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First published online December 16, 2008
Journal of Experimental Biology 212, 95-105 (2009)
Published by The Company of Biologists 2009
doi: 10.1242/jeb.016428
Influence of flexibility on the aerodynamic performance of a hovering wing
Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA
* Author for correspondence (e-mail: balab{at}umd.edu)
Accepted 21 October 2008
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Summary |
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Key words: flapping wing, fluid–structure interactions, finite-difference method, wing flexibility, non-linear resonance, low Reynolds numbers
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INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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;
Dickinson et al., 1999) as well
as computational investigations (e.g. Liu
and Kawachi, 1998; Sun and
Tang, 2002; Ramamurti and
Sandberg, 2002; Ramamurti and
Sandberg, 2006; Wang,
2000a; Wang,
2000b; Wang et al.,
2004) have been reported in the literature. The aim of most of
these studies has been to understand the complex, unsteady mechanisms that
enable the generation of aerodynamic forces for hovering and maneuvering.
Insect wings are complex structures that during flapping undergo deformations
due to aerodynamic forces and elastic forces, as well as inertial forces due
to the accelerations experienced by the system mass. The wing structural
behavior depends, to a large extent, on the internal distribution of compliant
components and mechanisms (Wootton,
1999). It is important to note that insect wings lack internal
muscles and, hence, there are no actuators to realize internal control forces
(Wootton et al., 2003).
In a variety of species, the roles of inertial, elastic and aerodynamic
forces during flapping flight have been the focus of many investigations (see,
for example, Ellington, 1984b;
Ennos, 1989; Lehman and
Dickinson, 1997; Sun and Tang,
2002; Daniel and Combes,
2002; Combes and Daniel,
2003; Song et al.,
2001). It is difficult to make direct comparisons between the
different studies, not only because the studies usually involve different
species but also because different approaches have been used to compute the
forces. For example, Combes and Daniel assessed the relative contributions of
aerodynamic, inertial and elastic forces to the wing deformation of the
Manduca sexta hawkmoth (Combes and
Daniel, 2003). They concluded that the wing motion of this
particular insect is mostly determined by the wing inertial and elastic forces
with the aerodynamic loads providing damping. During hovering, the typical
ratio of wing inertial force to aerodynamic force was found to be about seven.
This result was obtained by using scaling arguments and assuming a weight
balance to get a fluid–force estimate. In other species, this ratio has
been found to be much lower. Ennos, for example, showed that for several
species of Diptera, the magnitudes of inertial bending moments are about twice
the aerodynamic moments during harmonic flapping
(Ennos, 1989). Also, in this
case, the analysis was based on the weight-balance assumption and harmonic
kinematics. However, unlike Combes and Daniel
(Combes and Daniel, 2003),
Ennos considered the effect of the virtual or added mass of the surrounding
fluid. It should be noted that in the studies of Ennos
(Ennos, 1989) and Combes and
Daniel (Combes and Daniel,
2003), the aerodynamic forces were underestimated, since the drag
component of the fluid force was neglected.
With increases in computational power, computational models of insect
flight have become more sophisticated. Two-dimensional computations (e.g.
Wang, 2000a;
Wang, 2000b;
Miller and Peskin, 2005;
Miao and Ho, 2006) and
three-dimensional computations (e.g. Liu
and Kawachi, 1998; Ramamurti
and Sandberg, 2002; Ramamurti
and Sandberg, 2006; Sun and
Tang, 2002) with various degrees of complexity have been reported.
In general, for low ratios of inertial to aerodynamic forces, one can expect
complex aeroelastic interactions to occur. An interesting open question within
this context is the following: how does structural flexibility affect the
aerodynamic performance of a given flapping wing and what is the effect of the
Reynolds number? The present study attempted to address this question by using
computational investigations. To the best of our knowledge, no prior
computational studies addressing this question have been carried out.
In the present study, in order to explore a fairly wide parametric regime
in a cost-efficient manner, we limited ourselves to studies in two dimensions.
