The aerodynamics of insect flight
Department of Biology, University of Washington, Seattle, WA 98195, USA
(e-mail: sane{at}u.washington.edu)
Accepted 12 August 2003
![]() |
Summary |
---|
Key words: insect flight, aerodynamics, Kramer effect, delayed stall, quasi-steady modeling, flapping flight, kinematics, forces, flows, leading edge vortex
![]() |
Introduction |
---|
Insects have also stimulated a great deal of interest among physicists and engineers because, at first glance, their flight seems improbable using standard aerodynamic theory. The small size, high stroke frequency and peculiar reciprocal flapping motion of insects have combined to thwart simple `back-of-the-envelope' explanations of flight aerodynamics. As with many problems in biology, a deep understanding of insect flight depends on subtle details that might be easily overlooked in otherwise thorough theoretical or experimental analyses. In recent years, however, investigators have benefited greatly from the availability of high-speed video for capturing wing kinematics, new methods such as digital particle image velocimetry (DPIV) to quantify flows, and powerful computers for simulation and analysis. Using these and other new methods, researchers can proceed with fewer simplifying assumptions to build more rigorous models of insect flight. It is this more detailed view of kinematics, forces and flows that has led to significant progress in our understanding of insect flight aerodynamics.
![]() |
Experimental challenges |
---|
Even more challenging than capturing wing motion in 3-D is measuring the
time course of aerodynamic forces during the stroke. At best, flight forces
have been measured on the body of the insect rather than its wings, making it
very difficult to separate the inertial forces from the aerodynamic forces
generated by each wing (Cloupeau et al.,
1979; Buckholz,
1981
; Somps and Luttges,
1985
; Zanker and Gotz,
1990
; Wilkin and Williams,
1993
). In addition, tethering can alter the wing motion, and thus
forces produced, as compared with free-flight conditions. Researchers have
overcome these limitations using two strategies. The first method involves
constructing dynamically scaled models on which it is easier to directly
measure aerodynamic forces and visualize flows
(Bennett, 1970
;
Maxworthy, 1979
;
Spedding and Maxworthy, 1986
;
Dickinson and Götz, 1993
;
Sunada et al., 1993
;
Ellington et al., 1996
;
Dickinson et al., 1999
). A
second approach is to construct computational fluid dynamic simulations of
flapping insect wings (Liu et al.,
1998
; Liu and Kawachi,
1998
; Wang, 2000
;
Ramamurti and Sandberg, 2002
;
Sun and Tang, 2002
). The power
of both these approaches, however, depends critically on accurate knowledge of
wing motion.
![]() |
Conventions and terminology |
---|
![]() | (1) |
|
From one stroke to the next, insects rapidly alter many kinematic features
that determine the time course of flight forces, including stroke amplitude,
angle of attack, deviation from mean stroke plane, wing tip trajectory and
wing beat frequency (Ennos,
1989b; Ruppell,
1989
), as well as timing and duration of wing rotation during
stroke reversal (Srygley and Thomas,
2002
). Moreover, they may vary these parameters on each wing
independently to carry out a desired maneuver. Hence, it is misleading to lump
all patterns of insect wing motion into a single simple pattern. Mindful of
this vast diversity in wing kinematics patterns, the wing motion of insects
may be divided into two general patterns of flapping. Most researchers have
restricted their studies to hovering because it is more convenient
mathematically to calculate the force balance by equating lift and weight in
this case. While hovering, most insects move their wings back and forth in a
roughly horizontal plane, whereas others use a more inclined plunging stroke
(Ellington, 1984c
;
Dudley, 2000
). Despite the
predominance of the back-and-forth pattern, the terms `upstroke' and
`downstroke' are used conventionally to describe the ventral-to-dorsal and
dorsal-to-ventral motion of the wing, respectively. It is important to note
that as insects fly forward, their stroke plane becomes more inclined forward.
The term `wing rotation' will generally refer to any change in angle of attack
around a chordwise axis. During the downstroke-to-upstroke transition, the
wing `supinates' rapidly, a rotation that brings the ventral surface of the
wing to face upward. The wing `pronates' rapidly at the end of the upstroke,
bringing the ventral surface to face downward
(Fig. 1C).
In the present review, `linear (or non-flapping) translation' will refer to airfoils translating linearly (Fig. 1D), whereas `flapping translation' will refer to an airfoil revolving around a central axis (Fig. 1E). Since much of the theoretical literature addresses the aerodynamic performance of idealized 2-D sections of wings, it is important to distinguish between finite and infinite wings. The term `finite wing' refers to an actual 3-D wing with two tips and thus a finite span length. From the perspective of fluid mechanics, the importance of the wing tips is that they create component of fluid velocity that runs along the span of the wing, perpendicular to the direction of far-field flow during linear translation. By contrast, `infinite wings' are theoretical abstractions of 2-D structures that can only create chord-wise flow. Such wings are experimentally realized by closely flanking the tips of the wings with rigid walls that limit span-wise flow, thus constraining the fluid to move in two dimensions. It is also important to note that, by definition, a 2-D wing cannot perform flapping motions. Nevertheless, 2-D formulations based on an infinite wing assumption have often proved very useful in the study of animal flight and are particularly relevant in cases where wings have a high aspect ratio.
Within the context of force and flow dynamics, the term `steady' signifies explicit time independence, whereas the word `unsteady' signifies explicit temporal evolution due to inherently time-dependent phenomena within the fluid. In flapping flight, steady does not necessarily imply time invariant. Forces on airfoils may change with time without being explicitly dependent on time, simply because the underlying motion of the airfoils varies. If the forces at each instant are modeled by the assumption of inherently time-independent fluid dynamic mechanisms, then such a model is called `quasi-steady', i.e. steady at each instant but varying with time due to kinematic time dependence.
![]() |
Background theory for thin airfoils |
---|
Unless otherwise mentioned, the theory in this section applies to 2-D
airfoils moving in incompressible fluids. Also, in the analysis that follows,
most key physical parameters appear as non-dimensional entities.
Non-dimensional forms of equations are scale-invariant, thereby making it
possible to compare flows across a wide range of scales. Although any
reasonable scheme of non-dimensionalizing parameters is valid for the purpose
of this review, the scheme conventionally used is the one developed by
Ellington (1984b,
1984c
,
1984d
,
1984e
) for the purpose of
insect flight aerodynamics. For more detailed treatments of the physical
concepts, the reader is referred to classic fluid dynamics texts written by
Lamb (1945
), Landau and
Lifshitz (1959
), Milne-Thomson
(1966
) and Batchelor
(1973
) and books focusing on
thin airfoil theory, such as Glauert
(1947
) and Prandtl and
Tietjens (1957b
).
