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First published online December 28, 2007
Journal of Experimental Biology 211, 239-257 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.008649
Research Article, Biomechanics of Flight |
Near- and far-field aerodynamics in insect hovering flight: an integrated computational study
1 Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-cho,
Inage-ku, Chiba 263-8522, Japan
2 Next-generation computation research group, RIKEN, 2-1 Hirosawa, Wako-Shi,
Saitama 351-0198, Japan
3 Graduate School of Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku,
Chiba 263-8522, Japan
* Author for correspondence (e-mail: hliu{at}faculty.chiba-u.jp)
Accepted 12 June 2007
Summary
We present the first integrative computational fluid dynamics (CFD) study of near- and far-field aerodynamics in insect hovering flight using a biology-inspired, dynamic flight simulator. This simulator, which has been built to encompass multiple mechanisms and principles related to insect flight, is capable of `flying' an insect on the basis of realistic wing–body morphologies and kinematics. Our CFD study integrates near- and far-field wake dynamics and shows the detailed three-dimensional (3D) near- and far-field vortex flows: a horseshoe-shaped vortex is generated and wraps around the wing in the early down- and upstroke; subsequently, the horseshoe-shaped vortex grows into a doughnut-shaped vortex ring, with an intense jet-stream present in its core, forming the downwash; and eventually, the doughnut-shaped vortex rings of the wing pair break up into two circular vortex rings in the wake. The computed aerodynamic forces show reasonable agreement with experimental results in terms of both the mean force (vertical, horizontal and sideslip forces) and the time course over one stroke cycle (lift and drag forces). A large amount of lift force (approximately 62% of total lift force generated over a full wingbeat cycle) is generated during the upstroke, most likely due to the presence of intensive and stable, leading-edge vortices (LEVs) and wing tip vortices (TVs); and correspondingly, a much stronger downwash is observed compared to the downstroke. We also estimated hovering energetics based on the computed aerodynamic and inertial torques, and powers.
Key words: computational fluid dynamics (CFD), far-field flow, fruit fly, hawkmoth, hovering, leading-edge vortex (LEV), near-field flow, unsteady aerodynamics
Introduction
Flapping-flying insects employ unsteady aerodynamic mechanisms to keep them
afloat, and there have been many studies on this topic
(Ellington, 1984a
;
Ellington, 1984b;
Ellington, 1984c;
Ellington, 1984d;
Ellington et al., 1996;
van den Berg and Ellington,
1997a; van den Berg and
Ellington, 1997b; Dickinson et
al., 1999; Liu,
2002; Liu, 2005;
Sane, 2003;
Lehmann, 2004a;
Lehmann, 2004b;
Wang, 2005). A general
conclusion from these studies is that insects obtain enough lift force to
support their weight through the sophisticated vortices generated by the
flapping wings. Among the many mechanisms involved in insect flight, delayed
stall (Ellington et al.,
1996), which is featured by prolonged attachment of a leading-edge
vortex (LEV) on a wing, has been widely recognized as an important unsteady
aerodynamic mechanism contributing to the enhancement of lift force generation
in flapping-flying insects. However, the delayed stall has been deduced based
exclusively on experiments on the hawkmoth Manduca sexta, which is
one of the largest insects (wing span 5 cm). The question still remains as to
whether the delayed stall presents and plays similar roles in smaller insects
such as a fruit fly (wing span 0.2 cm). Birch et al.'s dynamically scaled
robotic wing model experiments (Birch et
al., 2004) offered some indirect evidence towards answering the
question. Their experimental results suggested that the transport of vorticity
from the leading edge to the wake, which permits prolonged vortex attachment
and enhanced force production, might take different forms at different
Reynolds numbers (Re; i.e. in insects of different sizes).
Besides the aforementioned LEV, other unsteady mechanisms might contribute
to force production in insect flight. Dickinson et al. suggested that
rotational circulation and wake capture increased aerodynamic force during the
rotational phase of wing motion (Dickinson
et al., 1999). The correct angular difference between two
counter-lateral wings during dorsal stroke reversal (`clap-and-fling') was
found to increase total lift force by up to 17%, which indicated the possible
role of wing–wing interaction in insect flight
(Lehmann et al., 2005). The
flow around a real hawkmoth in forward flight was visualized using digital
particle image velocimetry (DPIV) (Bomphrey
et al., 2005; Bomphrey et al.,
2006). The visualization results described an apparent vortex
structure across the insect thorax at the end of the downstroke, suggesting a
potential effect of wing–body interaction on aerodynamic force
generation.
Overall the above-mentioned studies, together with other relevant studies
(Birch and Dickinson, 2001;
Birch and Dickinson, 2003;
Lehmann, 2002;
Fry et al., 2003;
Fry et al., 2005;
Wang et al., 2004), have made
significant contributions to understanding of the different aspects of the
aerodynamic mechanisms involved in insect flight. However, most of the
previous studies have been focused exclusively on the near-field flow and its
correlation with aerodynamic force production. Therefore the challenging
problem to quantify the near- and far-field flow structures and to correlate
them to force-production still remains.
