## SUMMARY

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*.

## Introduction

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.

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.

## Materials and methods

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 ρ is the density of the fluid, μ is the dynamic viscosity of the
fluid, ν 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 × 0.4 m in width. In our simulations, the size of the
computational fluid domain was increased to 1 m in length × 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*(τ) is the translational velocity at dimensionless time τ
defined by Eq. 3, *t* is
the actual time, *c* is the chord length of the wing,τ
_{accel} is the dimensionless time when translational
acceleration begins, and Δτ_{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 τ_{decel} is the dimensionless time when translational
deceleration begins, and Δτ_{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), τ_{accel} andτ
_{decel} were taken to be 0.86, Δτ_{accel} andΔτ
_{decel} were taken to be 1.3, and *V* ranged from
about 0.00375 to 0.06 m s^{–1}.

The angles of attack were similarly defined using a set of equations
describing the angular velocity during the rotational phase of the stroke. Letα
be defined as the angle of attack of the wing relative to the
horizontal plane. In all fling simulations, the wings were rotated fromα
=90° to α=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 θ 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 ω_{rot} is a constant determined by the total angle of
rotation and by the duration of the rotational phase in
Eq. 6, ω(τ) is the
angular velocity as a function of dimensionless time, τ_{turn} is
the dimensionless time wing rotation begins, Δτ_{rot} is the
dimensionless duration of the rotational phase, and Δθ is the
total angle through which rotation occurs. Unless otherwise noted,Δθ
was set to π/4 and Δτ_{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, ρ 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*), δ(**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 δ, 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. **f _{targ}**(

*r,t*) is the force per unit length,

*k*

_{targ}is a stiffness coefficient,

*c*

_{targ}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.

**f**(

_{beam}*r,t*) is the force per unit length and

*k*

_{beam}is a stiffness coefficient. Eq. 13 describes the force applied to the fluid as a result of the resistance to stretching of the boundary [

**f**(

_{str}*r,t*)].

*k*

_{str}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 × 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 *C*_{L} is the lift coefficient, *C*_{D}
is the drag coefficient, *F*_{D} is the drag force per unit
spanwise length, *F*_{L} 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 ρ 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), *C*_{D} and *C*_{L}
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 ×1 m to
0.5 m ×0.5 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.

### 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 *A*_{0} is stroke amplitude, *x*(*t*)
describes the horizontal position of the center of the wing as a function of
time, and α(*t*) describes the angle of attack relative to the
*x*-axis as a function of time, φ sets the timing of rotation andβ
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, *A*_{0}/*c* was set to 2.8,φ
was set to 0, and β was set to π/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
75ν/π*cA*_{0}. 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 α=90°. The angle between the two wings, Φ,
was initially set to 0°, and the distance between the wings was set equal
to 0.10*c*. The wings were then rotated apart along their trailing edge
until Φ=180°. Spedding and Maxworthy measured lift forces during this
simplified fling motion at *Re*=3.0 ×10^{3}. 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×
10^{3} by calculating the lift coefficient as a function of
time, and setting the scaled force equal to
*F*_{scaled}=1/2ρ*C*_{D}*SŪ*^{2}_{max},
where *S* is the surface area of the experimental wing (0.03
m^{2}), *Ū*_{max} is the maximum velocity at the
midpoint of the wing (0.018 m s^{–1}), and ρ is the density
of the fluid (1030 kg m^{–3}). The numerical simulation of Sun
and Yu was also performed at *Re*=3.0 ×10^{3}, using two
elliptic wings with a thickness of 0.04*c* and placed 0.08*c*
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.

## Results

### 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.

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.

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.

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.

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.

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*.

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.

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.

### 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.

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.

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.

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.

Drag coefficients for a range of *Re* during one-winged clap and
fling are shown in Fig. 19. In
general, drag coefficients are larger at lower *Re* throughout the
entire stroke. In comparison to the two-winged case, drag coefficients per
wing are lower throughout the stroke for the one-winged case at all
*Re*. The differences between the one and two-winged cases are greatest
during wing rotation (particularly during fling) and acceleration at lower
*Re*. The drag coefficients over time in these simulations are also
strikingly similar to those measured experimentally by Dickinson
(1994) using one wing moving in
a two-dimensional motion.

