The Pennsylvania State University
The Graduate School
Department of Aerospace Engineering
COMPUTATIONAL AERODYNAMICS OF
FLAPPING FLIGHT USING AN
INDICIAL RESPONSE METHOD
A Thesis in
Aerospace Engineering
by
Karl R. Klingebiel
c 2006 Karl R. Klingebiel
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
August 2006
I grant The Pennsylvania State University the nonexclusive right to use this work for
the University’s own purposes and to make single copies of the work available to the
public on a not-for-profit basis if copies are not otherwise available.
Karl R. Klingebiel
The thesis of Karl R. Klingebiel was reviewed and approved* by the following:
Lyle N. Long
Professor of Aerospace Engineering
Thesis Advisor
Edward C. Smith
Professor of Aerospace Engineering
George A. Lesieutre
Professor of Aerospace Engineering
Head of the Department of Aerospace Engineering
*Signatures are on file in the Graduate School
iii
Abstract
The recent success of unmanned aerial vehicles (UAVs) in military reconnaissance
applications has fueled interest in the development of micro air vehicles (MAVs) and nano
air vechicles (NAVs) with the capability to deliver micro-payloads to highly confined or
indoor locations. The desired traits of these vehicles such as small size, as well as the capability for hovering, maneuvering and dash are exhibited by many creatures in nature.
With the use of flapping wings, insects and birds such as the hummingbird routinely
hover, gracefully maneuver and dash in light-to-moderate wind conditions. These creatures exploit unsteady and low Reynolds number effects to achieve high performance at
a scale where conventional airplane configurations do not. For this reason, flapping wing
MAV and NAV configurations are considered to be highly promising. Prediction of the
aerodynamic performance of these systems is complicated by the fact that they fly in the
highly unsteady, low Reynolds number regime. A computational tool has been developed
to estimate the aerodynamic coefficients of flapping wing configurations with very low
computational expense. Validation of this tool against experimental and computational
data on flapping wings has shown an order of magnitude lower runtime than other popular computational methods for flapping flight, with good agreement in the results for
reduced frequencies of up to 0.4. For these reasons, this tool would be particularly useful
in the optimization of wingbeat patterns and wing geometry for MAV and NAV designs,
which may require a large number of possible configurations to be evaluated quickly.
iv
Table of Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Chapter 2. Overview of Previous Research . . . . . . . . . . . . . . . . . . . . .
9
2.1
Analytical and Computational Approaches to Flapping Flight . . . .
9
2.2
Observation and Experimentation . . . . . . . . . . . . . . . . . . . .
14
2.3
Flapping Wing Vehicles . . . . . . . . . . . . . . . . . . . . . . . . .
17
Chapter 3. Blade Element Approach . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.1
Blade Element Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.2
Unsteady Aerodynamic Modeling . . . . . . . . . . . . . . . . . . . .
29
3.2.1
Non-Circulatory Lift . . . . . . . . . . . . . . . . . . . . . . .
30
3.2.2
Circulatory Lift . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.2.3
Time Varying Incident Velocity . . . . . . . . . . . . . . . . .
33
3.2.4
Solution of the Duhamel Integral . . . . . . . . . . . . . . . .
35
Nonlinear Numerical Lifting Line Method . . . . . . . . . . . . . . .
39
Chapter 4. Object Oriented Implementation . . . . . . . . . . . . . . . . . . . .
45
3.3
v
4.1
Implementation Details
. . . . . . . . . . . . . . . . . . . . . . . . .
46
4.2
FlapIR Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . .
54
Chapter 5. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.1
Steady Plunging . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.2
Simple Harmonic Oscillations . . . . . . . . . . . . . . . . . . . . . .
62
5.3
Flapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
5.4
Flapping and Twisting . . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.4.1
Comparison to Three-Dimensional Euler Solutions . . . . . .
86
5.4.2
Comparison to Unsteady Vortex Lattice Method Results . . .
90
Chapter 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
vi
List of Tables
4.1
FlappingMAV constructor arguments . . . . . . . . . . . . . . . . . . . .
49
5.1
Summary of constant parameters used in flapping parameter studies . .
90
vii
List of Figures
1.1
AeroVironment WASP micro air vehicle prototype (Ref [1]) . . . . . . .
3
1.2
Honeywell micro air vehicle (Ref [2]) . . . . . . . . . . . . . . . . . . . .
4
1.3
AeroVironment/Caltech Microbat flapping wing MAV (Ref. [3])
. . . .
5
1.4
Scope of aeronautical knowledge (Ref. [4]) . . . . . . . . . . . . . . . . .
7
2.1
Lilienthal’s rotating test apparatus (Ref. [5]) . . . . . . . . . . . . . . .
10
2.2
DeLaurier’s remotely piloted ornithopter (Ref. [6]) . . . . . . . . . . . .
19
2.3
Jones and Platzer’s biologically-inspired MAV (Ref. [7]) . . . . . . . . .
20
3.1
Body axes and spherical coordinate system for flapping wing . . . . . .
28
3.2
MAV body axes relative to the freestream . . . . . . . . . . . . . . . . .
44
4.1
FlapIR UML class diagram . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.2
FlapIR UML sequence diagram . . . . . . . . . . . . . . . . . . . . . . .
55
5.1
Transient lift coefficient for rectangular wing in steady plunging . . . . .
61
5.2
Comparison to experimental lift data for pure pitching (α = 6.74 ) . .
5.3
Comparison to experimental lift data for pure pitching (α = 13.48 ) . .
5.4
Comparison to experimental lift data for pure plunging (h = 1 ) . . . .
5.5
Comparison to experimental lift data for pure plunging (h = 2 ) . . . .
5.6
Comparison to experimental moment data for pure pitching (α = 6.74 )
5.7
Comparison to experimental moment data for pure pitching (α = 13.48 )
5.8
Comparison to experimental moment data for pure plunging (h = 1 ) .
◦
0
◦
0
00
0
00
0
◦
0
◦
0
00
0
64
65
66
67
69
70
71
viii
5.9
00
Comparison to experimental moment data for pure plunging (h = 2 ) .
72
5.10 XFOIL prediction of NACA 8318 lift coefficient at Re = 107,000 . . . .
75
5.11 Instantaneous lift coefficient versus position in flapping cycle . . . . . .
76
5.12 Instantaneous thrust coefficient versus position in flapping cycle . . . . .
77
0
◦
5.13 Average lift coefficient versus advance ratio (δ = 0 ) . . . . . . . . . . .
◦
5.14 Average thrust coefficient versus advance ratio (δ = 0 ) . . . . . . . . .
◦
5.15 Average lift coefficient versus advance ratio (δ = 15 ) . . . . . . . . . . .
◦
5.16 Average thrust coefficient versus advance ratio (δ = 15 ) . . . . . . . . .
◦
5.17 Average lift coefficient versus advance ratio (δ = 30 ) . . . . . . . . . . .
◦
80
81
82
83
84
5.18 Average thrust coefficient versus advance ratio (δ = 30 ) . . . . . . . . .
85
5.19 Instantaneous vertical force coefficient versus position in flap cycle . . .
88
5.20 Instantaneous streamwise force coefficient versus position in flap cycle .
89
5.21 Average lift coefficient for various values of α
. . . . . . . . . . . . . .
92
. . . . . . . . . . . . . . . . . .
93
5.23 Average lift coefficient for various values of β . . . . . . . . . . . . . . .
94
5.24 Average thrust for various values of β . . . . . . . . . . . . . . . . . . .
95
5.25 Average lift coefficient for various values of ω . . . . . . . . . . . . . . .
96
5.26 Average thrust for various values of ω . . . . . . . . . . . . . . . . . . .
97
tip
5.22 Average thrust for various values of α
tip
0
0
ix
Acknowledgments
I would like to thank my advisor, Dr. Lyle Long for his guidance in my research
project, as well as the opportunity to study an exciting topic in our field. I would
also like to thank Dr. Edward Smith and Dr. George Lesieutre for their feedback and
assistance on my thesis. I also greatly appreciate the National Science Foundation and
their financial support for my research through the Consortium for Education in ManyBody Applications (CEMBA), under grant number NSF-DGE-9987589.
Finally, I would like to thank my family for all of their support through both
undergraduate and graduate school, as well as their help with my many moves across
the country over the past five years.
1
Chapter 1
Introduction
In just over 100 years from the first piloted flight of a powered aircraft in 1903, the
field of aeronautics has produced an astonishing set of advancements. A great number of
these advancements have been driven by the desire for aircraft with unique capabilities,
or high performance for military applications. The most advanced military organizations
in the world today have been using unmanned aerial vehicles (UAVs) for decades to provide reconnaissance, and are rapidly approaching the use of UAVs for combat missions.
The popularity and increasing dependence on UAVs can be attributed to their great
operational success, which has provided surveillance data of equal or higher quality to
previous methods, while reducing the need to put humans in high risk situations. A
wealth of civilian organizations are beginning to embrace the UAV as a tool for applications such as law enforcement, search and rescue, border patrol, and agriculture
[8].
The earliest UAVs were of comparable size and speed to manned aircraft, with
the main innovation being simply the elimination of a human pilot onboard. Because
the electronics industry has advanced at such a high rate, the weight and volume of high
quality surveillance equipment has decreased dramatically, and significant processing
power is now available in very small packages. As a result, small UAVs with wingspans
2
of approximately two to six feet such as the AeroVironment Raven [1], and the LockheedMartin Desert Hawk [9] are currently in service with the United States military. These
vehicles are easily carried and operated by small teams of soldiers to rapidly provide
“over-the-hill” surveillance.
The government’s interest in decreasing the size of UAVs even further is evidenced
by the Defense Advanced Research Projects Agency (DARPA) creating a program in
1996 to encourage the development of what they termed Micro Air Vehicles (MAVs)
[10]. DARPA specified that MAVs should have no dimension larger than 15.2 cm, and
be capable of deploying a micro payload to a remote location. To the knowledge of the
author, no vehicles meeting the DARPA MAV specifications are currently in service.
However, products in current production (such as the AeroVironment WASP (Figure
1.1) with a 33 cm wingspan [1]) and in active development (such as the Honeywelldeveloped MAV (Figure 1.2) under the Boeing Future Combat Systems Program [2])
are based upon research conducted for the DARPA MAV program. In 2005, DARPA
created another program to encourage development of what they have called Nano Air
Vehicles (NAVs) [11]. The specifications for this program include a gross take-off weight
of 10 grams or less, a maximum dimension of 7.5 cm, and the ability to hover as well as
dash 1000 m at a speed of 5 to 10 m/s.
3
Fig. 1.1. AeroVironment WASP micro air vehicle prototype (Ref [1])
4
Fig. 1.2. Honeywell micro air vehicle (Ref [2])
5
The resources expended by DARPA on the MAV and NAV projects demonstrates
the perceived need for vehicles capable of deploying sensors indoors, or in highly confined
outdoor areas. The ability to fly with very low ground speed and/or hover is crucial for
success in these types of missions. In the minds of most designers, these requirements
eliminate the conventional fixed wing aircraft configuration from consideration because
they are unable to generate flow over their wings without the vehicle having some ground
speed. It is thought that rotary wing and flapping wing configurations will be significantly more effective designs for such applications. At least one design has already
demonstrated that flapping wing flight on the MAV scale is possible with our current
level of electronics and manufacturing technology. This design, shown in Figure 1.3,
was built for DARPA by AeroVironment in the late 1990s, and has successfully flown in
outdoor conditions under radio control.
Fig. 1.3. AeroVironment/Caltech Microbat flapping wing MAV (Ref. [3])
6
Flapping wing designs hold great interest in the community, as their capability for
complex motion is thought to hold great potential for the exploitation of unsteady aerodynamic effects. It is known that the configuration is effective, since it is the one which
has evolved in hovering and slow flying creatures in nature. However, due to their size
and speed, hummingbirds, insects, MAVs and NAVs operate in an aerodynamic regime
which is not well understood: low Reynolds number, unsteady flows. The Reynolds number is defined as Re = ρV L/µ, where V is the freestream velocity, L is a characteristic
length, ρ is the density, and µ is the coefficient of viscosity of the fluid. The Reynolds
number is a representation of the ratio of inertial forces to viscous forces in the flow,
therefore lower Reynolds number flows are influenced more heavily by viscous forces.
The level of unsteadiness in a problem is often measured using the reduced frequency,
which is defined as k = ωc/(2V ), where c is the chord length, and ω is the angular velocity of the unsteady motion. Problems having 0 ≤ k ≤ 0.05 are considered quasi-steady,
with small contributions from unsteady effects. Values of k larger than 0.05 indicate
unsteady flow, and the contributions from unsteady effects may not be neglected. Flows
with k > 0.2 are considered highly unsteady, and will be dominated by the unsteady
effects [12, p. 306]. As illustrated in Figure (1.4), small flapping wing vehicles will operate at reduced frequencies for which unsteady effects are non-negligible, and at Reynolds
numbers below those where designers have the most experience. Following the success
of Lilienthal’s gliders in the 19th century and the famed flight of a powered airplane
by the Wright brothers in 1903, there has been great emphasis on the development of
human piloted aircraft. The high demand for recreational, transportation and military
applications of aircraft have driven the majority of aeronautical engineering research to
7
focus on achieving higher flight speeds and capacity for payloads at least as large as a
human being. For these reasons, relatively little attention has been placed on the study
5
of flight at chord-based Reynolds numbers below 10 . While the rotorcraft community
has addressed the issue of unsteady flow, the majority of its focus has also been on higher
Reynolds number applications.
Fig. 1.4. Scope of aeronautical knowledge (Ref. [4])
The ability to predict the aerodynamic behavior of flapping wings in low Reynolds
number, unsteady flows may help to answer to many questions in MAV/NAV design and
the study of bird and insect flight. For example, are the complex wingbeat patterns of
the hummingbird optimal for hovering and forward flight performance? What is the ideal
wing planform geometry? When is it beneficial to retract the wings during the upstroke?
8
For practical use in the conceptual and preliminary design of flapping wing vehicles,
an efficient and easy-to-use tool providing estimates of aerodynamic force and moment
coefficients would be of great benefit. Such a tool would allow for selection of design
variables through trade studies and use in conjunction with optimization algorithms,
without a large time investment. What follows is the evaluation of such a tool for its
suitability to the design of small flapping wing aerial vehicles. An efficient computational
tool based on blade element theory (BET) has been developed to approximate the forces
and moments generated by flapping wing vehicles. While the basis of this unsteady
model is an inviscid approach, it allows for the modeling of some low Reynolds number
effects. This tool accommodates complex wing motions with spanwise varying twist,
and produces force and moment time history predictions. The assumptions involved
in this approach should allow for accurate prediction of attached flow situations (in
forward flight or hovering) and with the inclusion of dynamic stall modeling, should also
provide useful predictions for higher reduced frequencies, where separated flow would be
expected.
