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The control of flight force by a flapping wing: lift and drag production
Department of Integrative Biology, University of California, Berkeley, CA 94720, USA
*e-mail: sane{at}socrates.berkeley.edu
Accepted May 18, 2001
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Summary |
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(i) For a short, symmetrical wing flip, mean lift was optimized by a stroke amplitude of 180° and an angle of attack of 50°. At all stroke amplitudes, mean drag increased monotonically with increasing angle of attack. Translational quasi-steady predictions better matched the measured values at high stroke amplitude than at low stroke amplitude. This discrepancy was due to the increasing importance of rotational mechanisms in kinematic patterns with low stroke amplitude.
(ii) For a 180° stroke amplitude and a 45° angle of attack, lift was maximized by short-duration flips occurring just slightly in advance of stroke reversal. Symmetrical rotations produced similarly high performance. Wing rotation that occurred after stroke reversal, however, produced very low mean lift.
(iii) The production of aerodynamic forces was sensitive to changes in the magnitude of the wing’s deviation from the mean stroke plane (stroke deviation) as well as to the actual shape of the wing tip trajectory. However, in all examples, stroke deviation lowered aerodynamic performance relative to the no deviation case. This attenuation was due, in part, to a trade-off between lift and a radially directed component of total aerodynamic force. Thus, while we found no evidence that stroke deviation can augment lift, it nevertheless may be used to modulate forces on the two wings. Thus, insects might use such changes in wing kinematics during steering maneuvers to generate appropriate force moments.
(iv) While quasi-steady estimates failed to capture the time course of measured lift for nearly all kinematic patterns, they did predict with reasonable accuracy stroke-averaged values for the mean lift coefficient. However, quasi-steady estimates grossly underestimated the magnitude of the mean drag coefficient under all conditions. This discrepancy was due to the contribution of rotational effects that steady-state estimates do not capture. This result suggests that many prior estimates of mechanical power based on wing kinematics may have been grossly underestimated.
Key words: flapping flight, quasi-steady force, unsteady aerodynamics, fruit fly, Drosophila melanogaster, added mass, delayed stall, rotational circulation, wake capture.
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In a few cases, researchers have attempted to capture the free-flight kinematics of maneuvering insects (Ennos, 1989; Ruppell, 1989). While such analyses are essential because they reveal what insects actually do with their wings when steering, free-flight studies are limited because it is not yet feasible to relate the changes in wing kinematics directly to changes in instantaneous aerodynamic forces. An alternative approach is to measure instantaneous forces on tethered insects (Cloupeau et al., 1979; Wilkin, 1990; Zanker, 1990b; Zanker and Götz, 1990; Dickinson and Götz, 1996). However, forces and stroke kinematics measured on tethered insects may not accurately represent those generated in free flight. Further, since tethered flight force transducers measure whole-body forces, it is not possible to resolve the instantaneous aerodynamic forces generated by individual wings. A third approach is to calculate the aerodynamic forces generated by arbitrary stroke kinematics using computational fluid dynamics (Liu et al., 1998; Wang, 2000). However, because of the critical role of unsteady mechanisms and three-dimensional flow structure in insect flight aerodynamics (Ellington et al., 1996; Dickinson et al., 1999), theoretical or numerical approaches have, as yet, offered only limited insight into the aerodynamics of steering.
Given the current limitations in studies of both real animals and numerical simulations, we have chosen to study the problem of maneuverability using a dynamically scaled model of a flapping insect. Aerodynamic models have proved valuable in the study of insect flight, particularly in the identification and analysis of unsteady aerodynamics (Bennett, 1977; Maxworthy, 1979; Spedding and Maxworthy, 1986; Dickinson and Götz, 1993; Ellington et al., 1996; Dickinson et al., 1999). In large part through the use of mechanical models, researchers have identified an array of mechanisms that collectively account for the elevated aerodynamic performance of flapping wings. These include the clap and fling (Spedding and Maxworthy, 1986), dynamic stall (Dickinson and Götz, 1993; Ellington et al., 1996), rotational lift (Bennett, 1970; Dickinson et al., 1999) and wake capture (Dickinson, 1994; Dickinson et al., 1999). Now that the various mechanisms responsible for the elevated aerodynamic performance of insect wings have been identified, it is possible to tackle the question of how animals manipulate such mechanisms to steer and maneuver.
In this study, we use a dynamically scaled mechanical model of Drosophila melanogaster to investigate how changes in wing kinematics affect the production of aerodynamic forces. In particular, we explore the influence of five behaviorally relevant kinematic parameters: stroke amplitude, angle of attack, the timing and duration of wing rotation and stroke plane deviation. We chose this particular set of parameters because fruit flies actively vary them during flight maneuvers (Götz et al., 1979; Zanker, 1990a; Dickinson et al., 1993; Lehmann and Dickinson, 1998). However, the goal of this project is not to replicate the precise kinematics of free flying insects per se, but rather to map aerodynamic forces within a broad parameter space that encompasses the variation seen among insects.
