Aquatic wing flapping at low Reynolds numbers: swimming kinematics of the Antarctic pteropod, Clione antarctica
1 Department of Integrative Biology, University of California, Berkeley, CA
94720, USA
2 Smithsonian Tropical Research Institute, P.O. Box 2072, Balboa, Republic
of Panama
* Author for correspondence (e-mail: bborrell{at}berkeley.edu)
Accepted 6 June 2005
![]() |
Summary |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Key words: Clione antarctica, drag, flapping flight, kinematics, lift, pteropod, swimming
![]() |
Introduction |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
A general problem for lift generation by such small organisms operating at
lower Reynolds number (Re) is the rapid viscous dissipation of
kinetic energy imparted to the surrounding fluid
(Fuiman and Batty, 1997;
Podolsky and Emlet, 1993
;
Vogel, 1994
). Under such
circumstances, the efficiency of flapping locomotion declines because bound
circulation decreases and viscous drag increases, resulting in impractical
lift-to-drag ratios (Vogel,
1994
). Rowing locomotion will also be hindered at low Re
because fluid will fail to separate and translate away from oscillating
appendages at the end of the power stroke. Consequently, drag during the power
stroke may be only slightly greater than that during the recovery stroke
(Walker, 2002
). Furthermore,
as the relative importance of inertial forces declines at low Re,
reaction forces generated from unsteady motion of the rowing appendages will
also diminish (Daniel, 1984
;
McHenry et al., 2003
;
Williams, 1994b
).
In recognizing the effects of scale in biological fluid dynamics,
researchers have often partitioned the dynamic spectrum into an inertial
regime (Re>1000), a viscous regime (Re<10), and an
intermediate regime (1000>Re>10;
Daniel et al., 1992;
Fuiman and Batty, 1997
;
Webb and Weihs, 1986
). Fuiman
and Batty (1997
) manipulated
the viscosity of water to show that fish larvae follow hydrodynamic
predictions of the viscous regime up to Re=300, and also suggested
that viscous effects could be significant even at Re as high as 450.
Other studies of invertebrate larvae also supported the conclusion that
inertial forces do not play a major role in propulsion at Re<100
(McHenry et al., 2003
;
Williams,
1994a
,1994b
).
It is clear that we are coming closer to a mechanistic understanding of force
generation at low to intermediate Re, and thus it is germane to
explore stroke kinematics of organisms that may not fit established locomotor
paradigms.
Clione antarctica, a pteropod mollusc found in Antarctic waters,
is one of the smallest known aquatic flappers and uses its paired oscillating
appendages known as pteropodia (henceforth, wings) to ascend in the pelagic
water column. Morton (1954,
1958
) suggested that
thecosomatous (i.e. shelled) pteropods operating at about the same Re
as C. antarctica use their wings strictly as paddles, producing
thrust only once per stroke cycle, but also suggested that larger gymnosome
(i.e. shell-less) pteropods such as Clione limacina produce thrust
during both the upstroke and downstroke via a sculling motion. In a
detailed kinematic study of pteropod locomotion, Satterlie et al.
(1985
) concluded that C.
limacina produces thrust via lift forces and a clap-and-fling
mechanism. However, the smaller size and lower wingbeat frequency of C.
antarctica, when coupled with the higher viscosity associated with
near-freezing polar waters, suggests that in spite of morphological and
kinematic similarities with C. limacina, the underlying mechanisms of
thrust production may differ. Thus, we examined wing stroke parameters and
body kinematics of C. antarctica to develop a model for flapping
force production in these diminutive swimmers. Our results suggest that C.
antarctica modulates thrust in vertical ascent by altering the
orientation of the power phase of each half-stroke, and that thrust at high
velocities is generated primarily via steady-state drag, although
rotational lift, the acceleration reaction and wake recapture (see
Sane, 2003
) may also be
important.