Both near- and far-field flows are strongly three-dimensional (3D) in space
and unsteady in time. A reasonable investigation of the interaction between
them entails obtaining sufficient information on the unsteady 3D flows.
Obtaining such a large amount of information might be extremely difficult
experimentally; at this point, computational fluid dynamics (CFD)-based
simulation may provide a relatively more effective method. In fact, during the
last decade CFD has been widely applied to studies concerning insect flight
(Liu and Kawachi, 1998;
Liu et al., 1998;
Wang, 2000;
Ramamurti and Sandberg, 2002;
Sun and Tang, 2002a;
Sun and Tang, 2002b;
Wang et al., 2004;
Miller and Peskin, 2004;
Miller and Peskin, 2005;
Liu, 2005). These
simulation-based studies had varying degrees of success in dealing with the
specific issues under consideration, but most of them were still either
limited to two-dimensional (2D) computations, or lacking in quantitative
evaluation of the effect of wing–wing and wing–body
interactions.
In this study, we present the first integrative CFD study of the unsteady 3D near- and far-field vortex wake dynamics in a hovering fruit fly and their relation to lift force generation using a biology-inspired dynamic flight simulator (see Movies 1 and 2 in supplementary material; H.L., manuscript in preparation). To reproduce a hovering fruit fly on a computer using the simulator, both the wing–body morphological and kinematic models were built based faithfully on measurements from a real fruit fly, Drosophila melanogaster. The large amount of information on flow field and aerodynamic force offered by the computation enabled us to perform in-depth analysis of the correlation of vortex flow structure with force generation, the interaction between near- and far-field flows, and the effects of wing–wing and wing–body interactions. We also simulated a hovering hawkmoth and, by comparing the computed results for the two insects, we elucidate the marked dependence of the spanwise flow and the delayed stall on Re.
Materials and methods
A biology-inspired dynamic flight simulator
This study employs a biology-inspired, dynamic flight simulator, consisting
of in-house software developed recently for quantitatively investigating the
aerodynamics of a flapping-flying insect (see Movies 1 and 2 in supplementary
material; H.L., manuscript in preparation). This simulator is able to `fly' an
insect with realistic body–wing morphologies and flapping-wing and body
kinematics, and to evaluate unsteady aerodynamics including detailed vortex
flow fields and flying energetics involving aerodynamic and inertial torques
and powers. Details of the simulator can be found in the Appendix.
Morphological and kinematic model
We constructed a realistic wing–body morphological model
(Fig. 1) for the fruit fly
Drosophila melanogaster, body length 2.78 mm and wing length 2.39 mm.
We traced the borderlines of the wings and the body from pictures of a fruit
fly taken in two perpendicular views, and then reconstructed the shape of the
fruit fly on a computer based on the borderlines, by assuming that the cross
sections of both the body and the wings are in an ellipse shape. We assumed a
uniform wing thickness (1.2% of the mean chord) for the two wings, which
resembles the wing geometry of a real fruit fly. Finally, curve smoothing was
performed at the leading and trailing edges, and at the wing tip. Note that to
avoid the attachment of the wing on the body surface we added a virtual
portion of the wing length (approximately
/32, where
is the mean wing chord length) at
the wing base, which could largely improve the numerical convergence but
seldom affect the results in the hovering flight. More details can be found in
the Appendix (morphological modeling).
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The wing–body kinematic model was established based on the
measurements of a hovering fruit fly, Drosophila melanogaster
(Fry et al., 2005)
(Fig. 2). Reynolds number was
defined by
Re=Uref/
,
where is the kinematic viscosity of air (1.5x10–5
m2 s–1). According to the measured data
(=0.78 mm, R=2.39 mm,
=2.44 rad (140°), f=218 s–1) of the fruit
fly (see List of symbols and abbreviations for definitions) in this study,
Re was estimated to be 134 with a reduced frequency K=0.212.
More details can be found in the Appendix (kinematic modeling).
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Far-field flow: vortex wake structures and downwash
The absolute iso-vorticity surfaces around the hovering fruit fly and the
velocity vectors at the plane perpendicular to the stroke plane (see
Fig. 2B) are illustrated in
Fig. 3Ai,ii,Bi,ii,Ci,ii,Di) at
four typical moments over a flapping cycle (see
Fig. 2Ca,b,e,g). Note that the
absolute iso-vorticity surfaces around the hovering fruit fly without the body
are also illustrated in Fig.
3Aiv,v,Biv,v,Civ,v,Diii to evaluate the effect of the body. In
addition, to understand the relationship between the flow fields and the
intensity of the downwash, the downward speed contours and the velocity
vectors in the stroke plane are depicted in
Fig. 3Aiii,Biii,Ciii,Dii and
Fig. 4.
