First published online October 27, 2003
The Journal of Experimental Biology 206, 4191-4208 (2003)
Copyright © 2003 The Company of Biologists Limited
doi: 10.1242/jeb.00663
The aerodynamics of insect flight
Sanjay P. Sane
Department of Biology, University of Washington, Seattle, WA 98195,
USA
(e-mail:
sane{at}u.washington.edu)
Accepted 12 August 2003
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Summary
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The flight of insects has fascinated physicists and biologists for more
than a century. Yet, until recently, researchers were unable to rigorously
quantify the complex wing motions of flapping insects or measure the forces
and flows around their wings. However, recent developments in high-speed
videography and tools for computational and mechanical modeling have allowed
researchers to make rapid progress in advancing our understanding of insect
flight. These mechanical and computational fluid dynamic models, combined with
modern flow visualization techniques, have revealed that the fluid dynamic
phenomena underlying flapping flight are different from those of non-flapping,
2-D wings on which most previous models were based. In particular, even at
high angles of attack, a prominent leading edge vortex remains stably attached
on the insect wing and does not shed into an unsteady wake, as would be
expected from non-flapping 2-D wings. Its presence greatly enhances the forces
generated by the wing, thus enabling insects to hover or maneuver. In
addition, flight forces are further enhanced by other mechanisms acting during
changes in angle of attack, especially at stroke reversal, the mutual
interaction of the two wings at dorsal stroke reversal or wing–wake
interactions following stroke reversal. This progress has enabled the
development of simple analytical and empirical models that allow us to
calculate the instantaneous forces on flapping insect wings more accurately
than was previously possible. It also promises to foster new and exciting
multi-disciplinary collaborations between physicists who seek to explain the
phenomenology, biologists who seek to understand its relevance to insect
physiology and evolution, and engineers who are inspired to build
micro-robotic insects using these principles. This review covers the basic
physical principles underlying flapping flight in insects, results of recent
experiments concerning the aerodynamics of insect flight, as well as the
different approaches used to model these phenomena.
Key words: insect flight, aerodynamics, Kramer effect, delayed stall, quasi-steady modeling, flapping flight, kinematics, forces, flows, leading edge vortex
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Introduction
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Insects owe much of their extraordinary evolutionary success to flight.
Compared with their flightless ancestors, flying insects are better equipped
to evade predators, search food sources and colonize new habitats. Because
their survival and evolution depend so crucially on flight performance, it is
hardly surprising that the flight-related sensory, physiological, behavioral
and biomechanical traits of insects are among the most compelling
illustrations of adaptations found in nature. As a result, insects offer
biologists a range of useful examples to elucidate both
structure–function relationships and evolutionary constraints in
organismal design (Brodsky,
1994
; Dudley,
2000).
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.
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Experimental challenges
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Because of their small size and high wing beat frequencies, it is often
quite difficult to quantify the wing motions of free-flying insects. For
example, an average-sized insect such as the common fruit fly Drosophila
melanogaster is approximately 2–3 mm in length and flaps its wings
at a rate of 200 Hz. Just the mere quantification of motion for such small and
fast-moving wings continues to pose significant challenges to current
technology. Early attempts to capture free-flight wing kinematics such as
Ellington's comprehensive and influential survey
(Ellington, 1984c) relied
primarily on single-image high-speed cine. Although quite informative,
especially because film continues to offer exceptional spatial resolution,
single-view techniques cannot provide an accurate time course of the angle of
attack of the two wings. More recent methods have employed high-speed
videography (Willmott and Ellington,
1997b), which offers greater light sensitivity and ease of use,
albeit at the cost of image resolution. A further consideration is that
insects rely extensively on visual feedback, and hence care must be taken to
ensure that lighting conditions do not significantly impair an insect's
behavior.
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.
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Conventions and terminology
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Because most literature on flapping flight has adopted standard terminology
borrowed from fixed wing aerodynamics, it is necessary to first develop a
nomenclature that allows us to unambiguously distinguish between these two
types of flight. As in fixed wing aerodynamics, `wing span' refers to the
length between the tips of the wings when they are stretched out laterally
(Fig. 1A), whereas `wing
length' refers to the base-to-tip length of one wing. Wing span is often given
as twice wing length, thereby ignoring the width of the animal's thorax. `Wing
chord' refers to the section between the leading and trailing edge of the wing
at any given position along the span (Fig.