The fluid motion around an insect wing, like any other submersed object, is
adequately described by the incompressible NavierStokes equation, a
non-dimensional form of which can be written as:
![]() | (2) |
![]() | (3) |
The NavierStokes equation provides the fundamental theoretical basis
for simulating forces and flows from arbitrary or measured kinematics. It is
not, however, easy to use in an experimental context, because it is quite
difficult to measure the pressure field in the space around a wing. An
alternative and sometimes more convenient form of the NavierStokes
equation may be derived by taking the curl of both sides in equation 2. This
eliminates the pressure term because the curl of a gradient vanishes, and the
equation simplifies to:
![]() | (4) |
![]() | (5) |
Under completely inviscid conditions, one would expect the fluid to deflect
only minimally by the presence of an airfoil, thereby generating a flow field
around the wing similar to the one described in
Fig. 2A. Under such conditions,
the rear stagnation point (where velocity is zero) would be present not at the
tip of the trailing edge but on the upper surface of the wing. However, to
maintain this flow profile, the fluid must turn sharply around the trailing
edge causing a singularity or `kink' in the flow at the trailing edge. Such a
flow profile necessitates a high gradient in velocity at the trailing edge,
thereby causing high viscous forces due to shear. The viscous forces in turn
will eventually eradicate this singularity. Thus, the presence of even the
slightest viscosity in the fluid functions to smooth out sharp gradients in
flow. This phenomenon may be incorporated into an otherwise inviscid
formulation by adding a circulatory component to the flow field
(Fig. 2B). At a unique value of
the additional circulation, the stagnation point is stationed exactly at the
trailing edge. When this condition is met, the fluid stream over the plate
meets the fluid stream under the plate smoothly and tangentially at the
trailing edge (Fig. 2C). This
phenomenon is called the `Kutta condition', which ensures that the slopes of
the fluid streams above and below the wing surface are equal, and thus the
vorticity (i.e. curl of the velocity) at the trailing edge is zero. In
addition, when satisfied, the Kutta condition ensures that the inclined plate
imparts a downward momentum to the fluid. This, in essence, is the classic
KuttaJukowski theory of thin airfoils
(Kuethe and Chow, 1998). For
ideal fluids, the net force acts perpendicular to the direction of motion with
no component in the plane of motion. Thus, this theory predicts zero
resistance in the direction of motion (or `drag') for airfoils moving through
fluids at small angles of attack (called D'Alembert's paradox). However, in
the presence of even the smallest amount of shear, the net force vector is
tilted backward, i.e. normal to the wing. Even at reasonably high Re,
the net aerodynamic force on the wing surface is usually perpendicular to the
surface of the inclined wing rather than to the direction of motion. The
non-zero component of this force normal to fluid motion is defined as `lift',
and the component parallel to the fluid motion is defined as `profile drag'.
The component of drag due to viscous shear along the surface on an airfoil is
called `viscous drag'.
|
Far from the airfoil, the behavior of the fluid is similar to that expected
by potential flow theory. For this reason, although the fluid is not actually
irrotational, potential theory can be used to conveniently describe such
situations as long as the Kutta condition is satisfied. For steady inviscid
flows, the KuttaJukowski theorem relates circulation, and therefore
vorticity, around an airfoil to forces by the equation:
![]() | (6) |
Note that lift can also be related to vorticity using equation 5. In
equation 6, ' is the lift per
unit span non-dimensionalized with respect to the product of density of the
fluid (
), mean chord length, and the square of free-stream velocity of
the fluid (U
). This quantity (conventionally
multiplied by two) is called the `lift coefficient' and is usually denoted in
literature by CL. Similarly, the non-dimensional drag is
called the `drag coefficient' and is usually denoted by
CD. For inviscid fluids undergoing steady
(non-accelerated) flows,
![]() | (7) |
![]() | (8) |
When an airfoil starts from rest, the net circulation in the fluid before the start of the motion is zero. Thus, equation 8 is simply a mathematical expression for Kelvin's law, which states that the total circulation (and the total vorticity) in an ideal fluid must remain zero at all times. In other words, if new vorticity (or circulation) is introduced in an inviscid fluid (e.g. through an application of the Kutta condition), then it must be accompanied by equal and opposite vorticity.
Physically, because the presence of viscosity disallows infinite shear, the
fluid immediately abounding the airfoil is stationary with respect to the
airfoil. This condition, called the `no-slip condition', is an important
boundary condition in most analytical treatments of airfoils. Due to the
no-slip condition, a continuous layer forms over the airfoil across which the
velocity of the fluid goes from zero (for the stationary layer adjoining the
body) to its maximum value (corresponding to the free-stream flow). Such
regions are called `boundary layers' and their thickness depends on the
Reynolds number of the flow (Schlichting,
1979). Another boundary condition arises from the requirement that
the normal velocity of the fluid on the surface of the airfoil must be zero.
This condition is sometimes called the `no penetration' condition. These
boundary conditions apply at the interface of solids and fluids. In free
fluids, however, conditions may often arise where the tangential, but not the
normal, component of velocity is discontinuous across two adjacent layers.
Such interfaces have high vorticity and are called `vortex sheets', or `vortex
lines' for the two-dimensional case.
When a volume element dV of the fluid has non-zero vorticity
, it induces a velocity v at a distance r in the
neighboring region. The expression to calculate this velocity is given by (in
dimensional form):
![]() | (9) |
The solenoidal (i.e. zero divergence) nature of vorticity fields enables
vorticity-based methods to define very useful kinematic quantities called
`moments of vorticity'. These quantities are useful because their values are
independent of the conditions in the interior of a boundary surrounding the
region of interest since no new vorticity can be generated within a fluid
subject to conservative external forces. Instead, vorticity is generated at
the solidfluid boundary and diffuses into the fluid medium
(Truesdell, 1954). Of
particular utility is the first moment of vorticity because it can be related
to aerodynamic forces. This quantity is given by:
![]() | (10) |
![]() | (11) |
![]() |
Theoretical challenges |
---|
![]() |
Analytical models of insect flight |
---|
Despite the caveats presented in the last section, a few researchers have
been able to construct analytical near-field models for the aerodynamics of
insect flight with some degree of success. Notable among these are the models
of Lighthill (1973) for the
Weis-Fogh mechanism of lift generation (also called clap-and-fling), first
proposed to explain the high lift generated in the small chalcid wasp
Encarsia formosa, and that of Savage et al.
(1979
) based on an idealized
form of Norberg's kinematic measurements on the dragonfly Aeschna
juncea (Norberg, 1975
).
Although both these models were fundamentally two dimensional and inviscid
(albeit with some adjustments to include viscous effects), they were able to
capture some crucial aspects of the underlying aerodynamic mechanisms.
Specifically, Lighthill's model of the fling
(Lighthill, 1973
) was
qualitatively verified by the empirical data of Maxworthy
(1979
) and Spedding and
Maxworthy (1986
). Similarly,
the model of Savage et al.
(1979
) was able to make
specific predictions about force enhancement during specific phases of
kinematics (e.g. force peaks observed as the wings rotate prior to supination)
that were later confirmed by experiments
(Dickinson et al., 1999
;
Sane and Dickinson, 2002
). In
studies on dragonflies and damselflies, the `local circulation method' was
also used with some degree of success
(Azuma et al., 1985
;
Azuma and Watanabe, 1988
;
Sato and Azuma, 1997
). This
method takes into account the spatial (along the span) and temporal changes in
induced velocity and estimates corrections in the circulation due to the wake.
The more recent analytical models (e.g.