9
Chapter 2
Overview of Previous Research
For thousands of years, the graceful flight of insects and birds has captivated
those who have witnessed it. From those who dream of tasting the freedom of flight in
man-made vehicles, to those who hope to further our understanding of some of nature’s
most fascinating creatures, a great number of individuals have documented their study
of flapping wing flight. There has been great variety in the study of flapping flight,
since those pursuing the topic have come from fields such as biology, aerodynamics
and aviation. These studies have been undertaken from analytical, experimental and
computational standpoints. The following chapter highlights some of the work which
forms the basis of our current understanding of flapping flight.
2.1
Analytical and Computational Approaches to Flapping Flight
Although it is clear that humans have observed birds in flight and envisioned
flying machines mimicking them for thousands of years, it was not until the nineteenth
century that systematic scientific study of flapping flight was well documented. Otto
Lilienthal, the great pioneer in human flight, began his flying experiments with his
brother Gustav as a child in the mid-1800s. While Lilienthal may be most famous for
his glider experiments in the 1890s which proved that heavier than air flight is possible
without the use of flapping wings, much of his knowledge in aeronautics was derived
10
from experimentation with flapping wings and observation of birds. In 1889, Lilienthal
published a book describing his experiments and detailing his predictions for the energy
required for flapping wing flight [5]. Perhaps the most significant of his findings was the
benefit of using cambered airfoils instead of flat plates, which he discovered by testing
models on a rotating apparatus as shown in Figure 2.1.
Fig. 2.1. Lilienthal’s rotating test apparatus (Ref. [5])
Shortly after his death in a gliding accident in 1896, Lilienthal served as an inspiration to the Wright brothers in their successful effort to develop powered human
flight. Following the sustained flight of the Wright Flyer in 1903, the majority of those
researching aeronautics were focusing their efforts on the development of fixed wing aircraft. The one notable attempt to advance the study of flapping flight before the 1930s
11
was that of Sir Gilbert Walker. In 1925, Walker published a theoretical analysis of flapping flight, which was essentially a simplified blade element theory [13]. Walker’s theory
assumes constant angular velocity of the flapping wings, a flapping plane that is normal
to the freestream velocity, and a constant pitch angle of the blade elements over the
each half-stroke (upstoke and downstroke). Steady state airfoil data are then used to
calculate average lift and thrust forces over a flapping cycle. No correction is made in
Walker’s theory for the induced effects of a finite wingspan. Due its simple geometric
analysis combined with steady state airfoil data, Walker’s theory provides reasonable lift
and thrust predictions only for low reduced frequency flapping flight.
By the 1930s, fixed wing aircraft had improved greatly in performance. At this
point, the development of unsteady aerodynamic theory became important for the investigation of flutter. In 1935, Theodorsen published an analytical approach for estimating
the unsteady lift and moment on harmonically oscillating airfoils [14]. This method was
derived for inviscid, incompressible flow, and assumes that the wake extends downstream
to infinity from the airfoil trailing-edge in a single plane. Theodorsen’s theory has been
widely used in the study of flutter, rotorcraft aerodynamics and flapping flight from the
1930s up to the present day. Shortly after Theodorsen developed his theory for unsteady
lift, it was extended by Garrick [15] to determine the propulsion effect of a flapping wing.
Garrick drew upon the work of Theodorsen, as well as von Karman and Burgers [16] in
the development of his propulsion formulas. Although unsteady aerodynamic theory was
developing in parallel, the quasi-steady approach to the study of flapping flight remained
popular in the next few decades. A quasi-steady method similar to that of Walker was
12
proposed in 1942 by von Holst and Kuchemann [17]. This approach also greatly simplified the wing motion, and produces reasonable estimates of aerodynamic quantities
only in the case of low frequency flapping. The previously neglected issues of induced
flow and arbitrary flapping planes were addressed in the method proposed by Osborne
in 1951 [18]. With the study of insects in mind, Osborne assumed rigid wings which did
not bend or twist, but did allow for tilting of the wing flapping plane with respect to the
freestream.
In the late 1960s, a new approach for the analysis of power requirements for
flapping flight was devised. Noticing common traits between helicopters and most small
to medium-sized birds, Pennycuick began using general helicopter theory for the study
of bird flight. In 1968, he published an analysis of the power required by the pigeon
(Columba livia) and ruby-throated hummingbird (Archilocus colubris) for hovering and
forward flight [19]. Pennycuick’s method estimated the induced power, profile power,
and parasite power separately. The induced power was calculated by treating the area
swept out by the flapping wings as an infinitely thin actuator disk, across which the flow
is accelerated (momentum theory). Profile power, or the power required to overcome the
profile drag of the wings was estimated by applying steady state airfoil data to the wing
with strip theory. The parasite power, or the power required to overcome the drag of the
bird’s body was based on a simple frontal area calculation. Pennycuick concluded that
this approach produced results consistent with experimental data available at the time.
Although this approach was used in numerous studies of avian flight in the 1970s, it is
dismissed by Rayner in 1979 [20], despite his admission that it produced sensible results.
Rayner felt that flapping wing flight must be treated as an unsteady problem, making
13
quasi-steady and actuator disc solutions inadequate. His approach to the problem utilizes
stacks of vortex rings, where each ring has been created by a single wing stroke. A
detailed description of the application of vortex ring theory to hovering and forward
flight was also published by Rayner in 1979 [21].
Due to the growth of computers in power and accessibility, computational studies
of flapping flight became very common in the 1990s. In hopes of improving predictive
capabilities over quasi-steady and momentum theory approaches, many of these computational models account for the unsteady nature of the problem. In 1993, DeLaurier
developed a computational model intended to be used in the design of ornithopter wings
[22]. This model is based on strip theory, with the use of the modified Theodorsen
function for finite wings as proposed by Jones in 1940 [23] to account for the unsteady
wake effects. DeLaurier’s model also includes approximations for viscous effects in the
form of a leading-edge suction model, as well as a dynamic stall model. In combination
with a computational structural model, DeLaurier used this tool to design an efficient
ornithopter wing in 1993 [24]. Subsequent wind tunnel testing of the wing showed good
agreement with the computational results. Another approach was taken in 1996 by
Vest and Katz [25]. Their computational tool utilizes an unsteady potential flow panel
method, and was shown to produce reasonable results for the attached flow regime.
A number of more recent studies of flapping flight have utilized the vortex lattice
method (VLM). A description of this method is given by Katz and Plotkin [26, p. 613].
In response to the common argument that flapping flight is more efficient than propellerdriven flight at low Reynolds number, Hall and Hall [27] analyzed the efficiency of a rigid
rectangular wing flapping about a longitudinal axis parallel to the freestream. Their
14
analysis was based on minimum induced loss propeller theory, and utilized an unsteady
VLM with estimated profile (viscous) losses to predict the minimum power required for
the generation of lift and thrust with flapping wings. Hall and Hall reported that their
study did not support the argument that flapping flight is more efficient than propellerdriven flight at low Reynolds number, and asserted that biological systems most likely
utilize flapping flight due to limitations on their actuators. In 2004, Fritz and Long [28]
implemented the unsteady VLM with free-wake relaxation, vortex stretching, and vortex
dissipation effects. Their approach allows for the complex motion of finite flapping wings
including dynamic twisting, which makes it well suited for the study of bird flight. Fritz
and Long also provide a detailed survey of much of the work that has been performed
by biologists in the study of bird flight. This tool was used to illustrate the effect of
varying flapping parameters such as flapping amplitude, dynamic twisting amplitude,
and flapping frequency.
2.2
Observation and Experimentation
Many other attempts at furthering our understanding of flapping flight have fo-
cused on the observation of birds and experimental investigation of flapping systems.
Many of these studies have come from outside the field of engineering. For example,
one of the best sources of information on hummingbirds was published by Greenewalt
in 1960 for the American Museum of Natural History [29]. In his quest for high quality photographs of hummingbirds, Greenewalt compiled not only a large set of detailed
photographs, but a very useful data set. He compiled data on the size, weight, flapping frequency and energy output of many species of hummingbirds, as well as other
15
flying creatures. From the wealth of photographic hummingbird data taken in both the
wild, and in a wind tunnel, Greenewalt produced illustrations of the wingbeat patterns
used for hovering, maneuvering, and forward flight, and recorded the angle of the wing
flapping plane for a variety of flight speeds.
Beginning in the 1950s, and continuing for a number of decades, Weis-Fogh carried
out experimental studies of flying creatures. Based on a survey of data in the field, as
well as his own experiments, Weis-Fogh made a distinction in 1975 between creatures
which require “unusual” aerodynamic mechanisms for flight, and those that do not [30].
His observations indicated that most hovering creatures hover in a manner very similar
to the hummingbird. These creatures generally hover with their longitudinal axis nearly
vertical, utilize rapid and extensive wing twisting, and without bringing their wingtips
close to each other at the end of the upstroke or downstroke. Weis-Fogh asserted that the
hovering of these creatures could be accounted for with lift coefficient values predicted
by steady-state aerodynamics at the appropriate Reynolds numbers. However, a few
notable exceptions such as butterflies, wasps and hover flies seemed to generate average
lift coefficient values much higher than could be explained by conventional aerodynamics.
This prompted Weis-Fogh to propose his famous clap-fling theory to explain the flight of
these insects. In the same paper, considering the biological constraints on the mechanical
power that can be delivered by muscles and applying simple momentum theory, he
estimated that no animal with a mass greater than 100 grams should be able to hover
continuously.
Since the introduction of Weis-Fogh’s clap-fling theory, understanding of the
mechanisms responsible for the exceptionally high lift produced by insects has been
16
greatly advanced with experiments utilizing flow visualization. Flow visualization has
often been used in the study of birds, most notably by Spedding, who has done a great
deal of work in wake visualization and measurement of forces using wake measurements
[31], [32], [33]. Confirmation of the presence of a prominent leading edge vortex (LEV) in
a wide variety of low Reynolds number insects has been provided by experiments using
modern flow visualization techniques such as digital particle image velocimetry (DPIV).
Notable examples include the studies of Drosophila (fruit flies) by Dickinson and Götz
[34], and of hawkmoths by Ellington et. al [35]. These studies show that a spiral LEV is
created by leading-edge separation caused by flapping the wings at high angle of attack.
This mechanism produces both high lift and high drag. However, Ellington observes that
high drag is not detrimental to insects, as they are able to orient themselves such that
the resultant force vector is pointed in a direction which provides the desired lift and
thrust. Approaching the problem from an engineering perspective (with MAV design in
mind), a number of experiments have recently been performed on insect-based wings by
Singh et al. at the University of Maryland [36], [37]. They highlight the fact that insect
muscles do not extend past the wing root, making insects unable to actively control their
wing shape as birds do. Their results confirm the generation of high lift with the LEV,
and also indicate that the elastic modes of insect wings have significant effects on the
vertical force generated. Maryland researchers have also recently performed experiments
to investigate the relative efficiency of flapping wings and rotary wings at low Reynolds
numbers. Tarascio and Chopra [38] showed that flapping configurations hovered more
efficiently than rotary wing configurations over the Reynolds number range of approximately 5,000 to 21,000. Using these experimental studies, they also proposed a thrust
17
augmented entomopter design which utilizes flapping wings for efficient hover, and a
ducted propeller for propulsion in forward flight.
Despite the advances in understanding the high lift generated by insects, there
remains uncertainty in the field regarding the flight performance of the hummingbird. In
2006, Ellington summarized insect LEV studies, as well as force measurement and flow
visualization studies performed on hummingbird wings in a propeller rig [39]. An important observation was that the hummingbird wing produced lift coefficients comparable
to the insect wings, but with significantly less drag. Flow visualization showed evidence
that the LEV on hummingbird wings is much weaker than that of the insects. These results have prompted further study on the effect of details such as the effect of protruding
feathers, and the possibility of flow through the porous feathers of the hummingbird.
2.3
Flapping Wing Vehicles
Although we do not have a complete understanding of all the mechanisms which
govern flapping wing flight, a number of man-made flapping wing vehicles have been successfully flown. Free flight ornithopters powered with rubber bands are routinely flown
by model aircraft enthusiasts, and have their own category in the national indoor competition held yearly by the Academy of Model Aeronautics [40]. A similar rubber-powered
model has been built by Michelson [41] and his research team as a proof of concept for
their entomopter. The final entomopter will utilize a reciprocating chemical muscle for
power, rather than a rubber band. The wing design was based on the hawkmoth wing,
and represented a collaborative effort between Michelson’s team and Ellington’s group,
18
due to their extensive study of hawkmoth aerodynamics. Based on their research experience, Michelson and Naqvi offer a set of guidelines for the design of flapping wing aerial
vehicles. They stress the distinction between “biomimetic” and “biologically-inspired”
vehicles, pointing out that direct copying of nature does not always produce the optimal
solution to problems.
The most successful remotely piloted flapping wing vehicles are examples of
biologically-inspired designs, in that they share some traits of birds and insects, but
also exhibit differences which were driven by the necessity for practical implementation.
In 1991, DeLaurier flew a remotely piloted ornithopter for 2 min 46 s [6]. This aircraft
was a quarter-scale proof-of-concept model for DeLaurier’s piloted ornithopter design,
and is shown in Figure 2.2. In the late 1990s, DARPA funded the development of a flapping wing MAV as a joint project between Caltech and AeroVironment [3]. The product
of this research was the Microbat, shown in Figure 1.3. The Microbat has gone through
a number of design iterations, being originally operated in free-flight using a capacitor
power source. The final iteration includes radio control, and a Lithium-polymer battery.
The improvement in power source technology allowed for flight durations to jump from 9
seconds to greater than 6 minutes. In 2002, Jones and Platzer [7] completed a 15 minute
flight of their biologically-inspired, radio-controlled MAV using a Lithium-polymer battery. This MAV is unique in that it utilizes a biplane flapping configuration to provide
propulsion, while deriving lift from a more conventional fixed wing. Jones and Platzer
elected to use this configuration in order to gain the benefit of flapping over the whole
wing, as opposed to bird-like flapping configurations which have zero amplitude at the
wing root.