From the instantaneous force records, we calculate time-averaged aerodynamic force coefficients, lift-to-drag ratios and other measures of aerodynamic performance. The resultant data set is useful in identifying the kinematic parameters that most influence the magnitude and direction of aerodynamic forces generated by flapping wings. In a companion paper (S. P. Sane and M. H. Dickinson, in preparation), we will extend the analysis by considering the instantaneous and time-averaged force moments generated about the yaw, pitch and roll axes. The comprehensive parameter maps generated in these studies should be of help to biologists who wish to know the aerodynamic consequences of observed changes in wing kinematics as well as to engineers who wish to optimize the performance of small biomimetic flying robots. In addition, these data provide experimental validations for numerical simulations of the fluid motion around flapping wings.
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strain gauges wired in full-bridge configuration. This design rendered the sensor nearly insensitive to the position of the force load on the wing as well as to moments around its central axis. Forces generated by calibration weights placed at the tip, base, trailing edge and leading edge differed by less than 5%. The final calibration was based on a point load at the wing’s center of area. The proximal end of the force transducer was attached to a gearbox capable of three degrees of rotational motion (Fig.1A). The distal tip of the wing was located 25cm from the center of the gearbox. The gearbox was driven via three coaxial shafts by three stepper motors. The stepper motors were attached to the shafts by pulleys and timing belts with a 1:10 gear reduction, such that each 4.5° step of the motor produced a 0.45° rotation of the wing. The wings, force sensor and gearbox were immersed in a tank of mineral oil with a viscosity of 120cSt at room temperature (approximately 25°C). The viscosity of the oil was chosen to achieve a Reynolds number in the range of 102, although the exact value varies according to the kinematics for each trial. Since the forces on the wing are directly proportional to the density of the surrounding medium, the oil also serves to increase forces on the wings and to decrease the signal-to-noise ratio of the force measurements. Mineral oil provides an additional advantage of electrically and thermally isolating the sensor and thus reducing noise fluctuations.
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-1,
where 
is stroke amplitude, n is wingbeat frequency,
R is wing length, 
is kinematic viscosity, aspect ratio
is
and
S is the surface area of a wing pair) (Ellington, 1984c) and the reduced
frequency parameter (body velocity/wing velocity) constant (Spedding, 1993). For
hovering animals as well as the model fly, the reduced frequency
parameter is zero by definition, since their body velocity is
zero. The free-flight cruising velocity of D. melanogaster is
approximately 20cms-1 (David, 1978), while the mean velocity
of the wing tip is 280cms-1 (Lehmann and Dickinson,
1997). Thus, even while flying forwards, the reduced
frequency parameter is less than 0.1, indicating that the effect of
free-stream velocity on force generation should be of secondary
importance to the velocity generated by flapping. For these reasons,
our experiments in still fluid, which match the hovering case, should
also serve as a fair approximation for moderate forward speeds. Thus,
the reduced frequency parameter is not significantly different for
cruising D. melanogaster and our static, hovering model
fly. To obtain the correct range of Reynolds numbers, we used an
isometrically enlarged wing planform of an actual
D. melanogaster wing to ensure that the shape parameters
(Ellington,
1984a) were identical to those of
D. melanogaster. Using available data on
D. melanogaster morphology and kinematics (Lehmann and Dickinson,
1997), we estimated that a wing length of 0.25m, a surface
area of the wing pair of 0.0334m2 (or 0.0167m2
for a single wing), a kinematic viscosity of 120cSt and a wingbeat
frequency of 0.168Hz would allow us to achieve Reynolds numbers in the
same range as those of D. melanogaster.
Stroke kinematics
In the absence of wing deformation, the kinematics of the wings may be uniquely described by specifying the time course of three angles: stroke position, 
(t), angle of attack, 
(t), and stroke deviation, 
(t) (Fig.1B). In all experiments, the angular position of the wing within the stroke plane was described by a triangular waveform, which maintains a constant translational velocity throughout each half-stroke. The waveform was smoothed to minimize inertial accelerations during stroke reversal and to match more closely published stroke kinematics from a variety of insects (Ellington, 1984b; Zanker, 1990a). For smoothing, we filtered the triangular waveform using a zero-phase-delay low-pass two-pole Butterworth filter with a cut-off frequency equal to 10 times the stroke frequency of 0.17Hz. The peak-to-peak amplitude of the stroke angle waveform could be varied in each experiment. The angle of attack was described by a trapezoidal wave function, which maintained a constant angle of attack during each half-stroke and constant rotational velocity during stroke reversal. The shape of this waveform in each experiment was determined by setting the mid-stroke angles of attack during the upstroke and downstroke and by specifying the starting and stopping points for wing rotation. The resulting function was then smoothed using a low-pass filter with identical characteristics to that used for the stroke position waveform. We used two functions to describe stroke deviation: an ‘oval’ pattern in which the wing tip deviated from the stroke plane according to a half-sine-wave per stroke period and a ‘figure-of-eight’ pattern in which the stroke deviation varied as a full sine-wave. These patterns were chosen because they roughly approximate patterns described for a variety of insects (Ellington, 1984b; Zanker, 1990a).