Zbikowski, 2002
;
Minotti, 2002
) have been able
to incorporate the basic phenomenology of the fluid dynamics underlying
flapping flight in a more rigorous fashion, as well as take advantage of a
fuller database of forces and kinematics
(Sane and Dickinson,
2001
).
![]() |
Computational fluid dynamics (CFD) |
---|
One such approach involved modeling the flight of the hawkmoth Manduca
sexta using the unsteady aerodynamic panel method
(Smith et al., 1996), which
employs the potential flow method to compute the velocities and pressure on
each panel of a discretized wing under the appropriate boundary conditions.
Also using Manduca as a model, Liu and co-workers were the first to
attempt a full NavierStokes simulation using a `finite volume method'
(Liu et al., 1998
;
Liu and Kawachi, 1998
). In
addition to confirming the smoke streak patterns observed on both real and
dynamically scaled model insects
(Ellington et al., 1996
), this
study added finer detail to the flow structure and predicted the time course
of the aerodynamic forces resulting from these flow patterns. More recently,
computational approaches have been used to model Drosophila flight
for which force records exist based on a dynamically scaled model
(Dickinson et al., 1999
).
Although roughly matching experimental results, these methods have added a
wealth of qualitative detail to the empirical measurements
(Ramamurti and Sandberg, 2002
)
and even provided alternative explanations for experimental results
(Sun and Tang, 2002
; see also
section on wingwake interactions). Despite the importance of 3-D
effects, comparisons of experiments and simulations in 2-D have also provided
important insight. For example, the simulations of Hamdani and Sun
(2001
) matched complex
features of prior experimental results with 2-D airfoils at low Reynolds
number (Dickinson and Götz,
1993
). Two-dimensional CFD models have also been useful in
addressing feasibility issues. For example, Wang
(2000
) demonstrated that the
force dynamics of 2-D wings, although not stabilized by 3-D effects, might
still be sufficient to explain the enhanced lift coefficients measured in
insects.
![]() |
Quasi-steady modeling of insect flight |
---|
Although quasi-steady models had been used with limited success in the past
(Osborne, 1950;
Jensen, 1956
), they generally
appeared insufficient to account for the necessary mean lift in cases where
the average flight force data are available. In a comprehensive review of the
insect flight literature, Ellington
(1984a
) used the logic of
`proof-by-contradiction' to argue that if even the maximum predicted lift from
the quasi-steady model was less than the mean lift required to hover, then the
model had to be insufficient. Conversely, if the maximum force calculated from
the model was greater than or equal to the mean forces required for hovering,
then the quasi-steady model cannot be discounted. Based on a wide survey of
data available at the time, he convincingly argued that in most cases the
existing quasi-steady theory fell short of calculating even the required
average lift for hovering, and a substantial revision of the quasi-steady
theory was therefore necessary (Ellington,
1984a
). He further proposed that the quasi-steady theory must be
revised to include wing rotation in addition to flapping translation, as well
as the many unsteady mechanisms that might operate. Since the Ellington
review, several researchers have provided more data to support the
insufficiency of the quasi-steady model
(Ennos, 1989a
;
Zanker and Gotz, 1990
;
Dudley, 1991
). These
developments have spurred the search for specific unsteady mechanisms to
explain the aerodynamic forces on insect wings.
![]() |
Physical modeling of insect flight |
---|
![]() |
Unsteady mechanisms in insect flight |
---|
|
Clap-and-fling
The clap-and-fling mechanism was first proposed by Weis-Fogh
(1973) to explain the high
lift generation in the chalcid wasp Encarsia formosa and is sometimes
also referred to as the Weis-Fogh mechanism. A detailed theoretical analysis
of the clap-and-fling can be found in Lighthill
(1973
) and Sunada et al.
(1993
), and experimental
treatments in Bennett (1977
),
Maxworthy (1979
) and Spedding
and Maxworthy (1986
). Other
variations of this basic mechanism, such as the clap-and-peel or the
near-clap-and-fling, also appear in the literature
(Ellington, 1984c
). The
clap-and-fling is really a combination of two separate aerodynamic mechanisms,
which should be treated independently. In some insects, the wings touch
dorsally before they pronate to start the downstroke. This phase of wing
motion is called `clap'. A detailed analysis of these motions in Encarsia
formosa reveals that, during the clap, the leading edges of the wings
touch each other before the trailing edges, thus progressively closing the gap
between them (Fig. 4A,B). As
the wings press together closely, the opposing circulations of each of the
airfoils annul each other (Fig.
4C). This ensures that the trailing edge vorticity shed by each
wing on the following stroke is considerably attenuated or absent. Because the
shed trailing edge vorticity delays the growth of circulation via the
Wagner effect, Weis-Fogh
(1973
; see also
Lighthill, 1973
) argued that
its absence or attenuation would allow the wings to build up circulation more
rapidly and thus extend the benefit of lift over time in the subsequent
stroke. In addition to the above effects, a jet of fluid excluded from the
clapping wings can provide additional thrust to the insect
(Fig. 4C;
Ellington, 1984d
;
Ellington et al., 1996
).
|
At the end of clap, the wings continue to pronate by leaving the trailing
edge stationary as the leading edges `fling' apart
(Fig. 4DF). This process
generates a low-pressure region between them, and the surrounding fluid rushes
in to occupy this region, providing an initial impetus to the build-up of
circulation or attached vorticity (Fig.
4D,E). The two wings then translate away from each other with
bound circulations of opposite signs. Although the attached circulation around
each wing allows it to generate lift, the net circulation around the two-wing
system is still zero and thus Kelvin's law requiring conservation of
circulation is satisfied (Fig.
4F; Spedding and Maxworthy,
1986). As pointed out by Lighthill
(1973
), this phenomenon is
therefore also applicable to a fling occurring in a completely inviscid fluid.
Collectively, the clap-and-fling could result in a modest, but significant,
lift enhancement. However, in spite of its potential advantage, many insects
never perform the clap (Marden,
1987
). Others, such as Drosophila melanogaster, do clap
under tethered conditions but only rarely do so in free flight. Because
clap-and-fling is not ubiquitous among flying insects, it is unlikely to
provide a general explanation for the high lift coefficients found in flying
insects. Furthermore, when observed, the importance of the clap must always be
weighed against a simpler alternative (but not mutually exclusive) hypothesis
that the animal is simply attempting to maximize stroke amplitude, which can
significantly enhance force generation. Several studies of peak performance
suggest that peak lift production in both birds
(Chai and Dudley, 1995
) and
insects (Lehmann and Dickinson,
1997
) is constrained by the roughly 180° anatomical limit of
stroke amplitude. Animals appear to increase lift by gradually expanding
stroke angle until the wings either touch or reach some other morphological
limit with the body. Thus, an insect exhibiting a clap may be attempting to
maximize stroke amplitude. Furthermore, if it is indeed true that the Wagner
effect only negligibly influences aerodynamic forces on insect wings, the
classically described benefits of clap-and-fling may be less pronounced than
previously thought. Resolution of these issues awaits a more detailed study of
flows and forces during clap-and-fling.