19
Fig. 2.2. DeLaurier’s remotely piloted ornithopter (Ref. [6])
20
Fig. 2.3. Jones and Platzer’s biologically-inspired MAV (Ref. [7])
21
While all of the remotely piloted flapping wing aircraft mentioned above draw on
concepts inspired by nature, they exhibit differences from birds and insects as dictated by
the technology available at the time of their construction. It is known that birds utilize
asymmetric wing flapping for lateral-directional control. However, the aerodynamics of
symmetric wing flapping is a difficult problem in itself, therefore asymmetric flapping has
not yet received a great deal of study. As a result, none of these aircraft derive lateraldirectional control from their flapping wings. DeLaurier’s ornithopter and the Microbat
both utilize conventional horizontal and vertical tail surfaces for longitudinal and lateraldirectional control, while Jones and Platzer use a vertical surface for lateral-directional
control.
While this review of previous work on the topics of flapping flight and flapping
wing MAVs does not represent an exhaustive survey of the field, it attempts to cite
the most influential developments. Although our knowledge of the aerodynamics of
flapping flight has advanced tremendously since the mid-1800s, there remains much to
be learned. The development of small, lightweight electronics and batteries, along with
the efforts of the many biologists and engineers mentioned above have made possible
the flight of simple remotely piloted MAVs. The opportunity for significant performance
improvement exists though research and development based on the solid foundation
which has been provided for us.
22
Chapter 3
Blade Element Approach
Many computational tools are available for aerodynamic analysis, but unfortunately, the majority of them will not adequately capture the behavior of the flow in
the highly unsteady, low Reynolds number regime of interest for flapping wing vehicles.
While computational fluid dynamics (CFD) tools solving the Navier-Stokes equations
would be capable of accurately capturing the physics of these flows, they require manual creation of grids, and are too computationally expensive for use in conceptual or
preliminary design. A well validated tool based on blade element theory (BET) offers
the potential to model the unsteady effects in the flow around flapping wing MAVs with
enough accuracy for use in design applications, at low computational expense.
3.1
Blade Element Theory
According to Leishman [12, p. 78], BET was first described by Drzwiecki at the
beginning of the 20th century as a method of analyzing airplane propellers. To this
day, computational tools based on BET are commonly used for aerodynamic design and
analysis by the rotorcraft community. While propellers and helicopter rotors may have
geometric similarities such as high aspect ratio and spanwise varying twist and chord
distributions, they differ in their interaction with the flow. Airplane propellers operate
almost exclusively with the inflow direction perpendicular to the propeller plane, whereas
23
helicopter rotors may encounter the freestream at an angle (in forward flight for example). In addition, helicopter rotors may operate in much closer proximity to their own
wake, and can intersect some significant wake features such as wingtip vortices. Outside
of a few aerobatic applications, airplane propellers are operated such that their wake is
quickly translated downstream, and will not be re-encountered by the propeller. Because
rotors are also used as control surfaces on helicopters, they include significantly more
complex mechanics than propellers, which allow them to pitch, lead-lag and flap as they
vary their loading. Although BET was developed for propellers, it has enough flexibility
to model the complex motions of helicopter rotors and the flow they encounter. Many
of the features which make this approach attractive for use in rotor design are similarly
advantageous for flapping wing design. These features include the ability to treat spanwise varying wing geometry distribution, arbitrary wing motion, and low computational
expense.
The basis of BET is the representation of a propeller or rotor blade as a set of
two-dimensional elements distributed radially outward from the hub. The airfoil section,
twist angle, and chord length may vary from element to element. Given this information,
the local angle of attack of each element may be determined for a prescribed inflow
velocity and blade angular velocity. The lift, drag and moment characteristics of each
airfoil section may then be used to calculate the forces on each blade element, and the
total force on the blade is the integrated effect of the elements. Tip loss models are
usually incorporated to account for the effects of the vortex generated at the tip of the
blade.
24
While the problems of flapping wing and propeller design have a number of similarities, they also exhibit differences which dictate the appropriate structure of BET tools
for each application. In order to create a useful tool for the analysis of flapping flight
and design of flapping wing configurations, the most basic form of BET described above
has been augmented specifically for these applications. Due to its intended use for the
study of flapping flight, and its basis in the indicial response (IR) method, the resulting
computational tool has been named FlapIR, and will be described in the remainder of
this chapter.
The most salient difference between propellers and flapping wings lies in the movement of the aerodynamic surfaces. The blades of a propeller in a steady state flight condition will be rotating at a fixed angular velocity about an axis fixed with respect to the
aircraft. In the case of a flapping wing configuration, even in this simplest mode of flight,
the angular velocity and axis of wing rotation may vary with time inside of a single flapping cycle. Therefore, an analysis tool for flapping configurations must allow convenient
specification of the complex flapping motions, and produce time-accurate predictions of
the aerodynamic forces and moments. For this reason, BET has been implemented in
the form of a simulation in the time domain. This allows for the prediction of time
histories of force and moment on the vehicle for analysis, and access to average force and
moment values over a number of flapping cycles for use in design and optimization.
Modern construction techniques for both manned and unmanned aircraft are unable to produce perfectly rigid wings. In fact, some man-made flapping wing aircraft,
as well as the flapping wing flyers in nature utilize highly flexible wing surfaces. Given
the lack of perfectly rigid wing surfaces, the flapping flight problem is necessarily an
25
aeroelastic problem. Real flapping wing systems will encounter interactions between the
aerodynamic loads and structural deformation of the wings. The intent of the FlapIR
approach is to produce a useful tool for the aerodynamic design of flapping wing configurations. This may include optimization of wing geometry, or wingbeat patterns. In
the interest of speed and simplicity, the FlapIR approach will only model the unsteady
aerodynamics of flapping wings having prescribed motion and deformation.
In order to simplify the problem of describing the wing flapping motion, it was
assumed that the root of each wing is fixed to the main vehicle body at a single point,
about which the wing may rotate arbitrarily. The wing is then represented as a single
line extending from the wing root to the wing tip, upon which the blade elements are
distributed evenly. Each blade element may have a time-varying angle of rotation about
this axis as a function of its spanwise location. Therefore, the FlapIR method models a
rigid wing with dynamic spanwise varying twist. The position of the wing twisting axis is
described with respect to the vehicle body axes by two parameters which are meaningful
to the designer: flap angle, β, and sweep angle, Λ, as shown in Figure 3.1. While this
model allows for approximate representation of the flapping wing creatures found in
nature, as well as many man-made flapping configurations, it excludes the treatment of
flapping configurations in which the entire wing translates with respect to the body, such
as those investigated by Jones et al [42].
Description of the wing motion is achieved through specification of angles β and Λ
as functions of time, as well as the blade element twist relative to the freestream, α, as a
function of both time and spanwise location. While these functions may incorporate any
desired level of complexity, they have been defined as the following sinusoidal functions
26
for all studies presented here:
β(t) = β
base
Λ(t) = Λ
α(t, y) = α
base
+α
pitch
sin(ω
base
pitch
+ β sin(ω
0
+ Λ sin(ω
0
t+φ
pitch
f lap
t+φ
sweep
f lap
t+φ
0
pitch
0
and α are the amplitudes, ω
0
are the angular velocities, and φ
f lap
,φ
sweep
),
) + (r(y)/R)α sin(ω
base
0
(3.1)
sweep
where R is the distance from wing root to wing tip, β
offsets and β , Λ , α
),
,φ
pitch
f lap
and φ
,Λ
,ω
(3.2)
base
sweep
twist
t+φ
), (3.3)
and α
are the
twist
twist
base
,ω
pitch
and ω
twist
are the phase angles of
the flapping, sweeping, uniform pitching and twisting motions, respectively.
The position and velocity vectors of the n
th
blade element with respect to the
vehicle body may then be determined with a transformation from spherical to Cartesian
coordinates as follows:
x = r cos (β) sin (Λ) ,
n
n
y = r cos (β) cos (Λ) ,
n
n
z = −r sin (β) ,
n
ẋ = r
n
ẏ = r
n
n
h
i
Λ̇ cos (β) cos (Λ) − β̇ sin (β) sin (Λ) ,
h
i
−Λ̇ cos (β) sin (Λ) − β̇ sin (β) cos (Λ) ,
n
n
ż = −r
n
h
n
i
β̇ cos (β) ,
(3.4)
(3.5)
27
where r is the distance from the wing root to the blade element. The time derivatives
n
of flap and sweep angle are determined with fourth-order accurate central differences:
β
− 8β
+ 8β
−β
dβ
4
t−∆t
t+∆t
t+2∆t
= t−2∆t
+ O(∆t ),
dt t
12∆t
(3.6)
Λ
− 8Λ
+ 8Λ
−Λ
dΛ
4
t−∆t
t+∆t
t+2∆t
= t−2∆t
+ O(∆t ),
dt t
12∆t
(3.7)
where ∆t is the timestep length. The plunge velocity, ḣ, is the component of the airfoil
velocity normal to the chord line:
~ · ~n
ḣ = −V
n
n
n
(3.8)
E
D
~ = ẋ , ẏ , ż , and ~n is defined as a unit vector perpendicular to the chord
where V
n
n
n
n
n
line, in the direction of the upper airfoil surface. The remaining kinematic parameters
(α̇ , α̈ , and ḧ ) are determined with fourth order central differences similar to those
n
n
n
in Equation (3.6).
28
Fig. 3.1. Body axes and spherical coordinate system for flapping wing
29
3.2
Unsteady Aerodynamic Modeling
Prediction of the unsteady aerodynamic behavior of two-dimensional airfoil sec-
tions is a topic that has received serious attention since the early half of the 1900s. Such
capability is essential to the field of aeroelasticity, which is of great importance to the
fixed wing aircraft community in the study of flutter, and to the rotary wing community
for the analysis of rotorcraft blades. Due to the high speeds often achieved by modern
fixed wing aircraft, and the high tip speeds of rotorcraft blades, designers in these fields
must consider the effects of compressibility. In the case of small scale flapping wing
aircraft, the effects of compressibility will not commonly be encountered. These vehicles
will be designed for flight speeds well below even one-third the speed of sound, and given
the small dimensions of their wings, would require extremely high flapping frequencies
to generate tip speeds in the compressible regime. For this reason, FlapIR will treat the
flow around flapping wing configurations as incompressible potential flow.
The unsteady force and moment on an airfoil in abritrary motion is the sum of
contributions from two sources: the non-circulatory or apparent mass effect and the
creation of circulation around the airfoil. The former is a result of acceleration of the
airfoil with respect to the flow surrounding it, while the latter is the familiar mechanism
responsible for the generation of lift in steady flows.
30
3.2.1
Non-Circulatory Lift
Because FlapIR models incompressible flow, the problem of determining the ap-
parent mass effect is simplified. As explained by Lomax [43], the sudden upward displacement of a wing element generates a compression wave above the wing, and a rarefaction
wave below the wing. In reality, these waves will propagate away from the airfoil at
the speed of sound, however, the speed of sound for an incompressible flow is effectively
infinite. As a result, the change in boundary conditions is propagated instantaneously
to all fluid particles surrounding the airfoil. Therefore, the apparent mass effect in incompressible flow is a function only of the instantaneous motion of the airfoil, and does
not need to account for time history effects. A more complete discussion of the apparent mass effect for incompressible flow is given by Bisplinghoff [44, pp. 200-204, 263].
Long [45] and Long and Watts [46] describe a method for time-domain simulations of
compressible arbitrary motion aerodynamics. The apparent mass contributions to the
lift and moment on a thin airfoil as given by Bisplinghoff are
l (t) = πρb
nc
2
h
ḧ + V α̇ − baα̈
i
∞
1
2 1
2
m (t) = πρb baḧ − V b
− a α̇ − b
+ a α̈
nc
∞
2
8
2
where V
∞
(3.9)
(3.10)
is the freestream velocity encountered by the vehicle, b is the semichord, a is
the location of the pitch axis relative to the midchord measured in semichords, and ḧ is
the plunge acceleration.
31
3.2.2
Circulatory Lift
In contrast to the non-circulatory effects, the circulatory lift and moment contri-
butions are influenced by the shed wake. The presence of circulation in the wake induces
flow components normal to the freestream velocity at the location of the airfoil. The
effect of this induced flow is to alter the magnitude and direction of the relative wind
encountered by the airfoil, and hence change the effective angle of attack. One common
approach to modeling these time-history effects in rotorcraft aerodynamics is the use
of indicial functions, which is easily adapted for use in conjunction with BET. Indicial
functions are used to model the response to a step change in some quantity, which is
applied instantaneously at time zero and held constant afterward. These functions may
be derived analytically, or determined from computational or experimental data.
One well known example of an indicial function was derived by Wagner [47] in
1925. Wagner’s function, φ(s), models the circulatory lift response of a thin airfoil
undergoing a step change in angle of attack in incompressible flow as a function of
reduced time, s, where
s=
Z
2 t
V dt.
c 0
(3.11)
The reduced time parameter is commonly used in unsteady aerodynamics, as it represents
the distance the airfoil has traveled through the flow in terms of semichords. It is a
useful indicator of the relative position of the airfoil and features of its shed wake. The
application of Wagner’s function to a time domain simulation of the arbitrary motion of
an airfoil is accomplished by treating each timestep as a step change in the angle of attack
and pitch rate of the airfoil. The effect of the shed wake over time may then be captured
32
through the superposition of these indicial responses with the Duhamel integral. For a
general linear time-invariant system, the Duhamel integral can be written as:
Z t
df
y(t) = f (0)φ(t) +
φ(t − σ)dσ
0 dt
(3.12)
where y(t) is the system output, f (t) is the forcing function and φ is the indicial response
of the system. In the case of an airfoil undergoing arbitrary changes in angle of attack
and pitch rate, the effective angle of attack incorporating the time-history effects of the
shed wake is given by
Z s dα (σ)
eq
α
(t) = α (0)φ(s) +
φ(s − σ)dσ
unsteady
eq
dt
0
(3.13)
where α (t) is the equivalent geometric angle of attack at the three-quarter chord point
eq
of the airfoil. Under arbitrary motion, α (t) is determined (with small angle approxieq
mation) by
α (t) = tan
eq
w(t)
V
∞
!
≈
w(t)
V
(3.14)
∞
where w(t) represents the instantaneous vertical velocity of a fluid particle in contact
with the three-quarter chord point of the airfoil. Airfoil plunging velocity, pitch angle
and induced camber from pitch rate contribute to w(t) as given by Bisplinghoff [44, p.
