First published online January 5, 2005
Journal of Experimental Biology 208, 195-212 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01376
A computational fluid dynamics of `clap and fling' in the smallest insects
Laura A. Miller1,* and
Charles S. Peskin2
1 Department of Mathematics, University of Utah, 155 South 1400 East, Salt
Lake City, UT 84112, USA
2 Courant Institute of Mathematical Sciences, New York University, 251
Mercer Street, New York, NY 10012, USA
*
Author for correspondence (e-mail:
miller{at}math.utah.edu)
Accepted 8 November 2004
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Summary
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In this paper, we have used the immersed boundary method to solve the
two-dimensional Navier–Stokes equations for two immersed wings
performing an idealized `clap and fling' stroke and a `fling' half-stroke. We
calculated lift coefficients as functions of time per wing for a range of
Reynolds numbers (Re) between 8 and 128. We also calculated the
instantaneous streamlines around each wing throughout the stroke cycle and
related the changes in lift to the relative strength and position of the
leading and trailing edge vortices.
Our results show that lift generation per wing during the `clap and fling'
of two wings when compared to the average lift produced by one wing with the
same motion falls into two distinct patterns. For Re=64 and higher,
lift is initially enhanced during the rotation of two wings when lift
coefficients are compared to the case of one wing. Lift coefficients after
fling and during the translational part of the stroke oscillate as the leading
and trailing edge vortices are alternately shed. In addition, the lift
coefficients are not substantially greater in the two-winged case than in the
one-winged case. This differs from three-dimensional insect flight where the
leading edge vortices remain attached to the wing throughout each half-stroke.
For Re=32 and lower, lift coefficients per wing are also enhanced
during wing rotation when compared to the case of one wing rotating with the
same motion. Remarkably, lift coefficients following two-winged fling during
the translational phase are also enhanced when compared to the one-winged
case. Indeed, they begin about 70% higher than the one-winged case during pure
translation. When averaged over the entire translational part of the
stroke, lift coefficients per wing are 35% higher for the two-winged case
during a 4.5 chord translation following fling. In addition, lift enhancement
increases with decreasing Re. This result suggests that the Weis-Fogh
mechanism of lift generation has greater benefit to insects flying at lower
Re. Drag coefficients produced during fling are also substantially
higher for the two-winged case than the one-winged case, particularly at lower
Re.
Key words: insect flight, Reynolds number, aerodynamics, computational fluid dynamics, clap and fling
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Introduction
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While analyzing the hovering motion of the tiny wasp Encarsaria
formosa, Weis-Fogh (1973
)
proposed a novel aerodynamic mechanism that enhanced lift during flight. This
mechanism became known as the Weis-Fogh mechanism, and the corresponding
motion has been termed `clap and fling'. Lighthill
(1973) described analytically
how this motion is thought to augment lift using two-dimensional inviscid
theory. Later studies revealed that clap and fling is also used by insects
such as the greenhouse white-fly Trialeurodes vaporariorium
(Weis-Fogh, 1975), thrips
(Ellington, 1984) and
butterflies (Srygley and Thomas,
2002). Although most, if not all, tiny insects use `clap and
fling', the majority of insects do not
(Ellington, 1999). Moreover,
clap and fling could be merely a result of the insect maximizing stroke
amplitude rather than an independently evolved behavior to maximize lift. As a
result, clap and fling is not considered a general method of lift generation
in insect flight. There has not, however, been a rigorous study comparing the
effects of `clap and fling' for different Reynolds numbers (Re). It
is not known, therefore, if the lift-enhancing effects of clap and fling are
greater for the smallest insects in comparison to larger insects.
During clap and fling, the wings `clap' together at the end of the upstroke
(ventral to dorsal) and then fling apart at the beginning of the downstroke
(dorsal to ventral). The tiny wasp Encarsaria formosa and presumably
other tiny insects fly with their bodies inclined almost vertically
(Weis-Fogh, 1973). The wings
are translated back and forth along a nearly horizontal plane
(Fig. 1A). At the beginning of
the downstroke, the wings initially fling apart by rotating about the common
trailing edge (Fig. 1B). During
this rotation, large attached leading edge vortices form on each wing
(Maxworthy, 1979;
Spedding and Maxworthy, 1986).
The leading edge vortex of one wing acts as the starting vortex of the other
wing. Since these vortices are mirror images of each other, the circulation
about the pair of wings remains zero. As a result, trailing edge vortices are
not needed to conserve circulation, and indeed they are not initially formed.
This is significant because both leading and trailing edge vortices are formed
by a single wing in pure translation, resulting in smaller lift forces. This
vortical pattern leads to larger lift forces when compared to similar wing
kinematics without clap and fling
(Lighthill, 1973;
Sun and Yu, 2003). Towards the
end of rotation, the two wings begin to translate away from each other along a
horizontal plane.
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Fig. 1. `Clap and fling' (redrawn from
Weis-Fogh, 1973). The
three-dimensional motion (top) and the corresponding two-dimensional
approximation (bottom). In this drawing, the insect flies with its body
oriented almost vertically, and the wings move in a horizontal plane. At the
beginning of the upstroke (A), the wings move from the ventral to the dorsal
side of the body, and rotate together about the leading edges. At the
beginning of the downstroke (B), the wings rotate apart about the trailing
edges. Towards the end of rotation, the wings translate away from each
other.
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The Weis-Fogh mechanism of lift generation has been verified by a number of
experimental and computational studies. Maxworthy
(1979) confirmed the basic
premise of the Weis-Fogh mechanism using flow visualization on model wings.
Essentially, this study showed that two large leading edge vortices are formed
during fling. However, his results showed that the magnitude of the
circulation about each wing generated during fling is much larger than that
predicted by Lighthill. This result was also confirmed by Haussling
(1979) who determined the
instantaneous streamlines and vorticity lines by solving numerically the full
Navier–Stokes equations. Spedding and Maxworthy
(1986) measured the
instantaneous lift forces on model wings during fling and found that the
forces were larger than those predicted by Lighthill. Sunada et al.
(1993) characterized the
effects of `near fling' on lift generation using a series of three-dimensional
experiments. Near fling describes the case where the wings are only partially
clapped together. Using computational fluid dynamics, Sun and Yu
(2003) found that lift is also
enhanced for some time during the translational phase of the stroke following
a simple fling at Re=17. They did not, however, consider this effect
for different Re.