A representative section of the wing (two-dimensional foil) was used, and
spanwise bending and torsion flexibility were discarded. A two-link structure
connected with a torsion spring was used to account for deformation in the
chordwise direction. This system has four degrees of freedom, which are
effectively reduced to one by prescribing harmonic hovering motions of one of
the links. The links were considered to be rigid in the present work, and they
are currently being extended to flexible beams in ongoing efforts. The large
angular deformations of the links gave rise to cubic and higher order odd
non-linearities in the governing equations like those seen in equations
governing a pendulum as well as flexible beams (e.g.
Anderson et al., 1994). In a
sense, one could consider the two links as a double pendulum with a torsion
spring. Fluid non-linearities were also considered here. Different values of
the torsion spring stiffness were considered at the Reynolds numbers of 75,
250 and 1000, and the results obtained are reported in the form of mean lift
force, mean drag force, ratio of lift to drag, and ratio of mean lift
coefficient to total power input. The performance of the hovering wing when
excited at a non-linear resonance of the structural system was also
examined.
In the following section, a description of the system is provided along with the computational formulation. Then, the parametric space and system kinematics are detailed. Next, Results and Discussion sections follow, with a Conclusions section at the end.
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MATERIALS AND METHODS |
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 | (1) |
 | (2) |
fLcUc/µ,
where f and µ are the fluid density and viscosity,
respectively.
The dimensionless form of the equations governing the motion of the
structural system shown in Fig.
1 can be derived as:
 | (3) |
 |
(t)
are, respectively, the joint horizontal motion, joint vertical motion and
orientation angle of link B measured from an inertial reference frame
as shown in Fig. 1A, and
(t) is the deflection angle between links A and
B. Here, mi is the total mass of the ith
link (i=A,B), 
i is the distance
from the junction to the center of mass of bar i as shown in
Fig. 1A, and
Ii is the moment of inertia of link i with
respect to the hinge point b. Also, Qx and
Qy are the fluid forces along the x and
y directions, respectively, and Q and
Q are the fluid moments associated with the
generalized coordinates (t) and (t),
respectively. The quantities gx, gy,
g and g are the
corresponding contributions of centrifugal, elastic and gravitational forces.
The reference length scale and velocity defined above, together with the fluid
density are used to make Eqn 3
dimensionless. The fluid forces and moments are determined from Eqns
1 and
2.
In the numerical experiments conducted in this study, the translational
motions of the junction as well as the orientation of link B are
prescribed. With these prescribed motions, the four degrees of freedom of the
system can be effectively reduced to one; that is, the deflection angle
(t) between plates A and B. Thus, the
overall deformation of the wing section is determined by the deflection angle
(t), which is governed by the following reduced form of
Eqn 3:
 | (4) |
). The non-linearities arise from the
sin(+) term due to the kinematics and the fluid forcing. For
this particular study, we only take into account the fluid damping which
arises through the fluid moment Q. It should be
noted that selecting a proper structural damping model is far from trivial,
and this is an active research topic in structural biomechanics. Damping
models for insect wings are relatively few [e.g. classical viscous damping
model used by Herbert (Herbert,
2002) and the viscoelastic model used by Bao and colleagues
(Bao et al., 2006)] and the
existing models require a fair amount of empirical information.
Prescribed kinematics, parameter values and computational formulation
To prescribe the translational motions of the junction and the orientation
of link B, we define the states x(t),
y(t) and (t) as:
 | (5) |
 | (5) |
0 is the mean orientation angle for link B, 
is the rotation amplitude, 
f is the frequency of the
prescribing or forcing oscillation and 
is the phase angle between
x(t) and (t). The exponential terms were
used in order to reduce transient effects
(Combes and Daniel, 2003). The
time constant was chosen as 
=1.6x
/f because
99.8% of the prescribed amplitude was reached after a time length of 5
periods. The following parameters corresponding to symmetric hovering were
selected (Wang et al., 2004):
 | (6) |
Based on the adopted normalization, the problem is completely defined by
the density ratio b/f, the frequency
ratio f/n and the Reynolds number
Re. Here, b is the density of the wing's
material, and n=
(k/IA)
is the linear natural frequency of the oscillator
(Eqn 4). The frequency ratio
f/n is used to characterize the
flexibility of the wing section.