282]:
1
− a α̇ .
w(t) = ḣ + V α + b
∞
2
(3.15)
33
These three contributions may be combined into a single term for use in the Duhamel
integral because the indicial response functions for angle of attack and pitch rate disturbances are identical in the case of incompressible flow, according to Leishman [12, p.
348]. The indicial response function φ(s) used by FlapIR is Wagner’s function. While
Wagner’s function is known exactly, its form is not convenient for use in numerical simulations. As a lower computational expense alternative to a table lookup of the exact
Wagner’s function value, the R.T. Jones [48] exponential approximation is used:
−0.0455s
φ(s) ≈ 1.0 − 0.165e
3.2.3
−0.3s
− 0.335e
.
(3.16)
Time Varying Incident Velocity
One assumption inherent in Equation (3.13) is the convection of the shed wake
behind the airfoil with the direction and magnitude of V . However, the lead-lag or
∞
sweep oscillation of the wings will create a non-uniform variation in the distance between
the airfoil sections and shed vorticity behind them. It was shown by Leishman [12, pp.
333-336] that inclusion of the freestream velocity in the angle of attack derivatives in both
the non-circulatory and circulatory terms can model the effect of non-uniform incident
velocity. This modification is applied to the FlapIR approach by using the local relative
wind magnitude at each blade element, V , resulting in the following expressions for lift
n
34
and moment coefficient at the n
th
element:


d V α
πb
n n
− b aα̈ 
c
(t) = 2n ḧ +
n
n n
l n
dt
V
n

Z
d
V
α
(σ)
sn


n
eq n
1
 α
(0)φ(s
)
+
φ(s − σ)dσ 
 eq n
,
n
n
V 0
dt

!
dc
+
l
dα
steady
n
|
{z αunsteady
}
n
(t)
(3.17)
c
πb
m n
(t) =

n b aḧ − b
2
n n
n
V
n
dc
+
!
m
dα
steady

d V α
1
1
2
2
n n
−a
−b
+ a α¨ 
n
2
dt
8
n


Z s d(V α
)(σ)
n
1
n
eq n


φ(s − σ)dσ 
 αeq (0)φ(sn ) +
n
V 0
dt
n
n
|
π
+
2
"
b α˙ #
1
−a n n ,
2
V
{z αunsteady
}
n
(t)
(3.18)
n
where the derivatives of c and c
l
m
with respect to α represent the steady state lift and
moment slopes for the airfoil based on experimental or computational data. Internally,
FlapIR determines the circulatory lift and moment contributions using lookup tables
with α
unsteady
as the input parameter.
35
3.2.4
Solution of the Duhamel Integral
Numerical solution of the Duhamel integral is accomplished in FlapIR using the
recurrence solution outlined by Leishman [12, pp. 323-340]. Using the exponential
approximation to Wagner’s function of the form:
−b1 s
φ(s) = 1.0 − A e
1
−b2 s
−A e
2
,
(3.19)
the Duhamel integral may be expanded to
Z s d Vα
(σ)
1
eq
φ(s − σ)dσ
α
(s) = α (s )φ(s) +
unsteady
eq 0
V s
dt
0
−b1 s
= α (s ) − A α (s )e
eq
0
1 eq
0
Z
1 s (s)
d Vα
eq
V s
0
Z s d Vα
−b (s−σ)
eq
(σ)e 2
dσ.
ds
s0
−b2 s
− A α (s )e
2 eq
0
A
A Z s d V αeq
−b (s−σ)
− 1
(σ)e 1
dσ − 2
V s
ds
V
0
+
(3.20)
Because the unsteady behavior near the beginning of the first flapping cycle is not of pri−b1 s
mary concern, the short term transients due to the initial angle of attack, A α(s )e
1
−b2 s
and A α(s )e
2
0
0
, may be neglected, and the Duhamel integral rewritten as
α
unsteady
(s) = α (s) − X(s) − Y (s),
eq
(3.21)
36
where
A Z s d V αeq
−b (s−σ)
X(s) = 1
(σ)e 1
dσ,
V s
ds
(3.22)
A Z s d V αeq
−b (s−σ)
(σ)e 2
dσ.
Y (s) = 2
V s
ds
(3.23)
0
0
Taking s = 0 and a possibly nonuniform timestep ∆s, X may be written at the next
0
timestep s + ∆s:
A Z s+∆s d V αeq
−b (s+∆s−σ)
(σ)e 1
.
X (s + ∆s) = 1
V 0
ds
(3.24)
Splitting this integral at the current timestep gives
Z
d
V
α
A −b ∆s s
−b (s−σ)
eq
X (s + ∆s) = 1 e 1
(σ)e 1
V
ds
0
A Z s+∆s d V αeq
−b (s+∆s−σ)
(σ)e 1
dσ
(3.25)
+ 1
V s
ds
Z
d
V
α
s+∆s
A
−b ∆s
−b (s+∆s−σ)
eq
+ 1
= X(s)e 1
(σ)e 1
dσ
V s
ds
−b1 ∆s
= X(s)e
+ I.
(3.26)
The above recursive formula may be used to determine X at the next timestep, (s + ∆s),
by adding the increment I to the current value of X. Rearranging the expression for I
37
gives:
Z
d
V
α
s+∆s
A
−b (s+∆s−σ)
eq
I= 1
(σ)e 1
dσ
V s
ds
A −b (s+∆s) Z s+∆s d V αeq
b σ
= 1e 1
(σ)e 1 dσ.
V
ds
s
The derivative of V α
eq
(3.27)
with respect to s is evaluated with a backward-difference approx-
imation. Because ∆s may vary between timesteps, a second-order accurate backwarddifference which accounts for unequal stepsizes as described by Hoffmann and Chiang
[49] is used:
2
ζ(ζ
+
2)
α
−
(1
+
ζ)
α
+
α
dα eq i
eq i−1
eq i−2
eq ,
=
ds ζ(1 + ζ)∆s
(3.28)
i
where i denotes the current timestep, i − 1 and i − 2 are the two previous timesteps, and
ζ is the ratio of changes in s:
s
−s
i−2
ζ = i−1
.
s −s
i
(3.29)
i−1
In order to minimize error in the numerical integration of Equation (3.27) from the use
of larger timesteps, Simpson’s rule is used to approximate the integral. Locally, the error
38
5
in this integration is on the order of (∆s) . Using Simpson’s rule:
−b1 (s+∆s)
dα
I=A e
1
A
=
ds 1
b s
b (s+∆s/2)
b (s+∆s)
e 1 + 4e 1
+e 1
6
eq s+∆s
2
ζ(ζ + 2) α
−b1 ∆s/2
−b1 (s+∆s)
eq s+∆s
6ζ(1 + ζ)
× 1 + 4e
− (1 + ζ)
α
eq s
!
∆s
+ α
eq s−ζ∆s
×
+e
.
(3.30)
Following a similar procedure for Y (s), the full recurrence formulas are found to be
−b1 ∆s
X(s) = X(s − ∆s)e
A
+
1
ζ(ζ + 2) α
2
eq s
6ζ(1 + ζ)
− (1 + ζ)
α
eq s−∆s
+ α
×
eq s−(1+ζ)∆s
−b ∆s/2
−b s
× 1 + 4e 1
+e 1 ,
(3.31)
−b2 ∆s
Y (s) = Y (s − ∆s)e
A
+
2
6ζ(1 + ζ)
ζ(ζ + 2) α
−b2 ∆s/2
× 1 + 4e
2
eq s
−b2 s
+e
− (1 + ζ)
α
eq s−∆s
+ α
eq s−(1+ζ)∆s
×
,
(3.32)
The overall accuracy of the numerical evaluation of the Duhamel integral is of order
2
(∆s) . According to Leishman, this method will produce results within 0.05% of the
exact answer, provided that both b ∆t and b s are less than 0.5.
1
2
39
3.3
Nonlinear Numerical Lifting Line Method
The force and moment on a wing of infinite span would be accurately determined
by the numerical integration of the unsteady force and moment values on the set of
two-dimensional wing sections. However, wings of practical interest will have finite
wingspans, and low or moderate aspect ratios. It is well known from the analysis of
fixed wing aircraft that the lift and drag characteristics of finite wings differ significantly
from those of an infinite wing, or airfoil section. The lift force generated by a wing is a
byproduct of the pressure difference between its upper and lower surfaces. At the tip of
a finite wing, this pressure difference causes air to flow from the higher pressure region
below the wing to the lower pressure region above it in a curling motion. In combination
with the freestream velocity, the flow curling around the wingtips generates vortices
which trail the lifting wing. The presence of these vortices induces a flow velocity on the
wing which is described by the Biot-Savart law for inviscid, incompressible flow. Due to
the location of their centers at the wingtips and the direction of their circulation, the
velocity induced by the vortices takes the form of a downward component known as the
downwash, as well as spanwise flow from root-to-tip on the lower surface of the wing, and
from tip-to-root on the upper surface. The vector addition of the induced velocity and
the freestream velocity results in a spanwise variation in relative wind. The effect of the
downwash is to rotate the relative wind downward, which in turn rotates the local lift
vector toward the downstream direction. As a result of this rotation, the local lift vector
will have a diminished component normal to the freestream, and a non-zero component
in the direction of the freestream, known as induced drag.
40
In order to account for the induced effects caused by finite wingspan, the nonlinear
numerical lifting line approach proposed by Anderson, Corda and Van Wie [50] has
been implemented in FlapIR. This iterative approach is based on Prandtl’s classical
lifting line theory as presented by Anderson [51, pp. 360-367], with modification to
model nonlinear lift characteristics. The approach begins with the division of a wing
into a set of spanwise stations, which allows for simple integration into FlapIR since
this division is fundamental to the BET approach. Next, an elliptical starting input
circulation distribution is assumed. The iterative portion of the method is then applied
in the following five steps:
1. The induced angle of attack, α , due to the input circulation distribution is calcui
lated for the n
th
wing section by evaluating the following integral:
1
α (r ) =
i n
4πV
∞
Z b/2
(dΓ/dr)dr
.
r −r
0
(3.33)
n
It should be noted that the limits of integration have been modified to include only
the induced effects from one half span, since the wings may be flapping about the
centerline. Equation (3.33) is integrated numerically with Simpson’s rule by the
following summation:
α (r ) =
i n
1
4πV
∞
∆r
3
k
X
j=2,4,6
(dΓ/dr)
j−1
(r − r
n
j−1
)
(dΓ/dr)
+4
j
(r − r )
n
j
(dΓ/dr)
+
j+1
(r − r
n
j+1
)
(3.34)
where ∆r is the distance between wing sections j = 1 and j = k + 1 correspond to
the wing root and tip, respectively. As suggested by Anderson, for sections where
41
r =r
n
j−1
, r or r
j
j+1
the term is replaced by its average value based on the two
adjacent sections.
2. The effective angle of attack, α
ef f
α
ef f
, is determined at each section from
(r ) = α
n
unsteady
(r ) − α (r )
n
i n
(3.35)
3. The sectional lift coefficient at each station is determined using the effective angle
of attack from step 2, and a lookup table based on analytical, computational, or
experimental airfoil data.
4. The new circulation distribution is determined by using the Kutta-Joukowski theorem and the definition of sectional lift coefficient:
1
2
0
L (r ) = ρ V Γ ew(r ) = ρ V c (c )
∞
n
∞ ∞ n
n
2
∞ n l n
(3.36)
which may be simplified to
Γ
1
(r ) = V c (c )
2 ∞ n l n
new n
(3.37)
where c is the local wing section chord.
n
5. If the new circulation values all fall within 0.01 percent of the values fed in to step
1 (defined as Γ
old
), the method is considered to have converged. Otherwise, a new
42
input to step 1, Γ
input
, is generated with the following equation:
Γ
input
=Γ
old
+ D(Γ
new
−Γ
old
)
(3.38)
where D is a damping factor. Anderson suggests that a damping factor on the order
of 0.05 in necessary. All FlapIR results in the following chapters were produced
with D equal to 0.05.
Once the numerical solution for the circulation distribution on the wing has converged,
the local relative wind vectors for each blade element are rotated by the induced angle
of attack from unsteady and finite span effects to produce the effective angle of attack
as described by Equation (3.35).
The resultant force and moment per unit span on a blade element is calculated
as follows:
i
1 2 h
~ × V~ + (c ) V~
,
f~ = ρ V~ c (c ) R
n
d n
n
l n
n
n
n
2
1 2 2
~
m
~ = ρ V~ c (c ) R,
n
n m n
n
2
(3.39)
(3.40)
43
with


d V α
πb
n n
c
(t) = 2n ḧ +
− b aα̈  +
n
l n
n n
dt
V
dc
dα
n
πb
c
m n
(t) =
n b aḧ − b
n n
n
n
V2
dc
+

!
dα
steady
"
steady
ef f
(t),
(3.41)
b α˙ #
1
−a n n .
2
V
(3.42)
n
and c
(t) from a lookup table with α
d n
α

d V α
1
1
2
2
n n
−a
−b
+ a α¨ 
n
2
dt
n 8
π
α
(t) +
ef f
2
m
!
l
ef f n
as the input. The total force and mo-
ment on the vehicle are found by integration of the sectional values across the span using
Simpson’s rule. For all results shown here, lift, L, is defined as the force perpendicular
to the freestream, in the direction opposing gravity, and thrust, T , is the force directly
opposite the freestream velocity. Once the forces are known in the body axes, the lift
and thrust forces are given by:
L = F sin(γ) − F cos(γ),
(3.43)
T = F cos(γ) + F sin(γ),
(3.44)
x
x
z
z
where γ is the angle between the freestream and the MAV longitudinal axis, as shown
in Figure 3.2. Force coefficients are found by dividing the dimensional force values by
(1/2)ρV
2
∞
by (1/2)ρV
S, where S is the wing area. Moment coefficients are non-dimensionalized
2
∞
Sc̄, where c̄ is the mean aerodynamic chord of the wing.