Delayed stall and the leading edge vortex
As the wing increases its angle of attack, the fluid stream going over the
wing separates as it crosses the leading edge but reattaches before it reaches
the trailing edge. In such cases, a leading edge vortex occupies the
separation zone above the wing. Because the flow reattaches, the fluid
continues to flow smoothly from the trailing edge and the Kutta condition is
maintained. In this case, because the wing translates at a high angle of
attack, a greater downward momentum is imparted to the fluid, resulting in
substantial enhancement of lift. Experimental evidence and computational
studies over the past 10 years have identified the leading edge vortex as the
single most important feature of the flows created by insect wings and thus
the forces they create.
Polhamus (1971) described a
simple way to account for the enhancement of lift by a leading edge vortex
that allows for an easy quantitative analysis. For blunt airfoils, air moves
sharply around the leading edge, thus causing a leading edge suction force
parallel to the wing chord. This extra force component adds to the potential
force component (which acts normal to the wing plane), causing the resultant
force to be perpendicular to the ambient flow velocity, i.e. in the direction
of lift (Fig. 5A). At low
angles of attack, this small forward rotation due to leading edge suction
means that conventional airfoils better approximate the zero drag prediction
of potential theory (Kuethe and Chow,
1998
). However, for airfoils with sharper leading edge, flow
separates at the leading edge, leading to the formation of a leading edge
vortex. In this case, an analogous suction force develops not parallel but
normal to the plane of the wing, thus adding to the potential force and
consequently enhancing the lift component. Note that in this case, the
resultant force is perpendicular to the plane of the wing and not to ambient
velocity. Thus, drag is also increased
(Fig. 5B).
|
For 2-D motion, if the wing continues to translate at high angles of
attack, the leading edge vortex grows in size until flow reattachment is no
longer possible. The Kutta condition breaks down as vorticity forms at the
trailing edge creating a trailing edge vortex as the leading edge vortex sheds
into the wake. At this point, the wing is not as effective at imparting a
steady downward momentum to the fluid. As a result, there is a drop in lift,
and the wing is said to have stalled. For several chord lengths prior to the
stall, however, the presence of the attached leading edge vortex produces very
high lift coefficients, a phenomenon termed `delayed stall'
(Fig. 6A). The first evidence
for delayed stall in insect flight was by provided by Maxworthy
(1979), who visualized the
leading edge vortex on the model of a flinging wing. However, delayed stall
was first identified experimentally on model aircraft wings as an augmentation
in lift at the onset of motion at angles of attack above steady-state stall
(Walker, 1931
). At the lower
Reynolds numbers appropriate for most insects, the breakdown of the Kutta
condition is manifest by the growth of a trailing edge vortex, which then
grows until it too can no longer stay attached to the wing
(Dickinson and Götz,
1993
). As the trailing edge vortex detaches and is shed into the
wake, a new leading vortex forms. This dynamic process repeats, eventually
creating a wake of regularly spaced counter-rotating vortices known as the
`von Karman vortex street' (Fig.
6A). The forces generated by the moving plate oscillate in
accordance to the alternating pattern of vortex shedding. Although both lift
and drag are greatest during phases when a leading edge vortex is present,
forces are never as high as during the initial cycle.
|
The leading edge vortex may be especially important because insects flap
their wings at high angles of attack. An experimental analysis of delayed
stall in 2-D showed that flow separates to form a leading edge vortex at
angles of attack above 9°, a threshold well below those used by insects
(Dickinson and Götz,
1993). This study also directly measured time-variant force
coefficients and showed that the values created by the presence of the leading
edge vortex were at least sufficient to account for the `missing force' in
quasi-steady models. However, direct evidence that insect wings actually
create leading edge vortices came from Ellington et al.
(1996
), who used smoke to
visualize the flow around both real and 3-D model Manduca sexta at a
Reynolds number in the range of 103. In contrast to 2-D models, the
leading edge vortex was not shed even after many chords of travel and thus
never created a pattern analogous to a von Karman street. Thus, the wing never
stalls under these conditions (Fig.
6B). These observations have been confirmed at lower Reynolds
numbers in experiments on model fruit fly wings, which showed that forces,
like flows, are remarkably stable during constant flapping
(Dickinson et al., 1999
). What
causes this prolonged attachment of the leading edge vortex on a flapping wing
compared to the 2-D case? In their model hawkmoth, Ellington and co-workers
observed a steady span-wise flow from the wing hinge to approximately
three-quarters of the distance to the wing tip, at which point the leading
edge vortex detaches from the wing surface. This spanwise flow is entrained by
the leading edge vortex, causing it to spiral towards the tip of the wing
(Fig. 7). A similar flow was
observed by Maxworthy (1979
)
during early analysis of the 3-D fling. Because this flow redirects momentum
transfer in the spanwise direction, it should correspondingly reduce the
momentum of the flow from the chordwise direction, causing the leading edge
vortex to remain smaller. A smaller leading edge vortex allows the fluid to
reattach more easily and the wing can sustain this reattachment for a longer
time. Thus, axial flow appears to serve a useful role by maintaining stable
attachment of the leading edge vortex. As pointed out by Ellington, a similar
leading edge vortex is stabilized by an axial flow generated due to the
back-sweep of wings in delta aircraft such as the Concorde, creating one of
the more remarkable analogies between the biological and mechanized
worlds.
|
Recently, using DPIV to map the flow structure on a model fruit fly wing
(Re=115), Birch and Dickinson
(2001) reported stable
attachment of the leading edge vortex in the absence of a prominent helical
vortex. Whereas the axial flow within the core of the vortex was nearly an
order of magnitude lower than on the Manduca model in
Re=103 range, they observed a prominent axial flow within
a broad sheet of fluid on top of the wing behind the leading edge vortex that
rolls into a prominent tip vortex. These results from model hawkmoths and
fruit flies suggest that the 3-D flow structure may be quantitatively
different at high and low Reynolds numbers.
Interestingly, the observed differences in the 3-D flow structures do not
seem to be reflected in the measured forces. CFD simulations in 2-D
(Wang, 2000;
Hamdani and Sun, 2000
) and 3-D
(Ramamurti and Sandberg, 2002
)
airfoils show a remarkable similarity in forces calculated at Re=100
and those calculated using the inviscid Euler equation corresponding to an
infinite Re (or Re=100 000 in the case of
Hamdani and Sun, 2000
). These
results suggest that although viscosity is necessary for vorticity generation,
its contribution to net forces is very small beyond Re=100 and the
forces may be predominantly due to the dynamic pressure gradients across the
wing. The above conclusions from CFD models are also supported by empirical
data (Usherwood and Ellington,
2002b
). Together, these results present a somewhat paradoxical
conclusion that forces remain relatively unaffected even when flow structures
vary substantially with an increase in Reynolds number above 100.