There is reason to believe that the lift enhancing effects of the Weis-Fogh
mechanism could increase with decreasing Re. Using two-dimensional
computational fluid dynamics, we have determined that the lift coefficients
generated during translation are lower for Re<32 than for
Re>64 (Miller and Peskin,
2004). Wu and Sun
(2004) also found that lift
coefficients were greatly reduced for Re<100 in three-dimensional
simulations without clap and fling. This drop in lift corresponds to a change
in the behavior of the vortex wake. For Re=64 and above, a leading
edge vortex is formed and at least initially remains attached to the wing. The
trailing edge vortex is formed and shed from the wing. The stability of the
attached leading edge vortex appears to vary with several factors, one of
which is the dimensionality of the flow. In two dimensions, leading and
trailing edge vortices are alternately shed forming the von Karman vortex
street (Dickinson and Götz,
1993; Birch et al.,
2004; Miller and Peskin,
2004). The real situation of insect flight differs from the
two-dimensional model in at least two ways: the insect wing has finite span,
and its motion involves rotation about the dorsal–ventral axis of the
insect. In the three-dimensional rotating motion, the leading edge vortex
appears to remain attached for all time
(Usherwood and Ellington,
2002). Birch et al.
(2004) also observed a stable
attached leading edge vortex for Re=120 and Re=1400 using a
dynamically scaled robotic insect. For Re=32 and below, both leading
and trailing edge vortices are formed and remain attached to the wing
(Miller and Peskin, 2004), and
the leading edge vortex is more diffuse than the higher Re case
(Wu and Sun, 2004). The drop
in lift for lower Re is related to a loss of asymmetry in the
vortical pattern behind the wing. A similar transition has been observed for
thrust generation in flapping flight
(Childress and Dudley, 2004;
Vandenberghe et al.,
2004).
For these lower Re (32 and below), lift might be enhanced during
translation by `regaining' vortical asymmetry through clap and fling. In the
case of pure translation, equally sized leading and trailing edge vortices (by
the principle of conservation of vorticity) are formed at the beginning of the
stroke and remain attached to the wing until stroke reversal. During clap and
fling, two equally sized large leading edge vortices are formed, and no
trailing edge vortices are formed initially. Presumably, trailing edge
vortices will form and grow in strength during translation, reaching the same
strength as the leading edge vortices after a sufficient amount of time.
However, transient asymmetry between the leading and trailing edge vortices
should be produced by fling. This asymmetry in the vortical field should lead
to larger lift forces than in the case of pure translation.
In this paper, a two-dimensional version of clap and fling is studied for
Re ranging from 8 to 128, using the immersed boundary method. Two
motions are considered: `clap' and `fling'. Clap was modeled as a motion
similar to that of Fig. 1A, and
was divided into three stages: acceleration from rest at constant angle of
attack, translation, and rotation about the leading edge. Fling was modeled as
a motion similar to that of Fig.
1B, and was divided into two stages: rotation about the trailing
edge and translation. The first set of simulations corresponds to fling and
the downstroke. The second set of simulations corresponds to the upstroke,
clap, fling, and the subsequent downstroke. The lift forces generated per
wing for each Re were compared to the lift forces generated in
the case of one wing moving with the same motion.
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Materials and methods
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The basic design of this study is similar to that of our previous
computational work (Miller and Peskin,
2004), which was modeled after a physical experiment of Dickinson
and Götz (1993). In this
particular experiment, Dickinson and Götz immersed a robotic wing in
sucrose solution to study flight dynamics similar to that of Drosophila
melanogaster. This experiment was dynamically scaled such that the
Re of the model was approximately equal to that of Drosophila
melanogaster flight. The Re basically describes the ratio of
inertial to viscous forces in fluid flow and is given by the equation:
 | (1) |
where 
is the density of the fluid, µ is the dynamic viscosity of the
fluid, 
is the kinematic viscosity of the fluid, l is a
characteristic length of the immersed structure, and U is a
characteristic velocity of the flow. In our case, l is the chord
length of the wing (c), and U is the velocity of the wing
during the translational phase of the motion. The parameters in our
computational study were chosen to match those of the Dickinson and Götz
experiment, except that we varied the velocity of the wing to change the
Re. Their experiment used an aluminum wing with a chord of 5 cm
immersed in a sucrose solution with a dynamic viscosity of 0.0235 N s
m–2, about 20 times that of water. The two dimensions of the
experimental tank relevant to our two-dimensional simulations were 1 m in
length x 0.4 m in width. In our simulations, the size of the
computational fluid domain was increased to 1 m in length x 1 m in
width. This was done to reduce wall effects, which become more significant at
lower Re.
The full `clap and fling' motion studied here is a two-dimensional
idealization of one complete three-dimensional stroke. The wings are
translated towards each other from rest at a constant angle of attack during
the initial translational phase. Near the end of this initial half-stroke, the
wings rotate along the leading (upper) edge and are nearly clapped together. A
distance of 1/6 chord lengths is left between the wings, however. This
half-stroke is called the upstroke since its three-dimensional counterpart
describes the motion of the wing from the ventral to the dorsal side of the
body. At the beginning of the downstroke, the wings are held parallel and then
rotated apart about the trailing (lower) edge. By convention, the downstroke
is defined as the motion of the wing from the dorsal to the ventral side of
the body. The translational phase of the motion, which begins towards the end
of the rotational phase, is defined as the translation of the wings away from
each other along the horizontal axis. In a three-dimensional version of this
simulation, the translational phase would correspond to the motion of the
wings from the dorsal to the ventral side of the body. In the case of `fling'
(downstroke only) the wings translate through a distance of about 4.5 chord
lengths. In the case of `clap and fling' (one entire stroke) the wings
translate through a distance of about 3.5 chord lengths.
At present, no detailed quantitative description of the clap and fling
motion in small insects is available in the literature. Therefore, a
`reasonable' fling motion was constructed based on the normalized angular
velocities and translational accelerations used to model the flight of
Drosophila melanogaster. To construct a smooth motion with positive
lift generated throughout the stroke, wing rotation began before wing
translation ended during the upstroke. This motion was constructed such that
translation ended halfway through the first rotational phase. The wings were
rotated at the end of the upstroke about the leading edges (clap). At the
beginning of the downstroke, the wings were rotated apart about the trailing
edges (fling). The translational phase of the downstroke also began halfway
through the second rotational phase.
The kinematics of the left wing are described here. The right wing (when
present) was the mirror image of the left wing at all times during its motion.