Fig. 20 shows the average
lift and drag coefficients generated during a fling half-stroke started from
rest and a fling half-stroke that follows a clap half-stroke. The force
coefficients were averaged over wing rotation and subsequent translation of
about 3.5 chord lengths. For all *Re* considered, lift and drag
coefficients were higher for fling half-strokes that followed clap
half-strokes. This `wake effect' decreases with decreasing *Re*.

## Discussion

The main result of this paper is that the lift-enhancing effects of clap
and fling are larger for lower *Re* than for higher *Re*. Not
only are large lift forces generated during wing rotation, but they are also
transiently enhanced during the translation of the wing following fling. This
can be shown by comparing the lift forces generated by two wings to one wing
performing the same motion. In practice, the part of the translational phase
during which lift is enhanced lasts long enough to comprise most or all of the
actual wing motion. For *Re*=64 and higher, clap and fling enhances
lift during wing rotation, but does not enhance lift significantly during
translation. For *Re*=32 and below, lift is significantly enhanced
following clap and fling when compared to the one-winged case. This
lift-enhancing effect increases with decreasing *Re*. Moreover, these
*Re* differences in lift enhancement could explain why most tiny
insects have converged upon clap and fling while the vast majority of larger
insects do not use this mechanism.

Another result of potential significance is that the relative drag forces
required to rotate the wings apart during fling increase drastically for lower
*Re*. In these simulations, the wings begin fling at a distance of 1/6
chord lengths apart. In reality, the wings are pressed together, and it is
reasonable to assume that the drag forces generated during fling would be even
greater.

Some of the difference between the higher and lower *Re* cases can
be related to an aerodynamic transition observed in this *Re* range
(Childress and Dudley, 2004;
Vandenberghe et al., 2004;
Miller and Peskin, 2004). For
*Re*=64 and above in three-dimensional strokes, lift is generated in
part when an attached leading edge vortex is formed and a trailing edge vortex
is shed during each wing stroke. This vortical asymmetry generates negative
circulation around the wing and, consequently, creates positive lift. Lift is
also produced similarly in two-dimensional simulations at higher *Re*.
However, the leading edge vortex is only transiently attached to the wing. At
these high *Re*, lift enhancement by clap and fling during translation
is minimal (at least in two dimensions). For *Re*=32 and below in two
dimensions, leading and trailing edge vortices form and remain attached to the
wing during each stroke. This vortical symmetry greatly reduces the
circulation around the wing and the lift produced when compared to the
three-dimensional case of flight at higher *Re*. A recent study by Wu
and Sun (2004) also suggests
that lift is greatly reduced at these lower *Re* in three dimensions.
During clap and fling, two large attached leading edge vortices are formed on
each wing, and no trailing edge vortices are formed initially. When the wings
begin to translate away from one another, two weak trailing edge vortices are
formed on each wing and grow during translation. The leading edge vortices,
however, are much stronger than the trailing edge vortices throughout the
stroke. This vortical asymmetry produced by clap and fling recovers some of
the lift lost from this transition.

To understand the aerodynamic mechanism of lift generation, consider the
case of fling shown in Fig.
21. The two wings were rotated apart along their trailing edges
and are now translating away from each other along a horizontal plane. By
convention, positive motion is defined from right to left so that the
circulation around the left wing is negative and the lift is positive. During
rotation, two large leading edge vortices (**R _{n}1** and

**R**) of equal strength and opposite sign were formed and remain attached to the wing. During translation, two small trailing edge vortices of equal strength and opposite sign begin to form and grow in strength (

