a fraction of the time it would take to perform
an experiment or a 3D CFD simulation. Additionally, The computational cost of
calculating the entire database is negligible when compared to fully 3D calculations.
This makes the current approach ideal for preliminary vehicle design where basic
configuration parameters need to be chosen prior to the fabrication of a prototype.
183
5.7 Summary
This chapter presented the methodology and validation of a rotor design tool that
integrates a BEMT rotor model with a CFD calculated airfoil database. Using a
blade parameterization that couples blade planform with spanwise twist and airfoil
shape (restricted to circular arcs), the effect of geometric and operational parameters
on hover performance were modeled. The approach was validated first on rectangu-
lar blades with uniform airfoils along the span having elliptical leading edges. The
model was able to predict within the margins of experimental error the thrusts pro-
duced by the rotors below stall having cambers of 6% and smaller. However power
predictions were not as satisfactory, agreeing with experimental results only over a
limited collective range that corresponded to the regions where maximum FMs are
achieved.
The effect of the spanwise Reynolds number variation was studied in rect-
angular blades. It was found that assuming representative airfoil characteristics
corresponding to 75% span had a minor effect on the model results. However this
method can only be applied in rectangular blades with uniform airfoils.
Errors in the CFD calculations are responsible in large part for the inaccuracies
of the model. At lower collectives the underestimation of the drag coefficients by
the CFD solver produced FMs that exceed the experimental values, and at higher
collectives, when airfoils with cambers larger than 6% were used, early lift stall
resulted in lower thrust and FM values.
The beneficial effects of using sharp leading edge airfoils were not captured
by the model. Even though the right trends in airfoil sectional characteristics were
present at Re = 60,000, at lower Re lift coefficients were underestimated producing
inadequate rotor efficiency results. CFD methodology for the sharp edged airfoils
needs to be reviewed and may need the use of time accurate solutions replacing the
steady state assumption.
The model?s predictive capability using non-rectangular blades was satisfac-
tory. Planform changes followed the trends identified experimentally; however, sim-
184
ilar difficulties as the ones observed with rectangular blades were faced. A simple
case study that used a grid search algorithm to find the optimal blade configuration
within a well defined design space was presented.
185
Chapter 6
Coaxial Rotor Performance and Vehicle Design
6.1 Introduction
The modeling and design of small-scale single isolated rotors has been addressed in
the previous chapters of this document. Understanding the operational and geo-
metric parameters that govern single rotor performance was a required step before
looking into the issues involved in coaxial rotor design. This chapter shows the
methodology and results of experiments designed to measure the hover performance
of small-scale coaxial rotors, and extends BEMT to the coaxial rotor case. The last
sections of the chapter discuss the implementation of a fully functional coaxial MAV.
Subsystems design, integration and overall vehicle configuration are presented.
6.2 Experimental Setup
The hover test stand and data acquisition system used for single rotor tests and
described in section 2.2 is a flexible system that can handle up to 16 input channels
and 2 output channels. In the coaxial tests performed, the number of inputs channels
increased by 3, adding to a total of 8. An additional output control signal was
required to control the rotational speed of the second rotor. The input channels
acquired the following signals:
? Coaxial thrust
? Coaxial torque
? Rotational speed upper rotor
? Rotational speed lower rotor
186
? Current upper rotor
? Voltage upper rotor
? Current lower rotor
? Voltage lower rotor
For a coaxial rotor system at zero torque condition measuring the aerodynamic
power consumed by the system is not as straight forward as for a single rotor. The
ideal approach would be to mount each rotors on independent transmissions each
attached to thrust and torque sensors. This would allow to simultaneously measure
the aerodynamic power required by each rotor and to determine the thrust distribu-
tion between rotors. However, this is difficult to implement and requires additional
hardware that was not available at the time the experiments were performed. An-
other discarded option was to use the electrical power consumed by each rotor -each
driven by a different motor- to calculate the torque. However, current consumption
is a function of motor temperature, which changes constantly. It was finally decided
to perform two sets of tests that would allow to determine the power and thrust of
each rotor.
A first set of tests measured the total coaxial thrust, rotational speed, and
electrical power consumption of each rotor. The key characteristic of this setup
is that measurements are performed at a zero torque condition with a very small
margin of error. The setup was designed as follows:
? Rotors were mounted inverted, directing their thrust toward the ground.
? Each rotor was driven by an independent motor.
? Rotors were attached to a single transmission on top of the test stand. The
rotor assembly was free to rotate within a certain range due to a bearing at
the bottom of the mounting stem.
187
? A rate gyro controlled the rotational speed of the upper rotor such that a stable
zero torque condition is achieved (upper and lower locations are referenced to
the flow, upper rotor is the one closer to the stand).
A second set of tests was used to measure the upper rotor thrust and torque, rota-
tional speed and electrical power of both rotors. In this case measurement can be
performed at any torque condition. The setup had the following characteristics:
? Rotors were mounted inverted, directing their thrust toward the ground.
? Each rotor was mounted on an independent transmission such that no torque
can be transferred form one transmission to the other.
? Upper rotor is attached as a single rotor would be to the test stand.
? rotational speeds for zero torque conditions found in the first set of experiments
are used to set the operating conditions in these tests.
The two test configurations are shown in Figure 6.1. In both cases rotor spacing
can be changed from a minimum distance of a few mm up to 8 cm in the coaxial
transmission. By combining the information provided by each setup the thrust and
torque of upper and lower rotors at zero torque condition can be obtained. The
details of this procedure are explained in a following sections.
6.3 Coaxial Tests Objectives
The main motivation of the tests performed is to determine the coaxial rotor param-
eters that affect hover efficiency. This information is crucial for the implementation
of an efficient coaxial rotary wing MAV. The main questions to answer were:
? How does rotor spacing affect rotor performance at MAV scale.
? How should the collective pitch of rotors be set to maximize hover performance.
? What is the thrust distribution between rotors.
188
Rate gyro
Bear ing
Gap
Bear ing
No
Coax 
Transmission
(a) (b)
Figure 6.1: Coaxial experimental setups a) single coax transmission b) independent
single rotor transmissions.
189
Upper Rotor Col. (deg) lower Rotor Col. (deg)
16 14
16 15
16 16
16 17
16 18
Table 6.1: Test matrix for coaxial tests using 6% camber rectangular blades.
Radius 112 (mm)
Root cut-out 13.5% Radius
No of blades 2
Blade planform rectangular
Blade chord 22.5 mm
solidity 0.1279
Table 6.2: Third generation rotor geometric characteristics
To answer these questions the following series of experiments were performed:
? Rotor spacing tests in coaxial transmission
? Rotor spacing tests in independent transmissions
? Hover performance tests on coaxial transmission with different collective pitch
combinations over a range of rotational speeds.
6.4 Coaxial Test Results
This section presents the methodology and results of the various tests performed.
Third generation rotors as defined in Table 2.3 with baseline blade chord of 22.5 mm
were used in all the experiments. The results in this section correspond to the test
matrix of Table 6.2, were the various collective pitch combinations for rectangular
blades with 6% camber airfoils are specified.
190
80
100120140160180200
1800
2000
2200
2400
2600
2800
Upper rotor RPM
Coax Thrust (g)
Figure 6.2: Average coaxial thrust at 6 different rotor separations vs. upper rotor
rotational speed. Error bars cover 3 standard deviation, calculated using all data
points at each spacing.
6.4.1 Rotor Spacing
Using the coaxial transmission (Setup (a) of Figure 6.1) rotor spacing was increased
from 20 mm to 70 mm (0.178 < h/R < 0.625) keeping the zero torque condition.
The upper and lower rotors were identical and had a rectangular blade planform.
Results for the configurations of table 6.1 showed similar trends, this is why only a
representative case corresponding to blades with upper and lower collectives of 16
deg is presented. Figure 6.2 shows the total coaxial thrust vs. the upper rotor?s
rotational speed. The plotted values correspond to the average thrust of all the
values acquired at the six different rotor spacings, and the error bars cover three
standard deviations of all the values at all the rotor spacings tested. Figure 6.3 shows
the coaxial rotor thrust as function of the normalized rotor spacing h/R (where h is
rotor spacing and R is the rotor radius). In these figures error bars extend for three
standard deviations.
In these tests, upper rotor rotational speed was fixed, and the lower rotor
speed was set such that the zero torque condition was achieved. Consistently it
was observed that there was minimal or no need at all to retrim the system for
the different rotor separations. In other words, torque required by each of the two
191
90.5
91
91.5
92
92.5
93
93.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalized spacing, h/R
Coax Thrust (g)
(a) Coax thrust vs. rotor spacing at 2000 RPM
138140142144146148150
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalized spacing, h/R
Coax Thrust (g)
(b) Coax thrust vs. rotor spacing at 2500 RPM
Figure 6.3: Variation of coaxial thrust with rotor spacing for rectangular 6% camber
blades. upper and lower collectives set at 16deg.
rotors did not change as rotor spacing was modified. However, small but measurable
changes on the total rotor thrust occurred. At the lower thrust levels the average
values were practically constant as shown in Figure 6.3(a); but at higher rotational
speeds and disk loadings coaxial rotor thrust had a tendency to increase with rotor
spacing as shown in Figure 6.3(b). The percentile variation of thrust with rotor
spacing measured at the highest thrust was of the order of 5%. In general thrust
values suffered minor changes beyond a rotor separation of 4 cm (h/R=0.357).
The main conclusion that can be drawn from the previous results is that rotor
spacing has a limited effect on the coaxial rotor performance and is not a critical
parameter that has a dramatic effect on performance. Only at small vertical sepa-
rations, thrust reductions of about 5% occur. In order to maximize the performance
of a coaxial system at the scale and thrust coefficients investigated, rotor separa-
tions should be equal or larger than h/R= 0.357 (4 cm). The following bullet list
summarizes the trends observed:
? Reduced rotor spacing affects the coaxial rotor performance by reducing coax-
ial thrust.
? Percentile coaxial thrust reduction increases with disk loading.
192
? Beyond a certain rotor spacing coaxial thrust reduction stabilizes becoming
independent of rotor separation.
6.4.2 Rotor Interaction
Now that the effects of rotor spacing have been quantified the interaction between
rotors can be explored. Using the coaxial setup with separate transmissions shown
in Figure 6.1(b) the variation in thrust and torque of the upper rotor (rotor at-
tached to the test stand) as function of the lower rotor?s operating conditions was
measured. Rotational speeds were set based on tests performed on the single coaxial
transmission and corresponded to a zero torque condition at a rotor separation of
h/r=0.4464 (5cm). It was found that lower rotor had a negligible effect on the upper
rotor. At the rotor separation studied changes in thrust and torque with respect
to the single isolated rotor case were within the range of the experimental error
(variations < 2%). Hence it can assumed that the induced flow produced by the
lower rotor at the upper rotor plane is not strong enough to produce measurable
changes in the upper rotor performance. This is an important result that can be
used to simplify the experimental characterization and design of coaxial rotors.
? Rotor design is simplified since upper rotors can be designed independently
of lower rotors. A good performing single rotor will perform equally well in a
coaxial setup.
? Experimentation is simplified since all the relevant upper and lower rotor pa-
rameters (thrust, power and efficiency) can be obtained by combining the
information provided by single isolated rotor tests with tests on the coaxial
transmission (Fig. 6.1(a)). Lower rotor thrust is obtained by subtracting the
thrust of the isolated upper rotor to the total coaxial thrust, and lower rotor
torque is equal in both rotors (zero torque condition).
The only obvious requirement to implement this approach is that operating condi-
tions (collective, rotational speed) of the upper rotor in the isolated and coaxial tests
193
y = 1.52176E-05x
2  - 3.13150E-03x + 4.91340E-01
R
2
 = 0.9997
0
20406080
100120140
0
500
1000
1500
2000
2500
3000
3500
RPM
T(g)
Experimental 16 deg Quadratic fit 16
d
(a) Trust vs. RPM
y = 2.98430E-09x
2
 - 5.68359E-07x + 2.06546E-04
R
2
 = 0.9998
0.0000.0050.0100.0150.0200.0250.030
0
500
1000
1500
2000
2500
3000
3500
RPM
Q ( N .m )
Experimental 16 deg Quadratic fit 16
d
(b) Torque vs RPM
Figure 6.4: Experimental thrust and torque vs. RPM of single rotor with rectangular
blades and 6% camber airfoil with at 16 deg collective. Equations of best quadratic
fit shown.
have to be identical. For a few cases this is not a problem, however if a coaxial rotor
is to be characterized over a range of rotational speeds and collectives, the testing of
the upper rotor becomes a very time consuming task. However, this can be avoided.
For any rotor configuration thrust and torque are proportional to the square of the
rotational speed (at a fixed collective). Hence, if at least three measurements of
thrust and torque are available over a rotational speed range, a quadratic fit can
be used to interpolate intermediate values with a very small margin of error. The
coaxial tests performed covered rotational speeds roughly between 2000 and 3000
RPM, so single rotor tests at 2000 RPM, 2500 RPM, and 3000 RPM are sufficient
to implement the proposed approach.