The translational velocities over time were constructed with a set of
equations describing the acceleration and deceleration phases of wing
translation (Fig. 2). The
translational velocity during the acceleration phases of the wing is given by:
 | (2) |
 | (3) |
where V is the maximum translational velocity during the stroke,
v(
) is the translational velocity at dimensionless time
defined by Eq. 3, t is
the actual time, c is the chord length of the wing,
accel is the dimensionless time when translational
acceleration begins, and 
accel is the dimensionless
duration of translational acceleration. After acceleration, the translational
velocity of the wing is fixed as V. The translational velocities
during deceleration are given as:
 | (4) |
where decel is the dimensionless time when translational
deceleration begins, and decel is the dimensionless
duration of translational deceleration. In these simulations, the
dimensionless duration of an entire clap and fling stroke was taken to be 10.8
(this gives a translational distance of about 3.5 chord lengths). The
dimensionless duration of a fling half-stroke was taken to be 6 (this gives a
translational distance of about 4.5 chord lengths), accel and
decel were taken to be 0.86, accel and
decel were taken to be 1.3, and V ranged from
about 0.00375 to 0.06 m s–1.
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Fig. 2. Dimensionless translational and angular velocities of the wing as a
function of dimensionless time for one `clap and fling' stroke. The total
motion was used for all `clap and fling' simulations. For `fling' simulations,
the angular and translational velocities follow the second half of the graph.
Note that the wing begins to rotate before the end of translation during the
upstroke (first half-stroke). Translation during the downstroke (second
half-stroke) also begins before wing rotation has ended.
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The angles of attack were similarly defined using a set of equations
describing the angular velocity during the rotational phase of the stroke. Let
be defined as the angle of attack of the wing relative to the
horizontal plane. In all fling simulations, the wings were rotated from
=90° to =45° at the beginning of the downstroke. After
rotation, the angle of attack was held constant for the remainder of the
stroke. In all clap and fling simulations, the wings were translated at
constant angle of attack of 45° during the upstroke and rotated to 90°
at the end of the upstroke. The downstroke was constructed exactly as the
fling case. Let 
be defined as the angle between the left wing and the
positive x-axis (the origin is defined as the intersection of the
wing with the x-axis at the initial time). The angular velocity of
the left wing during the rotational phase at the end of the upstroke is given
by:
 | (5) |
and
 | (6) |
where 
rot is a constant determined by the total angle of
rotation and by the duration of the rotational phase in
Eq. 6, () is the
angular velocity as a function of dimensionless time, turn is
the dimensionless time wing rotation begins, rot is the
dimensionless duration of the rotational phase, and is the
total angle through which rotation occurs. Unless otherwise noted,
was set to 
/4 and rot was set to
1.74 in all simulations. Rotation at the beginning of the downstroke was
constructed similarly.
The numerical method
For the simulations presented here, a `target boundary' version of the
immersed boundary method was used to calculate the flow around the wing.
Basically, we wanted the wing to move with small deformations in a prescribed
motion. To achieve this, a target boundary that does not interact with the
fluid is attached with virtual springs to the actual immersed boundary. This
target boundary moves with the desired motion, and the spring attachments
apply a force to the actual boundary which is proportional to the distance
between corresponding points of the two boundaries. In other words, an
external force is applied that is proportional to the distance between the
wing and its desired trajectory. The force applied to the actual immersed
boundary by the target boundary and the deformation of the actual boundary are
then used to calculate the force applied to the fluid.
The two-dimensional incompressible Navier–Stokes equations describing
the motion of the fluid are as follows:
 | (7) |
and
 | (8) |
where u(x,t) is the fluid velocity,
P(x,t) is the pressure, F(x,t) is the
force per unit area applied to the fluid by the immersed wing, is the
density of the fluid, and µ is the dynamic viscosity of the fluid. The
independent variables are the time t and the position vector
x=[x,y]. Note that bold letters represent vector
quantities.
The interactions between the fluid and the boundary are described by the
following equations:
 | (9) |
and
 | (10) |
where f(r,t) is the force per unit length applied by the wing
to the fluid as a function of Lagrangian position (r) and time
(t), 
(x) is a two-dimensional delta function,
X(r,t) gives the Cartesian coordinates at time t of
the material point labeled by the Lagrangian parameter r.
Eq. 9 describes how the force is
spread from the boundary to the fluid. Eq.
10 evaluates the local velocity of the fluid at the boundary. In
this numerical scheme, the boundary is moved at the local fluid velocity at
each time step, and this enforces the no-slip condition. Each of these
equations involves a two-dimensional Dirac delta function , which acts
in each case as the kernel of an integral transformation. These equations
convert Lagrangian variables to Eulerian variables and vice
versa.
The equations that describe the force the boundary applies to the fluid are
given as:
 | (11) |
 | (12) |
 | (13) |
and
 | (14) |
Eq. 11 describes the external
force applied to the fluid that is proportional to the distance between the
boundary and its desired trajectory. ftarg(r,t) is
the force per unit length, ktarg is a stiffness
coefficient, ctarg is a damping coefficient, and
Y(r,t) is the prescribed position of the target boundary.
Eq. 12 describes the force
applied to the fluid by the boundary as a result of its elastic deformation in
bending. fbeam(r,t) is the force per unit length
and kbeam is a stiffness coefficient.
Eq. 13 describes the force
applied to the fluid as a result of the resistance to stretching of the
boundary [fstr(r,t)]. kstr is
the corresponding stiffness coefficient in tension or compression. Finally,
Eq. 14 describes the total force
applied to the fluid per unit length [f(r,t)] as a result of
both the external force and the deformation of the boundary.
The system of differentio-integral equations given by Eqns
9,
10,
11,
12,
13,
14 was solved on a rectangular
grid with periodic boundary conditions in both directions, as described by
Peskin and McQueen (1996). In
this case, a skew symmetric operator was used to discretize the nonlinear term
in the Navier–Stokes equations (Lai
and Peskin, 2000). The velocity near the outer boundary of the
domain was kept near zero on the edges of the domain by inserting four walls
that were 30 spatial steps away from the edges of the fluid domain. The
Navier–Stokes equations were discretized on a fixed Eulerian grid, and
the immersed boundaries were discretized on a moving Lagrangian array of
points. Unless otherwise stated, the fluid domain was 1230 x 1230 mesh
widths in all simulations. At this mesh width, the two wings were separated by
10 mesh widths at their closest approach. The wings were each discretized into
120 spatial steps.
Lift and drag forces were calculated as functions of time by taking the
opposite sign of the force applied to the fluid by one wing at each time step.