To create the kinematic patterns used in this study, we varied any or all of six parameters: (i) the stroke amplitude, (ii) the mid-stroke angle of attack during upstroke and downstroke, (iii) the timing of wing rotation at dorsal and ventral reversal, (iv) the duration of the stroke reversal, (v) the shape of the wing tip trajectory (‘oval’ or ‘figure-of-eight’) and (vi) the angular deviation from the mean stroke plane during the upstroke and downstroke (Fig.1C). In most of the experiments, the deviation amplitude was set to zero, such that the wing tip remained within the stroke plane throughout the cycle. Under these conditions, the kinematics of the wing stroke were symmetrical such that the upstroke and downstroke were mirror images of one another. Only in trials using ‘oval’ stroke deviations were the kinematics of the two strokes not identical. The frequency of the wing stroke (0.17Hz) remained constant in all experiments, as did the upstroke-to-downstroke duration ratio, which was fixed at 1. We constructed the kinematic patterns using a custom-designed MATLAB program (Mathworks) to convert the angular trajectories into a series of stepper motor commands.
Force measurements
Signals from the two-dimensional sensor were acquired using a National Instruments data-acquisition board (model BNC 2090) in a PC running custom-designed software written in MATLAB. Data were filtered on-line with an active four-pole Bessel filter with a cut-off frequency of 10Hz and off-line with a zero-phase-delay low-pass digital Butterworth filter with a cut-off frequency of 3Hz, which was 17.6 times the wing stroke frequency. Apart from increasing the high-frequency components resulting from motor jitter, increasing the cut-off frequency of the filter did not alter the time course of the force traces.
Each experiment consisted of one burst of four consecutive wing strokes following pre-programmed kinematics. The wing begins the first downstroke in still fluid, whereas during the subsequent strokes it moves through a wake created by the preceding strokes. As a result, the time course of forces generated during the first stroke is markedly different from those of subsequent strokes. For this reason, the data from the first stroke were excluded from this analysis, while those from the three subsequent strokes were averaged. Thus, each presented trace represents an average of three force records. After subtracting gravitational forces, the forces measured from the normal and parallel channels were transformed into lift, drag, thrust and radial components.
Added mass
The measured force at the wing base consists of gravitational, inertial and aerodynamic components. The gravitational contribution of the sensor and wing mass to the total force signal was easily calculated and subtracted from the measured force traces. The inertial components represent the acceleration forces on the mass of the sensor and wing as well as the added mass of the fluid around the wing. To examine the contribution of the inertial effects of the wing mass and sensor, we replaced the wing surface with an aerodynamically neutral inertial model of the wing. The aerodynamically neutral model was essentially a brass knob with the same mass and center of mass inside the oil as the Plexiglas wing. Because of its low surface area, the brass knob generated negligible aerodynamic forces compared with the Plexiglas wing. For any arbitrary kinematic pattern, the resulting force traces for the brass knob could be entirely accounted for by gravity. Thus, the inertial forces generated by flapping this brass model, and therefore the Plexiglas wing, were negligible and have been ignored. Compared with gravity and wing inertia, the non-circulatory forces due to added mass are more difficult to measure because the fluid acceleration induced by a moving wing changes dynamically as the wing rotates, decelerates or accelerates (Daniel, 1984).
To estimate the magnitude of added mass, we used an approximation derived for motions of an infinitesimally thin two-dimensional plate in an inviscid fluid (Sedov, 1965). Using blade element method, we adapted it to the case of a three-dimensional wing rotating around an axis located at one-quarter chord length from the leading edge. The force contribution normal to the wing surface due to the added mass inertia is given by:
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where 
is the fluid density, R is the wing length,
is the mean chord length,
and
() are the non-dimensional
radial position along the wing and non-dimensional chord length,
respectively (for nomenclature, see Ellington, 1984a), is the
angular position of the wing and is the angle of attack. Using
equation 1, we calculated an estimate of added mass inertia for each
set of kinematics. As illustrated by the representative trace in
Fig.2, the absolute
contribution of added mass to net forces on the wing was quite small
in all cases. Further, by comparing equation 1 and equation 3 (see
below), it can be seen that the added mass forces
(
R2
2) scale in proportion to aerodynamic forces
(R3
) for geometrically similar wings. Thus, for identical kinematics and
geometry, added mass will have the same physical effect on a model
wing as on the wing of a fly, provided that the Reynolds number is the
same. Because both added mass and aerodynamic contributions are
biologically relevant, we chose not to subtract the inertial estimates
from the presented force traces.