Although a detailed explanation of above results awaits a more rigorous
quantification of simultaneous flow and force data, these differences should
not obscure the more salient general features of separated flow at high angles
of attack. In particular, the absence of periodic shedding in these recent
experiments indicates that the 3-D flow around a flapping wing may be
remarkably self-stabilizing over a Reynolds number range of 100. For the
existence of such stability, the creation of vorticity at the leading edge is
matched perfectly by the convection and diffusion of vorticity into the wake,
thus creating a stable equilibrium. This situation may be analogous to the
continuous attachment of vortices behind bluff bodies at Reynolds numbers
below the threshold for von Karman shedding (see, for example,
Acheson, 1990
). What maintains
the balance in creation and transport of vorticity and how does this change
with Reynolds number? Similarly, what determines the magnitude of the leading
edge vortex supported by a flapping wing at equilibrium? Given the importance
of the leading edge vortex, the answers to these questions are critical to
determining the limits of aerodynamic performance in insect flight.
Kramer effect (rotational forces)
Near the end of every stroke, insect wings undergo substantial pronation
and supination about a spanwise axis, which allows them to maintain a positive
angle of attack and generate lift during both forward and reverse strokes.
Furthermore, there is some evidence from both tethered
(Dickinson et al., 1993) and
free (Srygley and Thomas,
2002
) flight that insects alter the timing of rotation during
flight maneuvers. The aerodynamic significance of these rotations for flapping
flight has been studied by Bennett
(1970
), and more recently in
detail by Sane and Dickinson
(2002
), but it is well known
in the aerodynamic literature in the context of fluttering airplane wings due
to the extensive theoretical work of Munk
(1925b
), Glauert
(1929
), Theodorsen
(1935
), Fung
(1969
) and supporting
experimental evidence from Kramer
(1932
), Reid
(1927
), Farren
(1935
), Garrick
(1937
), Silverstein and Joyner
(1939
) and Halfman
(1951
).
When a flapping wing rotates about a span-wise axis while at the same time
translating, flow around the wing deviates from the Kutta condition and the
stagnation region moves away from the trailing edge. This causes a sharp,
dynamic gradient at the trailing edge, leading to shear. Because fluids tend
to resist shear due to their viscosity, additional circulation must be
generated around the wing to re-establish the Kutta condition at the trailing
edge. In other words, the wing generates a rotational circulation in the fluid
to counteract the effects of rotation. The re-establishment of Kutta condition
is not instantaneous, however, but requires a finite amount of time. If, in
this time, the wing continues to rotate rapidly, then the Kutta condition may
never be actually observed at any given instant of time during the rotation
but the tendency of the fluid towards its establishment may nevertheless
dictate the generation of circulation. Thus, extra circulation proportional to
the angular velocity of rotation continues to be generated until smooth,
tangential flow can be established at the trailing edge. Depending on the
direction of rotation, this additional circulation causes rotational forces
that either add to or subtract from the net force due to translation. This
effect is also often called the `Kramer effect', after M. Kramer who first
described it (Kramer, 1932),
or alternatively as `rotational forces'
(Sane and Dickinson,
2002
).
Using the conceptual framework described above, Sane and Dickinson
(2002) measured rotational
coefficients (Kramer effect) and, following the recommendation of Ellington
(1984d
,f
),
included them with the translational coefficients in the existing quasi-steady
models. The revised quasi-steady model was able to capture the corresponding
time history of the force traces, in addition to the stroke-averaged forces,
better than the quasi-steady models that take only translation into account.
Similar rotational force peaks were observed in CFD simulations by Sun and
Tang (2002
), who described
these peaks as arising from `fast pitching-up rotation of the wing near the
end of the stroke', but appear essentially complementary to the Kramer effect.
Both the revised quasi-steady model and CFD models show close agreement with
the experimental measurements.
It is important to note that the Kramer effect (or the rotational force of
Sane and Dickinson, 2002) is
fundamentally different from the Magnus force, to which a loose qualitative
analogy was drawn in the past literature
(Bennett, 1970
;
Dickinson et al., 1999
),
leading to some confusion. Magnus force arises from circulation generated by a
blunt body such as a spinning cylinder or sphere set into translational motion
with respect to the real fluid (see, for example,
Prandtl and Tietjens, 1957a
).
Although the Magnus force can be calculated from this circulation using
KuttaJukowski theorem (as in airfoils), it excludes either explicit or
implicit application of the Kutta condition because, by definition, blunt
bodies have no surface singularities where such a condition can hold. On the
other hand, the application of the Kutta condition is necessary and
fundamental to all calculations of aerodynamic forces on thin airfoils. As a
result, the mechanism of Magnus force applies only in relation to cylinders,
spheres and blunt objects, and extending it to complex surfaces such as thin
airfoils or other surfaces with sharp edges is, at best, problematic
(Schlichting, 1979
). Not
surprisingly, therefore, the Magnus effect does not provide an explanation of
the rotational forces during pronation or supination
(Sun and Tang, 2002
), nor is
it possible to apply it to calculations of forces or circulation on a flapping
thin airfoil without making severe assumptions
(Walker, 2002
).
![]() |
Added mass |
---|
Methods of calculating added mass have been outlined in various texts, most
notably in Sedov (1965), Denny
(1993
) and Lighthill
(1975
) or in research articles
by Ellington (1984d
), Sunada
et al. (2001
), Sane and
Dickinson (2002
), Zbikowski
(2002
) and Minotti
(2002
). The added mass force
is typically modeled in quasi-steady terms using a time-invariant added-mass
coefficient, and any time dependence is implicit due to the time course of
wing acceleration. In a computational study of a 2-D insect wing, Hamdani and
Sun (2000
) simulated a series
of impulsive starts at different accelerations. Their force predictions, based
on the time derivative of the moment of vorticity integral (equation 11) over
their simulated flow field, accurately matched prior experimental results. The
acceleratory forces at the start of translation corresponded to a rapid rise
in the moment of vorticity. At this early stage of motion, the rise in the
moment of vorticity is due to both the convection and growth of vorticity.
Thus, added mass forces are closely tied to the initial stages of flow
separation and fluid acceleration, and further experimental investigations
offer a promising area for further research.
![]() |
Wingwake interactions |
---|
|
Recently, Sun and Tang
(2002) performed CFD
simulations for the kinematics similar to those in Dickinson et al.
(1999
) and have proposed that
the initial force peak is due to acceleration of the wing rather than due to
wingwake interactions. To show that wingwake interaction
produces a negligible effect on forces, they started a wing from rest (in
still air) and compared the calculated forces with the forces on a wing
undergoing identical motion after stroke reversal. Interestingly, the forces
were nearly identical in the two cases and they did not observe any force
peaks due to wingwake interactions. Second, they varied the period of
acceleration of the wing immediately after stroke reversal. For higher values
of acceleration, they calculated force peaks similar in magnitude and dynamics
to the experiments of Dickinson et al.
(1999
). From these
experiments, they concluded that the forces related to the acceleration of the
wing, not wingwake interactions, fully account for the force peaks
immediately following stroke reversal.
These results are puzzling for two reasons. First, to verify their
hypothesis of wingwake interactions, Dickinson et al.
(1999) stopped the wing at
stroke reversal. They argued that if forces were augmented due to relative
velocity between the wing and the induced wake, then even a non-accelerating
wing should continue to generate forces as it encounters the wake from its
previous stroke. The results of these experiments
(fig. 4 of
Dickinson et al., 1999
),
strongly suggested that wingwake interactions can indeed contribute
significantly to the aerodynamic forces immediately after stroke reversal.