By convention, lift and drag coefficients were calculated as follows:
 | (15) |
and
 | (16) |
where CL is the lift coefficient, CD
is the drag coefficient, FD is the drag force per unit
spanwise length, FL is the lift force per unit spanwise
length, S is a characteristic area (chord length of the wing
multiplied by unit length), U is the speed of the wing, and is
the density of the fluid. In these definitions, `spanwise' refers to the
direction perpendicular to the plane for two-dimensional simulations. Since
these definitions are designed for the high Re case
(Re>>1), CD and CL
become functions of Re for intermediate values of Re.
Validation of the method
To test for the convergence of the numerical method, two simulations were
considered: one at the mesh width used for all of the simulations presented in
the Results and the other at half that mesh width. For the convergence test,
the size of the fluid domain was reduced in both cases from 1 m x1 m to
0.5 m x0.5 m in order to make the fine grid computation practical. This
pair of simulations modeled a two-winged fling half-stroke at Re=128.
The particular wing kinematics used here are the same as those described in
the case of a fling half-stroke. The resulting lift and drag coefficients are
plotted as functions of dimensionless time in
Fig. 3. In general, there is
good agreement between the two mesh widths. Small deviations appear during
rotational fling. This does not appear to introduce error for the rest of the
stroke.
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Fig. 3. Lift (blue) and drag (red) coefficients per wing during a two-winged fling
half-stroke are plotted for two mesh widths. The coarser grid (dotted line)
has a mesh width of about 8.33 x10–4 m (the same mesh
width as the other simulations in this paper), and the finer grid (solid line)
has a mesh width of about 4.17 x10–4 m. The two grid
sizes show good agreement with small deviations occurring during the
rotational part of fling.
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Comparison to experimental and numerical data for one-winged strokes
In order to check the method against recent experimental and numerical
data, four sinusoidal one-winged flapping strokes similar to that of Wang et
al. (2004) were modeled. The
equations of motion of the wing are as follows:
 | (17) |
and
 | (18) |
where A0 is stroke amplitude, x(t)
describes the horizontal position of the center of the wing as a function of
time, and (t) describes the angle of attack relative to the
x-axis as a function of time, 
sets the timing of rotation and
ß sets the change in angle of attack during stroke reversal. Basically,
the wing flaps back and forth along a horizontal plane with a frequency of
f. In this case, A0/c was set to 2.8,
was set to 0, and ß was set to /2. This provides a symmetric
stroke with a minimum angle of attack of 45°. In order to obtain
Re=75, f was set equal to
75/cA0. Lift and drag coefficients were normalized
in the same manner as the two-dimensional elliptic wing described in Wang et
al. (2004).
Lift and drag coefficients as functions of time for all cases are shown in
Fig. 4. The green lines show
the results of the immersed boundary simulation, the blue lines represent
numerical data for a two-dimensional elliptic wing, and the red lines describe
the experimental data for a three-dimensional model wing
(Wang et al., 2004). There is
excellent agreement between the two-dimensional immersed boundary simulation
and the numerical simulation of a rigid elliptic wing given by Wang et al.
(2004). In both simulations,
the leading edge vortex did not appear to separate during wing translation,
and lift coefficients agree well with the three-dimensional experiment. The
small differences between our simulation and that of Wang et al.
(2004) are probably due to a
combination of differences in design (a flexible plate vs. a rigid
ellipse) and numerical error. Differences between the simulations and the
experiment are most likely due to differences in two and three dimensions as
well as experimental and numerical error.
Comparison to experimental and numerical data for two-winged fling
In order to check the method for accuracy in describing wing-wing
interactions, fling simulations similar to those described by experimentally
by Spedding and Maxworthy
(1986) and numerically by Sun
and Yu (2003) were performed.
At the beginning of the simulation, two wings were held parallel to each other
at an angle of attack =90°. The angle between the two wings, 
,
was initially set to 0°, and the distance between the wings was set equal
to 0.10c. The wings were then rotated apart along their trailing edge
until =180°. Spedding and Maxworthy measured lift forces during this
simplified fling motion at Re=3.0 x103. This
Re is well above those considered in this paper
(8<Re<128), and is beyond the range for which the immersed
boundary method provides reasonable results. To make a comparison between the
immersed boundary simulation and the experiment, the simulation was performed
at Re=128. The forces were scaled up to Re=3.0
x103 by calculating the lift coefficient as a function of
time, and setting the scaled force equal to
Fscaled=1/2CDS
2max,
where S is the surface area of the experimental wing (0.03
m2), max is the maximum velocity at the
midpoint of the wing (0.018 m s–1), and is the density
of the fluid (1030 kg m–3). The numerical simulation of Sun
and Yu was also performed at Re=3.0 x103, using two
elliptic wings with a thickness of 0.04c and placed 0.08c
apart.
The exact wing motion used in this simulation and the experiment of
Spedding and Maxworthy (1986)
is shown in Fig. 5A. The wing
motion used by Sun and Yu
(2003) is nearly identical.
The lift forces as functions of time for the immersed boundary simulation
(broken line), the numerical simulation of Sun and Yu (solid line), and the
physical experiment of Spedding and Maxworthy (dotted line) are shown in
Fig. 5B. There is reasonable
agreement between all three methods, and there is excellent agreement between
the two-dimensional numerical simulations. Flow visualization and the
corresponding streamline plots of the numerical simulations are shown at five
stages during fling in Fig. 6.
In all cases, two large leading edge vortices form and appear to remain
attached to each wing during rotation. A second pair of small vortices also
forms along the trailing edge.
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Results
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Fling
In order to determine the effect of Re on the lift generated
during fling, simulations using either one wing or two wings following the
motion described in Materials and methods were performed for Re
ranging from 8 to 128. The Re was varied by changing the velocity of
the wing and holding all other parameters constant.
The streamlines of the flow around two wings and one wing at Re=8
and two wings at Re=128 performing the same fling motion are shown at
six selected times in Fig. 7.
Thestreamlines are curves which have the same direction at each point in the
fluid as the instantaneous fluid velocity u(x,t). The
density of the streamlines in each plot is proportional to the speed of the
flow. For more details on how the plots were generated see Miller and Peskin
(2004). Normalized force
vectors at each point in time were also drawn on the wing to display the
direction of the force that the fluid applies to the wing.
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Fig. 7. Streamlines of the fluid flow around two wings (A) and one wing (B) at a
Reynolds number of 8 and around two wings (C) at a Reynolds number
(Re) of 128 for a fling half-stroke. The arrow on the left wing shows
the direction of the normalized force acting on the wing at each time
(i–vi). The wings begin at an angle of attack of 90° and rotate
about the trailing edge to an angle of attack of 45°. (A) During rotation,
attached leading edge vortices are formed on each wing and no trailing edge
vortices are formed (i–iii). When translation begins, small attached
trailing edge vortices begin to form (iii–v). As the trailing edge
vortices grow in size relative to the leading edge vortices, lift is reduced.