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f, which describes when the mid-point of a flip occurs within the stroke, may be calculated from:
 |
where 0 is flip start and 
is flip duration. As before, we determined the values of flip timing and flip duration that maximized mean lift within the 11x9 sets of trials. In the third set of experiments, we explored how aerodynamic forces varied with changes in deviation from the mean stroke plane using the values of stroke amplitude, angle of attack, flip timing and flip duration that maximize mean lift. In a set of 22 experiments, the total stroke deviation was varied from -50° (-25° during the downstroke, +25° during the upstroke) to +50° (+25° during the downstroke, -25° during the upstroke) in steps of 10°. The deviation followed either a half-sine per stroke or a full-sine per stroke time course, yielding ‘oval’ and ‘figure-of-eight’ trajectories, respectively (Fig.1B).
Data analysis
Since the conventions for lift and drag existing in current aerodynamic literature address non-flapping and primarily two-dimensional kinematic patterns, it is necessary to modify them slightly for three-dimensional motions. These modifications are based on the following two considerations. First, it is convenient to use a reference frame based on the insect body rather than its wing so that the measurements relate directly to free-flight studies. Second, the standard convention should apply when the kinematics reduce to two-dimensional motion. With these constraints in mind, we adopted the following convention: net aerodynamic force, defined as the total force on the wing, is resolved into three components: lift, drag and radial force. In hovering flight, lift must offset the gravitational force on the animal’s body mass. Hence, we define lift as the component of the net aerodynamic force perpendicular to the mean stroke plane of the wing regardless of its actual instantaneous trajectory. Since the mean stroke plane was horizontal in all experiments, lift is always the vertical force component. Drag is defined as the force component in the horizontal direction, opposing the wing movement. The radial component accounts for the remaining force component in the horizontal plane. For motions with no stroke deviation, these definitions reduce to the standard convention: lift is orthogonal to drag, and the radial component vanishes. With stroke deviation, the total normal pressure force consists of orthogonal vertical and radial components, each orthogonal to the drag vector in the horizontal direction.
From the forces on each wing, we calculated the corresponding mean force coefficients using an equation derived from blade element theory (Ellington, 1984c; Dickinson et al., 1999):
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where 
is the magnitude of a specific force
component (lift, drag, radial, total) averaged over the stroke,
is stroke amplitude, n is wing beat frequency,
is the
mean non-dimensional angular velocity of the wing and
22(S)
is the non-dimensional second moment of wing area. The radial force
component changes sign when the wing crosses the mean stroke plane. As
a result, in the ‘oval’ as well as the
‘figure-of-eight’ patterns, the mean radial coefficient
often averages to zero and is uninformative. For this reason, we base
our measurement of the average radial force coefficient on the
absolute values. Lift-to-drag ratios were calculated by dividing the
mean lift coefficient by the mean drag coefficient. Similarly,
radial-to-drag ratios were obtained by dividing the mean absolute
radial coefficients by the mean drag coefficients. To calculate the
ratio of mean lift to profile power, we estimated mean profile power,
, based on the
time-averaged product of instantaneous drag, D(t),
and instantaneous velocity, vwing(t):
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where T is the stroke period. This calculation of mean
profile power ignores the power required to rotate the wing in
place. Mean lift 
was calculated as the time
average of instantaneous lift throughout the stroke.
Measures of the quasi-steady-state translational force
coefficients 
and 
were derived from 180° sweeps of wing motion with fixed angles of
attack, as described elsewhere (Dickinson et al., 1999). The
equations that best fit measured translational force coefficients as
functions of angle of attack, , for the model wing are
(Dickinson et al.,
1999):
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and
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These equations are used to generate quasi-steady translational estimates for comparison with measured values.