Second, they visualized the near-field wake structure at the instant of stroke
reversal using particle image velocimetry. These images revealed the
substantial wake induced by the previous stroke and also demonstrated its
physical interaction with the wing in the period immediately following stroke
reversal (Fig. 8AE). It
seems unlikely that this wingwake interaction would not be reflected in
the time history of the corresponding aerodynamic forces. At present, the
cause of this discrepancy between the CFD simulations
(Sun and Tang, 2002
) and the
particle image velocimetric observations
(Dickinson et al., 1999
)
remains unclear. In any case, these results strongly suggest that neither the
wing acceleration nor the wingwake interactions should be ignored when
modeling wake capture.
![]() |
Current status of quasi-steady modeling |
---|
Fig. 9 shows a comparison
among the lift-drag polars during steady translation in non-flapping finite
and infinite wings, as well as for flapping wings. The aerodynamic
coefficients measured on flapping wings are significantly higher than the
corresponding coefficients for non-flapping finite wings. There are several
explanations for the significant differences. First, the data on non-flapping
finite wings represent time-averaged forces collected on finite wings placed
at fixed angles of attack in a wind tunnel. If the non-flapping finite wings
exhibited von Karman vortex shedding, the force records would fail to reflect
the benefits of an initial temporary attachment of the leading edge vortex,
thus causing the force coefficients on flapping wings to be higher than the
force coefficients for non-flapping finite wings. Moreover, the steady
coefficients measured on flapping wings are similar in magnitude to the
`early', pre-stall force coefficients (measured at two chord length motion and
before von Karman shedding occurs) on an impulsively started 2-D plate whereas
the non-flapping finite wings are similar in magnitude to the `late',
post-stall force coefficients of the same profiles
(Dickinson and Götz,
1993). The instantaneous forces measured on a flapping wing at
constant angles of attack provide further evidence that flapping wings do not
show von Karman shedding. In addition, following the inertial transients
arising from the impulsive start of the airfoil, the instantaneous force
coefficients reach steady values that remain constant through a substantial
duration of flapping translation
(Dickinson et al., 1999
).
However, if the wing continues the flapping translation, the wings eventually
show some decline in performance. This may be explained by the fact that a
wing revolving in a propeller-like fashion eventually establishes a strong
downwash in the far-field, thus lowering the effective angles of attack
(Usherwood and Ellington,
2002a
). An alternative view is that, under steady flow conditions,
non-flapping finite wings with aspect ratios typical of insect wings do not
exhibit vortex shedding. Like flapping wings, they rapidly attain a stable
pattern of flow although the strength of the total circulation is
substantially lower than with a flapping motion at comparable angles of
attack. In this view, there is some feature of flapping motion such as
the span-wise gradient in chord-wise velocity that allows the leading
edge vortex to attain a greater strength than in the non-flapping case. In the
future, these issues might be resolved by simultaneous flow visualization and
force measurement under the two experimental conditions. Of special interest
are differences in convective processes such as axial flow
(Ellington et al., 1996
),
downwash induced by tip vortices or diffusion that might limit the growth of
vorticity at the leading edge to different degrees in the two cases.
|
When we revisit the proof-by-contradiction method using these higher values
of steady lift and drag coefficients appearing in the recent literature on
flapping wings, the calculated average forces are proportionally higher and
sufficient to explain hovering. This has led to a revival of the quasi-steady
models in recent years. Indeed, when rotational coefficients are included
along with the translational coefficients in the quasi-steady model, the time
course of aerodynamic force generation is also well captured
(Sane and Dickinson, 2002).
However, such a model cannot yet account for the force peaks resulting from
wingwake interactions.
![]() |
Future research and directions |
---|
These developments in the area of insect flight aerodynamics will prove
critical to biologists who seek to understand how flight and flight-related
adaptations have enabled insects to be so extraordinarily successful in the
course of their evolution. In addition, they also promise to be useful in
breaking new ground in technology. The recent interest in developing
insect-inspired micro air vehicles (MAVs; also called micro-mechanical flying
insects or MFIs) has fostered a number of strong collaborations between
analytical and computational fluid dynamicists, micro-robotics engineers and
insect flight biologists (Ellington,
1999; Zbikowski,
2002
). The combination of expertise from these different areas
promises to help insect flight biologists ask questions and devise experiments
that were previously inconceivable. From an academic standpoint, the success
of these projects will allow us to satisfactorily demonstrate our
understanding of the fundamental fluid dynamic mechanisms underlying insect
flight.
List of symbols
![]() |
Acknowledgments |
---|
![]() |
References |
---|
Acheson, D. (1990). Elementary fluid dynamics. In Oxford Applied Mathematics and Computing Science series (ed. R. Churchouse, W. McColl and A. Tayler), pp.157 -200. Oxford: Oxford University Press.
Azuma, A., Azuma, S., Watanabe, I. and Furuta, T. (1985). Flight mechanics of a dragonfly. J. Exp. Biol. 116,79 -107.
Azuma, A. and Watanabe, T. (1988). Flight performance of a dragonfly. J. Exp. Biol. 137,221 -252.
Batchelor, G. K. (1973). An Introduction to Fluid Dynamics. Cambridge, New York: Cambridge University Press.
Bennett, L. (1970). Insect flight: lift and the rate of change of incidence. Science 167,177 -179.
Bennett, L. (1977). Clap and fling aerodynamics an experimental evaluation. J. Exp. Biol. 69,261 -272.
Birch, J. and Dickinson, M. H. (2001). Spanwise flow and the attachment of the leading-edge vortex. Nature 412,729 -733.[CrossRef][Medline]
Brodsky, A. K. (1994). The Evolution of Insect Flight. New York: Oxford University Press.
Buckholz, R. H. (1981). Measurements of unsteady periodic forces generated by the blowfly flying in a wind tunnel. J. Exp. Biol. 90,163 -173.
Chai, P. and Dudley, R. (1995). Maximum right performance and limits to power output of vertebrate striated-muscle. FASEB J. 9,A353 .
Cloupeau, M., Devillers, J. F. and Devezeaux, D. (1979). Direct measurements of instantaneous lift in desert locust; comparison with Jensen's experiments on detached wings. J. Exp. Biol. 80,1 -15.
Daniel, T. (1984). Unsteady aspects of aquatic locomotion. Am. Zool. 24,121 -134.
Daniel, T. and Combes, S. (2002). Flexing wings and fins: bending by inertial or fluid dynamic forces? Int. Comp. Biol. 42,1044 -1049.
Denny, M. (1993). Air and Water: The Biology and Physics of Life's Media. Princeton, NJ: Princeton University Press.
Dickinson, M. H. (1994). The effects of wing
rotation on unsteady aerodynamic performance at low Reynolds numbers.
J. Exp. Biol. 192,179
-206.
Dickinson, M. H. and Götz, K. G. (1993).
Unsteady aerodynamic performance of model wings at low Reynolds numbers.
J. Exp. Biol. 174,45
-64.
Dickinson, M. H., Lehmann, F.-O. and Götz, K. G.