Figure 6.4showsthe experimental thrustandtorque of arotorwith rectangular
blades having 6% camber airfoils at a collective of 16 deg. Rotational speed was
varied from 700 to 3000 RPM dividing the range in 30 intervals. The best quadratic
fits are also plotted. The quadratic equations obtained were used for the calculations
of lower rotor thrust and torque.
194
1800 2000 2200 2400 2600 2800
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
 upper rotor, (RPM)
Thrust, (N)
Lower rotor equation
Coaxial experiment
Lower rotor experiment
Coaxial equation
Upper rotor equation
Figure 6.5: Thrust vs. Upper rotor rotational speed. Total, upper and lower rotor
contributions shown. 6% camber airfoils, rectangular blades.
6.4.3 Rotor Load Sharing
Rotor load sharing as function of rotational speed can now be determined following
the approach described in the previous section. Figure 6.5 shows the total coaxial
thrust and the thrust components as function of the upper rotor?s rotational speed.
Collectives were set at 16 deg. The lower rotor thrust prediction can be obtained
in two ways, as mentioned before by taking the difference of the coaxial and upper
rotor fits (lines in the plot) or by directly subtracting the upper rotor lift prediction
to the experimental coaxial thrust values, these are the asterisks in the plot.
It was found that over the entire range of rotational speeds and disk load-
ings, the upper rotor produced between 59% and 60% of the total thrust, this is
shown in Figure 6.6. This result is consistent with full-scale helicopter character-
istics where a load distribution of about 60% - 40% also occurs. For the other
collective combinations tested (Table 6.1) similar results were obtained, upper rotor
thrust contributions was consistently bounded between 59% and 61%.
195
1800 2000 2200 2400 2600 2800
58.6
58.8
59
59.2
59.4
59.6
59.8
60
60.2
60.4
 upper rotor, (RPM)
Upper rotor % of total thrust
Figure 6.6: Upper rotor percentile contribution to total thrust. Collectives set at
16 deg.
6.5 Maximization of Power Loading in a Coaxial Rotor
The next question to address is the optimal collective setting that a coaxial rotor
should have in order to maximize its power loading. Sunada in Ref. [44] addressed
that problem and found that the thrust-torque ratio of a coaxial rotor at zero torque
condition is maximized when both upper and lower rotors have the same rotational
speed. Sunada performed his experiments at a fixed thrust, and his experimental
variables were upper and lower collective pitch and rotational speeds. The experi-
mentsusedsmall-scalerotorslargerthantheonesusedinthisinvestigation; however,
Reynolds numbers were comparable. Sunada?s conclusion was drawn at very specific
operating conditions and with limited experimental data.
The experiments performed in the current investigation were used to verify
how robust was Sunada?s conclusion. Figure 6.7(a) shows the power loading vs.
coaxial thrust for the different collective combinations of Table 6.1, and Figure
6.7(b) shows the percentile rotational speed difference between upper and lower
rotor vs. coaxial thrust. The average rotational speed difference is a quantity that
shows how dissimilar are the rotational speeds of upper an lower rotor. It is defined
as the difference in rotational speeds (?u ? ?l) divided by the average rotational
196
speed (0.5(?u + ?l)) multiplied by 100 to obtain a percentage.
The results of Figure 6.7 show that the rotational speed difference is a function
of the collective settings. When the upper rotor collective is the highest between
the two rotors, the percentile rotational speed difference is negative (?L > ?U), and
when the lower rotor has the highest collective the situation reverses (?L < ?U).
When the two collectives are equal the differences in rotational speed are very small,
changing signs depending on the thrust produced.
Power loading is also dependent on the collective settings and disk loadings.
Figure 6.7(b) shows that the lowest power loadings occur when lower rotor collec-
tives are smaller than the upper ones. In the coaxial thrust range of 1 N and 1.55
N, the configuration with upper 16 deg pitch and lower 18 deg pitch offers the best
performance. For this case the percentile rotational speed differences are on the
order of 8%, contradicting Sunada?s conclusion. The results obtained showed that
equal rotational speeds at a zero torque condition do not guarantee a maximiza-
tion of the power loading. Differences in blade and airfoil geometries, rotational
speed, Reynold number, and disk loadings affect the inflow distribution and wake
structures, making very difficult to draw a general conclusion about the operating
conditions that maximize the performance of a generic coaxial rotor.
The available experimental data can be used to answer a fundamental question:
What is the penalty in terms of power of using a coaxial configuration instead of
two side-by-side (tandem) rotors. The power loading of a coaxial rotor with upper
and lower collectives set at 16 deg was compared to the power loading of a tandem
system with identical rotor configuration. The tandem system is idealized as two
isolated rotors not interacting with each other; hence its performance in terms of
power loading vs. disk loading is the one of a single rotor. The results obtained
are shown in Figure 6.8. Over the range of DL considered, the coaxial rotor has
a power loading about 20% lower than the tandem system. Experimentally it was
found that the upper rotor can be treated independently, hence the drop in PL is
explained by the interaction of the lower rotor and the upper rotor?s slipstream.
The experiments were all performed at a zero torque condition, hence the drop in
197
performance has to be linked to a reduction in thrust per unit power of the lower
rotor. The next section addresses the use of momentum theory and BEMT applied
to coaxial rotor systems.
198
0.8 0.91 1.11.21.31.41.51.61.71.8
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.21
0.22
Coaxial Thrust , (N)
Power Loading, (N/W)
16U-14L   
16U-15L   
16U-16L   
16U-17L   
16U-18L   
(a) Power loading vs. coaxial thrust.
0.80.9 1 1.11.21.31.41.51.61.71.8
-15
-10
-5
0
5
10
15
Percentile rot speed difference
Coaxial Thrust, (N)
percentile rot speed difference = 100*(
Up
 - 
Low
)/0.5*(
Up
 - 
Low
)
16U-14L   
16U-15L   
16U-16L   
16U-17L   
16U-18L   
16U-18L   
16U-17L   
16U-16L   
16U-15L   
16U-14L   
(b) Percentile rotational speed difference vs. coaxial thrust.
Figure 6.7: Power loading vs. coaxial thrust and percentile rotational speed differ-
ence vs. coaxial thrust, for the collective settings of Table 6.1
199
0.8 1 1.2 1.4 1.6 1.8 2 2.2
0.14
0.16
0.18
0.2
0.22
0.24
0.26
Total thrust, (N)
Power Loading, (N/W)
Coaxial
Tandem
Figure 6.8: Power loading vs. thrust for coaxial and tandem systems
200
6.6 Modeling of a Coaxial Rotor
Momentum theory can be used to predict the performance of coaxial rotors using
the flow model of Figure 6.9. In the diagram it is assumed that the rotor?s vertical
separation is large enough for the lower rotor to operate in the fully developed wake
of the upper rotor. Under ideal flow conditions this would make the inner half area
of the lower rotor to perceive an induced climb velocity.
Two cases can be considered in the analysis:
1. Load sharing of the rotors is equal i.e Tup = TLow
2. Zero torque condition exists i.e. Qup = QLow
Vvert + Vup
Vvert
Upper rotor
Vvert + 2Vup
Lower rotor
A = R
2
Ac = rc
2
Vvert + 2Vup + Vlow
Figure 6.9: Flow model of a coaxial rotor operating in the fully developed wake of
the upper rotor.
For case (1) momentum theory estimates an increase in induced power of 28%, while
for the most realistic case (2) an increase of 22% is predicted. The interference-
induced power factor kint is defined as the ratio of the induced coaxial power by
twice the induced power of a single isolated rotor:
201
Kint = (Pi)coax(2P
i)isolated
(6.1)
Hence,for equally loaded rotors kint = 1.28 and for balanced torque condition kint =
1.22. This type of analysis assumes that the coaxial rotor operate as two isolated
rotors with a mutually induced interference effect. Derivations of the previous results
can be found in Ref. [49].
6.6.1 Blade Element Momentum Theory for a Coaxial Rotor
Momentum theory gives acceptable predictions for single and coaxial rotors at full-
scale where profile power is obtained by using the airfoil?s zero lift drag coefficient
(Cdo) and the induced power is calculated using induced power coefficients (?=1.15
and ?int = 1.22?1.28). However, as discussed in section 3.2.2, momentum theory is
difficult to use for the types of rotors studied in the current research. Profile power
calculations need to include the change in sectional drag coefficient with angle of
attack, and induced power factors are highly dependent on rotor design.
The BEMT model presented in Chapter 5 can be extended to be used in
coaxial systems. The upper rotor system is treated as a single isolated rotor, while
the upper rotor?s wake can be modeled as an apparent climb speed acting on the
inner regions of the lower rotor. In order for the BEMT model to incorporate a
climb or descent speed the discretized inflow equation (3.6) was re-derived from
basic blade element (BET) and momentum theories as follows. The incremental
thrust from momentum theory is given by:
dCT = 4F?(???c)rdr. (6.2)
Where F is Prandtl?s tip loss factor, ? the inflow produced by the rotor, and ?c is
the inflow due to the climb speed. The incremental thrust from BET is given by:
dCT = 12 ?Cl r2 dr = ?Cl?2 (?r2 ??r)dr. (6.3)
Where ? is the solidity, Cl and Cl? are respectively the lift coefficient and lift co-
efficient slope, ? is the collective, and r is the non-dimensionalized radial station.
202
Equating the incremental thrust coefficients from momentum and blade element
theories:
?Cl?
2 (?r
2 ??r)dr = 4F?(???c)rdr. (6.4)
?2 +
parenleftbigg ?C
l?
8 ? ?c,
parenrightbigg
? ? ?Cl?8 ?r = 0. (6.5)
This is a quadratic equation in ?:
?2 +
parenleftbigg ?C
l?
8F ? ?c,
parenrightbigg
? ? ?Cl?8F ?r = 0. (6.6)
Solving for ?:
?(r, ?c) =
radicalBiggparenleftbigg
?Cl?
16F ?
?c
2
parenrightbigg
+ ?Cl?8F ?r ?
parenleftbigg ?C
l?
16F ?
?c
2
parenrightbigg
. (6.7)
By solving Equation 6.7 the discrete inflow distribution along the span of a climbing
rotor can be found. The lower rotor of a coaxial system is a special case since in
hover it will perceive an apparent climb speed only in the region determined by
the contraction ratio of the upper rotor?s slipstream. Ideally a wake contraction
A/Ac = 2 occurs (see Figure 6.9). In that case the average climb speed is two times
the induced velocity at the upper rotor?s plane (?up). However, in reality lower
wake contractions are to be expected. For a general case the average speed in the
streamtube can be defined as:
Vup = (A/Ac)?up = (A/Ac)
radicalbigg
CTup
2 ?upR. (6.8)
For the hovering case ?c is assumed equal to zero in the regions outside the upper
rotor?s streamtube. If that is not the case and the coaxial rotor has a vertical
velocity, the climb velocity will have two components Vvert and Vup. In that case ?c
is calculated using simple superposition where necessary.
Since BEMT only provides the inflow at the rotor planes, the radial velocity
distribution of the fully developed upper rotor?s wake has to be prescribed. The
203
simplest case is to assume a uniform distribution with a given contraction ratio based
on the upper rotor?s CT. For the case studied the radius of the fully contracted upper
rotor wake was set at 0.8R (A/Ac=1.56) instead of the ideal 0.707. This value was
determined based on a simple wake characterization experiments were the vertical
component of the velocity field of an isolated rotor was measured using a Pitot tube.
Figure 6.10 shows a contour plot of a cross section of the velocity field below the
rotor, and Figure 6.11 shows the normalized wake speed at two vertical locations.
Both figures show that the wake boundary is in the vicinity of r=0.8.
0.20.30.40.50.60.70.80.9 1
r
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
Vertical distance from rotor plane, h/R
0
-0.2
-0.5
-1.0
-1.5
-0.5
-1.0
Figure 6.10: Contour plot of normalized wake velocities with respect to tip speed
below single rotor plane. Collective at 16 deg, 2500 RPM.
The coaxial BEMT model was implemented and validated with the experi-
mental data for a coaxial rotor with both upper and lower rotor collectives set at
16 deg. The BEMT calculations were run at five different trim conditions. For each
case upper rotor speed was set to match experimental conditions, and the lower
rotor speed was found by the code to obtain torque equilibrium. Upper and lower
rotor calculations are shown in Figure 6.12. The predictions for the upper rotor
thrust have and excellent agreement with the experiments. This is no surprise since
it is treated in the model as an isolated rotor, a case already validated in Chapter 5.