1A). The ratio of span to mean chord is an important
non-dimensional morphological parameter termed `aspect ratio'. `Angle of
attack' refers to the angle that the wing chord makes with the relative
velocity vector of the fluid far away from the influence of the airfoil, i.e.
relative to the `far-field flow' or `free-stream flow'
(Fig. 1B). The restriction to
far-field flow in this definition is necessary because the presence of the
airfoil influences the fluid field immediately around it. In all real
airfoils, the process of generating lift creates an induced downwash in the
flow all around the wing. Although the magnitude of this downwash
(U') is small compared with the `free-stream velocity'
(U
), it can significantly alter the direction of
resultant velocity and thus attenuate the performance of the wing by lowering
the angle of attack (Fig. 1B;
Munk, 1925a;
Kuethe and Chow, 1998). For
this reason, it is important to qualify whether the angle of attack is
measured with respect to the gross flow in the immediate vicinity of the wing
or far away from it. The angle of attack relative to the direction of
free-stream velocity is called `geometric angle of attack' (
), whereas
the altered angle of attack relative to the locally deflected free stream is
called the `aerodynamic' or `effective angle of attack' ('),
where:
 | (1) |
Because it is difficult to physically measure the downwash-related deflection
of the free stream, most insect flight studies report geometric rather than
aerodynamic angles of attack.
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.
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Background theory for thin airfoils
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Before addressing the specific theoretical challenges posed by insect
flight aerodynamics, it is first necessary to introduce general equations and
physical principles that govern forces and flows created by moving objects
submersed in fluids. These formulations borrow extensively from methods used
by physicists and engineers for nearly 100 years to predict the forces created
by thin flat wings moving at very low angles of attack
(Prandtl and Tietjens, 1957b;
Milne-Thomson, 1966).
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 Navier–Stokes equation, a
non-dimensional form of which can be written as:
 | (2) |
where û, 
,
are, respectively, the velocity of
the flapping wing relative to its fluid medium, time and pressure. All these
quantities are non-dimensionalized (denoted by ^) with respect to their
corresponding `characteristic' measures. The choice of a characteristic
measure is somewhat arbitrary and often based on the physicist's intuition of
which constants of the system are physically meaningful. For example, when
modeling the flow around a section of a high aspect ratio wing, the chord
length is often used as the characteristic length measure. The operator:
 | (3) |
is a non-dimensional form of the vector `del' operator, and i, j and
k are unit Cartesian vectors. The left-hand side of equation 2
represents the Lagrangian (or material) derivative of the velocity,
incorporating both the implicit and explicit dependence on time. In the
Eulerian representation, the Lagrangian derivative is simply the temporal
derivative of the motion of a fluid particle as measured by an observer moving
with the fluid. The denominator of the last term in equation 2 is the Reynolds
number (Re), a non-dimensional parameter that describes the ratio of
inertia of a moving fluid mass to the viscous dissipation of its motion.
Reynolds number can be calculated by the relation
Re=(
UL)/
, where
is the density of the fluid medium, U is the
velocity of the fluid relative to the moving object, L is a
characteristic length measure and is the dynamic viscosity of the fluid
medium. This parameter roughly characterizes the fluid dynamic regime in which
an insect operates from laminar (for low values of Re) to turbulent
(for high values of Re). When viscosity is large, Re is
small and the last term in equation 2 becomes relatively more important than
the pressure term. When viscosity is negligible, the values of Re are
large and the last term can be dropped from the equation to obtain the
inviscid form (i.e. zero viscosity) of equation 2, usually called the `Euler
equation'. Equation 2 also provides the mathematical justification for the use
of dynamically scaled physical models. The non-dimensionalized forces and
flows generated by isometrically scaled objects are the same provided that the
Re are identical.
The Navier–Stokes 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 Navier–Stokes
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) |
The quantity
,
defined as the `vorticity' of the fluid, is very useful in the
conceptualization and characterization of the flows around airfoils. For the
case of steady inviscid flows, 
and the flows are said to be `irrotational'. When flows are irrotational over
all space, it is often convenient to express the velocity field as a gradient
of a scalar potential function 
(i.e.
.