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|
As seen in Fig. 3Ai,ii, a ring-shaped vortex wake structure, which results from the vortices generated and detached from the wings and body during the preceding upstroke, is evident during pronation (see Fig. 2Ca). This wake structure is the most prominent feature over a flapping cycle. Since the starting vortex (the detached TEV) balances the circulation of the bound vortex, the air within the vortex ring is given downward momentum, the downwash, which is observed flowing downward through the center of the vortex ring as a jet (Fig. 3Ai,ii).
In the first half of the downstroke (see
Fig. 2Cb,
Fig. 3Bi,ii), a
horseshoe-shaped vortex is observed, which wraps around the wing and comprises
a leading-edge vortex (LEV), a wing tip vortex (TV) and a trailing-edge vortex
(TEV) (or starting vortex). A similar horseshoe-shaped vortex was also
observed in an experimental DPIV study of an impulsively started, dynamically
scaled, flapping wing (Poelma et al.,
2006). During the downstroke, the LEV and the TEV grow steadily in
size and expand towards the wing base. Eventually the TEV begins to detach
from the trailing edge and joins the TV. Subsequently, the LEV, the TV and the
shed TEV in toto form a doughnut-shaped vortex ring for one wing, and
hence a pair of vortex rings for the wing-pair. During most of the downstroke,
the doughnut-shaped vortex ring pair has an intense, downward jet-flow through
the `doughnut' hole, which forms the downwash during the downstroke
(Fig. 3Bi,ii,iii). In the
second half of the downstroke, the TV gradually enlarges, subsequently the LEV
and the TV weaken and detach from the upper surface, and the doughnut-shaped
vortex rings of the wing pair break up into two downward circular vortex
rings, forming the far-field wake below the fly.
During the upstroke (see Fig. 2Ce,g), as illustrated in Fig. 3Ci,ii,Di, the wing also generates a ring-shaped vortex wake structure that resembles, but is more structured than, the wake during the downstroke. The exact shape of the wake depends on how well the vortices shed from the wing merge with those vortices still attached to the wing (Fig. 3Ai,ii,Ci,ii,Di). The upstroke's downwash also resembles that of the downstroke.
Fig. 4 illustrates that the downward flow (downwash) velocity maps at two cross sections around the hovering fruit fly during mid down- and up-stroke: one is parallel to the X-axis and the other to the Y-axis. The computed results demonstrate that, throughout a flapping cycle, despite the existence of downward flows beneath the body, the strong downwash is only present in a limited area swept by the flapping wing, which shows a maximum speed of approximately 0.8Uref over one flapping cycle and is largely attenuated within 1.5 times of the body length of the fruit fly far behind the wing (see Fig. 3Aiii,Biii,Ciii,Diii and Fig. 4).
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Upstroke
At the beginning of the upstroke immediately after the wing reversal, a
horseshoe-shaped vortex is visible wrapping around the wing and consisting of
a TV, a LEV and a TEV. Similar to the downstroke, the TEV subsequently
detaches from the trailing edge but connects to the TV; and eventually a
doughnut-shaped vortex ring is observed. As the LEV continues to grow, a
spanwise flow becomes indiscernible in its core, but the LEV is mostly in a 2D
structure (Fig. 7Bii,iii). The
spanwise flow is not, however, observed in the core of the LEV but along the
rearmost half of the wing. As the positional angle of the wing approaches zero
at mid upstroke, a large LEV and TV become visible
(Fig. 5B) and a corresponding
negative pressure region is observed on the upper surface of the wing. The
upstroke LEV continues to grow without breaking down or shedding throughout
the translational phase of the upstroke. When the wing starts pronation, the
LEV and the TV become unstable and a stopping vortex is generated, and
together form a complicated 3D vortex structure
(Fig. 6B).
|
), which generates two vortices between the left and right
wings. We can observe this process in our simulation in the form of a TV and
the initial horseshoe-shaped vortex. After the pronation, while the wing
translates downwards and rotates, and its angle of attack increases, the
starting vortex is shed off from the trailing edge.
Re dependence on the role of flapping wing motion in the LEV formation
To identify the Re dependence on the role of translational and
rotational motion of the flapping wing in the formation of the LEV, we
compared the computed velocity vector maps and the spanwise flow contours at
the plane transecting the wing at 60% span at Re=6300 (hawkmoth) and
Re=134 (fruit fly) (Fig.