Lower rotor thrust and rotational speed predictions have larger margins of error but
are still very good considering the simplicity of the approach. Figure 6.13 shows the
percentile errors with respect to the experimental measurements of the lower rotor
204
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Non-dimensional radial location, r
Wake velocity vertical component / tip speed
h/R =0.1 h/R=0.3
Figure 6.11: Radial position vs. normalized downwash velocities at two vertical
locations.
thrust and rotational speed, and the total coaxial thrust. For the three parameters
errors are below 5%. Lower rotor thrust was lower than the experimental values
while its rotational speed was over predicted as a result of the under-estimation of
the rotor torque. Lower power values are obtained for three main reasons:
? The airfoil characteristics from CFD under-predict the drag coefficients at the
lower angles of attack. Since the interaction with the upper wake lowers the
induced angles of attack over large regions of the lower rotor?s blades, profile
power will have a larger margin of error than for the isolated rotor case.
? Prandtl?s tip loss function is used to account for the tip and root induced
losses. However, there is no equivalent function to model the additional losses
produced by the sudden change in inflow at the boundary of the upper rotor?s
wake on the lower rotor, hence induced losses are underestimated in that
region.
? Lower rotor model predictions are dependent on the choice of the upper wake?s
velocity profile and contraction ratio. For the validation cases the assumption
of a uniform wake velocity profile produced good results; however, contrac-
tion ratio had to be determined empirically. In order to completely eliminate
205
0
20406080
100120
1800
2000
2200
2400
2600
2800
Omega, (RPM)
T (g)
Experimental upper BEMT upper Lower - uniform vel dist Experimental  lower
Figure 6.12: Thrust vs. rotational speed for upper and lower rotors. Experimental
and BEMT results compared.
the need of some type of wake characterization or visualization experiment
higher order analytical tools such as free vortex modeling can be used as long
as experimental validation is successful. Unfortunately, those tools were not
available for the current research.
Figure 6.14 shows the non-dimensional inflow, thrust, and torque distribution in the
spanwise direction for the upper and lower rotors. In the case shown, the balanced
torque conditions was achieved with both rotors running at very similar speeds
(2520 RPM upper, 2570 RPM lower), however the differences in thrust resulted in
operating thrust coefficients of CTup = 0.0205 and CTlo = 0.0134. The effects of the
upper wake and lower rotor interaction are evident from the spanwise thrust and
power distributions (Fig 6.14(d) and 6.14(f)). At the boundary of the wake (r=0.8)
a dramatic loss of lift and an increase in power occur. The relatively high upper
rotor CT, results in flow velocities large enough to produce negative induced angles
of attack at the inner most lower rotor regions. Over the regions were negative lift
is produced the inflow produced by the blades opposes the general flow direction,
trying to increase the induced angles of attack. This explains the change of slope at
r=0.45 of the lower rotor?s inflow distribution in Figure 6.14(b).
206
-5-4-3-2-1
0123456
1800
2000
2200
2400
2600
2800
Omega up, (RPM)
% Error 
Error lower thrust prediction Error lower RPM @ balanced torque Error coax thrust
Figure 6.13: Percentile error vs upper rotor rotational speed for: lower rotor
thrust,lower rotor rotational speed, and coaxial thrust.
207
0.10.20.30.40.50.60.70.80.91
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Non-dimensional radial location, r
Non-dimensional inflow, 
Upper
(a) Upper rotor - spanwise variation in inflow
0.10.20.30.40.50.60.70.80.91
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Non-dimensional inflow, 
Non-dimensional radial location, r
Lower
(b) Lower rotor - spanwise variation in inflow
0.10.20.30.40.50.60.70.80.91
0
2
4
6
8
x 10
-4
Non-dimensional radial location, r
dC
T
/dr
Upper
(c) Upper rotor - spanwise variation in thrust.
0.10.20.30.40.50.60.70.80.91
0
2
4
6
8
Non-dimensional radial location, r
dC
T
/dr
x 10
-4
Lower
(d) Lower rotor - spanwise variation in thrust.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x 10
-4
dC
Q
/dr
Non-dimensional radial location
Upper
(e) Upper rotor - spanwise variation in torque.
0.10.20.30.40.50.60.70.80.91
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x 10
-4
Non-dimensional radial location
dC
Q
/dr
Lower
(f) Lower rotor - spanwise variation in torque.
Figure 6.14: BEMT predictions for spanwise variation in inflow and thrust for upper
and lower rotors.
208
Considering the low computational cost and the simplifications employed, the results
obtained are very encouraging and show that at least for rectangular bladed rotors
thrust and power predictions are sufficiently accurate for the model to be used as
a design tool. Lower rotor design is an aspect that can potentially benefit the
overall system efficiency. This is a problem that has been largely overlooked for the
longest time, and in general experimental and analytical coaxial studies have been
consistently performed using identical upper and lower blade designs. Only over
the last few years this problem has been addressed in Refs. [86] and [49]. Both
publications use BEMT to show that lower rotor twist distribution can be tailored
to improve its efficiency. Reference [49] goes a step beyond and under idealized
flow conditions solves the BEMT equations to design the optimal twist distribution
that satisfies the requirements for minimum induced and profile power (uniform
inflow and all blade sections operating at maximum L/D ratio). At low Reynolds
numbers the highly viscous flows and very non-linear airfoil behavior is expected
to have a noticeable effect on an optimal twist distribution. Even tough the same
fundamental principles apply, a particular solution will be a function of the sectional
airfoil characteristics and upper wake topology.
A possible approach to find an improved lower rotor design for coaxial MAVs
is to perform a parametric blade planform study similar to the one used on the single
isolated rotor of section 5.6. Arbitrary twist distributions can be easily specified and
root blade can be tapered in order to introduce positive twist to reduce the losses
at the inner rotor regions. The results obtained need to be experimentally validated
to assess the need of higher order wake modeling tools considering the uncertainty
in the upper rotor wake characteristics.
6.7 Vehicle Design and Integration
This section presents the design process an issues considered in the development of
fully functional rotary wing MAV.
209
6.7.1 Design Requirements
At the time the work that led to the results presented in this dissertation was ini-
tiated back in 1999, MAVs were a fairly new concept which design requirements
were set arbitrarily by DARPA. The size, weight and functionality that were sought
required a leap in the understanding of the physics involved and large improvements
in the performance of all the vehicle?s subsystems. Basic DARPA design require-
ments addressed mainly the vehicle dimensions, weight, and endurance; however,
only vague mission scenarios were specified and there were no restrictions on ve-
hicle configurations. The unique characteristics of a rotary wing vehicles seemed
like a good match for a wide variety of missions. Rotary wing vehicles are cur-
rently the only configuration that can takeoff and land vertically, fly indoors and
outdoors, hover and navigate in cluttered spaces at low forward speeds. Research
in flapping-wing configuration has been motivated by the idea of matching these
strengths without sacrificing good forward flight performance. Progress in that area
has been slow due to the difficult aerodynamics and the high mechanical complex-
ity. Rotary wing vehicle?s poor forward flight performance limits its use to missions
where hover and slow forward flight account for most of the mission duration It was
anticipated that, in a situation where the target is far from the point of origin, an
external delivery method (e.g. a mothership UAV or large scale munition) might be
employed to transport the MAV. In order to facilitate delivery, the vehicle must be
compactly packaged and able to withstand high g-loading.
6.7.2 Concept Selection
In order to select a configuration for the vehicle, different concepts were systemati-
cally compared. Selection was made based on the criteria listed below:
? Hover efficiency
? Compactness of stored/transported vehicle
? Ease of payload packaging
210
? Simplicity of structure
? Controllability
? Maneuverability
The configurations considered can be broken into four categories: single rotor, twin
rotor, multi-rotor, and hybrid helicopter.
Single rotor configurations. These are conventional main rotor/tail rotor, rotors with
vanes in the slipstream for providing anti-torque, and tip-jet driven rotors. The first
two of these configurations have been successfully tested for UAVs and MAVs. The
conventional main rotor/tail rotor design provides good aerodynamic efficiency and
has good controllability and maneuverability. However, compactness in folding is
adversely affected by the tail boom and comparatively a large rotor size is required.
The tail rotor introduces additional power requirements that can be avoided using
other designs. Vanes in the wake of the rotor working as antitorque devices are an
aerodynamically clean option that offers an excellent hover performance (Ref. [36]).
The use of vanes as antitorque devices can be complemented with a duct to have a
further improvement in hover performance; however, forward flight characteristics
and compactness are adversely affected by the large wetted area of the vanes and
duct. Structural weight is also a concern since the vanes and/or duct have to be stiff
enough to avoid any dynamic instabilities which may lead to overly heavy system,
negating the aerodynamic gains achieved. Finally, tip-jets are attractive for their
simplicity and ease of payload packaging as a result of the absence of a powerplant
inside the fuselage. However, they have the drawback of poor controllability due to
the high blade inertia.
Twin rotor configurations. Three twin rotor configurations were considered: coax-
ials, tandems and ducted coaxial configurations. The coaxial design is favored by
most of the key design criteria: compactness of folding, simplicity of structure and
ease of packaging. Tandem configurations have a hover efficiency higher than that
of a coaxial design because of the the smaller interference effects between rotors.
However, compactness and the difficulty of folding are among their key drawbacks.
211
Ducted coaxial designs are well suited for MAVs, but have significant compactness
problems since shrouds and ducts cannot be stored efficiently. A ducted coaxial
configuration has been widely used for UAV design, for which Sikorsky?s Cypher
and Cypher II are good examples (Ref. [87]).
Multi-rotor configurations. Recently, there has been an interest in the rotorcraft
industry in designing rotorcraft with four or more lifting rotors. Such configurations
could be controlled by varying the RPM of different rotors to change the direction
of the thrust vector. Some small scale examples include the Mesicopter (Ref. [31]),
the Gyronsaucer and the Roswell Flyer. The last two are commercially available RC
helicopters and are reported to have very good controllability. The Mesicopter, a
meso-scale flying machine which is no larger than a penny, is still in the development
stage. A quad rotor design can have good hover efficiency as well as good handling
and control characteristics. However motor efficiency can be an issue. In order to
avoid a complex transmission instead of having a single power plant each rotor is
driven independently by a motor. Higher percentile power plant weight and lower
motor efficiency levels are to be expected, adding this to the obvious compactness
problems makes the multi-rotor configurations a non-ideal choice.
Hybrid helicopter configurations. The candidates in the compound helicopter cate-
gory are rotor-wing or stopped rotor, tilt-rotor, tilt-wing. All these designs prove
difficult to achieve a compact stored/transported state because of the large size of
their lifting surfaces. And require complex mechanical and electronic systems to
coordinate the transition between hover and forward flight. They are all well suited
for payload packaging and also very effective in high-speed forward flight. However,
since high speed forward flight is not a design priority these configurations are not
suitable options.
From this qualitative assessment it can be concluded that the conventional single
rotor/tail rotor configuration, the single rotor with vanes, and the coaxial design,
are the best candidates for the present design problem. The coaxial configuration
has the advantages of compactness of folding and ease of deployment while the
conventional configurations is superior from a controllability view point. However,
212
given the relative strength of the compactness requirement, the folding problems
associated with the tail boom preclude its use. Hence, the final configuration chosen
is that of a coaxial rotorcraft.
6.7.3 Vehicle Configuration
Several iterations were necessary until all issues of the vehicle design were properly
solved. Vehicle configuration and performance evolved and improved as the aero-
dynamic research presented in the previous chapters provided the answers to basic
design questions. Rotor and swashplate design were the subsystems that underwent
the most radical changes; however, general vehicle configuration and operation did
not go through a fundamental transformation.
Outer shaft
Inner shaft
Stationary
shaft
Transmission
Motors
Figure 6.15: Current generation coaxial vehicle transmission design.
The basic core structure of the vehicle is a central stationary shaft from which
transmission, motors, electronics, and battery packs are attached. Figure 6.15 shows
213
a picture of the current generation design. Structure was kept at minimum and light
materials such as plastic and carbon fiber composites were used where possible. Each
rotor is driven by an independent motor, and a simple one-stage gear reduction is
used as transmission. Each gear is attached to one of the coaxial shafts that drive the
rotors. Friction is minimized by using small ball bearings. Batteries and electronics
are at the bottom of the central shaft in order to have a center of gravity below the
lower rotor. This is a key factor that largely determines the handling characteristics
of the vehicle. Landing gear was manufactured out of carbon composite rods and
served as support structure while protecting the rotors from collisions. Figure 6.16
shows a picture of the working prototype. Rotor diameter was set at 22 cm which
corresponds to the third generation rotors studied in Chapter 2. The blades have
a 6% baseline camber with a 2:1 linear asymmetric taper starting at 80% of the
span. With a 22 cm rotor diameter vehicle hover endurance is about 10 min with a
payload of 10 additional grams. Rotor diameter was set arbitrarily and can be easily
changed without any hardware modifications to improve payload and endurance.
Five main subsystems were considered in the implementation of the coaxial vehicle:
? Propulsion
? Motor/transmission
? Lateral control system
? Electronics
? Power source
The total weight of the vehicle without batteries is 115 g, and depending on their
capacity batteries add between 25g and 45g. Table 6.3 shows the vehicle?s weight
break down. Only propulsion and lateral control systems were designed from scratch,
the other subsystems were carefully selected off-the-shelf products. The following
sections discuss in more detail some of the vehicle?s subsystems.