This approach, called the `potential theory', has proven very useful in the
elucidation of many basic aerodynamic theorems. Essentially, the technique
involves constructing specific forms of that best describe a given fluid
dynamic phenomenon under its appropriate initial and boundary conditions.
Vorticity arises from a combination of mutually orthogonal spatial derivatives
of velocity at a given point in space. Thus, its value at any given point does
not offer a complete picture of the related aerodynamic forces. To calculate
aerodynamic forces, small vorticity elements must be integrated over a surface
area around an airfoil. Using the Stokes theorem, which relates the area
integral of normal component of vorticity to a line integral of velocity
around a closed contour 
bounding a surface S:
 | (5) |
The quantity on the left-hand side of this equation is defined as
`circulation' (often denoted by 
). For potential flows, its value
around any closed contour not enclosing a wing section is zero because
vorticity is zero everywhere in accordance with the assumption of irrotational
flow. However, if the closed contour encloses a wing section, then the
presence of even the slightest viscosity, and therefore a finite amount of
shear at the wing-fluid interface, will give rise to finite vorticity and thus
non-zero circulation.
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
Kutta–Jukowski 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'.
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Fig. 2. Kutta condition and circulation. The Kutta condition arises from a sum of
the flow around an airfoil placed in an inviscid fluid (A) to additional
circulation arising from the presence of viscosity (B) to yield a smooth,
tangential flow from the trailing edge (C). When satisfied, the Kutta
condition ensures that the vorticity generated at the trailing edge is zero.
For the inviscid case, the net force on the airfoil (blue arrow) acts
perpendicular to the free stream.
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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 Kutta–Jukowski 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) |
or
 | (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) |
where r is the displacement vector or distance vector. This is called
the Biot-Savart's law (Milne-Thomson,
1966), which, like its electromagnetic analog, is an inverse
square relationship. This integral must be evaluated over the entire volume of
the fluid (V). Equation 9 is very useful in most vorticity-based
analyses of fluid dynamics, as well as in modeling the effects of vortex
dipoles on their surrounding medium.
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 solid–fluid 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) |
where r is the distance from origin of an arbitrary co-ordinate system
moving with the free stream, is the vorticity and R is the
area of region of interest encompassing all vorticity elements. For the
two-dimensional incompressible viscous case, the sectional aerodynamic force
F may be derived from the first moment of vorticity 
by the
equation:
 | (11) |
where is fluid density, A is the cross-sectional area of the
airfoil and 
is the velocity of a point within the airfoil
(Wu, 1981). The first term on
the right-hand side of this equation represents the temporal derivative of the
first moment of vorticity, which is equal to the force arising from the
vorticity created by the movement of the airfoil. The second term in the
equation represents the inertial force of the fluid displaced by the wing
section. For an infinitesimally thin wing, the sectional area is negligible
and force depends solely on the moment of vorticity. For the simple case of
any bound circulation, a stable distribution of vorticity moves with the wing,
and a constant growth of the moment of vorticity results solely from the
wing's motion. In agreement with the Kutta–Jukowski theorem, the
sectional lift is equal to the product of the circulation created by a wing
and its translational velocity (Wu,
1981). Equation 11 is more general, however, and can account for
forces generated when both the strength and distribution of vorticity around
the wing are changing, as might occur at the start of motion, during rapid
changes in kinematics or when the wing encounters vorticity created by its own
wake or that of another wing.
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Theoretical challenges
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The challenges in adopting the traditional methods described in the
previous section to insect flight are manifold and only briefly described
here. Determined primarily by their variation in size, flying insects operate
over a broad range of Reynolds numbers from approximately 10 to 105
(Dudley, 2000). For
comparison, the Reynolds number of a swimming sperm is approximately
10–2, a swimming human being is 106 and a
commercial jumbo jet at 0.8 Mach is 107. At the high Reynolds
numbers characteristic of the largest insects, the importance of the viscous
term in equation 2 may be negligible and, as with aircraft, flows and forces
may be governed by its inviscid form (the Euler equation). Such
simplifications may not always be possible for most species, whose small size
translates into low Reynolds numbers. This is not to say that viscous forces
dominate in small insects. To the contrary, even at a Reynolds number of 10,
inertial forces are roughly an order of magnitude greater than viscous forces.