8Aii,Bii). The computed results suggest that the translational
motion of the wings (rotation of positional angle of the wings) dominates the
development of the LEV, while wing rotation about the leading edge (angle of
attack of wings) benefits the stability of the LEV. The role of wing rotation
is more evident at the low Re (134) than at the high Re
(6300). At the high Re (6300), an axial flow at the core of the LEV
is much more pronounced, and the LEV forms a 3D structure near the leading
edge; by contrast, at the low Re (134), only a weak axial flow is
detected, indicating that the vortical structure near the leading edge might
be mostly in a 2D structure. Yet at such a low Re, a remarkable
spanwise flow is predicted along the rearmost half of the wing at mid down-
and upstroke (Fig.
7Aii,iii,Bii,iii).
|
 | (1) |
 | (2) |
 | (3) |
is the density of the air (1.23 kg m–3),
Uref is the reference velocity, Sw is
the planform area of the wing and CL,
CD, CS are the three non-dimensional mean
force coefficients of the vertical, horizontal and sideslip forces,
respectively. According to Eqn 1,
2,
3, with the values assigned to
the parameters defined in the CFD simulation (Uref=2.54 m
s–1), the mean lift force is calculated to be
9.60x10–6 N, which exceeds the weight of the fruit fly
(9.41x10–6 N) by 2%; the horizontal and sideslip forces
computed are less than 4% of the vertical force. These force predictions agree
very well with the situation expected for hovering flight and therefore
indirectly validate our simulations.
While these mean force coefficients provide a validation of our
time-averaged results, the time courses of computed vertical (lift) and
horizontal (drag and thrust) forces over one flapping cycle can be validated
against the experimental data (Fry et al.,
2005) (Fig. 9).
This comparison confirms the match between time-averaged computational and
experimental data. However, there are discrepancies between the time courses
of the lift force. The main discrepancy occurs in the phase of the periodic
force signature, especially in the early down- and upstroke. These phase-shift
differences might be caused by different locations of the wing's axis of
rotation which, in the computations, is assumed to lie at a quarter chord
length from the leading edge. For horizontal force (drag and thrust), we
obtain satisfactory agreements between the computation and the experiment, for
both the mean and instantaneous magnitudes. The lift force peaks twice, at mid
downstroke and mid upstroke (Fig.
9A). The two peaks are most likely due to the existence of a large
negative pressure area on the upper wing surface, induced by the LEV and the
TV (Fig. 5). The drag force is
produced exclusively during the downstroke while the thrust force is produced
only during the upstroke (Fig.
9B).
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).
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Powers
Based on computed instantaneous aerodynamic forces and wing velocities, we
calculate muscle-mass-specific inertial, aerodynamic and mechanical powers
(Fig. 11). For comparison, the
data measured by Fry et al. (Fry et al.,
2005) are also given in Fig.
11. The muscle-mass-specific inertial power
Piner denotes the power consumed to accelerate the mass of
the wing (Eqn A14). The time
course of the computed inertial power is in satisfactory qualitative, but not
quantitative agreement with the experimental results
(Fry et al., 2005), which is
thought to be due to the difference in the wing shape between the
computational and experimental models (Fry
et al., 2005). By integrating inertial power over a flapping
cycle, we find that an average 56.9 W kg–1 is needed to
accelerate the wing. Note that wing deceleration is assumed to accrue no cost
and, at the same time, there is no elastic power storage. The
muscle-mass-specific aerodynamic power Paero is the power
necessary to overcome air resistance (Eqn
A15). The comparison between computations and measurements shows
good agreement except for a slight phase lag between the computational and the
experimental data (Fry et al.,
2005). The computed mean aerodynamic power (89.3 W
kg–1) is very close to the experimental result of Fry et al.
(Fry et al., 2005). Maximum
aerodynamic power is reached at each mid stroke when the flapping wing speed
and the aerodynamic forces approach maximum values. The muscle-mass-specific
total mechanical power Ptotal is the power required to
move the wing. We simply calculate Ptotal using
Eqn A16 as the sum of the
muscle-mass-specific aerodynamic and inertial powers. It can be seen that the
curve of Ptotal turns to be negative in the second half of
the upstroke because more power is required to overcome aerodynamic forces
when the wing decelerates (Fig.
11C).
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Discussion
The leading-edge vortex
Our CFD analysis indicates that the LEV of fruit flies differs in shape
from that of hawkmoths. The main difference is in the position on the wing
where the LEV is attached and in the intensity of the axial flow in the LEV's
core. Birch and Dickinson's experimental results
(Birch and Dickinson, 2001)
indicate that the LEV of fruit flies exhibit a rather stable vortex structure
without separation during most of the down- and upstroke, and show no evidence
of axial flow in the vortex core. These observations are in stark contrast to
those obtained with hawkmoths, whose LEV detaches from the wing surface at
approximately 75% of the wing length and whose LEV shows a strong axial flow
in the core (van den Berg and Ellington,
1997a). The computed results confirm the LEV features described by
Birch and Dickinson (Birch and Dickinson,
2001), who speculated that the difference between hawkmoths and
fruit flies can be attributed to effects of size or Re (100–250
for fruit flies, >6000 for hawkmoths). Birch and Dickinson further
suggested that considering that a typical adult insect is closer in size to a
fruit fly and that therefore their observations of flow patterns are more
likely to be typical, the insects are more likely to prolong the attachment of
the LEV due to the attenuating effect of the downwash. They explained the
absence of the axial flow in the LEV core of fruit flies by pointing out that
the pressure gradients within the vortex core might be too small to drive a
substantial axial flow on the smaller wing
(Birch and Dickinson,
2001).