Three Reynolds numbers were considered (Re=75, 250 and 1000) to
investigate the effect of the reduction in viscous dissipation on the system
dynamics. The mass ratio was set to
A/f=25, as this value provided a
ratio close to 2 for the maximum translational inertial force over maximum
drag force at Re=75 for the chosen geometry and kinematics. The above
ratio was determined through numerical experiments with the rigid wing. To
compute the maximum horizontal translational inertial force, the total wing
mass was multiplied by the maximum acceleration determined from the second
derivative of x(t) in Eqn
5. The value of peak drag force, on the other hand, was obtained
from the rigid wing simulation at Re=75. The wing has a thickness of
10% of the undeformed chord length and circularly formed edges. For the
simulations conducted at Re=75, the frequency ratio
f/n was set to 1/2, 1/2.5, 1/3, 1/3.5, 1/4
and 1/6. For Re=250 and 1000, this ratio was set to 1/2, 1/3, 1/4 and
1/6. The resulting range of maximum deflection angles varied from 10 to 70
deg. Also, the rigid wing problem (no angular deformation between the links)
was run for all of these Reynolds numbers. It should be noted that for
frequency ratios below 1/2, the computations would fail since the two plates
collide during rotation. This limitation arises from the fact that the
flexibility in the present model is concentrated at the hinge point and the
distributed chordwise variations of stiffness and mass are not accounted for.
A flexible beam model and/or inclusion of structural damping may help to
address this issue and enable computations with frequency ratios of about
one.
Eqns 1,
2 and
4 governing the dynamics of the
fluid–structure system are numerically solved by using a strongly
coupled, embedded-boundary formulation. The overall approach is a mixed
Lagrangian–Eulerian formulation, where Eqns
1 and
2 governing the fluid flow are
solved on a fixed Cartesian grid, which is not aligned with the wing surface,
and the non-slip conditions are enforced via local reconstructions of
the solution near the solid interface (see, for example,
Balaras, 2004;
Uhlmann, 2005;
Yang and Balaras, 2006). The
fluid and the structure are treated as elements of a single dynamical system,
and all governing equations are integrated simultaneously and interactively in
the time domain by using a predictor–corrector scheme. Further details
on the coupling scheme and the overall fluid–structure interaction
algorithm can be found in the work of Yang and colleagues
(Yang et al., 2008).
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RESULTS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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).
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x=y=0.0038Lc, and the
total number of points was 1229x551 along the x and y
directions, respectively. Through grid refinement studies, we found that the
above resolution was sufficient to capture all flow features. In
Fig. 2, computationally
obtained aerodynamic forces are shown for approximately half the resolution
throughout the computational domain (total number of points was 664x400)
and the same forcing conditions (i.e. =0 in
Eqn 5) and Reynolds number as
that for the baseline rigid wing computation. The corresponding lift and drag
coefficients determined in the computations of Wang and colleagues
(Wang et al., 2004), where a
hovering ellipse with the same kinematics is considered instead of a plate,
are also included in the figure. The agreement between the results obtained
with the two different grids is good, with a maximum difference of around 3%.
Despite the differences in the wing-section shapes, after the initial
transients (t/T>2), good agreement is also seen with the
results obtained by Wang and colleagues
(Wang et al., 2004).
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Re, and in order to keep the
resolution within the above range, a grid with 1320x1038 points was
found to be sufficient for both Re=250 and Re=1000 cases.