However, viscous effects become more important in structuring flow and thus
cannot be ignored. Due to these viscous effects, the principles underlying
aerodynamic force production may differ in small vs large insects.
For tiny insects, small perturbations in the fluid may be more rapidly
dissipated due to viscous resistance to fluid motion. However, for larger
insects operating at higher Reynolds numbers, small perturbations in the flow
field accumulate with time and may ultimately result in stronger unsteadiness
of the surrounding flows. Even with the accurate knowledge of the smallest
perturbations, such situations are impossible to predict analytically because
there may be several possible solutions to the flow equations. In such cases,
strict static and dynamic initial and boundary conditions must be identified
to reduce the number of solutions to a few meaningful possibilities.
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Analytical models of insect flight
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The experimental and theoretical challenges mentioned in the previous
sections constrained early models of insect flight to analysis of far-field
wakes rather than the fluid phenomena in the immediate vicinity of the wing.
Although such far-field models could not be used to calculate the
instantaneous forces on airfoils, they offered some hope of characterizing
average forces as well as power requirements. Most notable among these are the
`vortex models' (Ellington,
1978,
1980,
1984e; Rayner,
1979a,b),
both of which are derived by approximating flapping wings to blades of a
propeller or, more accurately, to idealized actuator disks that generate
uniform pressure pulses to impart downward momentum to the surrounding fluid.
By this method, the mean lift required to hover may be estimated by equating
the rate of change of momentum flux within the downward jet with the weight of
the insect and thus calculating the circulation required in the wake to
maintain this force balance. A detailed description of these theories appears
in Rayner
(1979a,b)
and Ellington (1984e) and is
beyond the scope of this review, which will focus instead on near-field
models.
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).
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Computational fluid dynamics (CFD)
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With recent advances in computational methods, many researchers have begun
exploring numerical methods to resolve the insect flight problem, with varying
degrees of success (Smith et al.,
1996; Liu et al.,
1998; Liu and Kawachi,
1998; Wang, 2000;
Ramamurti and Sandberg, 2002;
Sun and Tang, 2002). Although
ultimately these techniques are more rigorous than simplified analytical
solutions, they require large computational resources and are not as easily
applied to large comparative data sets. Furthermore, CFD simulations rely
critically on empirical data both for validation and relevant kinematic input.
Nevertheless, several collaborations have recently emerged that have led to
some exciting CFD models of insect flight.
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 Navier–Stokes 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 wing–wake 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.
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Quasi-steady modeling of insect flight
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In the hope of finding approximate analytical solutions to the insect
flight problem, scientists have developed simplified models based on the
quasi-steady approximations. According to the quasi-steady assumption, the
instantaneous aerodynamic forces on a flapping wing are equal to the forces
during steady motion of the wing at an identical instantaneous velocity and
angle of attack (Ellington,
1984a). It is therefore possible to divide any dynamic kinematic
pattern into a series of static positions, measure or calculate the force for
each and thus reconstruct the time history of force generation. By this
method, any time dependence of the aerodynamic forces arises from time
dependence of the kinematics but not that of the fluid flow itself. If such
models are accurate, then it would be possible to use a relatively simple set
of equations to calculate aerodynamic forces on insect wings based solely on
knowledge of their kinematics.
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.
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Physical modeling of insect flight
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Given the difficulties in directly studying insects or making theoretical
calculations of their flight aerodynamics, many researchers have used
mechanical models to study insect flight. When constructing these models, the
Reynolds number and reduced frequency parameter (body velocity/wing velocity)
of the mechanical model is matched to that of an actual insect. This
condition, called `dynamic scaling', ensures that the underlying fluid dynamic
phenomena are conserved. Because it is relatively easier to measure and
visualize flow around the scaled models than on insect wings, such models have
proved extremely useful in identification and analysis of several unsteady
mechanisms such as the clap-and-fling
(Bennett, 1977;
Maxworthy, 1979;
Spedding and Maxworthy, 1986),
delayed stall (Dickinson and Götz,
1993; Ellington et al.,
1996), rotational lift
(Bennett, 1970;
Ellington, 1984d;
Dickinson et al., 1999;
Sane and Dickinson, 2002) and
wing–wake interactions (Dickinson,
1994; Dickinson et al.,
1999). These various mechanisms are discussed in the following
section.