To test this hypothesis, Fig.
12 illustrates the pressure gradient contours on the wing of a
fruit fly and a hawkmoth. Obviously, the computed pressure gradients on the
wing of the hawkmoth are much larger than those of the fruit fly. This
observation suggests that the pressure gradient very likely causes the strong
axial flow at the LEV core and enhances the LEV's stability as long as
Reynolds numbers are high enough – in the order of several thousands
(Ellington et al., 1996;
van den Berg and Ellington,
1997a; Liu and Kawachi,
1998; Liu et al.,
1998). At a low Re of 100–250, e.g. for the fruit
fly, the flapping wing cannot create a LEV strong enough to generate a steep
pressure gradient at the vortex core. Nevertheless, our results indicate that
a fruit fly can produce a stable LEV during most of the down- and upstroke.
While the LEV of the hovering hawkmoth breaks down roughly at mid downstroke
(van den Berg and Ellington,
1997a; Liu et al.,
1998), the LEV of the hovering fruit fly remains attached to the
leading edge and grows stably throughout the downstroke, eventually breaking
down during the subsequent supination (Fig.
6A). At low Re values, the LEV on a flapping or rotary
wing may translate over a longer distance before growing sufficiently to break
down. Still, it is valid to ask which forces or mechanisms are responsible for
enhancing LEV stability. As observed in other studies on the flapping wing of
the fruit fly (Birch and Dickinson,
2001; Birch et al.,
2004), we also find a fairly strong spanwise flow outside that of
the LEV, but in the rearmost half of the wing, during most of the down- and
upstroke. While we show that the pressure gradient is not the cause
(Fig. 12), centrifugal and
Coriolis forces just might be sufficient to create the spanwise flow outside
the LEV's core in the rearmost half of the wing that is indicted by our CFD
analysis.
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The computed results of our analysis indicate that the lift-boosting
mechanism of the LEV observed in hovering hawkmoths
(Ellington et al., 1996) is
also likely to enhance lift force in the hovering fruit fly, because the
delayed stall also strengthens the fruit fly LEV and hence augments lift
force. However, we find significant discrepancies in lift force production
during the down- and upstrokes: the fruit fly produces 62% of the total lift
force during the upstroke and 38% during the downstroke. In contrast,
hawkmoths produce approximately 30–40% of the total lift force during
the upstroke and 60–70% during the downstroke
(Liu and Kawachi, 1998;
Liu et al., 1998;
Aono and Liu, 2006), and
hummingbirds produce approximately 25% of the total lift force during the
upstroke and 75% during the downstroke
(Warrick et al., 2005). This
implies that the size differences in wing and body kinematics play a key role
in the aerodynamic force generation of hovering insects and birds.
The vortex structure and the downwash
During most of the down- and upstroke, as shown in
Fig. 3, the doughnut-shaped
vortex rings of the wing pair are observed, which eventually detach from the
wing and body during the subsequent supination and pronation, breaking up into
two circular vortex rings, with strong downward flow through the core (hole)
of the vortex ring. These phenomena, of a vortex ring and its breaking-up,
were also predicted in an experimental study based an analysis of the vortex
wake structures around a robotic hawkmoth model
(van den Berg and Ellington,
1997b). Note that the root vortex as observed by van den Berg and
Ellington is absent here at the wing base during most of the downstroke, very
likely because of the existence of the insect body. van den Berg and Ellington
also observed a vortex wake ring with an intensive downwash through its centre
in the wing wake during the downstroke, and called it a dumbbell-shaped vortex
structure (van den Berg and Ellington,
1997b), but they did not quantify the formation, development and
break-up of the vortex ring because of technical limitations with their
smoke-rake flow visualization. Nevertheless, they predicted that the
dumbbell-shaped vortex structure should eventually break up into two single
circular vortex rings rather than merge together into a single one. The
computed vortex wake structure during the downstroke of the fruit fly analyzed
here also differs from the descriptions of the shed vortex wake observed in
previous studies of insect flight
(Ellington, 1984d;
Grodnitsky and Morozov, 1993;
Dickinson and Götz, 1996;
Willmott et al., 1997;
Birch and Dickinson, 2001).