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The influence of rotational circulation is most easily seen when the angle of attack during the upstroke and downstroke is zero (Fig.3G,H). Under these conditions, the vorticity generated during the translational phase of the stroke is minimal and, thus, the magnitude of the wake capture effect should be small. The influence of the wake is not entirely absent, however, because the process of rotation generates and sheds vorticity through which the wing must translate at the start of each stroke. Further, the vorticity created by rotation is particularly strong when the translational angle of attack is zero, because the wing must flip over by 180° during stroke reversal, making the angular velocity of the wing particularly large. At the end of the each stroke in Fig.3G,H, the influence of rotational circulation is manifest as a transient increase in lift and drag that exceeds the quasi-steady prediction. After stroke reversal, the continuing rotation of the wing generates a pressure force with opposite polarity, resulting in negative lift. The time course of this rotational effect is complicated by the presence of an added mass inertia and a modest amount of wake capture at the start of each stroke. The influence of these multiple mechanisms is manifest by the positive peak in lift immediately following stroke reversal due to wake capture, which is followed by the negative peak due to rotational circulation. The pattern of an early positive peak in lift followed by a later negative peak is seen throughout the traces in Fig.3.
The rest of the traces in Fig.3 illustrate the complex interactions among delayed stall, rotational circulation and wake capture that result from changes in angle of attack and stroke amplitude. At angles of attack of 30 and 50°, the wing generates lift throughout the stroke due to delayed stall (Fig.3C–F). The influence of rotational lift is reduced as the angle of attack increases, however, because the wing flips over a smaller arc with lower angular velocity. This effect can be seen by comparing the relative magnitude of the force peaks at the end of each stroke in Fig.3B,D,F,H. In contrast, the influence of wake capture is greater at higher angles of attack because the vorticity shed into the wake at the end of the stroke is stronger. This effect can be seen by comparing the relative size of the force transients at the start of each stroke in the same panels. Thus, changing stroke amplitude and angle of attack has a complex but interpretable influence on the magnitude of the different unsteady mechanisms. The kinematics that optimize the aerodynamic performance of the wing will reflect these complex interactions. The maximum mean lift-to-drag ratio (0.8) occurred at an angle of attack of 30° and amplitude of 180°. The forces corresponding to this optimal condition are shown in Fig.3F.
To provide a more comprehensive picture of how force production
changes with kinematics, we calculated the mean lift, drag and net
force coefficients averaged throughout the stroke and plotted them for
all pairs of stroke amplitude and angle of attack values. Fig.4A,B depicts the mean total
aerodynamic force,
,
and force coefficient,
,
in pseudocolor maps. These two maps differ from one another because
the mean force coefficient is normalized with respect to the square of
stroke amplitude, which is a variable in these experiments. Thus,
while
increases with increasing stroke amplitude (Fig.4A),
decreases with increasing stroke amplitude (Fig.4B). Both parameters rise with increasing
angle of attack. The influence of stroke amplitude and angle of attack
on the mean lift coefficient
and the mean drag
coefficient
is shown in Fig.5A,C. For a fixed stroke amplitude,
exhibits a broad maximum ranging from 1.8 to 2.0 between angles of
attack of 40 and 50°. As expected,
rises monotonically with increasing angle of attack for any given
value of stroke amplitude. It is worth noting that the range of
values is much higher than has been previously reported for
Drosophila virilis wings under steady-state conditions
(Vogel, 1967) or
estimated on the basis of Reynolds number (CD0.7;
Ellington,
1984c). Although much less pronounced than the dependence
on angle of attack, lift tends to rise, whereas drag falls, with
increasing stroke amplitude.
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and
are shown in Fig.5B
and Fig.5D,
respectively. In general, the measured force coefficients are greater
than the predicted quasi-steady force coefficients. This discrepancy
is particularly large for
.
Further, at high stroke amplitude, the angle of attack that generates
the maximum
is shifted by approximately 10° relative to the quasi-steady
predictions and by as much as 20° for the lower stroke
amplitudes. The dependence of stroke amplitude on the measured lift
and drag coefficients is not predicted by the quasi-steady
estimates. The greater difference between measured and predicted
values for smaller stroke amplitudes underscores the increased
importance of rotational effects under these conditions. Fig.5E indicates how the mean
lift-to-drag ratio,
/
varies with angle of attack and stroke amplitude. The maximum value
of
/
(0.8)
occurred at stroke amplitude of 180° and an angle of attack
of 30°. The corresponding quasi-steady-state estimates of
/
are independent of stroke amplitude, with a maximum of 1.1 at an angle
of attack of 20°. Thus, the quasi-steady model significantly
overestimates aerodynamic efficiency and fails to account for its
dependence on stroke amplitude.