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Unsteady mechanisms in insect flight
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Wagner effect
When an inclined wing starts impulsively from rest, the circulation around
it does not immediately attain its steady-state value
(Walker, 1931). Instead, the
circulation rises slowly to the steady-state estimate
(Fig. 3). This delay in
reaching the steady-state values may result from a combination of two
phenomena. First, there is inherent latency in the viscous action on the
stagnation point and thus a finite time before the establishment of Kutta
condition. Second, during this process, vorticity is generated and shed at the
trailing edge, and the shed vorticity eventually rolls up in the form of a
starting vortex. The velocity field induced in the vicinity of the wing by the
vorticity shed at the trailing edge additionally counteracts the growth of
circulation bound to the wing. After the starting vortex has moved
sufficiently far from the trailing edge, the wing attains its maximum steady
circulation (Fig. 3). This
sluggishness in the development of circulation was first proposed by Wagner
(1925) and studied
experimentally by Walker
(1931) and is often referred
to as the Wagner effect. Unlike the other unsteady mechanisms described below,
the Wagner effect is a phenomenon that would act to attenuate forces below
levels predicted by quasi-steady models. However, more recent studies with 2-D
wings (Dickinson and Götz,
1993) indicate that the Wagner effect might not be particularly
strong at the Reynolds numbers typical of most insects. For infinite wings
translating at small angles of attack (less than 10°), lift grows very
little, if at all, after two chord lengths of travel. Similar experiments for
flapping translation in 3-D also show little evidence for the Wagner effect
(Dickinson et al., 1999).
However, because this effect relates directly to the growth of vorticity at
the onset of motion, both its measurement and theoretical treatment are
complicated due to interaction with added mass effects described in a later
section. Nevertheless, most recent models of flapping insect wings have
neglected the Wagner effect (but see
Walker and Westneat, 2000;
Walker, 2002) and focused
instead on other unsteady effects.
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Fig. 3. Wagner effect. The ratio of instantaneous to steady circulation
(y-axis) grows as the trailing edge vortex moves away from the
airfoil (inset), and its influence on the circulation around the airfoil
diminishes with distance (x-axis). Distance is non-dimensionalized
with respect to chord lengths traveled. The graph is based on fig. 35 in
Walker (1931). The inset
figures are schematic diagrams of the Wagner effect. Dotted lines show the
vorticity shedding from the trailing edge, eventually rolling up into a
starting vortex. As this vorticity is shed into the wake, bound circulation
builds up around the wing section, shown by the increasing thickness of the
line drawn around the wing section.
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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).
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Fig. 4. Section schematic of wings approaching each other to clap (A–C) and
flinging apart (D–F). Black lines show flow lines, and dark blue arrows
show induced velocity. Light blue arrows show net forces acting on the
airfoil. (A–C) Clap. As the wings approach each other dorsally (A),
their leading edges touch initially (B) and the wing rotates around the
leading edge. As the trailing edges approach each other, vorticity shed from
the trailing edge rolls up in the form of stopping vortices (C), which
dissipate into the wake. The leading edge vortices also lose strength. The
closing gap between the two wings pushes fluid out, giving an additional
thrust. (D–F) Fling. The wings fling apart by rotating around the
trailing edge (D). The leading edge translates away and fluid rushes in to
fill the gap between the two wing sections, giving an initial boost in
circulation around the wing system (E). (F) A leading edge vortex forms anew
but the trailing edge starting vortices are mutually annihilated as they are
of opposite circulation. As originally described by Weis-Fogh
(1973), this annihilation may
allow circulation to build more rapidly by suppressing the Wagner effect.
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At the end of clap, the wings continue to pronate by leaving the trailing
edge stationary as the leading edges `fling' apart
(Fig. 4D–F). 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).
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Fig. 5. Polhamus' leading edge suction analogy. (A) Flow around a blunt wing. The
sharp diversion of flow around the leading edge results in a leading-edge
suction force (dark blue arrow), causing the resultant force vector (light
blue arrow) to tilt towards the leading edge and perpendicular to free stream.
(B) Flow around a thin airfoil. The presence of a leading edge vortex causes a
diversion of flow analogous to the flow around the blunt leading edge in A but
in a direction normal to the surface of the airfoil. This results in an
enhancement of the force normal to the wing section.
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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
TOP
Summary
Introduction