All the results presented in this article were obtained with a 1229x551
grid for the Re=75 simulations and a 1320x1038 grid for the
Re=250 and Re=1000 simulations. The governing equations were
integrated for a time length of 14 periods, 21 periods and 15 periods, for
Re=75, 250 and 1000, respectively. The time-averaged quantities were
computed over the last 7 periods, 13 periods and 10 periods, for
Re=75, 250 and 1000, respectively.
Aerodynamic quantities
For the flexible wings considered in this study, the lift and drag
coefficients were defined as:
 |
 | (7) |
and
are dimensional quantities and
Qx(t) and Qy(t) are
non-dimensional quantities. Once the equation for the deformation
(t) is solved at each iteration, the driving forces in the
prescribed generalized coordinates x(t),
y(t) and (t) are computed from
Eqn 3. The total power input,
which is the sum of horizontal translational power and rotational power, can
be computed from:
 | (8) |
(t) is the driving moment
in the (t) angular direction. In an ideal case were the
driving mechanism is perfectly elastic and the negative power provided to the
mechanism can be stored as potential energy for later use, this power will
enhance the wing's aerodynamic efficiency. Here, following Berman and Wang
(Berman and Wang, 2007), a
conservative approach was used and it was assumed that the negative power was
not available for reuse; that is:
 | (9) |
 | (10) |
and
are dimensional quantities
and Ptr(t) and
Prot(t) are non-dimensional quanitites.
In Fig. 3, the variations of
the mean values of CL and CD and the
aerodynamic performance ratios CL/CD
and CL/CPW with respect to the
frequency parameter f/n are shown. The
rigid wing has zero torsion stiffness or equivalently n=0.
For all cases, the lift and drag coefficients exhibit a peak at a frequency
ratio of f/n=1/3. The performance ratio
CL/CD also exhibits a prominent peak
at this frequency ratio. For Re=75, 250 and 1000, increases of about
28%, 23% and 21% over those obtained for the rigid wing were observed,
respectively. The variations of the aerodynamic quantities with respect to the
frequency ratio show similar characteristics for all three Reynolds numbers.
However, it is interesting to note the striking difference between the graph
of CL/CD obtained for the
Re=75 case and those obtained for the higher Reynolds numbers. For
the lowest Reynolds number and f/n=1/4,
the above ratio is over 13% higher than that obtained for the rigid wing,
while for Re=250 it is only increased by 0.5%. In
Fig. 3C, it can be seen that
for f/n=1/3, the performance ratio
CL/CPW is 39% and 28% higher than that
obtained for the rigid wing for the Re=75 and Re=250 cases,
respectively. Interestingly, this measure is only about 13% higher than that
obtained for the rigid case at Re=1000.
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f/n=1/2, corresponding
to the most flexible foil, this peak is negative, while for the rigid case the
coefficient of lift peaks at 0.5. For all cases in between, the enhancement of
the mean lift force seen in Fig.
3 comes from the gradual increase of this peak, which is at 0.83
and 1.28 for f/n=1/4 and
f/n=1/3, respectively. For the latter
frequency ratio, where a structural non-linear resonance occurs, the lift peak
is also delayed and nearly coincides with the translational lift peak. This is
translated into a larger area under the lift curve per period and a larger
time-averaged CL value. The temporal variations of the
lift and drag coefficients for Re=250 and 1000 are more complex than
the variations seen at Re=75, and the periodicity is practically
lost. Still, in an average sense, negative lift peaks after stroke reversal
and larger translational lift peaks are seen when
f/n=1/2. Also, a widened two-peak lift
curve is observed when f/n=1/3.
Vortex structures
In order to relate the temporal variations of the lift and drag forces to
specific flow structures, we carefully examined several realizations of the
instantaneous flow fields. In Fig.
5, vorticity isolines are shown for the rigid and
f/n=1/3 cases at Re=75. For
clarity, the lift coefficient variation has been added
(Fig. 5K), together with the
temporal variation of the phase-averaged circulation of the most important
vortical structures generated during a flapping cycle. These are the leading
edge vortex (LEV) shown in Fig.