The leading edge vortices, however, remain larger than the trailing edge
vortices for most of the half-stroke (v–vi). (B) In the one wing case,
attached leading edge and trailing edge vortices are formed during rotation
(i–iii). When translation begins, equally sized leading and trailing
edge vortices are attached to the wing, creating substantially lower lift
forces in comparison to the two-winged case (iii–vi). (C) At a Reynolds
number (Re) of 128, attached leading edge vortices are formed on each
wing and no trailing edge vortices are formed initially (i–iii). When
translation begins, however, the leading edge vortices are shed, and trailing
edge vortices are formed (v–vi). The trailing edge vortex grows in size
and is subsequently shed from the wing as a new leading edge vortex begins to
form. (D) Flow visualization of fling at Re=30 by Maxworthy
(1979). Similar to case A, a
pair of large leading edge vortices is formed and remains attached to the wing
during rotation. A smaller pair of trailing edge vortices is formed and grows
during translation. (E) Flow visualization of fling at Re=1.3
x104. Similar to case C, a pair of large leading edge
vortices (1) forms during rotation and is shed during translation. A new pair
of leading edge vortices forms during translation (2).
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In the two-winged case with Re=8
(Fig. 7A), the streamlines of
the flow during wing rotation are qualitatively similar to those described by
Lighthill (1973), calculated
numerically by Haussling
(1979), and observed
experimentally by Maxworthy
(1979), as shown in
Fig. 7D. As the wings rotate
apart along the trailing edge, two large leading edge vortices are formed on
each wing (Fig. 7Ai–iii).
No trailing edge vortices are formed until the wings begin to translate apart.
At the beginning of translation, two weak trailing edge vortices begin to form
on each wing (Fig.
7Aiii–iv). As the wings continue to translate away from each
other, the attached trailing edge vortices grow in strength
(Fig. 7Aiv–vi). The
strengths of the trailing edge vortices, however, are much less than
the strengths of the leading edge vortices throughout the entire
stroke considered here. Fig. 7D
shows flow visualization of fling and subsequent translation at Re
about 32 given by Maxworthy
(1979). Similar to the
numerical results, two large leading edge vortices form during rotation, and
two smaller trailing edge vortices form and grow during translation. Both sets
of leading and trailing edge vortices do not appear to separate from the wing.
The streamline plots at Re=16 and Re=32 are similar to those
described here.
In the one-winged case with Re=8, both a leading and a trailing
edge vortex are formed at the beginning of rotation
(Fig. 7Bi–iii). This
phenomenon is consistent with the principle of total vorticity conservation.
Consider an infinite fluid domain with a finite number of immersed solids that
are a finite distance apart. If the fluid and solid bodies are initially at
rest, then the total vorticity in the system (including the solid bodies) must
remain zero for all time. In this case, the leading and trailing edge vortices
spin in opposite directions during rotation and translation. Since the wings
are infinitely thin, we can consider vorticity only within the fluid domain,
and the principle of vorticity conservation demands that leading and trailing
edge vortices cancel so that the total vorticity in the system is zero. This
implies that the leading and trailing edge vortices are of equal and opposite
strength. During translation, these two vortices remain attached to the wing
until wing reversal (Fig.
7Biv–vi). This situation is markedly different from the
two-winged case: leading and trailing edge vortices are formed during rotation
in the one-winged case, while two leading edge vortices and no trailing edge
vortices are formed in the two-winged case. During translation, leading and
trailing edge vortices of equal strength are attached to the wing in the
one-winged case, while a strong leading edge vortex and a weak trailing edge
vortex form and remain attached to each wing in the two-winged case.
At Re=128 (Fig.
7C), the aerodynamics during the two-winged fling differ from the
corresponding cases at Re=32 and below. At the beginning of the
half-stroke, two strong leading edge vortices are formed during wing rotation
(Fig. 7Ci–ii). As the
wings translate apart, weak trailing edge vortices are formed and begin to
grow (Fig. 7Ciii–iv).
Unlike the low Re case, the leading edge vortices are shed at the
beginning of translation (Fig.
7Civ–v). During translation, a second pair of leading edge
vortices are formed and begin to grow. This same phenomenon was observed by
Maxworthy (1979) at higher
Re (Fig. 7E). Two
large leading edge vortices (1) are formed during rotation. As translation
begins, the pair of rotational leading edge vortices is shed, and a second
pair of leading edge vortices (2) is formed. In the immersed boundary
simulation, the trailing edge vortices are shed as translation continues
(Fig. 7Cv–vi). The
alternate vortex shedding at higher Re corresponds to the formation
of the von Karman vortex street. It is important, however, to note that in
three-dimensional insect flight at higher Re, alternate vortex
shedding does not occur (Birch et al.,
2004). Instead, the leading edge vortex remains attached to the
wing until wing reversal and the trailing edge vortex is initially shed.
Presumably, the leading edge vortex would remain attached to the wing during
three-dimensional clap and fling at higher Re, generating larger lift
coefficients for both the two- and one-winged cases.
The lift coefficients as functions of dimensionless time (fraction of
stroke) for the one- and two-winged cases at Re=8 are plotted in
Fig. 8. The bar at the top of
the graph shows the number of chord lengths traveled. The first peak in the
lift coefficients corresponds to the large lift forces generated during wing
rotation. The second peak in the lift coefficients corresponds to the period
of translational acceleration. The average lift per wing generated during wing
rotation in the two-winged case is about twice that generated in the
one-winged case. In addition, the lift forces are about 70% greater in the
two-winged case than in the one-winged case at the beginning of constant
translation. Lift forces per wing in the two-winged case are on average about
35% higher than in the one-wing case during the entire 4.5 chord length
translation. Another interesting phenomenon seen in the one-winged case is
that lift is slow to develop over the first couple chord lengths of
translation. This is most likely due to the Wagner effect, in which the
proximity of the trailing edge vortex to the wing transiently reduces the lift
until it moves sufficiently downstream of the wing. This idea is supported by
the fact that the phenomenon is not observed in the two-winged case where the
trailing edge vortices are initially absent. Both one- and two-winged lift
forces approach the same steady lift values at the end of translation. These
force traces are very similar to those calculated by Sun and Yu
(2003) at Re=17 using
a similar two-winged fling motion. The wings in their simulation were 0.08
chord lengths apart at the beginning of the stroke rotated at a faster angular
velocity. The average lift coefficient over the entire 3-chord-length half
stroke (rotation and translation) in their simulation was 2.4 for two wings
and 1.0 for one wing.