44
Fig. 3.2. MAV body axes relative to the freestream
45
Chapter 4
Object Oriented Implementation
The FlapIR tool has been written in Java, and therefore is implemented with an
object oriented approach. The object oriented nature of FlapIR will allow for the code to
be easily read and understood by those not involved in its development, as well as easily
maintained and enhanced in the future. Traditionally, Java has not been the language
of choice in high performance computing, due to the ability of comparable codes to run
faster if written properly in C++ or FORTRAN [52]. While an in-depth explanation of
the causes for this speed difference is beyond the scope of this thesis, a few of them are
worth mentioning, as they can be viewed as both advantages and disadvantages. Java
offers automatic garbage collection, which can increase the memory usage of a program,
but also simplifies memory management for the programmer. In addition, some features
that allow a greater degree of control, such as pointers, have been intentionally left out
of Java. The intention of Java’s creators was to simplify the language, which can help
the programmer avoid mistakes which are difficult to track down. Finally, Java is an
interpreted language, meaning that the code is compiled to an intermediate file known
as bytecode. Java bytecode may then be run on a wide variety of computing platforms
without recompilation, as it is interpreted by a “virtual machine” written specifically for
each platform. This approach simplifies the transport and distribution of Java code to
numerous computers, but the need to run a virtual machine brings a penalty in speed.
46
Modern Java virtual machines include the option to operate with just-in-time (JIT)
compilers, which compile bytecode to native machine code as they run. JIT compilers
can cause an increase in memory usage and a slight slowdown in starting time of the
virtual machine, but also provide a significant gain in the performance of the code. In
the case of the FlapIR code, the effect of increased memory usage should be minimal
on modern computers, as the memory requirements will be small enough that swapping
to disk will not be necessary. The native support for multithreaded code in Java also
allows the possibility for FlapIR to take advantage of multiple core processors which
are becoming prevalent in desktop computers without the need for the use of Message
Passing Interface (MPI) or OpenMP. The standard libraries included with Java contain a
wide variety of useful functions and data structures, which allow for rapid development.
Given that the FlapIR code was intended to be a small-to-medium sized development
project with low memory usage, and short development time was valued, Java was a
sound programming language choice.
4.1
Implementation Details
The FlapIR tool is intended to be used in the evaluation of a large number of
flapping wing designs. Therefore, the tool was designed with ease of integration into
optimization schemes in mind. The most basic utilization of FlapIR requires only the
instantiation of a single class: FlappingMAV. The FlappingMAV class represents a single,
complete design for a flapping wing configuration. Due to the nonlinearity of the problem,
and the possibility that there exist a number of high performance configurations which
may be very different from each other, the optimization of a flapping wing configuration
47
is a good candidate case for the application of an evolutionary algorithm (EA). Although
such an optimization has not yet been performed using FlapIR, it is used here to illustrate
how the FlappingMAV class would be used. Object oriented EA codes are often designed
to create a separate instance of a class for every individual in the population. Each
individual will have its own set of values for the design variables, and as a result, have
its own fitness value. The process of optimization is then centered around finding the
individual design(s) with the highest value of fitness, as defined for the specific problem.
The FlappingMAV class is designed to accept the flapping wing design parameters as
arguments to its constructor. Table (4.1) details the arguments for the constructor used
in the studies presented here. The FlappingMAV constructor can be easily overloaded
to include more or less design parameters to suit the needs of the optimization problem.
Once instantiated, the FlappingMAV class contains a full description of a flapping wing
design, and the appropriate methods to run the simulation and determine the relevant
aerodynamic quantities.
A graphical description of all the classes used in FlapIR and their relationships
to each other is given by the Unified Modeling Language (UML) class diagram in Figure
(4.1). Each box in the diagram represents a class, and contains up to three sections:
class name, member variables, and member functions. Connections with solid diamonds
indicate containment relationships, where the class nearest the diamond contains the
indicated number of instances of the attached class. Containment relationships with an
arrowhead indicate that the relationship has only one direction. In this case, the class on
the side of the arrow has no reference to the class in which it is contained. Open triangle
connections indicate inheritance relationships with the arrow pointing to the parent class.
48
Dashed arrow connections indicate dependencies, such as imported utility classes which
are not included in the standard Java libraries. For clarity, the class diagram in Figure
(4.1) shows only a subset of the member variables and functions in FlapIR as examples.
49
Table 4.1.
FlappingMAV constructor arguments
Argument
Type
Description
0
integer
Data Logging (0 = none, 1 = force, moment, 2 = 1 + geometry)
1
double
number of simulation timesteps per flapping cycle
2
double
freestream velocity magnitude (dimensional)
3
double
freestream density (dimensional)
4
integer
number of blade elements per wing
5
double
total number of flapping cycles to simulate
6
double
number of flapping cycles to log (taken from simulation end)
7
double
chordwise location of pitching axis (as fraction of chord)
8
double
chordwise location for moment reporting
9
double
amplitude of flapping (deg)
10
double
angular velocity of flapping (rad/s)
11
double
phase angle of flapping (deg)
12
double
flapping axis angle with respect to freestream (deg)
13
double
amplitude of sweeping (deg)
14
double
angular velocity of sweeping (rad/s)
15
double
phase angle of sweeping (deg)
16
double
amplitude of uniform pitching (deg)
17
double
angular velocity of uniform pitching (rad/s)
18
double
phase angle of uniform pitching (deg)
19
double
base angle of uniform pitching (deg)
20
double
amplitude of twist at wingtip (deg)
21
double
angular velocity of twist at wingtip (rad/s)
22
double
phase angle of twist at wingtip (deg)
23
double
base twist at wingtip (deg)
50
Fig. 4.1. FlapIR UML class diagram
51
From the class diagram, it can be seen that a FlappingMAV contains a variable
number of wing objects. A single wing is used by default, under the assumption that the
flapping motion is symmetric about the flapping axis. Because the interactions between
wings are not modeled, the mirroring of one wing about the flapping axis will produce
the same result as modeling two wings, but at lower computational expense. In the case
of a configuration with multiple pairs of wings, such as a dragonfly, wings may be easily
added, although the results should not be expected to include the effect of aerodynamic
interaction between wings. Each Wing object contains all of the geometric data necessary
for its description, such as wingspan, chord distribution, area, and twist distribution.
Using the concept of encapsulation, these properties are not modified directly by outside
functions, but instead accessed through a set of public methods. The primary benefit
of this approach is that the inner workings of the Wing function could be drastically
changed at any time, but as long as the public methods are supported, these changes
will be completely transparent to code outside of the Wing class. For example, the
following three lines of code may be written in the FlappingMAV class:
wing_object.applyFiniteSpanCorrection();
resultant_force = wing_object.integrateForces();
resultant_moment = wing_object.integrateMoments();
Whether the finite span correction is accomplished with a simple multiplier on the lift values, or through a more complex numerical lifting line method, applyFiniteSpanCorrection()
may be called using identical code outside of the Wing class. Similarly, the details of
the integration of wing force and moment are hidden from the FlappingMAV.
52
Each Wing object contains a set of WingSection objects, representing the blade
elements. Blade elements maintain their own local parameters such as normal vector,
relative wind and reduced time. Evaluation of the Duhamel integral is performed inside
the WingSection.advance() method, which is called once per timestep on each element.
This recurrence solution for the unsteady effects due to the shed wake is of complexity
O(n), where the problem size, n refers to the number of blade elements used.
Two additional utility classes were developed for use in FlapIR: LookupTable2D
and AirfoilLookupTable. The LookupTable2D class provides the ability to read space
or tab delimited data from a file, and perform a two-dimensional table lookup between
any two of the columns. The current implementation provides only linear interpolation
between points. As shown in the class diagram, AirfoilLookupTable extends LookupTable2D. While the use of growable array objects is used for convenience in LookupTable2D for loading input data sets of size unknown at compile time, this approach will
increase runtime as compared to the use of standard Java arrays. Because airfoil data
table lookups are performed at each timestep for each blade element, the speed of their
operation is of greater importance. The AirfoilTableLookup class stores lift, drag and
moment coefficient data in standard Java arrays for faster access, and also provides additional methods such as getZeroLiftAlpha() which only make sense for data specific
to airfoils. The LookupTable2D class may be extended again for future applications, or
used without modification. In order to analyze wing kinematics which are not easily described by analytical functions, the LookupTable2D would be easily used to read motion
profiles from input files.
53
A set of utility classes (not developed by the author) are also used within FlapIR.
These four classes are shown at the top level of the class diagram. Vector3d and Quat4d
are part of the 3D vector math application programming interface (API) called Vecmath
[53], and represent three-dimensional vectors and four-element quaternions, respectively.
Quaternions are essentially generalized complex numbers, consisting of a scalar part and
a vector part: q = (s, ~v ). The conjugate, product and inverse of quaternions are defined
as:
q̄ = (s, −~v )
0
0
(4.1)
0
0
0
0
qq = (ss − ~v · ~v , ~v × ~v + s~v + s ~v )
q
−1
=
q̄
kqk
(4.2)
(4.3)
Vector3d objects are used to represent the body axes of the vehicle, freestream velocity, wing pitching axis orientation, and the normal vectors of each blade element. All
rotations of these vectors are performed using quaternion rotations, for example:
q = (cosθ, ~usinθ),
0
p~ = q (0, p~) q
−1
,
(4.4)
(4.5)
0
where p~ represents the result of rotating p~ by an angle of 2θ about axis ~u. Quaternions
are frequently used in computer graphics since they are more compact than rotation
matrices, and quaternion rotations are more easily interpolated than those using Euler
angles [54]. The FileIn and FileOut classes have been developed by Chapman [55].
54
Although the standard Java libraries provide a wide variety of useful functions, they are
lacking in methods for simple file input/output operations. The FileIn class is used by
the LookupTable2D and AirfoilLookupTable classes to read airfoil data from files, while
FileOut is used by FlappingMAV to write the simulation data to an output file.
4.2
FlapIR Dynamic Model
Once the FlappingMAV class has been instantiated by an outside class, the sim-
ulation is started by calling the FlappingMAV.simulate() method. The resulting interaction between the FlapIR classes is detailed in the UML sequence diagram shown in
Figure (4.2). The sequence diagram contains a block at the top for each class with a vertical line, known as a lifeline, extending downward. Lifelines can be considered timelines
for each class, showing their creation at the top, followed by the actions performed by or
on them in sequential order. Frames labeled “loop” indicate that the actions inside the
frame are repeated in a loop over the variable listed in square brackets. Frames labeled
“opt” indicate optional actions, such as if-then statements in the code.
55
Fig. 4.2. FlapIR UML sequence diagram
56
It should be noted that the sequence diagram shown here represents the standalone operation of FlapIR for analysis. Minor changes to this diagram would be made
if FlapIR were to be used for optimization. The details of each step on the sequence
diagram are as follows:
1. The FlappingMAV.simulate() method is called by the Main method in the case
of a standalone FlapIR run, or by an outside piece of optimization code. For standalone operation, results are written to disk and no values are returned. In an
optimization, logging to disk would be eliminated, and the aerodynamic characteristics returned to the optimizer through some representation of fitness, such as
propulsive efficiency.
2. The FlappingMAV.getPitchAxisVelocity() method is called once per simulation
timestep. It is this method which evaluates the kinematic functions of the wing,
and determines the velocity vector of the wingtip due to wing motion as described
in Chapter 3. This value may then be scaled to give the velocity of each blade
element.
3. The Wing.evaluateSections() method is called once per simulation timestep.
Within this function, a loop is used to apply dynamic spanwise varying twist to
the wing, and evaluate the unsteady aerodynamics of each blade element.
4. The WingSection.advance() method is called on each wing section at each timestep.
The Duhamel integral is evaluated inside this method, following the completion of
the method calls listed in steps 6 and 7.
57
5. The WingSection.updateReducedTime() method is called by the advance() method
of each wing section at each timestep. In this step, the reduced time value for the
wing section is determined by integration with Simpson’s Rule.
6. The WingSection.updateAlpha() method is called by the advance() method of
each wing section at each timestep. This method calculates the local values of
α, α̇, α̈, ḣ and ḧ.
7. The AirfoilLookupTable.liftLookup() method is called from each wing section
at each timestep. The lift coefficient is returned, and used to calculate the sectional circulatory lift force, which is then added to the non-circulatory lift force
to determine the total sectional lift on the blade element. This force acts in the
direction of the local lift vector, which is rotated to reflect the induced angle of
attack from wake and finite span effects.
8. The AirfoilLookupTable.dragLookup() method is called from each wing section
at each timestep. The profile drag coefficient of the airfoil is returned and used to
calculate the sectional drag force on the blade element. Adding this vector to the
lift vector gives the total force per unit span on the blade element.
9. The AirfoilLookupTable.momentLookup() method is called from each wing section at each timestep. The moment coefficient of the airfoil is returned and used to
calculate the moment per unit span for the blade element. The moment data used
in the AirfoilLookupTable are expected to be reported about the pitching axis.
58
10. OPTIONAL: The Wing.applyFiniteSpanCorrection() method may be called to
apply the nonlinear numerical lifting line method for modeling finite span effects.
Because this method requires an integration across the wingspan for each blade
2
element at each timestep, it is of complexity O(n ). In order to avoid capturing
the initial transients due to starting the flapping from rest, a large number of
flapping cycles may be simulated, and only the last few cycles considered in the
computation of averages, and plotting of time histories. Because the finite span
correction is not dependent on time history, but is computationally expensive, it
is applied only to the flapping cycles which will be logged and averaged.
11. OPTIONAL: The Wing.integrateForces() method may be called to find the
total force acting on the wing. This method requires a single integration along
the wing, using Simpson’s rule, which is of complexity O(n). Although less computationally expensive than the finite span correction, this step is also run only
for the flapping cycles of interest, as it is unnecessary for the simulation of the
aerodynamics.
12. OPTIONAL: The Wing.integrateMoments() method may be called to find the
total moment acting on the wing. This step is nearly identical to the force integration, and is therefore applied only to the flapping cycles of interest.
13. OPTIONAL: The FlappingMAV.logData() method may be called to write simulation data to output files. This method would not be called in the case of an
optimization run. For standalone runs, this method will only be called during the
flapping cycles of interest.
59
Chapter 5
Results
This chapter presents results from the FlapIR program as applied to a variety of
cases. For the purpose of validation, several cases for which experimental or computational data has been published were selected for analysis. Cases involving varying levels
of kinematic complexity were chosen in order to test the ability of FlapIR to predict
the behavior of wings in steady plunging, oscillatory pitching and plunging, flapping,
and flapping with dynamic twisting. Due to an almost non-existent body of published
aerodynamic time-history data on flapping wing configurations in hovering flight, all
examples presented here are for forward flight
5.1
Steady Plunging
Steady plunging of a rectangular wing in quiescent air has been simulated by
running the FlapIR code with an initial angle of attack of zero, impulsively moved to
five degrees and held fixed thereafter. Unsteady lifting-surface solutions by vortex ring
elements have been published for this problem by Katz and Plotkin [26, pp. 430], using
wings of varying aspect ratio. Simulations were run with wings of aspect ratio eight and
infinity in order to illustrate the effect of the finite span correction in FlapIR. In this
case, a lift curve slope of 2π was used with FlapIR. It can be seen from Figure (5.1)
that the FlapIR predictions do not match the initial transient predicted by Katz and
60
Plotkin (this is especially apparent in the case of the finite aspect ratio). This result is
expected from the use of Wagner’s function, and incompressible apparent mass effects.