5A, the end of stroke vortex (ESV) shown in
Fig. 5C and trailing edge
vortex (TEV) shown in Fig. 5E.
The circulation of each of these vortices was computed as a function of time
by direct integration of the vorticity within a given threshold contour around
each vortex. The selection of the threshold contour, although arbitrary, was
consistently taken to be the lowest closed vorticity isoline in the vicinity
of the given vortex.
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, and the lower link A
rotates with an angular speed
.
The added angular speed 
affects the overall dynamics at stroke reversal. First, the camber generated
by the angular deformation (t) at the end of the stroke (see
Fig. 5B) reorients the zero
lift direction on the wing and enhances wake capture effects. This enhancement
mechanism is analogous to the one produced by orientation advancement in rigid
wings (e.g. Wang et al.,
2004). It is important to note that an excessive degree of
flexibility (low frequency ratios) produces a large camber at stroke reversal,
which has a negative effect on the lift production (e.g. at
f/n=1/2 in
Fig. 4). The evolution and
strength of the LEV on the other hand (see
Fig. 5A) is only a weak
function of the wing's flexibility. The formation time as well as the maximum
circulation shown in Fig. 5L
are approximately the same for the rigid and the flexible wings.
Another effect of the higher rotation speeds at the trailing edge for the flexible wings is the formation of a stronger shear layer, which rolls up into a stronger ESV (see Fig. 5C). On examining the ESV circulation plots (Fig. 5K), one finds that the strength and life span are significantly enhanced when compared with those of a rigid wing. The ESV pinches off later, forming a pair of counter-rotating vortices together with the LEV. This vortex pair generates flow directed towards the wing, enhancing the wake-capturing effects. This is more clearly reflected in the lift coefficient evolution shown in Fig. 5K. In contrast to the rigid wing, where the lift curve reaches a maximum (point H) and starts to decrease, for the flexible wing the production of increased lift continues for longer (point C).
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). The time at which the TEV pinches off is correlated with
the translational force peak; this peak is advanced in the
f/n=1/3 case when compared with that for
the rigid wing.
In Fig. 6, a quantitative
comparison of the LEV, ESV and TEV dynamics is shown for different
flexibilities at Re=75 by determining their average circulations as a
function of time. The maximum averaged circulation for each vortex and the
time at which it occurs with respect to the stroke reversal are provided in
Table 1. As expected, from what
was observed in Fig. 5, the LEV
dynamics is similar for all frequency ratios in terms of both strength and
timing. The TEV on the other hand, attains a higher maximum circulation as the
wing becomes more flexible. However, the time it takes to reach this maximum
circulation is shortest for f/n=1/3, where
the best aerodynamic performance is seen. For the ESV vortex, the maximum
circulation increases for the frequency ratios
f/n=1/3 and 1/4. The peak circulation for
f/n=1/3 is 20% lower than that obtained
for f/n=1/4, but the deflection at stroke
reversal in terms of the maximum deformation angle is 90% larger when
f/n=1/3 (56 deg. for
f/n=1/3 and 29 deg. for
f/n=1/4). This is translated into a larger
projected area contributing to the lift force. Also, as seen from
Fig. 6, the time delay of the
peak circulation of the ESV vortex is increased as the wing becomes more
flexible.
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A more direct illustration of the above-mentioned vortex evolutions is
given in Fig. 7, where
instantaneous vorticity isolines are shown for eight characteristic instances
during a flapping cycle. For f/n=1/3 and
1/4, it is clear that the enhanced ESV vortices produce an oblique-shaped TEV
vortex. For f/n=1/2, an excessive negative
camber is produced at stroke reversal, which then generates a high suction
zone on the lower side of the wing leading to the negative peak in the
CL curve seen in Fig.
4. The Reynolds number effects on the temporal evolution of the
lift and drag forces seen in Fig.
4 can also be observed in Fig.