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Fig. 8. Lift coefficients per wing at a Reynolds number of 8 are plotted as
functions of time for the fling half-strokes shown in
Fig. 7A,B. The bar at the top
of the graph shows the number of chord lengths traveled. The first peak in the
lift coefficients corresponds to the large lift forces generated during wing
rotation. The second peak in the lift coefficients corresponds to the period
of translational acceleration. The lift forces per wing are on average about
35% greater during translation after clap and fling than during the steady
translation of a single wing with no clap and fling (this average was taken
over the fraction of the stroke from 0.37 to 1, after rotation had
finished).
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Lift coefficients for a range of Re are plotted as functions of
dimensionless time for two-winged fling in
Fig. 9. The first peak
corresponds to the lift generated during wing rotation, and the second peak
corresponds to the lift generated during translational acceleration. For
Re=32 and below, the different cases are similar. Lift coefficients
decrease during translation as the trailing edge vortex grows. Lift is also
enhanced for longer periods of time at lower Re (the relative
difference in strength between the attached leading and trailing edge vortices
persists longer for lower Re). The growth of the trailing edge vortex
during translation and resulting drop in lift was also observed by Sun and Yu
(2003) at a Re=17.
For Re=64 and higher, the leading edge vortex is shed at the
beginning of translation, and lift forces subsequently drop. Lift forces grow
again as a new leading edge vortex is formed and the trailing edge vortex is
shed. This may not be obvious in Fig.
9, but the growth in lift followed by force oscillations become
apparent when longer periods of time are plotted. As stated earlier, the
three-dimensional case of flight at higher Re does not involve
oscillating lift forces since alternate vortex shedding does not occur. It is
also important to note that the leading edge vortex is shed after about 1
chord length of travel. Other studies, including a two-dimensional experiment
(Dickinson and Götz, 1993)
and a two-dimensional numerical simulation
(Wang et al., 2004), show that
the separation of the leading edge vortex and subsequent lift drop does not
occur until about 2.5–3.5 chord lengths of travel. Flow visualization by
Maxworthy (1979), as well as
these simulations, show that the leading edge vortex is shed near the
beginning of translation after fling at higher Re. This suggests that
the separation of the leading edge vortex from the wing could depend upon
wing-wing interactions and the kinematics of rotation.
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Fig. 9. Lift coefficients are plotted as functions of time for a two-winged clap
and fling half-stroke. The bar at the top of the graph shows the number of
chord lengths traveled. The letters i–vi along the x axis
correspond to the times the streamlined plots labelled i–vi in
Fig. 7A,C were drawn. The
angles of attack during pure translation were set to 45°. Reynolds number
(Re) was varied by changing the translational velocity of the wing
from 0.00375 to 0.06 m s–1. The first peak corresponds to the
lift generated during wing rotation, and the second peak corresponds to the
lift generated during translational acceleration. The lift enhancing
mechanisms of fling decrease with increasing Re. For Re=32
and below, lift coefficients decrease during translation after fling as the
trailing edge vortex grows in strength. For Re=64 and above, lift
coefficients fall as the leading edge vortices separate from the wings.
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Lift coefficients for a range of Re are plotted as functions of
dimensionless time for one-winged fling in
Fig. 10. The first peak
corresponds to the lift forces produced during wing rotation, and the second
peak corresponds to the lift forces generated during translational
acceleration. Lift coefficients for periods of rotation and acceleration are
higher at lower Re. This phenomenon might be due to the larger effect
of added mass at lower Re. As the Re decreases, the width of
the boundary layer around the wing grows, and the mass of the fluid
`entrained' by the wing is larger. Lift coefficients are also substantially
lower at all times and for all Re considered when compared to the
respective two-winged cases (note the difference in scales between Figs
9 and
10). For Re=64 and
higher, lift begins to drop after about 2.5 chord lengths of travel during
translation due to the separation of the leading edge vortex. Presumably,
translational lift forces would be higher in three-dimensional flight since
the leading edge vortices would not be shed.
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Fig. 10. Lift coefficients are plotted as functions of time for a one-winged fling
half-stroke. The letters i–vi along the x axis correspond to
the times the streamlined plots labelled i–vi in
Fig. 7B were drawn. The angle
of attack during pure translation was set to 45°. The Reynolds number
(Re) was varied by changing the translational velocity of the wing
from 0.00375 to 0.06 m s–1. The first peak in lift
corresponds to the lift forces generated during wing rotation. The second peak
corresponds to the lift forces generated during translational acceleration.
For Re=64 and above, lift coefficients fall as the leading edge
vortices separate from the wings.
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Drag coefficients for a range of Re are plotted as functions of
dimensionless time for two-winged fling in
Fig. 11. The first peak in
each of the drag coefficient plots corresponds to the large drag forces
produced as the two wings are rotated apart. The maximum drag coefficient
produced during rotation increases significantly as the Re is
decreased. The smaller second peak in drag coefficients corresponds to the
forces generated during the translational acceleration of the wings at the
beginning of the half-stroke. This translational drag coefficient also
increases with decreasing Re, but the effect is substantially smaller
than that produced during wing rotation. Drag coefficients for all Re
gradually decrease during translation to steady values.
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Fig. 11. Drag coefficients are plotted as functions of time for a two-winged fling
half-stroke. The letters i–vi along the x axis correspond to
the times the streamlined plots labelled i–vi in
Fig. 7A,C were drawn. The
angles of attack during pure translation were set to 45°. Reynolds number
(Re) was varied by changing the translational velocity of the wing
from 0.00375 to 0.06 m s–1. The first large peak corresponds
to drag generated during wing rotation. The second smaller peak corresponds to
drag forces generated during translational acceleration. Drag coefficients
increase with decreasing Re. This inverse relationship is
particularly significant during the initial wing rotation.
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Drag coefficients for a range of Re are plotted as functions of
dimensionless time for one-winged fling in
Fig. 12. The first peak in the
drag coefficients corresponds to the drag forces produced during the rotation
of a single wing. These drag forces are significantly smaller than those
produced during rotation with two wings (note the difference in scales between
Figs 11 and
12). The second peak in the
drag coefficient corresponds to the drag forces produced during the
translational acceleration of the wing. After acceleration, the drag
coefficients gradually decrease to steady values. Similar to the two-winged
case, drag coefficients during one-winged fling increase with decreasing
Re.