Given the specifications of the problem, the values of α̇ and α̈ will be nonzero for only
the single timestep corresponding to V t/c = 0 on the plot. Because the apparent mass
∞
terms involve only the instantaneous motion of the airfoil, their value will be infinite
at this timestep and zero elsewhere, giving overall transient lift behavior dictated by
Wagner’s function alone. The steady state lift coefficient predictions from FlapIR are in
close agreement with those of the unsteady lifting surface method for both infinite and
finite aspect ratio. The use of only the closest half of the full wing in the calculation of
the FlapIR finite span correction will also contribute to the error, with greater effect at
lower aspect ratio.
61
0.55
α = 5°
V∞∆t / c = 1/16
0.5
0.45
CL
0.4
0.35
0.3
Katz and Plotkin: AR = 8
Katz and Plotkin: AR = ∞
FlapIR: AR = 8
FlapIR: AR = ∞
0.25
0.2
0
1
2
3
4
5
6
7
8
9
V∞t / c
Fig. 5.1. Transient lift coefficient for rectangular wing in steady plunging
62
5.2
Simple Harmonic Oscillations
The next step in validation was to consider a test case including oscillatory motion.
A set of cases were designed for FlapIR to replicate the experimental data on oscillating
airfoils published by Halfman [56]. The experimental setup used by Halfman included
a wing of rectangular planform with a chord of 29.5 cm, span of 60.9 cm and NACA
0012 airfoil. The wing was mounted between two vertical fairings in the wind tunnel
test section in order to maintain two-dimensional flow over the wing. Lift and moment
data were taken for pitching, plunging, and combined pitching/plunging oscillations at
varying reduced frequency over a small range of freestream airspeeds, averaging 42.5
m/s. In order to more closely represent the lift, profile drag and moment curves of the
wing in the experiment, FlapIR was run with airfoil lookup table data generated with
the XFOIL [57] program for the NACA 0012 airfoil at Re = 886, 000.
For the pitching cases, the wing was oscillated about the elastic axis at 37% chord
over reduced frequencies with an approximate range of 0.05 to 0.4. The oscillations
◦
are described by α = α sin(ωt) with values of pitching amplitude, α , at 6.74 and
0
0
◦
13.48 . For the plunging cases, the wing was oscillated in vertical position according
to h = h sin(ωt). The test included values of plunging amplitude, h , of 1 and 2 in.
0
0
Angular velocity, ω, was determined by rearranging the definition of reduced frequency:
ω = (2kV )/c.
∞
Although FlapIR does not support pure translational plunging motion of the
wings, the plunging of an airfoil section may be simulated by logging the force and
moment data for a single blade element on a flapping wing. The appropriate flap angle
63
was determined by matching the length of the arc swept by the flapping wing section
to 2h with β = h /r where β is expressed in radians, and r is the spanwise distance
0
0
0
0
from the wing root to the wing section being sampled.
The lift data from both FlapIR and the experiment are presented in Figures 5.2,
5.3, 5.4 and 5.5 in terms of complex lift, which is equivalent to the amplitude of the lift
coefficient oscillations. The FlapIR lift predictions show good agreement with experimental data over the range of reduced frequencies tested, for the case of plunging wings.
Agreement in the case of pure pitching shows larger error at low reduced frequency.
These results compare exactly with the predictions of Theodorsen’s theory [14], since
the incompressible indicial response method used by FlapIR is simply a different mathematical realization of the same aerodynamic system [12, p. 342]. Halfman recognized
this error in pure pitching at low frequencies, and includes a survey of other pure pitching studies, which show closer agreement between Theodorsen’s theory and experiment.
Halfman concluded that Reynolds number effects are significant in pure pitching cases,
and may account for some of the error. In addition, Halfman noted that the clearance
between the tips of the wing and the vertical end plates may allow for some finite span
effects, which could lower the lift magnitude measured in the experiment.
64
0.6
Halfman
FlapIR
0.5
lift magnitude
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
reduced frequency, k
◦
Fig. 5.2. Comparison to experimental lift data for pure pitching (α = 6.74 )
0
65
1.2
Halfman
FlapIR
1
lift magnitude
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
reduced frequency, k
◦
Fig. 5.3. Comparison to experimental lift data for pure pitching (α = 13.48 )
0
66
0.3
Halfman
FlapIR
0.25
lift magnitude
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
reduced frequency, k
00
Fig. 5.4. Comparison to experimental lift data for pure plunging (h = 1 )
0
67
0.6
Halfman
FlapIR
0.5
lift magnitude
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
reduced frequency, k
00
Fig. 5.5. Comparison to experimental lift data for pure plunging (h = 2 )
0
68
The moment coefficient predictions about the 37% chord location from FlapIR
are compared to experiment in Figures 5.6, 5.7, 5.8 and 5.9. The moment coefficient
is defined in the standard manner: M/(qSc̄), where M is the dimensional moment,
q is the dynamic pressure, S is the wing area and c̄ is the mean aerodynamic chord.
◦
The moment coefficient predictions for the 6.74 pitching case show better agreement
than the lift predictions for the same case. FlapIR predicted moment coefficient values
approximately 18% greater than those shown in the experiment for the higher amplitude
◦
(13.48 ) pitching case. Agreement between the simulation and experiment was good for
both pure plunging cases.
69
0.12
0.1
CM
0.08
0.06
0.04
Halfman M
FlapIR
0.02
0
0
0.1
0.2
0.3
0.4
0.5
0.6
reduced frequency, k
◦
Fig. 5.6. Comparison to experimental moment data for pure pitching (α = 6.74 )
0
70
0.24
0.2
CM
0.16
0.12
0.08
Halfman
FlapIR
0.04
0
0
0.1
0.2
0.3
0.4
0.5
0.6
reduced frequency, k
◦
Fig. 5.7. Comparison to experimental moment data for pure pitching (α = 13.48 )
0
71
0.06
Halfman
FlapIR
0.05
CM
0.04
0.03
0.02
0.01
0
0
0.1
0.2
0.3
0.4
0.5
0.6
reduced frequency, k
00
Fig. 5.8. Comparison to experimental moment data for pure plunging (h = 1 )
0
72
0.12
Halfman
FlapIR
0.1
CM
0.08
0.06
0.04
0.02
0
0
0.1
0.2
0.3
0.4
0.5
0.6
reduced frequency, k
00
Fig. 5.9. Comparison to experimental moment data for pure plunging (h = 2 )
0
73
5.3
Flapping
Following the validation of harmonic pitching and plunging for an infinite wing,
the flapping of finite wings was investigated. Fejtek and Nehera [58] have collected lift
and thrust data for a finite span rigid wing with a highly cambered airfoil (NACA 8318)
flapping over a range of advance ratios in a wind tunnel. Advance ratio, λ, is the ratio
of the speed of the freestream to the maximum speed of the wingtip: λ = V /(2πf β b),
∞
0
where f is the flapping frequency and b is the wingspan (root-to-tip). The wing was of
rectangular planform with a blunt wingtip, a chord of 76 mm and span of 305 mm. Tests
were performed at freestream velocities ranging from 14.3 m/sec to 25.2 m/sec, which
correspond to chord-based Reynolds numbers of 72,000 and 129,000, respectively. Trip
strips were placed at 10% chord on both the upper and lower wing surfaces in order to
produce turbulent flow over the wing surface and minimize the effect of the low Reynolds
number on lift and drag forces as well as the stall angle of attack. Because the majority
of tests were performed at Re = 107, 000, the aerodynamic coefficients of the NACA 8318
were estimated for use in FlapIR with XFOIL at that Reynolds number. The XFOIL
predictions were generated with boundary layer transition forced at 10% chord on both
the upper and lower airfoil surfaces. It can be seen in Figure 5.10 that XFOIL predicts
positive and negative stalls, but also shows recovery to high lift coefficient magnitudes
at high angles of attack. The accuracy of the XFOIL predictions could not be confirmed,
due to lack of experimental two-dimensional data for the NACA 8318 at Re = 107, 000.
74
Fejtek and Nehera present instantaneous lift and thrust data over a single flapping
cycle, and compare to the quasi-steady analytical approach of Holst and Kuchemann [17].
◦
The flapping parameters used in the comparison were: f =3.3 Hz, β = 45 and V
0
∞
=
21.4 m/s, which correspond to λ = 4.3. FlapIR lift coefficient predictions are compared
to these experimental and analytical data in Figure 5.11. The zero position in the flap
cycle corresponds to the beginning up the downstroke. Both the FlapIR and Holst and
Kuchemann predictions lead the experimental data slightly in phase. The FlapIR lift
magnitude compares well to experimental data on the downstroke, but exhibits more
flattening on the upstroke than the analytical and experimental data. Figure 5.12 shows
thrust coefficient versus position in the flapping cycle, with C
T
= T /(qS), where T is
the total force in the direction opposing the freestream. The FlapIR data appear to have
a small bias which shifts the thrust curve in the direction of higher thrust than shown in
the experiment. However, the FlapIR predictions agree more closely with experimental
data than the analytical theory, showing good agreement in amplitude and curve shape
on both the upstroke and downstroke.
75
1.4
1.2
1
0.8
Cl
0.6
0.4
0.2
0
-0.2
-0.4
-20
-15
-10
-5
0
5
10
15
20
α (deg)
Fig. 5.10. XFOIL prediction of NACA 8318 lift coefficient at Re = 107,000
76
2
Fejtek & Nehera (experimental)
Holst & Kuchemann (analytical)
FlapIR
1.5
CL
1
0.5
0
downstroke
-0.5
0
1
2
upstroke
3
4
5
6
position in cycle (rad)
Fig. 5.11. Instantaneous lift coefficient versus position in flapping cycle
77
0.3
0.25
Fejtek & Nehera (experimental)
Holst & Kuchemann (analytical)
FlapIR
0.2
0.15
0.1
CT
0.05
0
-0.05
-0.1
-0.15
-0.2
downstroke
-0.25
-0.3
0
1
2
upstroke
3
4
5
6
position in flapping cycle (rad)
Fig. 5.12. Instantaneous thrust coefficient versus position in flapping cycle
78
A variety of advance ratios were tested in the experiment, with three values of
flapping axis angle, δ. The flapping axis angle is defined as the angle between the plane
normal to the freestream, and the plane of the wing flapping motion. Non-zero flapping
axis angles introduce a forward and backward sweeping component to the wing motion.
The results of these tests were compared by Fejtek and Nehera to the analytical theory
of Walker [13]. A range of advance ratios between approximately 5 and 20 was achieved
in the experiment through variation of β , f and V . Because the exact values of these
∞
0
parameters were not specified for each data point, the FlapIR program was run using
constant values of f = 3.3 Hz and V
∞
= 21.4 m/s, and varying β to achieve the desired
0
advance ratio for all data points.
Figures 5.13 - 5.18 show lift and thrust coefficient data versus advance ratio for
◦
◦
◦
δ = 0 , 15 and 30 . While there is a good deal of scatter in the experimental data,
◦
the FlapIR predictions for average lift coefficient at δ = 0 are high. This may be
due to FlapIR estimating higher lift on the upstroke, as seen in Figure 5.11. Thrust
◦
◦
predictions show good agreement for δ = 0 . In the case where δ = 15 , the FlapIR lift
predictions show closer agreement to the experiment. The thrust predictions agree well
in magnitude at low advance ratio, but FlapIR predicts little variation with increased λ,
whereas the experiment shows a trend toward higher thrust. Fejtek and Nehera suggest
that viscous effects play a large role in the trend difference between thrust values for
zero and non-zero flapping angles. If this is indeed the case, it may account for the
difference in thrust coefficient trends between the inviscid theory used in FlapIR and the
◦
experiment. For δ = 30 , the FlapIR lift predictions are contained within the scatter
of the experimental data at low and moderate advance ratios, and slightly higher than
79
experiment for high advance ratios. The thrust predictions at this flapping angle show
good agreement at moderate and high advance ratios, while FlapIR predicts less negative
thrust than shown in the experiment for low advance ratios. It is interesting to note that
the FlapIR and Walker’s theory predict opposite trends in almost all cases, with the
largest discrepancy at low advance ratios. While the FlapIR lift predictions do not agree
◦
as closely with experiment as Walker’s theory for δ = 0 , they show better agreement
than the analytical theory for non-zero flapping axis angles.
80
0.6
0.55
0.5
0.45
average CL
0.4
0.35
0.3
0.25
0.2
0.15
Fejtek and Nehera (experimental)
Walker (analytical)
FlapIR
0.1
0.05
0
0
5
10
15
20
25
λ
◦
Fig. 5.13. Average lift coefficient versus advance ratio (δ = 0 )
81
0.05
Fejtek and Nehera (experimental)
Walker (analytical)
FlapIR
average CT
0
-0.05
-0.1
0
5
10
λ
15
20
25
◦
Fig. 5.14. Average thrust coefficient versus advance ratio (δ = 0 )
82
0.6
0.55
0.5
0.45
average CL
0.4
0.35
0.3
0.25
0.2
0.15
Fejtek and Nehera (experimental)
Walker (analytical)
FlapIR
0.1
0.05
0
0
5
10
15
20
25
λ
◦
Fig. 5.15. Average lift coefficient versus advance ratio (δ = 15 )
83
0.05
average CT
0
-0.05
Fejtek and Nehera (experimental)
Walker (analytical)
FlapIR
-0.1
0
5
10
15
20
25
λ
◦
Fig. 5.16. Average thrust coefficient versus advance ratio (δ = 15 )
84
0.6
0.55
0.5
0.45
average CL
0.4
0.35
0.3
0.25
0.2
0.15
Fejtek and Nehera (experimental)
Walker (analytical)
FlapIR
0.1
0.05
0
0
5
10
15
20
25
λ
◦
Fig. 5.17. Average lift coefficient versus advance ratio (δ = 30 )
85
0.05
Fejtek and Nehera (experimental)
Walker (analytical)
FlapIR
average CT
0
-0.05
-0.1
0
5
10
15
20
25
λ
◦
Fig. 5.18. Average thrust coefficient versus advance ratio (δ = 30 )
86
5.4
Flapping and Twisting
In order to most closely replicate the wing kinematics of birds, one more level
of complexity must be considered: dynamic twisting. Dynamic spanwise varying twist
is key to effective flapping wing flight because it allows for the reduction of the large
angles of attack generated by the flapping motion. Avoiding large regions of separated
flow is crucial to the efficient production of lift and thrust. Computational results of
varying fidelity for the aerodynamics of flapping wings with dynamic twisting have been
published. Neef and Hummel [59] approached the problem with CFD, solving the threedimensional Euler equations for compressible flow. Fritz and Long [28] developed a tool
for flapping flight based on the unsteady vortex lattice method (VLM). The computational expense of this vortex lattice approach is much less than that of the solution of
the Euler equations, allowing for a wider range of flapping configurations to be quickly
and easily studied.