(1993). The active control of wing rotation by Drosophila.J. Exp. Biol. 182,173
-189.
Dickinson, M. H., Lehmann, F.-O. and Sane, S. P.
(1999). Wing rotation and the aerodynamic basis of insect flight.
Science 284,1954
-1960.
Dudley, R. (1991). Biomechanics of flight in neotropical butterflies aerodynamics and mechanical power requirements. J. Exp. Biol. 159,335 -357.
Dudley, R. (2000). The Biomechanics of Insect Flight. Princeton, NJ: Princeton University Press.
Dudley, R. and Ellington, C. P. (1990a). Mechanics of forward flight in bumblebees. 1. Kinematics and morphology. J. Exp. Biol. 148,19 -52.
Dudley, R. and Ellington, C. P. (1990b). Mechanics of forward flight in bumblebees. 2. Quasi-steady lift and power requirements. J. Exp. Biol. 148, 53-88.
Ellington, C. P. (1978). The aerodynamics of normal hovering flight: three approaches. In Comparative Physiology: Water, Ions and Fluid mechanics (ed. K. Schmidt-Nielsen, L. Bolis and S. Maddrell), pp.327 -345. Cambridge: Cambridge University Press.
Ellington, C. P. (1980). Vortices and hovering flight. In Instationare Effekte an schwingended Fluegeln (ed. W. Nachtigall), pp.64 -101. Weisbaden: F. Steiner.
Ellington, C. P. (1984a). The aerodynamics of hovering insect flight. I. The quasi-steady analysis. Phil. Trans. R. Soc. Lond. B 305,1 -15.
Ellington, C. P. (1984b). The aerodynamics of hovering insect flight. II. Morphological parameters. Phil. Trans. R. Soc. Lond. B 305,17 -40.
Ellington, C. P. (1984c). The aerodynamics of hovering insect flight. III. Kinematics. Phil. Trans. R. Soc. Lond. B 305,41 -78.
Ellington, C. P. (1984d). The aerodynamics of hovering insect flight. IV. Aerodynamic mechanisms. Phil. Trans. R. Soc. Lond. B 305,79 -113.
Ellington, C. P. (1984e). The aerodynamics of hovering insect flight. V. A vortex theory. Phil. Trans. R. Soc. Lond. B 305,115 -144.
Ellington, C. P. (1984f). The aerodynamics of hovering insect flight. VI. Lift and power requirements. Phil. Trans. R. Soc. Lond. B 305,145 -181.
Ellington, C. P. (1999). The novel aerodynamics
of insect flight: applications to micro-air vehicles. J. Exp.
Biol. 202,3439
-3448.
Ellington, C. P., Van den Berg, C., Willmott, A. P. and Thomas, A. L. R. (1996). Leading-edge vortices in insect flight. Nature 384,626 -630.[CrossRef]
Ennos, A. R. (1989a). Inertial and aerodynamic torques on the wings of Diptera in flight. J. Exp. Biol. 142,87 -95.
Ennos, A. R. (1989b). The kinematics and aerodynamics of the free flight of some Diptera. J. Exp. Biol. 142,49 -85.
Farren, W. S. (1935). The reaction on a wing whose angle of incidence is changing rapidly. Rep. Memo. Aeronaut. Res. Comm. (Great Britain) 1648.
Fung, Y. C. (1969). An Introduction to the Theory of Aeroelasticity. New York: Dover.
Garrick, I. E. (1937). Propulsion of a flapping and oscillating airfoil. NACA report 567,419 -427.
Glauert, H. (1929). The force and moment on an oscillating airfoil. Rep. Memo. Aeronaut. Res. Comm. (Great Britain) no. 1561.
Glauert, H. (1947). The Elements of Aerofoil and Airscrew Theory. New York: Cambridge Science Classics.
Halfman, R. (1951). Experimental aerodynamic derivatives of a sinusoidally oscillating airfoil in two-dimensional flow. NACA TN 2465.
Hamdani, H. and Sun, M. (2000). Aerodynamic forces and flow structures of an airfoil in some unsteady motions at small Reynolds number. Acta Mechanica 145,173 -187.
Hamdani, H. and Sun, M. (2001). A study on the mechanism of high-lift generation by an airfoil in unsteady motion at low Reynolds number. Acta Mechanica Sinica 17, 97-114.
Jensen, M. (1956). Biology and physics of locust flight. III The aerodynamics of locust flight. Phil. Trans. R. Soc. Lond. B 239,511 -552.
Kramer, M. (1932). Die Zunahme des Maximalauftriebes von Tragflugeln bei plotzlicher Anstellwinkelvergrosserung (Boeneffekt). Z. Flugtech. Motorluftschiff. 23,185 -189.
Kuethe, A. and Chow, C.-Y. (1998). Foundations of Aerodynamics: Bases of Aerodynamic Design. New York: John Wiley & Sons.
Lamb, H. (1945). Hydrodynamics. New York: Dover Publications.
Landau, L. D. and Lifshitz, E. M. (1959). Fluid Mechanics. London: Pergamon Press.
Lehmann, F.-O. and Dickinson, M. H. (1997). The
changes in power requirements and muscle efficiency during elevated force
production in the fruit fly Drosophila melanogaster. J. Exp.
Biol. 200,1133
-1143.
Lighthill, M. (1973). On Weis-Fogh mechanism of lift generation. J. Fluid Mech. 60, 1-17.
Lighthill, M. (1975). Mathematical biofluiddynamics. In Regional Conference Series in Applied Mathematics, vol. 17. Philadelphia: Society for Industrial and Applied Mathematics.
Liu, H., Ellington, C. P., Kawachi, K., Van den Berg, C. and
Willmott, A. P. (1998). A computational fluid dynamic
study of hawkmoth hovering. J. Exp. Biol.
201,461
-477.
Liu, H. and Kawachi, K. (1998). A numerical study of insect flight. J. Comput. Physics 146,124 -156.[CrossRef]
Marden, J. H. (1987). Maximum lift production during takeoff in flying animals. J. Exp. Biol. 130,235 -258.
Maxworthy, T. (1979). Experiments on the Weis-Fogh mechanism of lift generation by insects in hovering flight. Part 1. Dynamics of the `fling'. J. Fluid Mech. 93, 47-63.
Milne-Thomson, L. M. (1966). Theoretical Aerodynamics. London, New York: Macmillan, St Martin's Press.
Minotti, F. (2002). Unsteady two-dimensional theory of a flapping wing. Phys. Rev. E 66, art. no. 051907.
Munk, M. (1925a). Elements of the wing section theory and of wing theory. NACA Report 191,141 -163.
Munk, M. (1925b). Note on the air forces on a wing caused by pitching. NACA TN 217.
Nachtigall, W. (1977). Die Aerodynamische Polare des Tipula-Flugels und eine Einrichtung zur halbautomatischen Polarenaufnahme. In The Physiology of Movement: Biomechanics (ed. W. Nachtigall), pp.347 -352. Stuttgart: Fischer.
Norberg, R. (1975). Hovering flight of the dragonfly, Aeschna juncea L., kinematics and aerodynamics. In Swimming and Flying in Nature, vol.2 (ed. T. Wu, C. Brokaw and C. Brennen), pp.763 -781. New York: Plenum Press.