214
Batteries & Electronics
Landing Gear 
Rotor Protection
Swashplate 
&
Transmission
Upper Rotor
with Stabilizer Bar
Figure 6.16: Coaxial MAV prototype.
6.7.4 Vehicle Control
A practical MAV must to be controlled in the three dimensional space without
couplings that limit its functionality. This means that pitch, roll, altitude, and
yaw have to be independent from each other. In order to achieve this the following
configuration was used:
? Altitude is controlled by simultaneously increasing the rotational speed of the
two rotors without affecting the heading. Hence there is no need to have
collective pitch control in the rotors, which would have a huge weight and
complexity penalty.
? Yaw is controlled by changing simultaneously the rotational speed of the two
rotors. A torque imbalance in both directions can be achieved without a
change in altitude if one rotor speeds up while the other slows down keeping
total thrust constant.
? Pitch and roll are controlled by tilting the tip path plane of the lower rotor. A
215
Element Mass (g)
Structure 34
Electronics 20
Motors 34
Stabilizer bar 4
Blades 4
Landing gear 9
Batteries 2 or 3 cell 24 - 36
Total Weight 129 - 141
Table 6.3: Coaxial vehicle weight breakdown.
swashplate that introduces a cyclic pitch input was custom designed for this
purpose.
At the early stages of the project different control schemes were considered for the
implementation of lateral control in the prototype. Some of the options consid-
ered were flaps in the downwash, a gimbaled transmission, thrust vectoring (using
a ducted fan at the base of the vehicle), and a swashplate. At first the swashplate
seemed prohibitive because of the mechanical complexity involved, especially consid-
ering the vehicles first generation had a three bladed rotors. Only after experimen-
tal studies identified the manufacturing and aerodynamic advantages of using two
bladed rotors the vehicle design was changed; and two bladed rotors were adopted.
With this new configuration the use of a swashplate became plausible.
Swashplate Configuration
The main advantage of having a two bladed rotor is that the cyclic pitch input
of one bladed is exactly the opposite of the other. In addition, if a teetering two
bladed rotor is considered rotor response will lag the cyclic pitch inputs by exactly 90
degrees. These two facts allow for great mechanical simplification. The swashplate
was conceived keeping in mind three main considerations: weight, complexity and
216
 
Servos 
Ball bearing 
Pitch link 
Teetering hub 
Floating 
plate 
Pitch link cam 
(a) Exploded view of the swashplate (b) Assembled swashplate
Figure 6.17: Exploded and assembled views of the custom designed swashplate.
robustness. Figure 6.17 shows an exploded view of the design, and Figure 6.19 shows
a picture of the first working prototype manufactured. The swashplate controls only
the lower rotor, and can introduce only a cyclic input to the blades. The design
was inspired by the spider system shown in Figure 6.18 from Ref. [88]. The leading
edges of the blades are connected to the legs of a ?spider?, which is connected in
turn to the cyclic pitch control axle. This can be tilted about a central universal
joint through the inputs of the stick in the cockpit. When the spider is tilted, the
lower spider arm gives a minimum pitch to its blade, while a maximum pitch is
given to the position diametrically opposite.
The swashplate consists of five main elements: a teetering hub that holds the
blades, a pitch link, a pitch link cam, two servos, and a ?floating plate? that holds
a ball bearing. Since the vehicle is coaxial, the inner space of the mast is already
being filled by the upper rotor shaft so only exterior components can be used. The
U shaped part labeled pitch link in Figure 6.17(a) is hinged to the main shaft. The
hub that holds the blades is hinged at 90 deg to the pitch link creating a universal
joint. The pitch link arms have a steel pin that protrudes at the bottom on each
side. These pins fit two slots in the pitch link cam. This assembly of elements is
equivalent to the spider system shown above.
217
Figure 6.18: Spider system fir changing cyclic pitch from Ref. [88].
In this design, pitch inputs are not given with a vertical displacement as in
a conventional swashplate. Instead, the servos move the floating plate that holds
the ball bearing in the horizontal plane around the mast of the helicopter. Cyclic
pitch input is given to the blades through the pitch link cam and is dependent
on its azimuth angle and offset position with respect to the rotational axis of the
rotors. Each of the two servos moves the floating plate in orthogonal axis on a plane
perpendicular to the main shaft, giving almost uncoupled longitudinal and lateral
cyclic inputs.
Swashplate Kinematics
The input to the swashplate is an angular deflection of the servos that will move
the floating plate around the rotor mast. In order to have accurate control of the
rotor response, it is required that the angular deflection of the servos for a given
pitch input is as large as possible. In the current prototype maximum servo angular
deflections are of the order of ? 5 deg, producing floating plate displacements of ?
1.5 mm on each axis.
Displacement of the floating plate will be translated in a displacement of the
pitch link cam, which will drive back and forth (in the horizontal plane) the pitch
link. The distance between the pitch link axis and the pins that fit in the pitch link
cam is 8.5 mm. This yields a maximum pitch input of ? 10 deg. Maximum cyclic
inputs are necessary only when aggressive maneuvering is required. In a conservative
218
maneuver only half the cyclic input is used.
Design Considerations
The tip path plane of the lower rotor tilts 6 deg when a maximum cyclic input
is given. This produces a lateral force at the hub of about 10% the lower rotor
thrust. In order for the vehicle to pitch or roll in the direction the tip path plane
tilts, the Center of Gravity (CG) of the vehicle has to be below the lower rotor.
The side force rotor component multiplied by the hub-CG separation gives the total
pitching/rolling moment. Location of the CG is about 10 mm below the lower hub
when no batteries are on the vehicle. and with batteries depending on their capacity
CG location ranges from 25 mm to 35 mm below the lower rotor hub. The moments
produced are more than sufficient to control the vehicle in hover and in slow forward
flight.
As was shown in a previous section rotor spacing has a very small influence
on the aerodynamic efficiency of the coaxial rotor, however it needs to be carefully
specified in order to avoid in-flight rotor collisions. The prototype has a teetering
upper rotor that incorporates a Bell-type stabilizer bar that works as a flywheel.
The stabilizer bar isolates the upper rotor from the fuselage motion, making the
helicopter passively stable in hover. The pitch and roll stability of the helicopter
depend in large part in the characteristics of the stabilizer bar. The two important
parameters are the paddle?s weight and their radial spacing. Increasing any of those
two variables increases the vehicle?s stability and reduces its maneuverability. No
formal mathematical study was performed for the dynamic modeling of the rotor
system and the stabilizer bar parameters were chosen based on flight testing in order
to achieve a balance between stability and maneuverability. Reference [89] addresses
the modeling form first principles and basic aerodynamics of a model helicopter that
has a Bell-Hiller type stabilizer bar. The approach followed in that work can be used
as a guide for the eventual dynamic modeling of the coaxial prototype.
219
Figure 6.19: Swashplate prototype.
6.7.5 Electronics
The vehicle used the following off-the-shelf electronics:
? A four channel RF receiver
? Two brushless speed controllers
? A rate gyro
? Two micro servos
? 2 or 3 cell LI-PO batteries
The four available receiver channels were used to control the two speed con-
trollers and the two swashplate servos. In order to allow a pilot to fly the helicopter
a programmable radio controller (Futaba 8UA) was used to mix the motor signals.
In the controller the throttle stick changed simultaneously the rotational speed of
both rotors keeping the vehicle in a trimmed zero torque condition. The yaw input
differentially changed the speed of the rotors (increasing speed of one and reducing
220
speed of the other) such that only a small variation in total thrust occurred. The
rate gyro was used to stabilize yaw and was connected in line to the upper rotor
control signal. The vehicle does not have a high-level control or stabilization system,
and the pilot is in charge of manually compensating for perturbations. However, the
vehicle has extremely good handling stability characteristics, which makes it easier
to fly than a conventional RC helicopter. These characteristics are very desirable
specially for the development and implementation of a control or stability augmenta-
tion system. The prototype is a very forgiving testbed for algorithms and hardware,
which require extensive testing and fine tuning before they can be integrated to the
vehicle.
Earlier prototypes used Lithium Ion batteries; however, their discharge rate
was not large enough to efficiently power the vehicle. The commercialization of
Lithium Polymer batteries with higher energy densities and much better discharge
rates (up to 13C continuous) provided an energy source that is a better match for
hovering MAVs. The vehicle can operate using a two cell or three cell battery packs
providing respectively 7.4 V and 11.1 V. Batteries used had a nominal capacity of
560 mAh and weighted 12g per cell.
6.7.6 Powertrain - Rotor Matching
An important consideration in the design of any aerial vehicle is the choice and
characteristics of the power plant. From airliners to MAVs not only the aerody-
namic efficiency of the design determines its range payload and endurance, the
motor/turbine/engine efficiency plays also a major role. For a rotor the relationship
between thrust and power can be easily derived from momentum theory to explore
the effects of aerodynamic and mechanical efficiency of the system. Starting from
the definition of thrust and power:
T = CT ?A(?R)2 P = CP ?A(?R)3. (6.9)
taking the ratio of
221
T3
P2 =
C3T ?3 A3 (?R)6
C3P ?2 A2 (?R)6 =
C3T
C2P ?A. (6.10)
Solving for the power
P = T
3/2
radicalBig
C3T
C2P ?piR
2
= T
3/2
R??piC3/2TC
P
(6.11)
The inflow ratio ?h is related to the thrust coefficient in hover by
?h = ?i?R =
radicalBigg
T
2?A(?R)2 =
radicalbigg
CT
2 . (6.12)
And the relationship between FM,?h, CT, and CP gives
FM = ?h CTC
P
= C
3/2
T?
2CP . (6.13)
Replacing Eq.6.13 into Eq.6.11
P = T
3/2
R?2?piFM (6.14)
The compound powerplant-transmission efficiency ? can be introduced in the equa-
tion
P = 1? T
3/2
R?2?piFM (6.15)
Or solving for the thrust
T = (?P Rradicalbig2?piFM)2/3 (6.16)
Using these simple equations the effects of the mechanical and aerodynamic efficien-
cies as well as rotor diameter on various vehicle characteristics can be plotted. Fig-
ures 6.20(a) and 6.20(b) show respectively the effect of FM and motor-transmission
efficiency (?) on the Maximum Take Off Weight (MTOW) of the vehicle. A fixed
maximum shaft power of 10W and no powertrain losses were assumed in Figure
6.20(a), and a fixed FM of 0.6 was assumed in Figure 6.20(b). From the plots it
is clear that mechanical and aerodynamic efficiencies are equally important for the
222
0
50
100150200250
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FM
MTOW (g)
Rotor dia = 9 in Rotor dia = 6 in Rotor dia = 3 in 
(a) MTOW vs. FM @ ? = 1.
0
20406080
100120140160180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Mechanical  efficiency,
MTOW (g)
Rotor dia = 9 in Rotor dia = 6 in Rotor dia = 3 in 
(b) MTOW vs. ? @ FM=0.6.
Figure 6.20: MTOW vs. FM and MTOW vs. ? for various rotor diameters.
performance of any rotary wing vehicle. Low levels of mechanical efficiency result in
a large power requirements regardless of how optimized the aerodynamics are. This
is why it is key for the implementation of any vehicle to simultaneously achieve the
highest possible level of mechanical and aerodynamic efficiency.
Motor Characterization
In order to match the operational conditions for a good mechanical and aerody-
namic performance the behavior of the powerplant over a wide range of operational
conditions is required. Manufacturers rarely provide the rotor performance curves
and if they do, overly optimistic results are usually given. This is why an electric
motor characterization procedure was developed. The same hover test stand used
for rotor testing (see section 2.2) was used for measuring current, voltage, torque and
rotational speed at various motor loads. The rotor is replaced by a solid aluminum
cylinder, and a no-contact symmetric load is applied to the system through and
array of strong neodymium magnets. The Eddie currents induced by the magnets
result in a torque that opposed the motion of the cylinder. magnitude of the torque
is a function of the rotational speed and the strength of the magnetic field. The
magnet array was mounted on a height gage such that vertical spacing between the
magnet array and the spinning cylinder could be accurately controlled. Figure 6.21
shows a picture of the setup.
223
Magnets
Aluminum
cylinder
Motor-Transmission
Figure 6.21: Motor characterization test stand showing Eddie-current break.
The vehicle prototype currently uses brushless motors manufactured by Feigao
(ref 1208436L). Brushless motors are triphasic and require an electronic speed con-
troller that supplies the commutated current to the motor windings. The current
and voltage required by the system were measured before the speed controller (Cas-
tle Creations Phoenix 10), hence the experiments provide the compound efficiency
of controller, motor and transmission. Experiments were performed at two fixed
nominal voltages of 7.4V and 11.1V. These are the voltages provided by two and
three lithium polymer cells (LI-PO) connected in series which are the batteries used
in the prototype. Figure 6.22 shows the results obtained.