The effects of flip timing and flip duration
In all subsequent experiments, we set the stroke amplitude to
180° and the mid-stroke angle of attack to 45°, which are the
values that maximized lift production in the first set of
experiments. Next, we systematically varied the timing and duration of
wing rotation to examine their effects on force production. Sample
force traces selected from 99 pairs of flip start and flip duration
are shown in Fig.6A–H, with the corresponding values
of flip timing, f, given in the upper left corner of
each panel. A comparison of Fig.6A,C,D illustrates that a long-duration
flip, =0.5, can produce quite different forces
depending on when the flip occurs. If the flip occurs symmetrically
about the stroke reversal (f=0, Fig.6A),
is quite large
(=1.54)
and the time course is well approximated by the quasi-steady
predictions. An advanced flip (f=-0.25,
Fig.6C) results in very
low mean lift
(=0.36),
but produces a fairly prominent wake capture peak. In contrast, a
rotation after stroke reversal (f=+0.25,
Fig.6D) results in mean
negative lift
(=-0.28)
because of the adverse effects of rotational circulation following
stroke reversal.
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=0.1) at different
flip times. As flip duration decreases, the aerodynamic performance of
the wing generally rises. Symmetrical and advanced flips yield nearly
identical mean lift
(=1.9,
symmetrical;
=1.87,
advanced), whereas a delayed flip generates somewhat lower lift
(=1.52).
These differences are due primarily to the amount of lift produced at
the start of each stroke. Early and symmetrical flips (Fig.6F,G) result in a substantial
wake capture peak at the start of translation. However, if the flip
occurs very early in the stroke (f=-0.45,
Fig.6E), the wing
translates through most of the stroke at negative angles of attack,
leading to a large value of negative lift
( =-1.41;
Fig.6E). When rotation
is delayed, the wake capture peak is missing, revealing two negative
peaks at the start of translation, an early small peak due to added
mass inertia and a later more prominent peak due to rotational
circulation (Fig.6H).
The maps of mean force coefficients as a function of flip duration and flip start are shown in Fig.7A,C,E, with comparable quasi-steady, translational predictions shown in Fig.7B,D,F. Flip timing, the non-dimensional time when the mid-point of the flip occurs, is indicated by the inclined parallel lines on each graph. Both
and
are strongly influenced by flip timing and duration. For example, at a flip duration of 0.1,
varies with flip timing from as low as -1.5 to as high as +2. The comparable values of the quasi-steady translational estimate,
, also vary, but over a smaller range (from -1 to 1.6).
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increases monotonically with flip duration (Fig.7C). Measured values range from 2.6 and 4.1, representing a somewhat smaller variation than was seen with
.
However, unlike the case with lift, the discrepancy between measured
and the quasi-steady estimate,
,
is substantial. In addition to generally underestimating the magnitude of drag, the quasi-steady predictions fail to observe the local rise in drag along the -0.25 flip timing iso-line. Since the range of variation for drag is less than that for lift, the measured lift-to-drag ratio map resembles the lift coefficient map (Fig.7A,C,E). Further, because the quasi-steady translational predictions underestimate lift and drag by approximately the same proportion, the predicted lift-to-drag ratio map is quite similar to the measured map (Fig.7E,F). The map for the net force coefficient (Fig.7G) resembles the drag map (Fig.7C), which is expected since the values for the mean drag coefficient are, on average, twice those for the lift coefficient at comparable points on the kinematic surface.
The effects of stroke plane deviation
Using kinematic values for stroke amplitude, angle of attack, flip duration and flip start that maximized lift production (=180°, =45°, =0.1 and f=-0.05), we tested how forces vary with deviation from the mean stroke plane in a set of 22 experiments. The peak-to-peak magnitude of stroke deviation was varied from -50 to +50° in 10° steps for both the half-sine (‘oval’) and full-sine (‘figure-of-eight’) patterns. It is worth noting that, in the oval pattern, an upward deviation at the start of the downstroke requires a downward deviation at the start of the upstroke and vice versa. This is not the case for the figure-of-eight pattern, in which the two half-strokes are mirror images of one another.
Fig.8 shows a selection of force traces resulting from different patterns of stroke deviation. In general, the figure-of-eight pattern had a more profound influence on the magnitude and time course of force production than did the oval pattern. In both cases, however, the direction of stroke deviation at the start of each translational phase greatly influenced the magnitude of the force transient at the start of the stroke. For example, in the figure-of-eight pattern shown in Fig.8C, each stroke begins with an upward motion, and the lift and drag transients at the start of each stroke are quite small. In contrast, the comparable kinematic pattern that starts with a downward motion (Fig.8D) generates sizeable force peaks. A similar trend is seen in the oval patterns (Fig.8A,B). The upstroke in Fig.8A and the downstroke in Fig.8B, which both start with a downward motion, are marked by sizeable force peaks at the start of translation, whereas the strokes that begin with upward motion are not. This dependence of the early force transient on the direction of deviation is explained in part by an increase in the aerodynamic angle of attack caused by the downward motion of the wing. However, the measured force peaks are much greater than the quasi-steady estimates, which take into account this effect, suggesting that there is a substantial wake effect at the start of each stroke. The influence of the wake is stronger if the wing moves downwards, towards the descending vorticity of the previous stroke, than if it moves upwards, away from the wake. In all cases, the significance of radial forces increases with increasing deviation. As expected, like the lift and drag forces, the time course of radial forces is also dependent on the shape of the wing trajectory.