5.4.1
Comparison to Three-Dimensional Euler Solutions
Because the Euler equations do not include viscous effects, Neef and Hummel
chose flapping parameters representative of a large bird in forward flight, for which the
flow is expected to remain attached. A rectangular wing with a root-to-tip length equal
to four times the chord length and a NACA 0012 airfoil was modeled with a rounded
◦
wingtip. The wing was flapped with β = 15 and k = 0.1. The dynamic twisting
0
occurred about the leading-edge and was varied linearly with spanwise position, with a
◦
maximum twist angle of 4 at the wingtip. The twisting motion led the flapping motion
87
◦
by a phase angle of 90 . In this configuration, the twisting serves to reduce the angle
of attack at the wingtips, keeping them below stall. In order to most closely match
the conditions of the Euler solutions, FlapIR was run using inviscid airfoil data for the
NACA 0012 as generated by the XFOIL program. Results of the two computational
approaches are compared in Figures 5.19 and 5.20 for mean angles of attack, α ase, of
b
◦
◦
0 and 4 . It should be noted that the coordinate system used by Neef and Hummel
defines the positive x-direction to be that of the freestream. As a result, positive values
of C represent drag, while negative values represent thrust. The FlapIR results compare
x
well with Euler solutions for both lift and thrust coefficients. Magnitudes, as well as the
complex shape of the thrust curve for a non-zero α are captured well by FlapIR. In all
0
cases, the FlapIR predictions exhibit a small phase shift, leading the Euler predictions.
The CFD solutions were generated by modeling a single wing, due to the symmetry of the problem, and used a grid of 204,800 points. Two flapping cycles were
simulated, with the second cycle being used in the analysis of aerodynamic coefficients
in order to avoid transient effects. Using 40 timesteps per flapping cycle, the CFD code
r
required about 17 hours on an HP -9000/J5000 to produce a solution. The simulation
of two flapping cycles, using 40 timesteps per cycle with the FlapIR program required
approximately 1 second of CPU time on a 1.8 GHz Pentium 4 M laptop computer. Although the FlapIR code was run on a slightly more modern processor, it is clear that the
BET approach is orders of magnitude less computationally expensive than the solution
of the Euler equations.
88
1
Euler solution, αbase = 0°
Euler solution, αbase = 4°
FlapIR, αbase = 0°
FlapIR, αbase = 4°
Cz
0.5
0
0
0.25
0.5
0.75
1
t/T
Fig. 5.19. Instantaneous vertical force coefficient versus position in flap cycle
89
0.05
Euler solution, αbase = 0°
Euler solution, αbase = 4°
FlapIR, αbase = 0°
FlapIR, αbase = 4°
Cx
0.025
0
-0.025
-0.05
0
0.2
0.4
0.6
0.8
1
t/T
Fig. 5.20. Instantaneous streamwise force coefficient versus position in flap cycle
90
5.4.2
Comparison to Unsteady Vortex Lattice Method Results
In order to determine the effect of varying flapping parameters, Fritz and Long [28]
performed a set of studies on flapping wings with dynamic twist using their unsteady
vortex lattice method. Three parameter sweeps were performed to show the effect of
wingtip twist amplitude, α
tip
, flapping amplitude, β , and flapping angular velocity,
0
ω. In all cases, the angular velocity of the flapping and twisting were equal, with the
◦
twisting leading the flapping motion by a phase angle of 90 . The remainder of the flight
parameters were fixed to the values shown in Table 5.1 for all cases. To most closely
Table 5.1.
Summary of constant parameters used in flapping parameter studies
Parameter
Value
V∞
11.0 m/s
b (root-to-tip)
0.445 m
c
0.076 m
α0
-3 deg
Camber
NACA 83XX
match the conditions used in the VLM code, FlapIR was run using inviscid airfoil data for
the NACA 8306, generated in XFOIL. Figures 5.21 and 5.22 show average lift coefficient
and thrust values versus α . Lift coefficient results of the two computational approaches
0
compare very closely. Thrust predictions compare reasonably well on the downstroke,
but differ significantly in magnitude on the upstroke. Over the majority of the α range,
0
91
FlapIR predicts positive thrust, while the VLM predicts negative thrust. Similar results
are shown for the β sweep in Figures 5.23 and 5.24. Once again, the lift predictions
0
of the two methods compare well over both the upstroke and downstroke. In this case,
the FlapIR predictions show a shallower slope of thrust values with respect to change
in β , with the closest agreement being at higher flapping angles. Figures 5.25 and 5.26
0
show the results of the sweep in flapping frequency, which corresponds to a range of k
between 0.035 and 0.19. The lift values agree well for k values of approximately 0.12
and below, with the VLM predicting higher average lift coefficient on the downstroke at
high reduced frequency. Again, the variation of average thrust with reduced frequency
is lower in the FlapIR results, with a smaller difference in average thrust between the
upstroke and downstroke than that predicted by the VLM at all values of ω.
92
1.4
VLM avg CL
VLM avg CL (downstroke)
VLM avg CL (upstroke)
FlapIR avg CL
FlapIR avg CL (downstroke)
FlapIR avg CL (upstroke)
1
CL
0.6
0.2
-0.2
-0.6
0
5
10
15
20
25
30
35
40
αtip (degrees)
Fig. 5.21. Average lift coefficient for various values of α
tip
45
93
1.5
VLM avg T
VLM avg T (downstroke)
VLM avg T (upstroke)
FlapIR avg T
FlapIR avg T (downstroke)
FlapIR avg T (upstroke)
1.1
Thrust (N)
0.7
0.3
-0.1
-0.5
-0.9
-1.3
0
5
10
15
20
25
30
35
40
αtip (degrees)
Fig. 5.22. Average thrust for various values of α
tip
45
94
1.6
VLM avg CL
VLM avg CL (downstroke)
VLM avg CL (upstroke)
FlapIR avg CL
FlapIR avg CL (downstroke)
FlapIR avg CL (upstroke)
1.2
CL
0.8
0.4
0
-0.4
-0.8
0
5
10
15
20
25
30
35
40
45
50
β0 (degrees)
Fig. 5.23. Average lift coefficient for various values of β
0
95
0.4
0
Thrust (N)
-0.4
-0.8
-1.2
VLM avg T
VLM avg T (downstroke)
VLM avg T (upstroke)
FlapIR avg T
FlapIR avg T (downstroke)
FlapIR avg T (upstroke)
-1.6
-2
0
5
10
15
20
25
30
35
40
45
β0 (degrees)
Fig. 5.24. Average thrust for various values of β
0
50
96
2
VLM avg CL
VLM avg CL (downstroke)
VLM avg CL (upstroke)
FlapIR avg CL
FlapIR avg CL (downstroke)
FlapIR avg CL (upstroke)
1.5
CL
1
0.5
0
-0.5
0
10
20
30
40
50
ω (1/s)
Fig. 5.25. Average lift coefficient for various values of ω
60
97
5
4
VLM avg T
VLM avg T (downstroke)
VLM avg T (upstroke)
FlapIR avg T
FlapIR avg T (downstroke)
FlapIR avg T (upstroke)
Thrust (N)
3
2
1
0
-1
-2
0
10
20
30
40
50
ω (1/s)
Fig. 5.26. Average thrust for various values of ω
60
98
As a method of decreasing the CPU time required, the VLM includes the option
to include only a portion of the wake in the aerodynamic calculations. The runtime of
the VLM was illustrated by Fritz and Long in an example calculation of a rectangular
wing undergoing three plunging cycles. Using 780 total timesteps, the VLM required
1.25 seconds to evaluate a single timestep when the entire wake was included, and 0.64
seconds when only the most recent 620 timesteps were used in the calculation of induced
velocities. A comparable simulation was run with the FlapIR program, using 50 blade
elements, and applying the finite span correction at each of the 780 timesteps. Running
on the exact 1.8 GHz Pentium 4 M laptop which was used to benchmark the VLM
code, the FlapIR program required 0.03 seconds to evaluate each timestep. It should be
noted that in optimization applications, as previously mentioned, a number of flapping
cycles would be simulated, but to avoid capturing the starting transients, only the last
few cycles would be used in determining aerodynamic quantities. Therefore, the most
computationally expensive component of FlapIR, the finite span correction, would only
need to be run on a subset of the simulated flapping cycles, further decreasing the CPU
time required.
99
Chapter 6
Conclusions
Flapping wing flight remains one of the least well-understood topics in aerodynamics today. Our understanding of these low Reynolds number, three dimensional,
unsteady flows has been improved through observation, experimentation, and computational study, but still has room for much improvement. Given the current trends in
small UAVs, interest in the development of predictive tools for flapping flight is rising.
The objective of this work was to produce a tool having low computational expense, and
the capability to approximate the aerodynamic forces and moments on flapping wing
configurations. The utility of such a tool would be in its application to the design and
optimization of flapping wing MAVs. Using established methods from the fields of aeroelasticity and rotorcraft aerodynamics, the FlapIR program has been developed to meet
the above objective.
The basis of the FlapIR program is blade element theory implemented with the
capability to simulate the complex wing kinematics of biological and mechanical flapping
wing systems. The indicial response of each blade element to angle of attack and pitch
rate is determined with Wagner’s function to model the time history effect of the shed
wake. The induced effects due to finite wingspan are then approximated with a numerical
nonlinear lifting line method. This simple method allows for the approximation of the
force and moment on deformable, finite span wings undergoing complex motions. The
100
user must be aware that although FlapIR can use viscous airfoil data to represent the lift,
drag and moment curves of the blade elements, the underlying unsteady aerodynamic
theory is formulated for incompressible potential flow, and is only valid for small angles
of attack. For this reason, in its current implementation, FlapIR is unable to model
situations in which there is significant separated flow. In addition, the assumption of
incompressibility remains valid only for M k << 1, where M is the freestream Mach
number [12, pp. 308].
In order to validate the method, the FlapIR program results were compared to
experimental and computational data for a variety of unsteady cases. Although the
numerical nonlinear lifting line method was validated by its creators, the effectiveness of
the method was confirmed in the study of two wings in steady plunging and compared to
computational data from Katz and Plotkin [26]. FlapIR was shown to agree well in both
lift and moment coefficient with the experimental data on pitching and plunging airfoils
from Halfman [56]. For the case of cambered wings flapping without twisting, FlapIR
was able to closely reproduce experimental time histories of lift and thrust published by
Fejtek and Nehera [58]. From the time history, as well as average lift measurements over
a range of advance ratios, it can be seen that the FlapIR method tends to underestimate
the minimum lift value achieved on the upstroke. It is likely that the lift deficiency
on the upstroke is caused by the airfoil data used in the FlapIR simulation. As shown
◦
in Figure 5.10, XFOIL predicts a negative stall around −6 angle of attack. If the
estimated negative stall angle of attack does not match the actual value for the NACA
8318 at Re = 107, 000 with top and bottom trips at 10% chord, the FlapIR results
would not be expected to match the experiment. At the Reynolds number in question,
101
it is very reasonable to expect that the XFOIL predictions do not identically match
reality, particularly in the neighborhood of stall. While the FlapIR results do not match
experimental lift data as closely as Walker’s theory [13] for zero flapping plane angle, they
provide better estimates for non-zero flapping plane angles. Thrust estimates show closer
agreement for all flapping plane angles. Flapping with dynamic twisting represents the
closest approximation to the wing kinematics of birds and flapping wing MAVs. FlapIR
results compare well to inviscid CFD results for a case representative of a large bird
in forward flight given by Neef and Hummel [59]. The magnitude and shape of both
lift and thrust time history curves from both computational approaches compared well
for both zero and non-zero base angles of attack. Flapping parameter studies with
an unsteady vortex lattice method by Fritz and Long [28] were replicated using FlapIR.
Although the lift values from these two methods agree very closely, the thrust predictions
show significant differences. The FlapIR results generally show less negative, or even
positive thrust predictions on the upstroke for cases in which the vortex lattice method
predicts negative thrust. Since neither approach includes the effects of viscosity, these
differences must be a result of the induced effects, or simply a difference in the geometric
representation of the flapping. The trends shown in the CFD results for flapping with
twisting in Figures 5.19 and 5.20 support the FlapIR results in that positive thrust is
◦
generated along with negative lift on the upstroke for a mean angle of attack of 0 . In
◦
the case of a 4 angle of attack, the upstroke shows only slightly negative thrust, with
a peak reaching almost zero thrust. For a real flapping system, one would expect the
thrust values to be lower, due to the addition of viscous skin friction drag on the wing,
which is not included by any of the three computational methods mentioned here.
102
Because FlapIR was written in Java using an object-oriented approach, the code
may be easily read, understood, and modified. The current implementation could be
easily integrated with an optimization code by the creation of a method to determine
the fitness of a configuration from the results of its simulation. Using the aerodynamic
force data which is currently generated, other useful quantities such as power may be
calculated. Using inviscid airfoil data, FlapIR will produce only an estimate of induced
power. The use of viscous airfoil data for the blade elements will add a simple estimate
of profile power, giving an overall estimate of total power. It was shown in the validation
cases that FlapIR is capable of producing reasonable estimates of aerodynamic coefficients with significantly less computational expense than CFD and VLM approaches.
Running on a 1.8 GHz Pentium 4 laptop, FlapIR required 1 second of CPU time, whereas
r
the Euler solutions from Neef and Hummel required 17 hours on the HP -9000/J5000
to solve two flapping cycles. The aerodynamic coefficient predictions of these two approaches showed close agreement for this case, which was representative of a bird in
forward flight. Evaluating a similar case in both FlapIR and the unsteady VLM from
Fritz and Long on the same Pentium 4 laptop showed that FlapIR required only 0.03
seconds per timestep, while the VLM required a minimum of 0.64 seconds per timestep
for the same flapping case. The low evaluation time required by FlapIR makes it an
ideal candidate for use in design optimization.