System efficiency improved monotonically with rotational speed. The highest
efficiency value measured was 65% at 24,000 RPM using a nominal voltage of 7.4 V.
The use of higher voltage to drive the system increased maximum output power at
all rotational speeds, however efficiency values were about 5% lower. For a coaxial
vehicle weighting 140g, an efficient rotor design needs to operate in the vicinity of
2750 RPM. Hence, in order to maximize mechanical efficiency an optimal transmis-
sion reduction ratio of 9 to 1 is desired. Before the motor characterization tests were
performed the vehicle?s transmission had a reduction ratio of 4 to 1. The mechanical
224
power required by each rotor is about 4 W in hover; hence, from Figure 6.22(a) the
system efficiency can be estimated to be ?43%. This means that a simple change in
the reduction ratio can potentially offer power savings in excess of 30% with respect
to the original vehicle power consumption.
The prototype?s transmission was upgraded to a reduction ratio of 6 to 1 such
that the motors were operating in hover at about 16500 RPM. This improved the
system efficiency up to ? 50%. Unfortunately, before larger reduction ratios can
be implemented, swashplate design needs to be modified such that the servos do
not interfere with the larger gears needed. The improvements in motor efficiency
translated into a noticeable increase in vehicle?s endurance.
6.8 Summary
This chapter explored the effects of basic coaxial rotor parameters on the system?s
performance. An experimental setup that allowed to measure the thrust and rota-
tional speed of a upper and lower rotors at a zero torque condition was developed.
The following conclusions were obtained from the tests:
? Rotor spacing does not have a large effect on coaxial rotor performance. At
vertical separations larger than h/R=0.4 system thrust does not undergo mea-
surable changes.
? At a trimmed zero torque condition the upper rotor carries ?60% of the total
thrust when identical blades are used. Load distribution is barely affected by
disk loading or by collective pitch settings as long as torque between rotors is
balanced.
? For the cases studied, the upper rotor is practically unaffected by the lower
rotor. Induced aerodynamic effects produced by the lower rotor at the upper
rotor plane are negligible.
? Power loading of the coaxial rotor was found not to be maximized when both
rotors were operating at the same rotational speed, contradicting Sunada?s
225
0
10203040506070
0
2
4
6
8
10
12
14
Output mech power (W)
% System Efficiency
8000 RPM 12000 RPM 16000 RPM 20000 RPM 24000 RPM
(a) System efficiency at 7.4V.
0
10203040506070
0             2             4              6             8
1
0
1
2
1
4
Output mech power (W)
 % System Efficiency
8000 RPM 12000 RPM 16000 RPM 20000 RPM 24000 RPM
(b) System efficiency at 11.1V.
Figure 6.22: Efficiency vs. shaft power of controller-motors-transmission system at
various rotational speeds.
226
conclusion from Ref. [44]. Sunada?s draw his conclusion using a limited amount
of experimental data and considered only a very specific case. The current
research covered a wider test space, finding that optimal rotor operating con-
ditions are a function of disk loading and that PL of a coaxial is maximized
in some cases when rotors operate at very different rotational speeds.
The first two bulleted results are consistent with the interaction of the lower rotor
with a fully developed wake. A rudimentary single rotor wake characterization
experiment using a Pitot tube supported this result. The experiments showed that
the wake of an isolated rotor was fully contracted at a vertical spacing h/R?0.2.
Basic blade element momentum theory equations were rederived to include
climb speed and were used to predict the performance of coaxial rotors. The upper
rotor was treated as a single rotor and the interaction between lower rotor and up-
per rotor?s slipstream was modeled as an apparent climb speed that extended over a
fraction of the rotor area. Lower rotor predictions are a function of the upper?s wake
radial velocity profile and its contraction ratio. From wake characterization exper-
iments upper rotor?s wake boundary was set at 0.8R and upper rotor?s inflow was
scaled according to the assumed contraction. Validation with experiments showed
that good thrust prediction were obtained (? 5% error), however lower rotor torque
was largely under-predicted because of two main factors:
? Sectional drag coefficient from CFD are under-predicted at the lower angles
of attack at which large sections of the lower rotor blades operate.
? Inducedpowerattheboundaryoftheupperrotorslipstreamisunder-predicted
since the sudden inflow change does not include a loss function.
A fully functional coaxial MAV was implemented and designed based on the aero-
dynamic coaxial studies performed. Vehicle configuration, rotors and swashplate
were designed from scratch, while batteries, motors and electronics were carefully
selected off-the-shelf components. A motor characterization test procedure was de-
veloped and used to determine the most efficient working regime for the prototype?s
227
motors and an optimal transmission reduction ratio. The prototype?s stability and
handling characteristics were superior to those of conventional RC helicopters, mak-
ing it a very forgiving platform for the development of control systems or stability
augmentations algorithms.
228
Chapter 7
Concluding Remarks
Micro air vehicles are an emerging technology that can potentially transform the way
a wide array of law enforcement, military, and civilian missions are approached. The
possibility of having small flying vehicles sending video or sensor data from remote
locations where it is either too dangerous, time consuming, or simply impossible for
a human to be has been the main motivation for this area of research. However,
before MAVs can be used as intended, two main aspects of the technology have to
be resolved: platform development and control system implementation.
Efficient vehicle configurations with enough endurance, range, and good ma-
neuverability have to be developed. Yet, this is just half the problem. What will
make MAVs practical is the integration of a good performing platform with the
necessary electronics to minimize or eliminate altogether human intervention. This
dissertation focused on the understanding of the aerodynamic issues related to ro-
tary wing MAVs, and on the development of a stable platform that can be used for
the testing and implementation of control systems.
7.1 Summary of Contribution and Results
Most of the experimental research performed addressed the effects on hover per-
formance of geometric and operational parameters of single isolated rotors. The
following are the most important results and contributions that relate to this aspect
of the investigation.
1. A computerized hover stand that directly measured thrust, torque, rotational
speed and electric power of small-scale rotors was developed. Hardware and
229
software were customized to achieve excellent levels of accuracy, minimizing
testing time and assuring repeatability of the results.
2. Systematic hover testing of single isolated rotors was performed. Experiments
can be divided into two main categories: tests with rectangular blades, where
the effects of airfoil where isolated. And tests with non-rectangular blades,
where geometric blade and airfoil parameters were coupled. From the rectan-
gular blade tests the following conclusions and results were obtained:
? The use of conventional streamlined airfoils historically used in full-scale
vehicles, results in extremely poor performance at MAV scale. The low
Reynolds number regime at which the blades operate has large adverse
effect on sectional airfoil characteristics.
? Thin cambered plates which are able to achieve higher lift coefficients and
lift-to-drag ratios are a better choice for small-scale rotor implementation.
? Having more than two blades in an MAV rotor is not a good design choice.
Even though theory predicts a reduction in tip losses as the number of
blades increase, other more significant aerodynamic and structural advan-
tages are achieved with two bladed rotors. For a given solidity a fewer
number of blades translates into wider chords, which at a fixed rotational
speed will have higher Reynolds numbers. Additionally, for a fixed blade
thickness wider chords results in curved plates with a lower thickness ra-
tios. Higher Reynolds numbers and low thickness ratios are factors that
benefit the aerodynamic airfoil performance, resulting in higher rotor ef-
ficiency.
? The effect of Reynolds number on airfoil performance also influences the
choice of rotor solidity. As opposed to full-scale helicopters where rotors
having the highest CT/? usually achieve the highest FMs, at MAV scale
low solidity translates into low chord-based Reynolds numbers and non-
optimal rotor performance. The lower blade area does not compensate for
theaddedviscouslossesincreasingtheoptimalchordwidth. Additionally,
230
similar structural limitations as for the choice of number of blades also
apply in this case.
? Rotor efficiency at MAV scale is largely determined by the airfoil char-
acteristic. The most relevant thin curved airfoil parameters are: camber
amount, camber location, thickness ratio, and leading edge shape.
? Circular arcs are outperformed by thin curved plates with a maximum
camber location shifted toward the leading edge. However, the ease of
manufacture and simple parameterization made circular arcs the airfoil
of choice in this dissertation.
? Experimentally the highest FMs were achieved with airfoil cambers rang-
ing from 6% to 9%.
? Measurement of the zero-lift drag on rotors with flat plate blades showed
that over the tip Reynolds numbers range covered (25,000<Re<65,000),
the flow in the rotor blades cannot be assumed laminar. Most likely the
flow is transitional and is separating at various at different regions of the
blade surface.
? Leading edge shape had a large influence on the zero-lift drag coefficient
of flat plates, the use of sharp symmetric leading edges resulted in the
lowest drag values. For the various leading edge shapes considered zero-
lift drag coefficients were adversely affected by a reduction in Reynolds
number and an increase in thickness ratio.
? Leading edge shape also played an important role in hover performance.
Maximum FMs were increased and rotor stall was delayed by the use of
sharp asymmetric leading edges.
Non rectangular blades were used to study the coupled airfoil and blade pa-
rameters on rotor hover performance. The tapered geometries were manufac-
tured by removing material from baseline rectangular blades. This resulted
in a coupling between blade planform, twist distribution and spanwise airfoil
231
geometries. From the experiments with non-rectangular blades the following
conclusions and results were obtained:
? when using flat plate airfoils, blade planform only matters to the extend
it affects solidity distribution. As long as the thrust weighted solidity is
kept constant, the effects on rotor performance will be minimal.
? Large improvements in hover rotor performance were achieved by taper-
ing blades with 6% and 9% airfoil cambers. Negative twist and gradual
camber reduction were introduced at the tapered sections resulting in the
simultaneous reduction of induced and profile power.
? The highest maximum FM achieved was 0.65 by rotors with 6% and 9%
baseline camber blades with asymmetric 2:1 linear taper starting at 80%
span. Maximum FM reached the same nominal value; however, they
occurred at different disk loadings.
3. Rotational speed at MAV scale, as opposed to full scale-helicopters, is a param-
eter that can be chosen over a wide range. Lower rotational speeds translate
into lower FM but higher PL. Since the final objective is to minimize the power
consumption, maximization of the power loading and not the FM should be
the design driver.
4. A Blade element momentum theory model was implemented and an algorithm
to calculate the equivalent sectional airfoil characteristics from hover rotor
tests was developed. This approach was used to compare the characteristics
of various airfoils, and to determine the relative contributions of induced and
viscous losses to the total power of some sample rotors. It was found that
viscous losses accounted for 40% of the total power on a representative rotor
with rectangular untwisted blades. This is larger than the 30% profile power
contribution typical of full-scale rotors. In order for MAV rotors to achieve
high levels of efficiency both induced and viscous losses have to be as low as
possible. This is achieved by having a good blade design coupled with well
232
performing airfoils.
5. CFD was used in a parametric study of a family of circular arc airfoils. The
effects on lift, drag and moment coefficients of a series of geometric parameters
was explored over a range of Reynolds numbers. The CFD solvers TURNS,
Fluent and INS2d were evaluated, validating the results with published wind-
tunnel experiments of a circular arc airfoil at a Reynolds number of 60,000.
Validation showed that TURNS produced the best lift coefficient predictions
followed by INS2d which under-predicted lift at the higher angles of attack.
Fluent was unable to capture the non-linearities of the lift measurements. All
three codes under-predicted drag; however, TURNS and INS2d qualitatively
captured the trends in L/D ratio. The solver INS2d was chosen over TURNS
for its ease of convergence at low Re and lower computational cost. The
parametric study produced a multidimensional airfoil database.
6. The CFD generated airfoil database was integrated to a BEMT rotor model. A
blade-airfoil parameterization was defined coupling blade planform with twist
distribution and airfoil shape. The algorithm interpolated the database to ob-
tain the spanwise airfoil characteristics. Prediction were validated with some
of the measurement for rectangular and non rectangular blades. Thrust and
maximum FM predictions were satisfactory for blades with maximum cambers
of 6% and below. For blades with higher cambers the early stall predictions of
the CFD solver resulted in lower thrust values than experimentally measured.
Similarly, power predictions at the lower collectives were underestimated be-
cause of the limited accuracy of the CFD drag coefficients at low angles of
attack. Nevertheless, the approach can be efficiently used in the preliminary
design of MAV rotors. Thrust and power calculations for a wide variety of
geometries and operational conditions are obtained in just a few minutes with
similar or even better precision than with the much more computationally
expensive three dimensional CFD simulations. A parametric grid search was
used to find the optimal tip geometry within a defined design space.
233
7. Experimental characterization of a representative coaxial rotor was performed
at a zero torque condition over a range of rotational speeds. The hover test
stand was used to measure the coaxial thrust,rotational speeds, and electric
power of each rotor. The following conclusions were obtained by exploring the
coaxial rotor configuration:
? Rotor spacing does not have a large effect on coaxial rotor performance.
At vertical separations larger than h/R=0.35 system thrust does not
undergo measurable changes.
? At a trimmed zero torque condition the upper rotor produces approx-
imately 60% of the total thrust when identical blades are used. Load
distribution is barely affected by disk loading or by collective pitch set-
tings as long as torque between rotors is balanced.