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Fig.9 summarizes the effects of stroke deviation on the time-averaged measured forces and quasi-steady predictions. While the influence of stroke deviation on the time course of the aerodynamic forces is large, its impact on the mean force coefficients is surprisingly small. This indicates that the differences in the dynamics of force production noted in Fig.8 tend to average out over the stroke. For both the ‘oval’ and ‘figure-of-eight’ deviation trajectories (Fig.9A), the mean lift and drag coefficients decreased with increasing absolute deviation (Fig.9C,D). The changes in average performance for the ‘oval’ deviation pattern should be symmetrical around zero deviation, since oval patterns with positive and negative deviations are mirror images of one another. Thus, the downstroke in Fig.8A should resemble the upstroke in Fig.8B, and the upstroke in Fig.8A should resemble the downstroke in Fig.8B. The asymmetry in these measurements results from the mechanical play in the gear mechanism of the robot. However, the asymmetry in the performance of the ‘figure-of-eight’ patterns around zero deviation represents, in part, a real aerodynamic effect (Fig.9B,C,D). In this case, a positive deviation will result in a downward motion at the start of both the upstroke and downstroke, whereas a negative deviation indicates upward motion at the beginning of both strokes. Downward deviation should enhance wake capture, as described above. Values of
and
fall off faster with increasing positive deviation close to zero deviation. However, at large deviations, the coefficients for negative deviations are lower than the coefficients for positive deviations (Fig.9C,D). There is little effect on the
/
ratio because the influence of stroke deviation is nearly identical for both lift and drag. In contrast, because of the linear nature of sine functions at low angles,
/
ratios appear linear with the small range of stroke deviation in our experiments (Fig.9E).
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and
are higher than quasi-steady translational estimates due to unsteady
effects (Fig.9C,D). Lift and drag are almost equally
underestimated, which explains why the predicted lift-to-drag ratio
(Fig.9E) is only
slightly higher than the measured ratio. For oval trajectories, the
quasi-steady predictions for the mean force coefficients behave as
scaled-down versions of the measured traces. For example, both
quasi-steady and measured mean force coefficients are maximal at zero
stroke deviation and decrease for increasing absolute deviation
(Fig.9C,D). However,
for figure-of-eight trajectories, the quasi-steady translational model
is much less accurate in predicting the changes in the mean force
coefficients with stroke deviation. For example, the quasi-steady
model predicts that mean lift should exhibit a local maximum at a
stroke deviation of +20°, whereas the measured maximum occurs
at 0° deviation. Similarly, the estimated drag increases
monotonically with increasing positive deviation, whereas the measured
drag is maximal at 0° deviation.
Ratio of mean lift to mean profile power
Within the range of Reynolds numbers relevant for fruit flies, the
total mechanical power required to flap the wings is dominated by
profile power, , the cost to
overcome drag on the flapping wings (Lehmann and Dickinson,
1997). Fig. 10A–C shows how
, estimated using equation 6,
varies with changes in the kinematics parameters. Fig.10D–F shows the
corresponding ratios of mean lift to mean profile power for the same
kinematics. The estimates vary
extensively, even within subregions of the parameter maps in which the
values of
are high enough to support flight. This result suggests that it may
be difficult to estimate mechanical power in free or tethered flight
solely on the basis of measures of stroke amplitude. In particular,
varies extensively with the timing
of wing rotation and the angle of attack, parameters that are revealed
only by extensive three-dimensional reconstructions of wing
kinematics.
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Discussion |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionList of symbolsReferences |
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Comparison between the mechanical model and real flies
The maximum mean unsteady lift coefficients in this study are in
excellent agreement with measurements on tethered Drosophila
spp., which generated elevated flight force in response to optomotor
stimuli. Peak
values in tethered flight were 1.9 (Lehmann and Dickinson,
1998), which is precisely the same maximum value measured
on the model wing. Tethered flies generating just enough force to
support body weight produce a
value of 1.6. The tethered flight estimates relied on the assumption
that both up- and downstrokes generated lift, which appears to be
correct given the time course of forces generated when
Drosophila spp. kinematics are played through the model
(Dickinson et al.,
1999). Tethered Drosophila spp. flap with
morphologically maximum stroke amplitude when producing peak lift
(Lehmann and
Dickinson, 1998), which is consistent with the present
results. Unfortunately, apart from stroke amplitude, we do not have
adequate knowledge of other kinematic parameters during the peak
performance of real flies to compare them with the values that
maximized lift on the model.