In our fruit fly and hawkmoth simulations, we also find this vortex ring
pair (we call them a doughnut-shaped vortex ring pair). It has an intense
downwash or jet-stream at its core, and eventually breaks up into two single
circular vortex rings during the subsequent supination and pronation. A
follow-up study on the details of a lattice wake structure and dynamics of the
downwash will be conducted in the near future. Moreover, our results also
reveal that the vortex wake structure is more compact than the wakes of other
insects and mechanical insect models reported by many previous studies
(Ellington, 1984d;
Grodnitsky and Morozov, 1993;
Dickinson and Götz, 1996).
However, whether the compact wake structure could benefit the flight
efficiency is not clear in the scope of this study.
Although the ring-shaped vortex wake structure is observed over a complete flapping cycle, the wake formed during the upstroke is more structured and stronger than the wake formed during the downstroke. Corresponding to the intensity of the vortex, the strongest downwash is present on the far-body side of the ring-shaped vortex core (Figs 3 and 4). The relationship between the downwash generated by the flapping wings and instantaneous aerodynamic force production is illustrated in Figs 4 and 9. At mid down- and upstroke, high-lift force is produced by the flapping wings (see Fig. 9), and then the remarkable downwash in the stroke plane is also observed. In attention to the intensity of the downwash, the downwash at mid upstroke is stronger than that at mid downstroke. However, more detailed quantitative study is required to understand how the downwash is linked to the aerodynamic force generation in flapping-flying insects.
For the fruit fly, the very weak axial flow in the LEV core implies that
the LEV does not feed significant amounts of vorticity into the TV, as is the
case for the hawkmoth. In other words, the LEV discharges little energy into
the TV and hence into the total vortex wake structure. However, adding flow
energy locally by prolonging the LEV's attachment to the wing does not
necessarily increase global flight efficiency because of wake effects, such as
an increased downwash. Based on the results of robotic experiments, Birch and
Dickinson suggested that the downwash had a potent inhibitory effect on the
magnitude of force production (Birch and
Dickinson, 2001). This is confirmed by our computed results as
shown in Fig. 13, where time
course of the lift force is plotted for three complete stroke cycles. The mean
lift force generated during the second cycle is reduced by 22.5% compared with
the first cycle, but almost no difference is observed between the second and
the third cycle. Considering that insects begin to flap in static air, the
drop in lift force between the first and second stroke cycle can be attributed
to the inhibitory effect of the downwash, which is fully established at the
end of the first cycle.
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The effect of the body on the wake dynamics of hovering flight
Fig. 3Aiv,v,Biv,v,Civ,v,Diii
shows that removing the body changes the vortex wake structure only slightly.
Specifically, the trace of the shed TEV seems to expand slightly now that the
barrier formed by the body is removed. This slight change in the flow field is
also confirmed by its effect, albeit small, on aerodynamic forces shown in
Fig. 14. Albeit fluctuations
of up to 100% in instantaneous lift and sideslip force coefficients are
observed at particular moments, the magnitudes of the mean lift and sideslip
force coefficients with versus without the presence of the body
differ by less than 2%. In short, the effect of the body on hovering
aerodynamics is very small and hence it can be neglected. Nevertheless, in
other flight modes, such as during forward flight and quick turn, the
body-effect-related instantaneous forces or the inertial torques of the body
may play important roles in the flight control.
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Appendix
A biology-inspired dynamic flight simulator
In the following we give a brief description of the methodology of the
simulator with a specific focus on three relevant modeling steps: (1)
morphological modeling, (2) kinematic modeling, and (3) multi-block- and
overset-grid-based computational fluid dynamic (CFD) modeling.
Morphological modeling
We obtain a morphological model of an object in four steps. First, we image
the object digitally; secondly, we segment the image to extract the object's
shape as a wire frame and/or skeleton model; thirdly, we smooth and
curve/surface-fit the wire frame to construct a morphological model; and
finally, we render the surface and/or volume to reconstruct the object and
then decompose the object in the computational domain to generate a grid. We
further develop an efficient computer-aided method that unifies a
morphological and kinematic modeling of 3D flyers
(Liu, 2002;
Liu, 2005) (H.L., manuscript
in preparation).
Kinematic modeling
A flying insect comprises a body and flapping-wing kinematics
(Fig. 2). The body movements
are quantified by the body angle 
(inclination of the body relative to
horizontal plane), and the stroke plane angle β (plane in which the wing
flaps). The wingbase-fixed coordinate system illustrated in
Fig. 2A,B has its origin at the
wing base, with the x-axis normal to the stroke plane, the
y-axis perpendicular to the body axis, and the z-axis
parallel to the stroke plane. The flapping-wing kinematics consist of three
basic motions within the stroke plane: (1) flapping about the x-axis
in the wingbase-fixed coordinate system, described by the positional angle
; (2) rotation of the wing about the z-axis, described by the
elevation angle 
; and (3) rotation (feathering) of the wing about the
y-axis by varying the angle of attack 
. Here, a general
definition of the positional angle, the elevation angle and the angle of
attack are given in degrees using the first three Fourier terms
(Fig. 2C):
 | (A1) |
 | (A2) |
 | (A3) |
f/2Uref,
where f is flapping wing frequency,
is the mean wing chord length
(reference length) and Uref is the reference velocity at
the wingtip defined by 2Rf, where is the wing beat
amplitude and R is the wing length. The Fourier coefficients
cn, sn, cn,
sn, cn and sn are
determined accordingly where n is integer varying from 0 to 3.