? Power loading of the coaxial rotor was found not to be maximized when
both rotors were operating at the same rotational speed. The current
research covered a wider test space, finding that optimal rotor operating
conditions are a function of disk loading and that PL of a coaxial is
maximized in some cases when rotors operate at very different rotational
speeds.
8. BEMT equations were derived to include a climb speed. The interaction of
the lower rotor with the fully contracted upper rotor slipstream was modeled
as an apparent climb speed over the inner regions of the rotor disk. Lower
rotor predictions are a function of the upper?s wake radial velocity profile and
its contraction ratio. From wake characterization experiments upper rotor?s
wake boundary was set at 0.8R and a uniform wake velocity distribution was
assumed. At fixed collectives of 16 degs, for a range of operating conditions,
themodelwas usedtopredictcoaxialthrustandlowerrotorrotationalspeedto
achieve torque equilibrium. Results were validated with experiments obtaining
satisfactory levels of accuracy in both thrust and power calculations (? 5%
error). The model tends to under-predict the power of the lower rotor because
234
of the following reasons:
? Sectional drag coefficient from CFD are under-predicted at the lower
angles of attack at which large sections of the lower rotor blades operate.
? Induced power at the boundary of the upper rotor slipstream is under-
predicted since the sudden inflow change does not include a loss function.
? Model calculations are a function of the assumed contraction ratio and
upper wake?s velocity profile. contraction ratio was empirically estimated,
and a uniform velocity profile was assumed.
9. A fully functional coaxial MAV was implemented and designed based on the
aerodynamic coaxial studies performed. Vehicle configuration, rotors and
swashplate were designed from scratch, while batteries, motors and electronics
were carefully selected off-the-shelf components.
10. A motor characterization test procedure was developed and used to determine
the most efficient working regime for the prototype?s motors. It was found that
motor efficiency and proper rotor-motor matching is as important as having
good aerodynamic performance.
11. The coaxial configuration chosen for the implementation of the prototype
proved to have large advantages in terms of compactness and stability. In fact,
the prototypes stability and handling characteristics are superior to those of
conventional RC helicopters, making it a very forgiving platform for the de-
velopment of control systems or stability augmentations algorithms.
7.2 Recommendations for Future Work
The experimental aspects of this dissertation established a solid foundation for the
characterization of small-scale rotors. The methods, software, and hardware used,
provide repeatable results within the margins of experimental error and can be used
with confidence in the future. However, some of the computational and analytical
235
aspects of this research need to be refined. Most of the inaccuracies in the rotor
thrust and power calculations can be linked to the early lift stall and the under-
prediction of drag coefficients from INS2d solutions. The use of a different CFD
solver with better predictive capabilities at the Reynolds numbers considered, or
empirical corrections to the database are possible approaches. Blade element mo-
mentum theory (BEMT) is an extremely inexpensive approach that coupled with
good sectional airfoil characteristics can be used as a powerful rotor design tool.
Based on the results obtained in this dissertation, it can be concluded that BEMT
capabilities are adequate for preliminary MAV-scale rotor design and basic rotor
optimization. The main appeal of the method implemented in this dissertation,
is that once the airfoil database has been calculated it can be used to predict the
performance of a wide variety of blade geometries and rotor configurations with a
minimum of additional effort. This is not the case for higher order aerodynamic
modeling tools such as Free Vortex Modeling (FVM), or three-dimensional CFD.
This is specially true for CFD simulations that require vasts amounts of computing
power. Higher order modeling tools might be used in the refinement of a previously
optimized design with the approach implemented in this research.
This dissertation focused on the use of circular arc airfoils for MAV rotor
implementation. Circular arcs offered unique advantages in terms of manufacturing
ease and blade-airfoil parameterization. However, better aerodynamic performance
can be achieved with thin curved plates having different camber lines. Detailed
experimental and computational parametric studies should be carried out to identify
an airfoil family with optimized sectional characteristics for the use in rotary wing
MAVs.
The coaxial MAV prototype developed through the course of the research
can undergo large performance and functionality gains with some basic component
redesign. First of all, the transmission reduction ratio is currently 6:1 and needs to
be increased to the optimal 9:1. Only then, motors will be able to work at their
most efficient rotational speed.
Outdoor vehicle maneuverability and control authority in the current proto-
236
type need to be improved. When facing wind gusts the lower rotor swashplate is
not able to produce large enough moments to fly into the wind. This can be solved
by introducing a cyclic input to the paddles of the upper rotor?s stabilizer bar, in-
creasing the vehicle?s control authority without sacrificing stability. The existing
swashplate can be modified to do this without a large weight penalty.
237
Appendix A
Low Reynolds Number Flow Physics
and Basic Rotor Aerodynamics
A.1 Flow Regimes, Reynolds and Mach Numbers
The nondimensional parameter known as the Reynolds number was introduced by
Irish engineer Osborne Reynolds in 1883. The Reynolds number is defined as the
ratio of inertial forces (?V?) to viscous forces (?/c) in the fluid, and it provides a
criterion for dynamic similitude. The Reynolds number of an airfoil with chord c
and free stream velocity V? is given by Eq. A.1.
Re = ??V?c? . (A.1)
Where ?? is the fluid density and ? the dynamic viscosity of the fluid. As Reynolds
number is reduced viscous forces become dominant and boundary layer behavior
becomes more complex. Phenomena such as transition separation and reattach-
ment can occur within a short chordwise distance having a large effect on pressure
distribution and airfoil performance.
Mach number is another key non-dimensional parameter used to characterize
aerodynamics flows. Free stream Mach Number is defined as the ratio of the free
stream convection velocity V? and the local speed of the sound a? as shown in
Eq.A.2.
M? = V?a
?
. (A.2)
Mach number can be understood as the ratio of inertial forces in the flow to forces
resulting from compressibility effects. Three different aerodynamic regimes are de-
238
0.001
0.01
0.1
1
10 1010 10 10 10
4 5 6 7 8 9
M
Re
Birds
UAVs
Human 
Powered
Vehicles
Handgliders 
&
Ultralights
General Aviation
Large 
Aircraft
Lighter than Air
Vehicles
Range of Interest
Insects
10
3
Figure A.1: Reynolds Vs. Mach number for various types of vehicles
fined based on Mach number. When M < 1 the flow is subsonic, if M = 1 the flow
is sonic and if M > 1 the flow is supersonic.
Figure A.1 shows Reynolds number vs. Mach number for various types of
vehicles and animals. The dashed rectangle defines the range of interest for MAV
development. Micro Air Vehicles fly in the lower subsonic range at Reynolds num-
bers below 70,000. This is the regime where most birds and insects are also en-
countered. For the rotors studied in this dissertation a maximum Mach number of
approximately 0.14 at the blade?s tips was attained, making compressibility effects
negligible.
A.2 Boundary layer Behavior
The boundary layer concept was introduced by Prandtl in 1904 and was used to
explain the differences between the observed fluid behavior and inviscid flow theory.
This was a problem that could not be solved in the 19th Century and was known
as d?Alembert?s paradox. When there is relative motion between a solid body and
fluid, the fluid adheres to the surface, creating a thin region where the flow velocity
will increase from zero up to the free stream velocity. This region is known as the
239
boundary layer, and it is responsible for the frictional drag forces on an airfoil. The
flow within the boundary layer can be either laminar or turbulent. A laminar flow is
characterized by being organized and uniform, while a turbulent flow is disorganized
and has abundant mixing between the fluid layers.
Boundary Layer thickness is defined as the distance from the body surface
up to where 99% of the free stream velocity is recovered. Due to the momentum
transfer within the turbulent boundary layers, these have fuller velocity profiles,
and are thicker than laminar ones. When laminar flow is disturbed and changes
into turbulent,the boundary layer undergoes what is known as transition.
The drag produced by the two types of boundary layer varies with the inverse
of the Reynolds number. However laminar flow produces lower values of drag. For
flows over a flat plate, Blasius found the expressions for the laminar boundary
layer thickness (?) and the skin friction coefficient (cf) given by Eq.A.3 and Eq.A.4
respectively.
? = 5.2x?Re
x
. (A.3)
cf = 0.664?Re
x
. (A.4)
There is no exact analytical solutions for the turbulent boundary layer over a flat
plate. However, numerical solutions have been found using empirical models of
turbulent diffusion. The approximate thickness is given by Eq.A.5.
? = 0.37xRe0.2
x
. (A.5)
Where Rex is the Reynolds number based on the distance from the leading edge
of the plate. The total shear stress drag can be obtained by integrating the skin
friction drag coefficient along the surface, as shown in Eq.A.6
Cd = 1c
integraldisplay c
0
cf dx (A.6)
240
By replacing the skin friction coefficient in the previous expression with Eq.A.4 the
laminar shear drag coefficient can be obtained as shown in Eq.A.7.
Cdlam = 1.328Re?0.5x . (A.7)
Since no exact solution for the turbulent boundary layer has been obtained, there
are various proposed models for the turbulent shear drag coefficient. Two of the
most widely used are presented in Eqs. A.8 and A.9 from Refs. [72] and [73]
respectively.
Cdtur = 0.074Re?0.2x . (A.8)
Cdtur = 0.455(logRex)?0.258. (A.9)
These expressions can be used to check experimental data. For example, the laminar
shear drag coefficient can be considered an absolute lower bound for the minimum
drag coefficient of an airfoil, while turbulent drag coefficient are usually close to the
minimum drag coefficients of airfoils at Reynolds number of one million and above.
The variation of pressure in the direction of the flow, or pressure gradient,
has a large influence over the boundary layer. The well known Bernoulli?s principle,
which states that an increase in velocity occurs simultaneously with decrease in
pressure (with various assumptions), can be used to give a simple explanation of the
boundary layer?s velocity profile behavior.
When the pressure decreases in the direction of the flow, a favorable pressure
gradient exists. The boundary layer?s velocity profile becomes fuller as energy is
gained, this corresponds to case (a) of Figure A.2. When pressure increases in the
direction of the flow an adverse pressure gradient is obtained, gradually distorting
the boundary layer?s velocity profile As shown in case (b) of Figure A.2. Since the
boundary layer has limited energy, if the adverse gradient is sustained long enough,
the flow will eventually reverse its direction. When this occurs the external flow
pattern breaks down and the flow separates from the surface forming a turbulent
241
Favorable pressure gradient
Pressure decreasing - Velocity increasing
Adverse pressure gradient
Pressure increasing - Velocity decreasing
(a)
(b)
Figure A.2: Effect on boundary layer of favorable and adverse pressure gradients.
wake. The difference in pressure between the front and back of a body produced by
separation introduces another source of drag, known as form drag.
The lower shear drag forces produced by laminar boundary layers come at a
price. There is a trade-off between lower drag and boundary layer stability. Laminar
boundary layer are comparatively unstable, being very sensitive to flow disturbances
and able to withstand only small adverse pressure gradients. However stability tends
to increase as Reynolds number is reduced.
When a laminar Boundary layer faces an adverse pressure gradient there are
three possible outcomes. It can separate stalling the airfoil, it can separate and
eventually reattach as a turbulent boundary layer, or it can transition into a tur-
bulent boundary layer without the occurrence of separation. What exactly occurs
depends on a series factors, such as the intensity of the adverse pressure gradients,
surface roughness, surface curvature, freestream turbulence and of course, Reynolds
number.
242
A.3 Lift Generation and Boundary Layers
An airfoil generates lift by producing a pressure difference between the upper and
lower surfaces. This can be achieved by having low pressure in the upper surface,
high pressure in the lower surface or a combination of the two. As discussed in
section A.2, if pressure increases in the direction of the flow, the boundary layer
faces an adverse pressure gradient, this means that no matter how lift is generated
adverse pressure gradients have to occur. Figure A.3 shows the velocity distributions
assuming inviscid flow for three different airfoils at 3 degrees angle of attack. Airfoil
a) is a NACA 0012, it is a symmetric airfoil that relies mostly on accelerating
the flow over the upper surface producing a strong suction peak. Airfoil b) is a 5
digit NACA 88012. In this case, both a positive pressure in the lower surface and
a negative pressure in the upper surface are used. Finally, airfoil c) is a 4 digit
NACA 6406, its key characteristics are a relatively small thickness and a concave
lower surface. This airfoil relies largely on producing a positive pressure in the lower
surface, such that if a laminar separation bubble or catastrophic separation occurs
in the upper surface, a considerable amount of lift can still be generated.
A.4 Low Reynolds Numbers Sub-Regimes
Reynolds number is a dominant factor that dictates airfoil performance and vehi-
cle design. Carmichael in his 1981 survey (Ref. [5]) divided the Reynolds number
realm into twelve segments covering ten orders of magnitude. For each range, ba-
sic flow characteristics and main aerodynamic issues were explained. Four of these
regimes are of interest for the current research. Some of Carmichael?s comments
complemented with additional information are presented here.