_{p}1**R**and

_{n}2**R**). Let the rest of the fluid domain (

_{p}2**R**) be of negligible vorticity. Note that the subscript

_{f}**n**denotes regions of negative (clockwise) vorticity, and

**p**denotes regions of positive (counterclockwise) vorticity. In the following discussion, an Eulerian frame of reference will be used. The total lift acting on both wings can then be defined as follows using a general viscous aerodynamic theory developed by Wu (1981): 19 where |ω| is the absolute value of the vorticity. The vortices in each pair are convected in opposite directions with each wing as the wings are translated apart. In an Eulerian framework, the vortices defined by

**R**and

_{n}1**R**move with negative velocity (the vortices are pulled leftward with the wing). The vortices defined by

_{p}2**R**and

_{p}1**R**move with positive velocity (the vortices are pulled rightward with the wing). The equation for total lift in this case can be rewritten as follows: 20 This equation basically states that the total lift on both wings is proportional to the difference between the magnitude of the time rate of change of the first moment of vorticity associated with the leading edge vorticity and the time rate of change of the first moment of trailing edge vorticity. Therefore, vortical asymmetry produced by clap and fling will transiently enhance lift forces during translation.

_{n}2

**List of symbols**

- A
_{o} - stroke amplitude
- c
- chord length of wing
*c*_{targ}- damping coefficient
*C*_{D}- drag coefficient
*C*_{L}- lift coefficient
- f
- flapping frequency
**f**(*r,t*)- force per unit length applied by the wing to the fluid
- f
_{beam} - force per unit length applied to fluid due to bending stiffness
- f
_{str} - force per unit length applied to fluid due to stretching stiffness
- f
_{targ} - force per unit length applied to fluid due to the target boundary
**F**(**x**,*t*)- total force per unit area applied to the fluid
*F*_{D}- drag force
*F*_{L}- lift force
*F*_{scaled}- scaled lift force
*k*_{beam}- bending stiffness coefficient
*k*_{str}- stiffness coefficient in tension/compression
*k*_{targ}- stiffness coefficient of target boundary
- l
- characteristic length
*P*(**x**,*t*)- fluid pressure
- r
- Lagrangian position
- R
_{f} - two-dimensional fluid space
- R
_{n} - region of negative vorticity
- R
_{p} - region of positive vorticity
- S
- characteristic surface area
- t
- time
**u**(**x**,*t*)- fluid velocity
- U
- characteristic speed
*U*_{max}- maximum velocity during rotation at the wing's midpoint
**U**(**x**,t)- velocity of the boundary
- v(τ)
- translational velocity at dimensionless time?
- V
- maximum translational velocity
- x
- position vector
- x(t)
- horizontal position of the center of the wing
**X**(*r,t*)- position vector of boundary at Lagrangian position
*r* **Y**(*r,t*)- position vector of target boundary at Lagrangian position
*r* - α
- angle of attack
- β
- parameter to set change in angle of attack
- δ(x)
- delta function
- Δθ
- total angle through which rotation occurs
- Δτaccel
- dimensionless duration of translational acceleration
- Δτdecel
- dimensionless duration of translational deceleration
- Δτrot
- dimensionless duration of rotational phase
- θ
- angle between wing and positive
*x*-axis - μ
- dynamic viscosity
- ν
- kinematic viscosity
- ρ
- fluid density
- τ
- dimensionless time
- τaccel
- dimensionless time when translational acceleration begins
- τdecel
- dimensionless time when translational deceleration begins
- τturn
- dimensionless time when wing rotation begins
- parameter for the timing of wing rotation
- Φ
- angle between two wings
- ω
- vorticity
- |ω|
- absolute value of vorticity
- ω(τ)
- angular velocity at dimensionless time?
- ωrot
- rotational constant
- φ
- parameter to set timing of rotation

## ACKNOWLEDGEMENTS

We wish to thank Michael Shelley, Stephen Childress and Michael Dickinson for their perceptive comments on our work. We also thank the reviewers for their excellent suggestions which have led to a greatly improved paper. This work was supported by the National Science Foundation under Grant Number DMS-9980069.

- © The Company of Biologists Limited 2005