Regridding for a flapping wing and a moving body
The 3D movements of flapping wings and a body cause large wing deformations
and 6 d.f. (degrees of freedom) displacements of the body. Modeling such
movements requires an efficient and robust grid generator that fits the
instantaneously deforming wing surface as well as the moved body and other
boundaries. To model the 3D movements of a flapping wing
(Fig. 2), we employed a
previously described method (Liu and
Kawachi, 1998; Liu et al.,
1998; Liu, 2005)
that uses the initial grid and the wing kinematics to analytically regenerate
the wing-fitted grid, while minimizing additional computational requirements.
The method is implemented in three steps: (1) rotating grids in the whole
wing-fitted sub-domain (the grid) according to the `rigid' feathering motions
of the wing; (2) rotating the feathering-based grids in the whole sub-domain
according to the `rigid' flapping motion; and (3) rotating the feathering- and
flapping-based grids in the whole sub-domain according to the elevation
motion.
Multi-blocked, overset grid method
Modeling the shape of an insect with two or four wings is a challenging
problem for CFD simulations. The wings are not only undergoing large-scale
movements relative to the body, but also flap rapidly, requiring us to model
highly unsteady vortical flows about multiple and moving bodies. To achieve
this, we develop a multi-blocked, overset grid method and incorporate it into
an in-house CFD solver (Liu,
2005; Aono and Liu,
2006; H.L., manuscript in preparation). This grid method uses
three individual structured grid systems, one for the body and the one for
each wing. Each grid system is made to fit the object (body or wing), moving
and deforming with the object. A trilinear interpolation technique ensures the
communication of velocities and pressures among overlapping grids
(Liu, 2005;
Aono and Liu, 2006; H.L.,
manuscript in preparation).
As shown in Fig. 1B, three grids are generated for the body and the two wings of the fruit fly. The wing grid comprises 45x45x31 cells with the outer boundary 2 mean chord lengths away from the wing surface; the grids for both wings are identical copies, using the relation of geometrical symmetry of the two wings about the body axis. The body grid is much larger because it is used as a background grid to envelope the two wing grids for the interpolation; it comprises 45x47x95 cells, and the grid is approximately 20 mean chord lengths wide (measured as the distance between outer boundary and body surface).
Solutions to the Navier-Stokes equations
The governing equations are the three-dimensional, incompressible unsteady
Navier-Stokes (NS) equations, written in a strong conservative form for
momentum and mass, and non-dimensionalized in an integral form, such that:
 | (A4) |
 |
is the pseudo-compressibility coefficient;
p is pressure; u, v, w are the x, y, z velocity
components in the Cartesian coordinate system; t denotes physical
time, 
is pseudo time; Re is the Reynolds number. Note that the
term q associated with pseudo time is designed for an inner-iteration
at each physical time step, and will vanish when the divergence of velocity is
driven to zero so as to satisfy the equation of continuity. Time-dependent
solutions of the incompressible NS equations are formulated in an ALE
(Arbitrary Lagrangian–Eulerian) manner with the FVM (Finite Volume
Method) and are performed in a time-marching manner with a
pseudo-compressibility method; we enforce conservation of mass and momentum in
both time and space. More details can be found elsewhere
(Liu and Kawachi, 1998).
Boundary conditions
As shown in Fig. 1, the
solutions to the NS equations with a multi-blocked, overset grid for a
flapping-flying insect require appropriate boundary conditions for the
overlapping zones among the different single grid block, the moving walls of
the wing and the body, and the far-field outside boundary.
For individual grid blocks and for the total wing and total body we use the
fortified solutions to the NS equations by adding a forcing term with
communication of a vector q* to offer the boundary
conditions for velocity and pressure in the overlapping zones of the two grids
(H.L., manuscript in preparation). The fortified equations are solved inside
the computational domain, except for holes and the single grid boundary. In
the case of a fruit fly, each time step requires that we solve the fortified
NS equations three times, once for each grid cell block. On the body surface,
the no-slip condition is applied to calculate the velocity components. To
account for dynamic effects due to the accelerations of the oscillating body
(moving and/or deforming body surface), pressure divergence at the surface
stencils is derived from the local momentum equation, such that
 | (A5) |
 | (A6) |
For the background grid of the insect body we need to define appropriate
boundary conditions at the outside boundary
(Fig. 1B). Consider that, when
an insect hovers or flies forward at a velocity Vf, the
boundary conditions for the velocity and the pressure may be given such as:
(1) at upstream V(u, v, w)=Vf while pressure
p is set to zero; (2) at downstream zero-gradient condition is taken
for both velocity and pressure, i.e. 