There is still debate amongst the biologists and engineers studying flapping flight
on which aerodynamic mechanisms are exploited by large hovering creatures such as
the hummingbird, and to what degree. If hovering can be achieved by larger vehicles
without separation and reliance on dynamic stall, then FlapIR may be appropriately used
103
in their design as well. Application of the tool to smaller vehicles, or systems with very
high flapping frequencies may be achieved with some modification. The ability to model
dynamic stall should improve the accuracy of the code for the study of smaller vehicles or
insects which rely heavily on leading-edge separation and the formation of leading-edge
vortices. Future work on the tool will include the addition of the Leishman-Beddoes [60]
dynamic stall model. Leishman and Beddoes detail the use of this semi-empirical model
in conjunction with a Duhamel superposition approach [61] very similar to that used in
FlapIR. This addition to the code will give greater consideration to low Reynolds number
effects, as it utilizes static airfoil data at the appropriate Reynolds number. Such data
will have to be acquired from experiments, or CFD results such as those produced by
Schroeder and Baeder [62].
As is illustrated by the validation cases, the FlapIR tool provides useful aerodynamic lift and moment estimates for flapping wing configurations with attached flow. As
currently implemented, the tool would certainly be of use to MAV designers in the selection of flapping parameters and wing geometry for forward flight. Future application
of FlapIR will include such an optimization. With multiple variables to be optimized,
such as flapping frequency, as well as the amplitudes of flapping, sweeping and twisting,
it is expected that the fitness space of the problem may have multiple peaks and valleys.
There may be more than one set of parameter values which produce good lift and thrust
performance. In order to investigate the vast number of possible choices, an evolutionary
algorithm called Differential Evolution (DE) will be used. This optimization technique
was developed by Storn and Price [63] in the late 1990s with the goal of reducing the
number of parameters controlling the optimization process in order to make the tool
104
easy to use. Storn has also made a Java implementation of DE available to the public
[64], which will allow for a rapid application to the optimization of flapping parameters
using FlapIR.
105
References
[1] “AeroVironment, Inc. - Unmanned Aerial Systems,” Website, http://www.avsuav.
com.
[2] “Boeing Future Combat Systems Program,” Website, http://www.boeing.com/
defense-space/ic/fcs/bia/index.html.
[3] Pornsin-Sirirak, T., Lee, S., Nassef, H., Grasmeyer, J., Tai, Y., Ho, C., and Keennon,
M., “MEMS Wing Technology for a Battery-Powered Ornithopter,” International
Symposium on Smart Structure and Microsystems 2000 (IS3M 2000), Hong Kong,
October 2000.
[4] Ames, R., Wong, O., and Komerath, N., “On the Flowfield and Forces Generated
by a Flapping Rectangular Wing at Low Reynolds Number,” Fixed and Flapping
Wing Aerodynamics for Micro Air Vehicle Applications, edited by T. J. Mueller,
Vol. 195, AIAA, Reston, VA, 2001, p. 289.
[5] Lilienthal, O., Birdflight as the Basis of Aviation, Markowski International Publishers, Hummelstown, PA, 2001.
[6] DeLaurier, J. and Harris, J., “A Study of Mechanical Flapping Wing Flight,” Aeronautical Journal , Vol. 97, No. 968, October 1993, pp. 277–286.
[7] Jones, K. and Platzer, M., “Bio-Inspired Design of Flapping Wing Micro Air Vehicles - An Engineer’s Perspective,” 44th AIAA Aerospace Sciences Meeting and
Exhibit, Reno, NV, January 2006, AIAA Paper 2006-0037.
106
[8] Newcome, L. R., Unmanned Aviation - A Brief History of Unmanned Aerial Vehicles, AIAA, Reston, VA, 2004.
[9] “Lockheed Martin,” Website, http://www.lockheedmartin.com/wms/findPage.
do?dsp=fec&ci=17051&rsbci=0&%fti=0&ti=0&sc=400.
[10] McMichael, J. M. and Francis, M. S., “Micro Air Vehicles - Toward a New Dimension
in Flight,” 1997, http://www.fas.org/irp/program/collect/docs/mav_auvsi.
htm.
[11] “DARPA Nano Air Vehicle Program,” Website, 2005, http://www.darpa.mil/
dso/thrust/matdev/nav.htm.
[12] Leishman, J. G., Principles of Helicopter Aerodynamics, Vol. 12 of Cambridge
Aerospace Series, Cambridge Univ. Press, Cambridge, England, UK, 2000.
[13] Walker, G. T., “The Flapping Flight of Birds,” Journal of the Royal Aeronautical
Society, Vol. 29, 1925, pp. 590–594.
[14] Theodorsen, T., “General Theory of Aerodynamic Instability and the Mechanism
of Flutter,” NACA Report No. 496, 1935.
[15] Garrick, I., “Propulsion of a Flapping and Oscillating Airfoil,” NACA Report No.
567, 1937.
[16] von Kármán, T. and Burgers, J., “General Aerodynamic Theory - Perfect Fluids,”
Aerodynamic Theory, edited by W. Durand, Vol. 2, Julius Springer, Berlin, 1935.
107
[17] von Holst, E. and Kuchemann, D., “Biological and Aerodynamical Problems of
Animal Flight,” Journal of the Royal Aeronautical Society, Vol. 46, 1942, pp. 39–
56.
[18] Osborne, M., “Aerodynamics of Flapping Flight with Application to Insects,” Journal of Experimental Biology, Vol. 28, 1951, pp. 221–245.
[19] Pennycuick, C., “Power Requirements for Horizontal Flight in the Pigeon Columba
livia,” Journal of Experimental Biology, Vol. 49, No. 3, 1968, pp. 527–555.
[20] Rayner, J., “A New Approach to Animal Flight Mechanics,” Journal of Experimental Biology, Vol. 80, No. 1, 1979, pp. 17–54.
[21] Rayner, J., “A Vortex Theory of Animal Flight,” Journal of Fluid Mechanics,
Vol. 91, No. 4, 1979, pp. 697–763.
[22] DeLaurier, J., “An Aerodynamic Model for Flapping-Wing Flight,” Aeronautical
Journal , Vol. 97, April 1993, pp. 125–130.
[23] Jones, R., “The Unsteady Lift of a Wing of Finite Aspect Ratio,” NACA Report
No. 681, 1940.
[24] DeLaurier, J., “The Development of an Efficient Ornithopter Wing,” Aeronautical
Journal , Vol. 97, May 1993, pp. 153–162.
[25] Vest, M. and Katz, J., “Unsteady Aerodynamic Model of Flapping Wings,” AIAA
Journal , Vol. 34, No. 7, July 1996, pp. 1435–1440.
108
[26] Katz, J. and Plotkin, A., Low-Speed Aerodynamics, Vol. 13 of Cambridge Aerospace
Series, Cambridge Univ. Press, Cambridge, England, UK, 2nd ed.
[27] Hall, K. and Hall, S., “A Rational Engineering Analysis of the Efficiency of Flapping
Flight,” Fixed and Flapping Wing Aerodynamics for Micro Air Vehicles, edited by
T. J. Mueller, American Institute of Aeronautics and Astronautics, Inc., 2001, pp.
249–274.
[28] Fritz, T. and Long, L., “Object-Oriented Unsteady Vortex Lattice Method for
Flapping Flight,” Journal of Aircraft, Vol. 41, No. 6, November-December 2004,
pp. 1275–1290.
[29] Greenewalt, C., Hummingbirds, Dover Publications, Inc., New York, NY, 1990.
[30] Weis-Fogh, T., “Unusual Mechanisms for the Generation of Lift in Flying Animals,”
Scientific American, Vol. 233, No. 5, 1975, pp. 81–87.
[31] Spedding, G., “The Wake of a Jackdaw (Corvus monedula),” Journal of Experimental Biology, Vol. 125, No. 1, 1986, pp. 287–307.
[32] Spedding, G., “The Wake of a Kestrel (Falco tinnunculus),” Journal of Experimental
Biology, Vol. 127, No. 1, 1987, pp. 45–78.
[33] Spedding, G., McArthur, J., and Rosén, M., “Deducing Aerodynamic Mechanisms
From Near- and Far-Wake Measurements of Fixed and Flapping Wings at Moderate
Reynolds Number,” 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV,
January 2006, AIAA Paper 2006-0033.
109
[34] Dickinson, M. and Götz, K., “Unsteady Aerodynamic Performance on Model Wings
at Low Reynolds Numbers,” Journal of Experimental Biology, , No. 174, 1993,
pp. 45–64.
[35] Ellington, C., VandenBerg, C., Willmott, A., and Thomas, A., “Leading-Edge Vortices in Insect Flight,” Nature, , No. 384, 1996, pp. 626–630.
[36] Singh, B. and Chopra, I., “Wing Design and Optimization for a Flapping Wing Micro Air Vehicle,” 60th Annual Forum of the American Helicopter Society, Baltimore,
MD, June 2004.
[37] Singh, B., Ramasamy, M., Chopra, I., and Leishman, J., “Insect-Based Flapping Wings for Micro Hovering Air Vehicles: Experimental Investigations,” American Helicopter Society International Specialists Meeting on Unmanned Rotorcraft,
Phoenix, AZ, January 2005.
[38] Tarascio, M. and Chopra, I., “Design and Development of a Thrust Augmented Entomopter - An Advanced Flapping Wing Micro Hovering Air Vehicle,” 59th Annual
Forum of the American Helicopter Society, Phoenix, AZ, May 2003.
[39] Ellington, C., “Insects Versus Birds: The Great Divide,” 44th AIAA Aerospace
Sciences Meeting and Exhibit, Reno, NV, January 2006, AIAA Paper 2006-0035.
[40] “Academy of Model Aeronautics,” Website, http://www.modelaircraft.org.
[41] Michelson, R. and Naqvi, M., “Beyond Biologically-Inspired Insect Flight,” RTOAVT von Karman Institute for Fluid Dynamics Lecture Series, November 2003.
110
[42] Jones, K., Lund, T., and Platzer, M., “Experimental and Computational Investigation of Flapping Wing Propulsion for Micro Air Vehicles,” Fixed and Flapping Wing
Aerodynamics for Micro Air Vehicles, edited by T. J. Mueller, American Institute
of Aeronautics and Astronautics, Inc., 2001, pp. 307–339.
[43] Lomax, H., “Indicial Aerodynamics,” AGARD Manual on Aeroelasticity, chap. 6,
1968.
[44] Bisplinghoff, R., Ashley, H., and Halfman, R., Aeroelasticity, Dover Publications,
Inc., Mineola, NY, 1996.
[45] Long, L., “The Compressible Aerodynamics of Rotating Blades using an Acoustic
Formulation,” NASA Technical Paper, No. TP-2197, December 1983.
[46] Long, L. and Watts, G., “Arbitrary Motion Aerodynamics using an Aeroacoustic
Approach,” AIAA Journal , Vol. 25, No. 11, November 1987, pp. 1442–1448.
[47] Wagner, H., “Über die Entstehung des dynamischen Auftriebes von Tragflügeln,”
Zeitschrift für Angewandte Mathematik und Mechanick , Vol. 5, No. 1, 1925, pp. 17.
[48] Jones, R., “Operational Treatment of the Non-Uniform Lift Theory in Airplane
Dynamics,” NACA Technical Note 667, 1938.
[49] Hoffman, K. A. and Chiang, S. T., Computational Fluid Dynamics, Vol. 1, Engineering Education System, Wichita, KS, 4th ed., 2000.
[50] Anderson, Jr., J. D., Corda, S., and Van Wie, D. M., “Numerical Lifting Line
Theory Applied to Drooped Leading-Edge Wings Below and Above Stall,” Journal
of Aircraft, Vol. 17, No. 12, December 1980, pp. 898–904.
111
[51] Anderson, Jr., J. D., Fundamentals of Aerodynamics, McGraw-Hill, New York, NY,
3rd ed., 2001.
[52] Jelovic, D., “Why Java Will Always be Slower Than C++,” Website, http://www.
jelovic.com/articles/why_java_is_slow.htm.
[53] “The Vecmath API,” Website, https://vecmath.dev.java.net/.
[54] Angel, E., Interactive Computer Graphics: A Top-Down Approach Using OpenGL,
Addison Wesley, New York, NY, 3rd ed., 2003.
[55] Chapman, S. J., Java For Scientists and Engineers, Pearson Prentice Hall, Upper
Saddle River, NJ, 2nd ed., 2004.
[56] Halfman, R. L., “Experimental Aerodynamic Derivatives of a Sinusoidally Oscillating Airfoil in Two-Dimensional Flow,” NACA Report No. 1108, 1952.
[57] Drela, M., “XFOIL: An Analysis and Design System for Low Reynolds Number
Airfoils,” Conf. on Low Reynolds Number Airfoil Aerodynamics, Univ. of Notre
Dame Press, South Bend, IN, June 1989.
[58] Fejtek, I. and Nehera, J., “Experimental Study of Flapping Wing Lift and Propulsion,” Aeronautical Journal , Vol. 84, January 1980, pp. 28–33.
[59] Neef, M. and Hummel, D., “Euler Solutions for a Finite-Span Flapping Wing,” Fixed
and Flapping Wing Aerodynamics for Micro Air Vehicles, edited by T. J. Mueller,
American Institute of Aeronautics and Astronautics, Inc., 2001, pp. 429–451.
112
[60] Leishman, J. and Beddoes, T., “A Semi-Emprical Method for Dynamic Stall,” J.
American Helicopter Soc., Vol. 34, No. 3, 1989, pp. 3–17.
[61] Leishman, J. and Beddoes, T., “A Generalized Method for Unsteady Airfoil Behavior and Dynamic Stall Using the Indicial Method,” 42nd Annual Forum of the
American Helicopter Society, Washington, DC, June 1986.
[62] Schroeder, E. and Baeder, J., “Using Computational Fluid Dynamics for Micro
Air Vehicle Airfoil Validation and Prediction,” 23rd AIAA Applied Aerodynamics
Conference, Toronto, Ontario, Canada, June 2005, AIAA Paper 2005-4841.
[63] Storn, R. and Price, K., “Differential Evolution - A Simple and Efficient Heuristic
for Global Optimization over Continuous Spaces,” Journal of Global Optimization,
Vol. 11, 1997, pp. 341–359.
[64] Storn, R., “Differential Evolution Homepage,” Website, http://www.icsi.
berkeley.edu/~storn/code.html.