As seen in Fig.5B, the maximum quasi-steady translational estimate of the mean lift coefficient,
was 1.6. Given that this value is sufficient to explain the forces required for a fruit fly to hover (Lehmann and Dickinson, 1998), it is tempting to claim that the quasi-steady estimates are sufficient to explain insect flight (Jensen, 1956). However, a simple comparison of time-averaged lift coefficients is not a robust test of the quasi-steady model. While the mean values might be similar, the time histories of measured forces and quasi-steady translational estimates differ greatly (Fig.3, Fig.6, Fig.8). Further, the precise time history of lift and drag is critical for calculating force moments and, thus, essential to considerations of stability and maneuverability. In addition, since many insects can fly while supporting forces nearly twice their body weight (Marden, 1987), any model must explain not only the forces required to hover, but also those required for maximal lift.
The inadequacies of the translational quasi-steady model are even more apparent when considering drag. In all experiments, we observed that the measured drag coefficients were greater in magnitude than those estimated from measured translational force coefficients. Also, the range of drag coefficients measured here (CD=0–6.5) is substantially higher than previously reported measurements from both real and model Drosophila virilis wings (CD=0.2–1; Vogel, 1967). The previous experimental values were based on steady-state measurements and thus excluded the contribution of rotational circulation and wake capture as well as added mass. However, even the range of steady-state coefficients in the previous study (0<CL<1, 0<CD< 1) (Vogel, 1967) is substantially lower than the quasi-steady estimates in this study (0<CL,t<1.91; 0.37<CD,t<3.47). At present, the reason for this discrepancy is not clear.
The relative importance of unsteady mechanisms
The relative contributions of delayed stall, rotational circulation and wake capture to total force production vary with the precise kinematics of the stroke. In general, the importance of delayed stall increases with stroke amplitude because the wing can integrate the influence of the leading edge vortex over a greater distance. In contrast, rotational effects become more important with lower stroke amplitude. For kinematic patterns mimicking those of hoverflies with a 60° stroke amplitude, rotational effects account for more than half the total force (Dickinson et al., 1999). Because of these differences, quasi-steady translational estimates should more closely resemble the measured values for kinematic patterns with large stroke amplitudes and deviate from measured values for those with lower stroke amplitudes. These predictions are borne out by the force traces shown in Fig.3 and the maps of
in Fig.5A,B. The quasi-steady translational model predicts maximum lift at an angle of attack of 45°. While this is close to the maximum value for the measured forces at high stroke amplitude, the measured lift maximum is shifted to higher angles of attack (60°) at low stroke amplitude.
There are two possible explanations for this shift. First, rotational circulation might make a greater contribution to mean lift at low stroke amplitude. However, this is unlikely to be true for symmetrical flips, for which rotational circulation enhances lift only before stroke reversal, but attenuates it after stroke reversal (Dickinson et al., 1999). Also, at stroke reversal, the wing rotates less for a 60° angle of attack than it does for a 45° angle of attack. Thus, the proportional contribution due to rotational lift is further minimized. Second, the shift in the lift maximum may reflect an increasing importance of wake capture at low stroke amplitude. The large contribution of wake capture can be easily seen in the forces generated by hoverfly-like patterns (Dickinson et al., 1999) as well as in the drag traces in Fig.3. The increasing importance of wake capture also explains the changes in drag coefficients at low stroke amplitude. While the translational quasi-steady model predicts that drag should be independent of amplitude (Fig.5D), the measured mean drag coefficient clearly increases at smaller stroke amplitudes.
Separating the effects of wing rotation from wake capture
Since rotational circulation, wake capture and added mass usually occur together during stroke reversal, it is often difficult to separate these effects. After estimating the effect of added mass on force traces before and after subtraction (Fig.2), we concluded that, at these Reynolds numbers, the magnitude of added mass is small compared with rotational circulation or wake capture. To isolate these two mechanisms, it is helpful to focus on kinematic patterns in which either the entire flip occurs prior to stroke reversal (Fig.6B,E,F) or the wing does not flip at all (Fig.3A,B). In the case of an advanced flip, the force peak that exceeds the translational estimate prior to stroke reversal may be attributed to rotational circulation, while the force peak after stroke reversal is due to wake capture. In the case of no flip, the large drag peak at the start of each stroke is due to wake capture, in which vorticity shed from the previous stroke elevates force by inducing an increase in flow velocity towards the wing (Dickinson, 1994; Dickinson et al., 1999). Because of the squared dependence of forces on relative velocity, even small changes in flow can cause a large elevation in force.
The influence of wake capture should be reduced when the wing translates at a 0° angle of attack during the previous stroke. Under these circumstances, the wing sheds less
, mid-stroke angle of attack, 