The first interval covers Reynolds numbers between 1,000 and 10,000. About
30% of the inner blade segments of propellers and rotors of existing MAVs are in this
range. However, its relevance is expected to grow as the next generation of smaller
vehicles is developed. The flow in this regime is characterized for being laminar
with a tendency to separate before undergoing transition. The research performed
243
0 0.2 0.4 0.6 0.8 1
1
2
0
v/Vinf
x/c
0 0.2 0.4 0.6 0.8 1
1
2
0
v/Vinf
x/c
0 0.2 0.4 0.6 0.8 1
1
2
0
v/Vinf
x/c
NACA 0012
NACA 88012
NACA 6406
a)
b)
c)
Figure A.3: Velocity distribution for three different types or airfoils at 3 deg aoa.
244
on the Mesicopter (Refs. [31, 52, 53, 54]) is in this range. Computational results
showed how airfoil design, specially camber distribution, still is an important factor
that affects airfoil performance.
The next interval defined by Carmichael covers Reynolds numbers between
10,000 and 30,000. Here the flow exhibits a similar behavior than in the immedi-
ately lower range, being laminar with a tendency to separate before transitioning
into turbulent. Very limited research has been performed in this regime. One of the
few sources, as reported by Carmichael, is an obscure report by Bauer (Ref. [70])
that claims the existence of 100% laminar-flow hand-launched gliders. Bauer deter-
mined experimentally that the drag coefficient of the model tested matched closely
what the laminar boundary layer theory predicts. However, this was only achieved
at low lift coefficents of the order of 0.5. When angles of attack were increased
separation without reattachment occured and the airplanes stalled. The lack of re-
liable experimental data on this Reynolds number range makes difficult to ascertain
general flow characteristics and behavior.
The third interval covers the Reynolds numbers between 30,000 and 70,000.
This is where most of the fixed-wing MAVs and the outer sections of MAV rotors
operate. Flow in this regime is characterized by the occurrence of laminar separa-
tion bubbles that introduce hysteresis in the experimental measurements. Below a
Reynolds number of about 50,000 the chordwise distance is generally not sufficient
for reattachment to occur; however, experiments have shown that the use of bound-
ary layer trips or turbulators to reduce the critical Reynolds number (Reynolds
number at which transition occurs) has been successful near the upper bound of
this range.
Finally the interval that covers Reynolds numbers between 70,000 and 200,000
is characterized by the ease to obtain extensive laminar flow that is able to transition
to turbulent without catastrophic separation. Turbulators can still be useful to
reduce the size of the separation bubbles, but only in the vicinity of the lower bound
of the interval. Most of the low Reynolds number experimental research covers this
regime, which is typical of fixed-wing radio control models, and small UAVs.
245
R
e
c
ir
c
ul
at
in
g
 Z
one 
Bubble Length
Laminar - T ransitional
 Region
Turbulent
Region
Developing
Turbulent 
Boundary Layer
Zero V elocity
Line
Dividing Streamline
Outer Layer
Inner Layer
Laminar
 Flow
Outer Edge of 
Boundary Layer
(2)
Reattachment
 Point
Turbulent
Flow
(1)
Separation
Point
Figure A.4: Sketch of laminar separation bubble.
A.5 Laminar Separation Bubble
In this section a description of the laminar separation bubble, conditions for its
formation and its effects on airfoil performance are presented.
Figure A.4 shows a diagram of a laminar separation bubble superimposed to
the expected boundary layer velocity profile. As previously discussed, when a lami-
nar boundary layer faces an adverse pressure gradient separation occurs as soon as
there is a change in sign along the velocity profile, this corresponds to point 1 in the
diagram. The bubble is contained within the outer edge of the boundary layer, and
has recirculating flow, easily identifiable in the pressure distribution plots as a seg-
ment of constant pressure. The recirculating region is divided in two distinct zones,
one with laminar-transitional flow and another with turbulent flow. Reattachment
occurs at point 2 in the diagram. The attached boundary layer is now turbulent
and exhibits its typical fuller velocity profile.
From the discussion in section A.4 it is clear that laminar separation bubbles
are constrained to a Reynolds number range. However, this range varies with air-
foil shape and environmental conditions. Carmichael (Ref. [5]) proposed a rule of
246
Cl
a 
Cl
Re 
Re_1
Re_2
_1 
Re _1Re _2
_1 
(a)  Angle of attack (b) Reynolds number
Figure A.5: Schematic of hysteresis in lift coefficient with variations in a) angle of
attack, and b) Reynolds number from data in (Ref. [1])
thumb that gives a rough estimate of the chances of a bubble to form. Considering
the chordwise distance from the point of separation to the point of reattachment,
the Reynolds number using the freestream velocity is approximately 50,000. This
means that airfoils with chord Reynolds numbers below 50,000 will undergo lami-
nar separation without bubble formation. At the lower Reynolds numbers, bubbles
will be long, extending for a substantial fraction of the chord and having a marked
effect on the airfoil?s pressure distribution. A smaller suction peak reduces the lift
coefficient slope while increasing pressure drag. Bubble length shortens as Reynolds
number increases, gradually reducing its adverse effects over the airfoil performance.
Laminar separation bubbles can occur at much higher Reynolds number -of about
one million- in those cases bubble length is very small extending just for a few per-
cent of the total chord length. Short bubbles do not have a noticeable effect on
the pressure distribution or on the airfoil performance, they just work as a tripping
mechanism that facilitates the transition of the boundary layer. Figure A.6 obtained
from Ref. [71] illustrates the previous point, showing how dramatic the influence of
a large bubble can be as opposed to the minute influence of a short one.
Long bubbles explain the airfoil behavior first identified by Schmitz (Ref. [1]).
In his wind tunnel experiments he documented hysteresis in both lift and drag when
angle of attack and Reynolds numbers were cycled, as shown in Figure A.5. When
247
Potential Flow
Experiment
Cp Cp
Figure A.6: Long and short bubble effect on pressure distribution (Ref. [71])
increasing angle of attack there is a point where turbulent reattachment does not
occur and the bubble bursts, producing thin airfoil stall. In order to reattach the
flow, it is not sufficient to go back to an angle immediately below to where stall
occurred. It is necessary to reduced it by several degrees more until a bubble forms
again as illustrated in Figure A.5 (a). A similar hysteresis loop is observed when
cycling the Reynolds number while keeping a fixed angle of attack. To reattach the
flow, Reynolds number has to be increased well beyond the point were stall initially
occurred as shown in Figure A.5 (b). This behavior needs to be considered when
performing rotor testing, care must be taken to avoid misleading measurements when
acquiring data for a given collective. If a hysteresis loop is identified, a consistent
experimental procedure needs to be followed to avoid erroneous measurements, this
issue is addressed with more detail in Section 2.2.3.
248
A.6 Reynolds Number Effects on Aerodynamic Airfoil Characteristics
One of the most relevant wind tunnel experimental investigations at low Reynolds
numbers was performed by Schmitz (Ref. [1]). He compared the characteristic of
five different airfoils over a Reynolds number range of 42,000 up to 420,000. Three
thick streamlined airfoils, a flat plate and a thin cambered plate were studied, these
are shown in Figure A.7.
Go 625
Flat Plate
Go 417a
N60
N60R
Figure A.7: Airfoils studied by Schmitz in Ref. [1] .
Schmitz results showed how airfoil behavior undergoes radical changes when
tested below and above their critical Reynolds number, this is the Reynolds number
at which transition occurs. He also identified some distinct characteristics of thin
flat and curved plates.
Figure A.8 shows the polar plots at different Reynolds numbers for each of the
airfoils Schmitz investigated. The first striking difference are the large hysteresis
loops present at the higher Reynolds numbers for the streamlined airfoil. These
249
loops are a consequence of the formation of long laminar separation bubbles. This
does not occur for the flat or the cambered plates. For all cases, as Reynolds number
is reduced, drag and lift undergo large variations, adversely affecting the lift to drag
ratio of the profiles.
The effect of Reynolds number over maximum lift coefficient is shown in Fig-
ure A.9. For the thin plates there is a gradual reduction in CLmax with no change is
slope, as opposed to the streamlined airfoils that below the critical Reynolds number
go through a dramatic and sudden drop in maximum lift. Minimum drag coefficients
have a similar behavior, linearly reducing as Reynolds number increases - for the
thin plates - and suddenly dropping - for the streamlined airfoils - as the critical
Reynolds numbers are reached, this is shown in Figure A.10.
If now the lift coefficient slope is considered, it can be observed in Figure
A.11 that the streamlined, airfoils such as the N60, N60R and Go625, undergo a
large reduction in CL? as Reynolds number is reduced. These airfoils achieve a
maximum Cl? of 1.8pi at a Re=420,000. At Re=42,000 there is no linear section
in the lift coefficient curve, and Cl? cannot be defined. On the other hand the
effect of Reynolds number on thin plates is more subtle. The flat plate is practically
not affected, keeping an ideal lift coefficient slope of 2pi over the whole Reynolds
number range. The Go417A airfoil achieves surprisingly a lift coefficient slope of
2.8pi at Re=420,000, having a slight reduction down to Cl? = 2.77pi at Re=42,000.
A.6.1 Thin Cambered Airfoil Characteristics
Schmitz results suggest that if an airfoil is required to work below their critical
Reynolds number, thin cambered airfoils have desirable aerodynamic characteristics
that are not present in conventional thick streamlined airfoils in this regime.
Laitone, following on Schmitz work, focused his investigation (Refs. [16, 17,
18]) on explaining the superior aerodynamic performance of a sharp-nosed thin plate
over any known round nosed airfoil at low Reynolds numbers. He performed wind-
tunnel force measurements as well as pressure distribution and flow visualization
experiments. Laitone?s result showed that sharp leading edges continually shed
250
Cd
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Cl
1.4
1.2
1.0
0.8
0.6
0.4
0.2
Cl
Cd
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Cl
1.4
1.2
1.0
0.8
0.6
0.4
0.2
Cd
0 0.05 0.10 0.15 0.20 0.25 0.30
Cl
Cd
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.4
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Go 417a
N60
Cl
Cd
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
-0.2
0.05 0.10 0.15 0.20 0.250
-0.4
0
Re=420,000
Flat plate
Re=84,000
Re=42,000
Re=105,000
N60R
Go 625
Re=42,000
Re=84,000
Re=63,000
Re=168,000
Re=420,000
Re=42,000
Re=126,000
Re=168,000
Re=42,000
Re = 105,000
Re=42,000
Re=168,000
Re=84,000
Figure A.8: Drag polars for the airfoils studied by Schmitz from data in Ref. [1] .
251
1.4
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
20 160140120100806040
417A
Flat Plate
N60
Go 625
N60R
Cl max
Re 10
-3
Figure A.9: Reynolds number vs. maximum lift coefficient for the airfoils studied
by Schmitz from data in Ref. [1] .
417a
Flat Plate
N60R
N60
Go625
N60R
N60
417a
Go625
0.08
0.06
0
0.02
0.04
0 16014012010080604020
Re 10
3
Cd min
= -3.4 deg
-3.45 deg
-0.7 deg
-1.03 deg
0 deg 0 deg
-6.64 deg
-3.85 deg
-6.64 deg
-7.0 deg
-4.58 deg
-1.21 deg
180
Figure A.10: Reynolds number vs. minimum drag coefficient for the airfoils studied
by Schmitz from data in Ref. [1] .
252
0.5
1.0
1.5
0 15105 0 15105 20
0
effective aoa, (deg)
Cl
0.5
1.0
1.5
0 15105 0 15105 20
0
Cl
effective aoa, (deg)
417a
Flat plate
Go625
N60
N60R
417a
Flat plate
N60
Go625
N60R
417a
Flat plate
Go625
N60
N60R
417a
Flat plate
Go625
N60
N60R
Re = 84,000 Re = 42,000
Re = 420,000 Re = 128,000
2.8 2 2.77 2
2.8 2 1.8 1.672.8 2
Figure A.11: Effective AoA vs. lift coefficient at different Reynolds numbers for
airfoils studied by Schmitz,(Eff. aoa=0 at Cl = 0), from data in Ref. [1].
253
0.5
0.8
0.9
10 20
0.1
0.2
0.3
0.4
0.6
0.7
0
251550
CL
, deg
Reversed NACA 0012
turbulence=0.02%
NACA 0012
turbulence=0.02%
NACA 0012, 
turbulence = 0.10%
Figure A.12: Angle of attack vs. lift coefficient for rectangular wings of AR 6 at
two different turbulence levels and Re=20,700 Ref. [17]
small vortices that rolled along the upper surface preventing flow separation over
a range of angles of attack. He also explored the effects of turbulence in the force
measurements, showing how it had a strong effect on the NACA 0012 airfoil (cf.
Fig. A.12) while thin plates were practically unaffected. Higher turbulence levels
improve NACA 0012 lift generation characteristic by lowering the critical Reynolds
number, and thus delaying separation.