8, where the instantaneous vorticity isolines are shown for
Re=250 for all frequency ratios. Clearly, as the viscous damping is
decreased, the system dynamics system ceases to be periodic and the important
vortices are stronger and are not dissipated as quickly as seen in the
Re=75 case (see Fig.
7). For the case of a rigid wing, for example, the LEV from a
given stroke interacts with the shear layer being generated in the next
stroke, and this induces a premature formation of the new LEV. This process is
not periodic, which is also reflected in the evolution of lift and drag
forces. A similar interaction is observed at
f/n=1/2 (see last three frames in
Fig. 8B).
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;
Wootton et al., 2003) has
shown that the wing's first natural frequency is close to the driving
frequency in normal flapping motion. This suggests that insects may be taking
advantage of a structural resonance to reduce energy consumption and enhance
aerodynamic performance. Despite the significance of such a hypothesis, only a
limited number of studies have addressed the problem due to the exceedingly
complex fluid–structure interactions that are encountered in
experimental or numerical work. Although three-dimensional computations of the
Navier–Stokes system coupled with a wing structural system are within
the reach of today's computers, one still needs to develop appropriate
mathematical models and tools to capture all important phenomena in this
complex system. In this regard, the present study extends computational work
that has been conducted before with simplified two-dimensional rigid wings to
include the effects of flexibility. The wing is represented by two rigid
links, which are joined at the center by a pin that contains a torsion spring.
The kinematics of one of the links is prescribed, while the motion of the
other link is determined by the fluid–structure interactions. Although a
fairly wide range of Reynolds numbers and frequency ratios was examined, we
found that the computations would fail at forcing frequencies close to the
linear resonance frequency due to an excessive degree of flexibility. This
limitation is due to the concentration of flexibility at a discrete point, and
replacement of rigid links with elastic links modeled as elastic beams is
expected to help in overcoming this limitation.
As mentioned earlier, the two-link structural system can be perceived as a
double pendulum with a common hinge. In particular, when one of the link
motions is prescribed, the other link behaves as a pendulum subjected to a
constraint arising from the prescribed motion and complex
fluid–structure interactions. Eqn
4 does resemble the equation of a pendulum driven in a fluid. A
straightforward perturbation analysis (e.g.
Nayfeh and Balachandran, 1995)
shows that the structural system can exhibit non-linear resonances at
f=1/3n and
f=3n. These resonances are expected in
systems with cubic non-linearities; for example, in the equations governing
local oscillations of a pendulum about an equilibrium position and elastic
systems such as beams (e.g. Anderson et
al., 1994). Operation of the flexible wing at the non-linear
superharmonic resonance f=1/3n was found
to be beneficial for aerodynamic performance. Inclusion of fluid effects will
give rise to quadratic non-linearities and additional non-linear
resonances.
For the specific set of kinematics that we considered, most of the benefits
of having a flexible wing are associated with the stroke reversal phase of the
cycle. Especially for the optimal flexibility cases
(f=1/3n), the strength and timing of the
ESV, as well as the dynamical changes of the wing's camber due to structural
deformations, are responsible for the performance enhancement. The overall
enhancement mechanism is analogous to that produced by orientation advancement
in rigid wings (see, for example, Wang et
al., 2004). It is noted that the present computations cover a wide
range of frequency ratios and, consequently, wing deflections range from a few
degrees to very large values. For example, in the case of highly stiff wings
(e.g. f/n=1/6), the maximum deflection
angle between the links was about 11, 13 and 16 deg. for Re=75, 250
and 1000, respectively, while for highly flexible wings (e.g.
f/n=1/2) the corresponding numbers were
67, 68 and 91 deg., respectively. In insects, the wing deformation magnitudes
increase as the body size and mass increase, and it is conceivable that
deformations seen in this study at the aerodynamically preferred frequency
ratio of f/n=1/3 could be possible in some
species. On the other hand, for small insects such as Drosophila only
small magnitude wing deformations have been observed. The computations of this
study show that as the wing is made stiffer, the performance enhancements are
marginal when compared with a rigid foil. For example, at Re=75 and
the highly stiff case f/n=1/6,
CL/CD is approximately 6% higher than
that of a rigid foil. For the higher Reynolds numbers Re=250 and 1000
there is actually no enhancement, and the performance is worse than that
obtained with a rigid foil. The above results indicate that low Reynolds
number regimes might benefit performance even at small chordwise
distortions.