Osborne, M. F. M. (1950). Aerodynamics of flapping flight with application to insects. J. Exp. Biol. 28,221 -245.
Polhamus, E. (1971). Predictions of vortex-lift characteristics by a leading-edge suction analogy. J. Aircraft 8,193 -199.
Prandtl, L. and Tietjens, O. K. G. (1957a). Applied Hydro- and Aeromechanics; Based on Lectures of L. Prandtl. New York: Dover Publications.
Prandtl, L. and Tietjens, O. K. G. (1957b). Fundamentals of Hydro- and Aeromechanics. New York: Dover Publications.
Ramamurti, R. and Sandberg, W. C. (2002). A
three-dimensional computational study of the aerodynamic mechanisms of insect
flight. J. Exp. Biol.
205,1507
-1518.
Rayner, J. M. V. (1979a). A vortex theory of animal flight. Part 1. The vortex wake of a hovering animal. J. Fluid Mech. 91,697 -730.
Rayner, J. M. V. (1979b). A vortex theory of animal flight. Part 2. The forward flight of birds. J. Fluid Mech. 91,731 -763.
Rees, C. (1975). Aerodynamic properties of an insect wing section and a smooth aerofoil compared. Nature 258,141 -142.[Medline]
Reid, E. (1927). Airfoil lift with changing angle of attack. NACA TN 266.
Ruppell, G. (1989). Kinematic analysis of symmetrical flight manoeuvres of odonata. J. Exp. Biol. 144,13 -43.
Sane, S. P. and Dickinson, M. H. (2001). The
control of flight force by a flapping wing: lift and drag production.
J. Exp. Biol. 204,2607
-2626.
Sane, S. P. and Dickinson, M. H. (2002). The
aerodynamic effects of wing rotation and a revised quasi-steady model of
flapping flight. J. Exp. Biol.
205,1087
-1096.
Sato, M. and Azuma, A. (1997). Flight
performance of a damselfly Ceriagrion melanurum Selys. J.
Exp. Biol. 200,1765
-1779.
Savage, S., Newman, B. and Wong, D. (1979). The role of vortices and unsteady effects during the hovering flight of dragonflies. J. Exp. Biol. 83, 59-77.
Schlichting, H. (1979). Boundary-Layer Theory. New York: McGraw-Hill.
Sedov, L. I. (1965). Two-Dimensional Problems in Hydrodynamics and Aerodynamics (ed. C. Chu, H. Cohen and B. Seckler), pp. 20-30. New York: Interscience Publishers.
Silverstein, A. and Joyner, U. (1939). Experimental verification of the theory of oscillating airfoils. NACA Report 673.
Smith, M., Wilkin, P. and Williams, M. (1996).
The advantages of an unsteady panel method in modeling the aerodynamic forces
on rigid flapping wings. J. Exp. Biol.
199,1073
-1083.
Somps, C. and Luttges, M. (1985). Dragonfly flight novel uses of unsteady separated flows. Science 228,1326 -1329.
Spedding, G. R. and Maxworthy, T. (1986). The generation of circulation and lift in a rigid two-dimensional fling. J. Fluid Mech. 165,247 -272.
Srygley, R. and Thomas, A. (2002). Unconventional lift-generating mechanisms in free-flying butterflies. Nature 420,600 -664.[CrossRef][Medline]
Sun, M. and Tang, J. (2002). Unsteady
aerodynamic force generation by a model fruit fly wing in flapping motion.
J. Exp. Biol. 205,55
-70.
Sunada, S., Kawachi, K., Matsumoto, A. and Sakaguchi, A. (2001). Unsteady forces on a two-dimensional wing in plunging and pitching motion. AIAA J. 39,1230 -1239.
Sunada, S., Kawachi, K., Watanabe, I. and Azuma, A.
(1993). Fundamental analysis of 3-dimensional near fling.
J. Exp. Biol. 183,217
-248.
Theodorsen, T. (1935). General theory of aerodynamic instability and the mechanism of flutter. NACA Report 496.
Truesdell, C. (1954). The Kinematics of Vorticity. Indian University Publications Science Series 19. Bloomington: Indiana University Press.
Usherwood, J. R. and Ellington, C. P. (2002a).
The aerodynamics of revolving wings I. Model hawkmoth wings.
J. Exp. Biol. 205,1547
-1564.
Usherwood, J. R. and Ellington, C. P. (2002b).
The aerodynamics of revolving wings II. Propeller force coefficients
from mayfly to quail. J. Exp. Biol.
205,1565
-1576.
VandenBerg, C. and Ellington, C. P. (1997). The three-dimensional leading-edge vortex of a `hovering' model hawkmoth. Phil. Trans. R. Soc. Lond. B 352,329 -340.[CrossRef]
Vogel, S. (1967). Flight in Drosophila. III. Aerodynamic characteristics of fly wings and wing models. J. Exp. Biol. 46,431 -443.
Vogel, S. (1994). Life in Moving Fluids. Princeton, NJ: Princeton University Press.
Wagner, H. (1925). Über die Entstehung des dynamischen Äuftriebes von Tragflügeln. Z. Angew. Math. Mech. 5,17 -35.
Walker, J. (2002). Rotational lift: something different or more of the same? J. Exp. Biol. 205,3783 -3792.[Medline]
Walker, J. A. and Westneat, M. W. (2000). Mechanical performance of aquatic rowing and flying. Proc. R. Soc. Lond. Ser. B. Biol. Sci. 267,1875 -1881.[CrossRef][Medline]
Walker, P. B. (1931). Experiments on the growth of circulation about a wing and an apparatus for measuring fluid motion. Rep. Memo. Aeronaut. Res. (Great Britain) No1402.
Wang, Z. J. (2000). Two dimensional mechanism for insect hovering. Phys. Rev. Lett. 85,2216 -2219.[CrossRef][Medline]
Weis-Fogh, T. (1973). Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol. 59,169 -230.
Wilkin, P. J. and Williams, M. H. (1993). Comparison of the instantaneous aerodynamic forces on a sphingid moth with those predicted by quasi-steady aerodynamic theory. Physiol. Zool. 66,1015 -1044.
Willmott, A. and Ellington, C. (1997a). The mechanics of flight in the hawkmoth Manduca sexta. 2. Aerodynamic consequences of kinematic and morphological variation. J. Exp. Biol. 200,2773 -2745.
Willmott, A. P. and Ellington, C. P. (1997b).
Measuring the angle of attack of beating insect wings: robust
three-dimensional reconstruction from two-dimensional images. J.
Exp. Biol. 200,2693
-2704.
Wu, J. (1981). Theory for aerodynamic force and moment in viscous flows. AIAA J. 19,432 -441.
Zanker, J. M. and Gotz, K. G. (1990). The wing beat of Drosophila melanogaster II. Dynamics. Phil. Trans. R. Soc. Lond. B 327,19 -44.
Zbikowski, R. (2002). On aerodynamic modelling of an insect-like flapping wring in hover for micro air vehicles. Phil. Trans. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci. 360,273 -290.