These findings complement Schmitz results that attributed the good airfoil
performance to the large adverse pressure gradients produced by the sharp leading
edges at a very short chordwise distance. These work as boundary layer trips that
induce early transition, making the upper surface boundary layer more resilient to
perturbances.
Laitone also proved experimentally how at low Reynolds numbers, lift gener-
ation does not depend on the Kutta trailing edge condition. By testing a reversed
NACA 0012 (trailing edge working as leading edge) he showed that considerably
254
1.0
1.6
1.8
10 20
0.2
0.4
0.6
0.8
1.2
1.4
0
30251550
CL
, deg
5% camber
circular arc
Flat plate
Reversed NACA 0012
NACA 0012
Figure A.13: Angle of attack vs. lift coefficient for AR=6 rectangular wings at
Re=20,700, Ref [16]
.
higher lift coefficients could be achieved as shown in Figure A.13. Lift generation
was improved, however maximum lift-to-drag ratio was reduced from 7.45 to 6.7 due
to the additional drag produced by the separated flow at the thicker region of the
airfoil.
Thin cambered airfoils not only have desirable aerodynamic characteristics at
low Reynolds numbers but are also easily manufactured. Since they have a uni-
form thickness except at leading and trailing edges, mold design and manufacturing
techniques are simpler than for a streamlined airfoil. This is specially true when
considering circular arcs, which can be easily molded from tubes and cylinders. An
advantage unique to circular arcs is that when they are cut, the resultant segments
are still circular arcs. This unique feature is extensively used in the research pre-
sented in Chapters 4 and 5. As a Summary, the characteristics that make thin plates
preferable to streamlined airfoils at Reynolds numbers below 70,000 are:
? Higher maximum lift coefficients than conventional airfoils
255
? Lower minimum drag coefficients than conventional airfoils
? Linear behavior of lift and drag coefficients with Reynolds number, without
abrupt changes
? Independence of lift coefficient slope with Reynolds number
? Lack of hysteresis loops when cycling Reynolds numbers or angle of attack
? Low sensitivity to ambient turbulence
? Ease of manufacture
These characteristic make thin circular arc airfoils ideal for the systematic study
and implementation of rotary wing MAVs.
A.7 Fundamental Rotor Aerodynamics
Until this point issues related exclusively to low Reynolds number airfoil aerody-
namics have been addressed. This section introduces fundamental rotary-wing aero-
dynamic concepts.
A.7.1 Momentum Theory for a Hovering Rotor
The simplest analysis that can be performed on a hovering rotor involves the use of
the classic momentum theory developed by Rankine in the nineteen century. The
well known equations of conservation of mass, momentum and energy can be applied
to the hovering rotor problem using the flow model shown in Figure A.14. It is
assumed that thrust loading is uniform over an infinitely thin propeller disk (infi-
nite number of blades), and that the flow is quasi-steady, irrotational, inviscid and
incompressible (ideal fluid). Applying conservation of mass on the control volume
the mass flow rate ?m is given as
?m = ?A?? = ?A2vi = ?Avi. (A.10)
256
Vc +
Vc + vi
Vc
Rotor plane
Figure A.14: Flow model used by momentum theory for a single rotor in axial flight.
Where ? is the fluid density, A is the actuator disk area and vi is the induced velocity
in the plane of the rotor. By applying the conservation of momentum equation the
expression for the rotor thrust T as function of the induced velocity vi is found
T =
integraldisplayintegraldisplay
?
?(??V ?d??S )??V = ?m?. (A.11)
Now applying the conservation of energy concept, the power P consumed by the
hovering rotor is given by
P = Tvi =
integraldisplayintegraldisplay
?
1
2?(
??V ?d??S )??V 2 = 1
2 ?m?
2. (A.12)
From equations A.11 and A.12 the relationship between the flow velocities at the
rotor plane and at the far wake is easily obtained
vi = 12?. (A.13)
This means, following the conservation of mass principle, that the area of the slip-
stream at infinity is half the area of the rotor disk. From equations A.10, A.11
and A.13 the expression for the induced velocity as function of the thrust can be
257
obtained.
vi =
radicalBigg
T
2?A =
radicalBiggparenleftbigg
T
A
parenrightbigg 1
2?. (A.14)
Where T/A is the disk loading, an important rotor operational parameter also de-
noted as DL. The disk loading has units of force per unit area, and it will be
discussed in more detail in the following section. Using equation A.14 the power
required by a hovering rotor as a function of its thrust is given by
P = Tvi = T
radicalBigg
T
2?A =
T3/2?
2?A. (A.15)
Considering that for this derivation it was assumed that the rotor was working in
an ideal fluid with potential flow, the power calculated does not include any viscous
loses. In fact this is known as ideal power, since it is the theoretical minimum that
the rotor will require for a given thrust.
A.7.2 Thrust and Power Coefficients
As it is common in engineering, non-dimensional coefficients are defined to facilitated
the analysis of the problems. In rotor analysis rotor thrust and power are non-
dimensionalised as follows
CT = T?A?2R2. (A.16)
CP = P?A?3R3. (A.17)
Where A is the rotor disk area, R is the rotor radius and ? is the rotational speed
of the rotor in rad/s. Another common nondimensional quantity, the inflow ratio ?,
can be defined now in function of the thrust coefficient.
?i = vi?R = 1?R
radicalBigg
T
2?A =
radicalBigg
T
2?A(?R)2 =
radicalbigg
CT
2 . (A.18)
The power coefficient con now be expressed as
258
CP = CT?i = C
3/2
T?
2 . (A.19)
A.7.3 Rotor Hover Efficiency Metrics
There are two main rotor efficiency metrics that are used in helicopter analysis, these
are the figure of merit or FM and the power loading or PL. The figure of merit
has the advantage of being a nondimensional quantity that has a clear theoretical
background, as opposed the power loading that has unit of thrust per unit power.
A.7.4 Figure of Merit
The figure of merit is defined as the ratio of the ideal induced power, as obtained
in equation A.19, by the actual power that the rotor is consuming (cf. Eq A.20).
The actual power can be experimentally measured to obtain a given rotor?s FM.
Since in most cases that is not an option, a standard approach has been adopted to
analytically calculate the power required for a given rotor.
FM = IdealpowerActualpower = IdealpowerInducedPower +ProfilePower. (A.20)
The power required to hover has two main components, induced and profile power.
Induced power is related to the acceleration of the fluid necessary to generate thrust,
and it also includes losses product of physical phenomena ignored by the assump-
tions made in momentum theory and the actuator disk model. These are tip losses,
nonuniform inflow, slipstream rotation and finite number of blades. The ideal in-
duced power can be corrected to include all the non-ideal effect by multiplying the
induced power coefficient by a constant, known as the induced power factor and
denoted by ?. The induced power factor needs to be empirically determined for
each rotor; however, a value of 1.15 gives in most cases a good approximation for
full-scale helicopters.
Profile power includes all the viscous effects that influence the blades perfor-
mance. Viscous effects on an airfoil section such as separation, pressure drag and
259
skin friction are modeled through the drag coefficient. Drag coefficient is a function
of angle of attack, Reynolds and Mach numbers. Since these parameters change
along the blade, profile power should be obtained on a blade element basis and inte-
grated along the span. For full-scale helicopters Reynolds numbers are high enough
such that it can be assumed that the coefficient is constant and equal to the zero lift
drag coefficient Cd0 of the airfoil at 75% of the span. Assuming this, profile power
for a rotor of diameter R with Nb rectangular blades of chord c and a drag force per
unit span D is given by
P0 = ?Nb
integraldisplay R
0
Dydy = 18 ?Nb ?3 cCd0R4. (A.21)
Dividing by ?A(?R)3 the profile power coefficient is obtained
CP0 = 18
parenleftbiggN
b cR
A
parenrightbigg
Cd0 = 18
parenleftbiggN
b cR
piR
parenrightbigg
Cd0 = 18?Cd0. (A.22)
Where the solidity ? is the ratio of the blade by the rotor areas. The figure of merit
can now be expressed as function of the thrust and power coefficients, the induced
power factor, the zero lift drag coefficient and the rotor solidity.
FM = IdealpowerInducedPower +ProfilePower =
C3/2T?
2
?C3/2T?2 + ?Cd08
. (A.23)
In order to have the highest possible figure of merit it is necessary to simul-
taneously minimize the profile and induced losses. The minimum induced power
is achieved when the inflow ratio over the rotor disk is uniform. The inflow radio
distribution along the blade span in hover is defined as
?h(r) = vi(r)?R . (A.24)
Where vi(r) is the induced velocity at a radial position r. Gessow in his classic book
(Ref. [76]) shows that a parabolic twist distribution, known as ideal twist results
in a theoretical uniform inflow. However, such a geometry cannot be physically
implemented. The minimum theoretical profile power is achieved when each blade
260
station is operating at its maximum lift to drag ratio (L/D). This is a simple concept
very difficult to implement considering that the angle of attack for maximum L/D
is not only a function of the airfoil shape but also depends on Reynolds number
and Mach number. In actual helicopter rotors a combination of twist and taper
is generally used to reduce the induced and profile powers to acceptable levels.
References [77] and [78] give an excellent overview of the theory.
In order to have a valid comparison of the hover efficiency of two rotors, it is
necessary for both rotors to have the same disk loading, if this is not the case, the
comparison is meaningless since the ideal power for the two rotors will be different.
As mentioned before, this expression for figure of merit makes the assumption that
the profile losses are accurately obtained by assuming that the drag coefficient is
constant along the blade span and equal to the zero-lift drag coefficient Cd0. This is
a reasonable approximation at Reynolds numbers of more than one million, where
the average drag coefficient at the operating collective angles is close to Cd0, but as it
is shown in Chapter 3, for low Reynolds numbers flows, this assumption introduces
large inaccuracies. For acceptable predictions, other values of the average drag
coefficient should be used.
A.7.5 Power Loading
Power loading is defined for a rotor as the ratio of the thrust produced by the power
required. It has units of thrust per unit power. When designing a vehicle it is wanted
to maximize the power loading such that energy requirements are minimized. This
will give the vehicle the best endurance or payload capabilities possible. Power load-
ing comparisons are not affected by operating conditions or vehicle configurations,
being always valid and able to clearly show which rotor is requiring less energy for
a given thrust.
A relationship between figure of merit and power loading can be obtained by
combining the figure of merit equation and the expression for induced velocity from
equation A.14
261
0 20 40 60 80 100120140160180
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Disk Loading N/m
2
FM=0.25 
FM=0.50 
FM=0.75 
FM=1 (Ideal rotor) 
Power Loading N/W
Figure A.15: Disk Loading vs. Power Loading for various Figures of Merit at sea
level (? = 1.225kg/m3).
FM = T viP
Act
= TP
Act
radicalBiggparenleftbigg
T
A
parenrightbigg 1
2? = PL
?DL
?2? . (A.25)
This equation can be used to plot the disk loading vs. power loading for different
figures of merit. Figure A.15 shows how PL varies with FM and DL. It is clear that
power loading increases as FM approaches the ideal value of 1, and as disk loading
is reduced. Full-scale helicopters have disk loadings that range from 185 N.m?2,
for small vehicle such as the Bell 206, up to 720 N.m?2 for the Sikorsky CH-53E.
Over that range the power loading curves have very shallow slopes, being almost
independent of the disk loading. This is not the case for MAVs, that may have disk
loadings that are in the region where larger effects on the power loading are present.
This is illustrated with experimental data in Chapter 2.
262
BIBLIOGRAPHY
[1] Schmitz, F. W., ?Aerodynamics of Model Aircraft wing Measurements I?,
R.T.P. Translation No 2460. Issued by Ministry of Aircraft Production. 1983.
[2] Miley, S.J.,?A Catalog of Low Reynolds Number Airfoil Data for Wind Turbine
Applications?. RFP-3387 VC-60, Rockwell International Feb. 1982. Us Dept.
of Energy, Wind Energy Technology Division, Federal Wind Energy Program.
[3] Bruining, A., ?Aerodynamic Characteristics of a Curved Plate Airfoil Section
at Reynolds Numbers 60,000 and 100,000 and Angles of Attack from -10 to
+90 Degrees,? Technische Hogeschool, Delft, Netherlands, Dept. of Aerospace
Engineering,Report Number VTHLR-281, May, 1979.
[4] Althaus, D., ?ProfilePolaren Fur den Modellflug,? Institut Fur Aerodynamik
u. Gasdynamik der Universitat Stuttgart Neckar-Verlag, Postfach 1820, 7730
Villingen-Schwenningen, Germany, 1980.
[5] Carmichael, B.H., ?Low Reynolds Number Airfoil Survey,? Volume I, NASA
Contractor Report 165803, November, 1981.
[6] Arena, A.V., and Mueller, T.J., ?Laminar Separation, Transition and Turbulent
Reattachment Near the Leading Edge of Airfoils,? AIAA Journal, Vol. 18, No.
7, pp. 747-753, July 1980.
[7] Mueller, T.J., and Batill, S.M., ?Experimental Studies of Separation on a Two-
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