The force histories, in particular for the low Reynolds numbers, appear to
reach a periodic steady state after the initial transients for all of the
frequency ratios that were considered, suggesting that quasi-steady models
might be able to reproduce this behavior. Such models have been reported in
the literature, and have been adapted for flapping flight based on models
developed for high Reynolds number fixed wing aeroelasticity studies by
including wing rotation along with translation
(Ellington, 1984a;
Ellington, 1999), and forces
due to added mass (Sane and Dickinson,
2002). In a more recent study by Wang and colleagues
(Wang et al., 2004) the
unsteady forces from experiments and with two-dimensional computations were
compared with the quasi-steady model predictions. They pointed out that the
force predictions, which were made by using models based on potential flow
theory (Munk, 1925) for a
constant pitching amplitude and constant translating speed wing, deviated
substantially from the experimentally determined unsteady forces. The force
predictions from a semi-empirical model based on numerical results from steady
translating wings at a fixed angle of attack were in broad qualitative
agreement with the unsteady forces. However, detailed comparisons revealed
that, depending on the kinematics, the unsteady effects can reduce the total
lift by a factor of 2 to 3. In the present case, due to the wing's
flexibility, the identification of the quasi-steady contributions is more
complex as additional new states have been included.
Conclusions
In the present work, the influence of flexibility on the aerodynamic
performance of a two-dimensional hovering wing section was numerically
studied. The wing model consists of two rigid links that are joined at the
center with a linear torsion spring. By prescribing the kinematics of the top
link, the structural system is effectively reduced to a single
degree-of-freedom non-linear oscillator. The viscous flow around this
structure is described by the incompressible form of Navier–Stokes
equations. The combined set of equations describing the fluid and structural
dynamics are integrated in time by using a scheme that can capture strongly
coupled fluid–structure interactions.
The results obtained in this study demonstrate that flexibility can be beneficial in terms of enhancing aerodynamic performance. Furthermore, we found that in the frequency range below the first natural frequency, the best performance is achieved when the wing is driven at a frequency close to one of the non-linear resonances (a superharmonic resonance of order three) of the system. This behavior is common to all of the Reynolds numbers studied. In terms of the flow physics, the wake capture mechanism is enhanced partially due to a stronger flow around the wing at stroke reversal. However, it needs to be noted that the cases where the wing is driven at or close to the first natural frequency of the system were not considered in this study, and it is possible that a better aerodynamic performance may be achieved with a linear resonance and this remains to be explored.
The study also leads to the following open questions. (i) Why is there a performance enhancement when the system is excited at a flapping wing's non-linear resonance and would one achieve a better performance with a non-linear resonance compared with a linear resonance? (ii) Which kinematics is preferable from an aerodynamic efficiency standpoint? The interplay between wing flexibility and kinematics together with qualitative changes (and bifurcations) in the system dynamics as a function of the Re number require further investigation.
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Footnotes |
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References |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
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Anderson, T. J., Balachandran, B. and Nayfeh, A. H. (1994). Nonlinear resonances in a flexible cantilever beam. J. Vib. Acoust. 116,480 -484.[CrossRef]
Balaras, E. (2004). Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations. Comput. Fluids 33,375 -404.[CrossRef]
Bao, L., Hu, J. S., Yu, Y. L., Cheng, P., Xu, B. Q. and Tong, B. G. (2006). Viscoelastic constitutive model related to deformation of insect wing under loading in flapping motion. Appl. Math. Mech. 27,741 -748.[CrossRef]
Berman, G. J. and Wang, Z. J. (2007). Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582,153 -168.[CrossRef]
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