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Fig. 12. Drag coefficients plotted as functions of time for a one-winged fling
half-stroke. The letters i–vi along the x axis correspond to
the times the streamlined plots labelled i–vi in
Fig. 7B were drawn. The angles
of attack during pure translation were set to 45°. Reynolds number
(Re) was varied by changing the translational velocity of the wing
from 0.00375 to 0.06 m s–1. The first peak corresponds to the
drag forces generated during translational acceleration. Note that the drag
forces per wing generated during rotation in the one-winged case are
significantly smaller than those generated per wing in the two-winged case
(see Fig. 11). In general,
drag coefficients increase with decreasing Re.
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The average lift per wing generated during translation after a two-winged
fling half-stroke divided by the average lift per wing generated during
translation after a one-winged fling half-stroke are plotted for a range of
Re in Fig. 13. The
average lift coefficients per wing were calculated as the average lift after
translational acceleration and during the steady translation of the wing at a
constant angle of attack (0.37–1.0 fraction of the half stroke). For
Re=8, the average lift generated during a 4.5 chord translation after
two-winged fling is on average 35% larger than the average lift generated
during translation after one-winged fling. Lift enhancement provided by
two-winged fling decreases with increasing Re. For a Re=128,
the average lift per wing produced during translation after two-winged fling
is about equal to the average lift generated during translation following a
one-winged fling. It is important to note that these ratios only consider the
effect of lift enhancement after rotation and acceleration.
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Fig. 13. The average lift coefficients per wing during translation following
two-winged fling divided by the corresponding average lift coefficients for
one-winged fling vs Reynolds number (Re). The average lift
coefficients following fling were calculated as the average lift coefficients
generated after rotation and translational acceleration and during translation
at a constant angle of attack. This value decreases with increasing
Re, suggesting that the lift enhancing effects of clap and fling are
more significant at lower Re.
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The maximum drag coefficients produced during rotation for two-winged fling
for a range of Re are plotted in
Fig. 14. The drag coefficients
produced during rotational fling sharply increase with decreasing Re.
This same phenomenon is also true during all periods of rotation and
acceleration. The Re effect is, however, most pronounced during
fling. This relationship suggests that tiny insects must apply large forces to
the fluid to turn and rotate their wings. Perhaps flexible wings and setal
fringing reduce this Re effect.
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Fig. 14. The maximum drag coefficient during the rotation of the wings at the
beginning of fling plotted against the Reynolds number (Re). The drag
coefficient significantly increases for decreasing Re. This result
suggests that relatively larger forces are needed for tiny insects to rotate
their wings and perform a `fling'.
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Clap and fling
The streamlines of the flow around two wings and one wing performing a
complete clap and fling stroke at a Re=8 are shown at eight points in
time in Fig. 15A,B. In each
case, the wings accelerate from rest and move towards each other at a constant
translational speed. At a Re=8, leading and trailing edge vortices
form and remain attached to each of the wings during the first half-stroke
(Fig. 15Ai–iii). As the
wings near each other, they begin to rotate and `clap' together
(Fig. 15Aiv–v). During
this rotation, the leading and trailing edge vortices are shed. The wings then
rotate apart and translate away from each other during `fling'
(Fig. 15Avi–viii). This
is similar to the previous case of simple `fling', except that the wings are
now translating through their wakes. During rotation
(Fig. 15Avi), new leading edge
vortices are formed on each wing, and no trailing edge vortices are formed
initially. As the wings translate away from each other, they move back through
their wakes, and weak trailing edge vortices are formed
(Fig. 15Avii–viii). In
the case of one wing, a pair of leading and trailing edge vortices is formed
during the initial translation (Fig.
15Bi–iii). These vortices are then shed during wing rotation
(Fig. 15Biv–v). During
the downstroke, a pair of large leading and trailing edge vortices is formed.
This is different from the two-winged case where only two large leading edge
vortices are formed initially.
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Fig. 15. Streamlines of fluid flow around two wings (A) and around one wing (B)
during a full clap and fling stroke at a Reynolds number (Re) of 8
and around two wings at Re=128 (C). The arrow on the left wing shows
the direction of the normalized force acting on the wing. (A) During
translation, leading and trailing edge vortices form and remain attached to
the wing (i–iii). During `clap', the wings rotate together at the end of
translation (iv–v). At this time, the leading and trailing edge vortices
are shed. During `fling', the wings rotate apart forming two new leading edge
vortices (vi). Towards the end of rotation, the wings are translated apart at
a constant angle of attack and speed (vi–viii). During translation, the
leading edge vortices remain attached to the wing, and weak trailing edge
vortices are formed. (B) Large leading and trailing edge vortices are formed
during the initial translation of the wing (i–iii). This pair of
vortices is shed during rotation (iv–v), and a new pair of leading and
trailing edge vortices is formed during the subsequent translation
(vi–viii). Note that in the two winged case, no trailing edge vortices
are formed during wing rotation, and much smaller trailing edge vortices are
formed during the subsequent translation. (C) At Re=128, leading edge
vortices are formed and the trailing edge vortices are shed (i–iii).
After a translation of about 2.5 chord lengths, the leading edge vortices
begin to separate from the wings (iii). During `clap', the wings rotate
together at the end of translation (iv–v). At this time, the leading and
trailing edge vortices are shed. During `fling', the wings rotate apart. Two
large leading edge vortices are formed, and no trailing edge vortices are
formed initially (v–vi). Towards the end of rotation, the wings are
translated away from each other and the pair of leading edge vortices formed
during rotation is shed. A second pair of leading edge vortices begins to form
near the end of translation (vi–viii).
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The streamlines of the flow around two wings performing a complete clap and
fling stroke at a Re=128 are shown in
Fig. 15C. The leading edge
vortices are formed and the trailing edge vortices are shed during the initial
translation of the wing (Fig.
15Ci–ii). The leading edge vortices begin to separate from
the wing after a translation of about 2.5 chord lengths
(Fig. 15Ciii). Lift drops as
the wings near each other and clap together
(Fig. 15Civ–v). During
this rotation of the wings, the vortices from the first half-stroke are shed.
The wings continue to rotate apart and then translate away from each other
during `fling' (Fig.
15Cv–vi). Similar to the previous case of a fling
half-stroke, two large leading edge vortices are formed during rotation, and
no trailing edge vortices are formed initially
(Fig. 15Cvi). As the wings
translate away from each other, the leading edge vortices are shed
(Fig. 15Cvii), and new
trailing edge vortices form and grow in strength. Later in the stroke, a
second pair of leading edge vortices forms and grows in strength
(Fig. 15Cviii).