(u, v, w,
p)/n=0, where n is the unit outward normal vector at
the outside boundary.
Evaluation of forces, torques and powers
Given the wing kinematics, according to the Newton's second law, the
non-dimensional inertial forces acting upon the wing may be calculated as the
sum of the inertial forces on all wing cells, such that:
 | (A7) |
w is assumed to be constant throughout
the wing, 
vi is the volume of the cell constructed
by eight grid points on the wing, and
(t) is the
computed wing velocity at each cell center at time t. The aerodynamic
forces, 
of lift and
drag coefficients acting on the wing surface can be calculated from the
pressure and stresses along its surface based on the solutions to the NS
equations. The resultant lift and thrust forces are calculated first in the
local wingbase-fixed coordinate system (x, y, z), and then
transformed into the earth-based coordinate system (X, Y, Z)
accounting for the stroke plane angle (Fig.
2B), yielding vertical, horizontal and sideslip forces. Both the
aerodynamic and inertial forces are non-dimensionalized with the reference
velocity Uref, the reference length
, the air density and the
planform area of the wing Sw, such that:
 | (A8) |
 | (A9) |
The non-dimensionalized inertial and aerodynamic torques of the wing are
calculated as the sum of the cross product of each force and the positional
vector at each cell center of the wing about the origin of body (O''),
such that:
 | (A10) |
 | (A11) |
denotes the
non-dimensionalized positional vector of the cell center on the wing surface,
and 
and
represent
the aerodynamic and inertial forces at each cell center, respectively.
The aerodynamic and inertial powers are calculated as the scalar products
of the velocity and the aerodynamic and inertial forces of the wing as:
 | (A12) |
 | (A13) |
is the computed
wing velocity at the cell i. Furthermore, the muscle-mass-specific
aerodynamic and inertial powers can be calculated as:
 | (A14) |
 | (A15) |
 | (A16) |
Verification and validation
A variety of benchmark tests for verification and validation have been
undertaken to assess the reliability of the single-grid NS solver for unsteady
flows about a moving and deforming body
(Liu and Kawachi, 1998;
Liu et al., 1998;
Liu, 2002). Verification and
validation for the present multi-block- and overset grid-based in-house NS
solver were further conducted through an extensive study of unsteady flows
past a single, rowing-feathering fin with a single BFC (body-fitted
coordinate) grid (81x31x31) and with a two-block grid consisting
of a single grid (case1: 51x31x25 and case2: 31x25x13)
fitted to the fin and a cubic background grid (81x31x31), at a
Reynolds number Re of 1.597x104 and the reduced
frequency K of 3.0 (Liu and Kato,
2004 see Fig. 5).
The computed results with the two-block grid match very well those of the
single-grid, and the computed time course of three force coefficients.
Cx, Cy and Cz,
show reasonable agreement with the measurements
(Liu and Kato, 2004) (see
Fig. 6).
Verification of the present integrated, computational system is performed
with a specific focus on its self-consistency in terms of grid refinement and
time step effect. A grid sensitivity analysis demonstrates that the wing grid
(33x35x19) and the body grid (33x35x35) achieved
reasonably accurate solutions for a hovering fruit fly. As illustrated in
Fig. A1A, a maximum difference
in lift and drag coefficients is obtained within 5% among this set of grids
and a set of fine grids (wing grid: 45x45x31, body grid:
45x43x95) and finest grids (wing grid: 45x45x31, body
grid: 57x55x121). The effect of time increment on the force
generation is further investigated using two time steps of 0.005 and 0.0025,
and the computed results show almost no difference between the two cases;
hence a physical time step of 0.005 is used throughout the simulations
(Fig. A1B). An extended study
of validation of the hovering fruit fly is further discussed in the Results by
comparing the vertical (lift) and horizontal (drag and thrust) forces as well
as the powers with the experimental data
(Fry et al., 2005).
|
List of symbols and abbreviations
f/2Uref
vI
cn, sn, cn, sn, cn, sn
w
Acknowledgments
We thank the anonymous referees for their valuable comments and suggestions on the manuscript. This work was partially supported by a PRESTO (Precursory Research for Embryonic Science and Technology) program of the Japan Science and Technology Agency (JST), and Grant-in-Aid for Scientific Research Nos. 18656056 and 18100002, Japan Society for the Promotion of Science (JSPS). The simulations were performed in a supercomputer [Super Combined Cluster (RSCC)], RIKEN (The Institute of Physical and Chemical Research), Japan.
Footnotes
Supplementary material available online at http://jeb.biologists.org/cgi/content/full/211/2/239/DC1
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