Lift coefficients for a range of Re are plotted as functions of
dimensionless time for two-winged clap and fling in
Fig. 16. The markers on the
time axis denote the points in time that the streamline plots in
Fig. 15A,C were drawn. The
lift coefficients naturally divide into two patterns, lift coefficients for
Re=64 and above and lift coefficients for Re=32 and
below.
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Fig. 16. Lift coefficients as functions of time for a two-winged clap and fling
stroke. The letters i–viii along the x axis correspond to the
times the streamlined plots labelled i–viii in
Fig. 15A,C were drawn. The
angle of attack during pure translation was set to 45°. The Reynolds
number (Re) was varied by changing the translational velocity of the
wing from 0.00375 to 0.06 m s–1. In general, lift
coefficients were larger at higher Re during the initial upstroke.
Lift coefficients, however, were smaller at higher Re during fling
and subsequent translation. For Re=64 and higher, lift coefficients
peak during translational acceleration and rotation. Lift coefficients drop
when the leading edge vortices separate from the wings (vii–viii). For
Re=32 and below, lift coefficients also peak during translational
acceleration and rotation. Lift coefficients are relatively constant during
translation in the first half-stroke (i–iii). Lift coefficients are
transiently augmented during translation after fling (vi–viii).
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The lift coefficients for Re=32 and below are characterized by
relatively constant forces during translation due to the attachment of the
leading and trailing edge vortices. The first peak in the lift coefficient
corresponds to the forces produced during wing acceleration at the beginning
of the stroke (i). During constant translation, lift coefficients initially
drop and then gradually increase as the wings approach each other
(ii–iii). Lift rapidly drops as the wings begin to decelerate (iii).
When wing rotation begins, lift forces increase again as the wings clap
together (iii–iv). Lift finally drops to about zero at the end of
rotation (v). At the beginning of the second half-stroke (fling), lift
coefficients peak as two large leading edge vortices are formed (v–vi).
The next peak in lift (vi–vii) corresponds to the lift generated during
translational acceleration. These translational lift coefficients are larger
than those produced during the first half stroke. This lift enhancing effect
is due to the asymmetry in the vortical field produced by the clap and fling
motion.
The lift coefficients for Re=64 and above are characterized by
unsteady lift forces due to vortex shedding. The initial peak in lift
coefficients corresponds to the lift forces produced during the translational
acceleration of the wings (i). During constant translation, the leading edge
vortex begins to separate but lift does not drop significantly until a
translation of about three chord lengths (ii–iii). Lift then drops as
the two wings decelerate at the end of the upstroke and clap together
(iv–v). At the beginning of the second half-stroke, lift is enhanced
when two large leading edge vortices are formed during rotation (v). As the
wings begin to translate away from each other, the leading edge vortices are
shed and trailing edge vortices grow (vi). Later during translation, the
trailing edge vortices begin to separate from the wing, a new pair of leading
edge vortices begins to grow, and the lift coefficient subsequently increases.
This is consistent with the pattern of vortex shedding and growth visualized
by Maxworthy (1979) at high
Re. In Maxworthy's flow visualization, the initial pair of leading
edge vortices is shed when translation begins, and a second pair of leading
edge vortices begins to grow during translation.
Lift coefficients for one wing moving in a clap and fling motion are shown
in Fig. 17 for a range of
Re. The initial peak in lift corresponds to the lift forces generated
during wing acceleration from rest. During upstroke translation, lift
coefficients generally increase with increasing Re. For higher
Re, the lift forces grow during the first three chord lengths of
translation (ii–iii). The leading edge vortex then begins to separate
from the wing, and the lift forces drop as the wing decelerates and begins to
rotate (iv–v). During wing rotation, lift coefficients are slightly
higher at lower Re. During translation, lift coefficients are higher
at higher Re. For Re=64 and above, lift drops after a
translation of about 2.5 chord lengths as the leading edge vortex separates
from the wing. Dickinson (1994)
measured lift forces experimentally on a wing moving with a two-dimensional
motion similar to the motion used in this simulation. The lift coefficients
measured over time in each case are remarkably similar. Lift forces peak
during rotation and acceleration, fall to values near 1.5 during the first
2–2.5 chord lengths of translation, and drop to values near 1 as the
leading edge vortex separates from the wing.
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Fig. 17. Lift coefficients as functions of time for one wing moving with the same
clap and fling motion as shown in Fig.
16. The letters i–viii along the x axis correspond
to the times the streamlined plots labelled i–viii in
Fig. 15B were drawn. The angle
of attack during pure translation was set to 45°. The Reynolds number
(Re) was varied by changing the translational velocity of the wing
from 0.00375 to 0.06 m s–1. In general, lift coefficients
increase with Re. The lift forces generated during the initial
upstroke are very similar to those shown in the two-winged case (i–iii).
During fling, lift coefficients during rotation and the subsequent translation
(vi–viii) are significantly less than the two-winged case for
Re=32 and below.
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Drag coefficients for a range of Re are plotted as functions of
dimensionless time for two-winged clap and fling in
Fig. 18. The letter markers
(i–vi) on the time axis denote the points in time when the streamline
plots in Fig. 15A,C were
drawn. In general, drag coefficients increase with decreasing Re. The
first peak in the drag coefficients corresponds to the drag forces generated
during the translational acceleration of the wing (i). Drag coefficients
remain relatively constant during the translational phase of the first
half-stroke (i–iii). The drag coefficients drop during wing
deceleration, but sharply increase again when the wings are rotated (clapped)
together (iv). At the beginning of the second half-stroke, the drag
coefficients peak again as the wings are rotated apart (v–vi). There is
a smaller peak during translational acceleration (vi–vii). Finally, the
drag coefficients approach steady values as the wings translate apart
(vii–viii). The Re differences in drag coefficients are most
pronounced during wing rotation.
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Fig. 18. Drag coefficients as functions of time for a two-winged clap and fling
stroke. The letters i–viii along the x axis correspond to the
times the streamlined plots labelled i–viii in
Fig. 15A,C were drawn. The
angle of attack during pure translation was set to 45°. The Reynolds
number (Re) was varied by changing the translational velocity of the
wing from 0.00375 to 0.06 m s–1. In general, drag
coefficients are larger for lower Re. This inverse relationship is
most significant during periods of wing rotation (iv–vi). In general,
drag forces peak during periods of acceleration and rotation and remain
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TOP
Summary
Introduction
3 x103. (B) Streamline plots of fling
calculated numerically for two rigid elliptic wings by Sun and Yu
(2003