![]() |
Materials and methods |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
We constructed a filming chamber by lining a 500 ml finger bowl with a sheet of black plastic rolled into a truncated cone. This setup funneled pteropods into the field of view of the camera and also provided flapping individuals with adequate space during their ascent to avoid both physical contact with the substratum and potential boundary effects. A high-speed video camera (Redlake Imaging Motionscope, San Diego, CA, USA) was positioned approximately 1.5 m above the filming chamber to provide a frontal view of ascending animals. We ensured that the line of sight of the camera was orthogonal to the water surface (and largely parallel with the body axis of swimming individuals) by use of a spirit-level mounted on the camera. To simultaneously record frontal and lateral views of wingbeat kinematics, a front-surface mirror (40 mmx40 mmx56.5 mm) was placed in the filming chamber at 45° relative to the water's surface. Because of corrosion during the course of the study, later trials were filmed using a commercial-grade back-surface mirror. The width of these mirrors provided the spatial calibration for both frontal and lateral views. The experimental chamber was filled with filtered seawater obtained from McMurdo Sound, and the entire chamber was placed inside a variable-temperature water bath (Frigomix 1495; Braun Biotech International, Goettingen, Germany) filled with 50% propylene glycol to prevent freezing. Water bath temperature was monitored using a copper-constantan thermocouple (Barnant 100; Barrington, IL, USA) with a resolution of 0.1°C, placed on one side of the chamber, and/or a platinum thermometer placed on the opposite side of the chamber. We periodically stirred the seawater within the filming chamber and averaged temperature readings from each thermocouple/thermometer when both were available. Experimental water temperatures ranged from 1.8°C to 2.0°C.
Individual pteropods were placed within the filming chamber and allowed to acclimate for at least 10 min prior to filming. They were otherwise held at 0°C during each trial run. Two fiber optic microscope lamps illuminated the filming chamber from above. Individuals ascended in the chamber of their own volition and, when doing so, were filmed at 125 frames s1. The relatively high framing rate of the video camera provided approximately 80 frames per wingbeat, corresponding to a high temporal resolution that was sufficient to determine the timing of kinematic events and to estimate the first and second derivatives of wing and body position.
Wingbeat geometry and body kinematics
To characterize representative kinematics, we selected ten ascent sequences
in which the wings of the animal were clearly visible and the transverse body
axis was orthogonal to the lateral filming view (i.e. <10° body roll
relative to the mirror's edge). Using Quickimage, a modification of NIH Image
distributed by J. Walker
(www.usm.edu/walker),
the wing base and wingtip were digitized in both the lateral and frontal
views. We measured wing length, R, as the maximum tipbase
distance exhibited during a cycle. We also calculated the area of a wing pair,
S, at midstroke by measuring the projected area in the top view and
dividing it by the cosine of the wing angle with respect to the horizontal
axis as measured simultaneously in the lateral view. This calculation of wing
pair area was carried out for both the upstroke and the downstroke of each
ascent sequence and was then averaged. From the wing angle at midstroke, we
calculated the angle of attack,
, as the angle between the wing surface
and ßps, the stroke plane during the power phase of a
half-stroke (see below for definition). The mean wing chord,
, is the wing area divided by the
span (=S/2R), and the aspect ratio (AR) equals
4R2/S. Stroke area is the area swept by the wing
during a half-stroke and is related to the total force generated by an
oscillating appendage.
Video sequences were then smoothed to reduce pixel noise and thresholded to
provide a binary image from which the pteropod's image could be manually
isolated from the background. The wings were then erased virtually from video
images, and a geometric center of area of the body was determined for both the
lateral and frontal views of the animal. Mean body velocity
() was expressed non-dimensionally as
the number of wing lengths travelled per wingbeat. The angle between the
horizontal x-axis and the mean path of ascent,
, was calculated
using the mean body velocity vector in the yz plane during the
course of an entire wingbeat (see below). A second ascent angle,
ps was calculated during the power phase only. Body angle
(
) was determined as the angle subtending the major axis of the animal
and the vertical z-axis (Fig.
1). For each ascent sequence, we calculated a mean body angle,
, and an angle of oscillation,
a, corresponding to the
difference between the maximum and minimum values of
during a single
wingbeat.
|
Finally, a spline function was fitted to the positional data derived from
wingtip coordinates, body center of area and body angle. Velocity and
acceleration estimates were then obtained using the program Quicksand
(Walker, 1998).
Time-standardized kinematics were produced from these functions by
interpolating the ensuing estimates to 100 equally spaced points. These
profiles were then aligned for eight ascent sequences based on the first
minimum in the x position of the wing. We defined
=0 as
the initiation of the power phase of the upstroke, when the wing reaches a
maximum in the z-axis. Averaging these profiles yielded mean
positional data, velocity, and acceleration estimates for these eight
sequences. Wingtip velocity in the transverse plane was calculated as the
vector sum of velocity components in the (x',
y') plane and an analogous calculation was made for foreaft
wingtip velocity in the sagittal plane (see
Fig. 2A).Because steady-state
lift and drag forces should be approximately proportional to the square of the
transverse and foreaft components of wingtip velocity, respectively,
body accelerations would be predicted to coincide temporally with maximum
wingtip velocities if unsteady mechanisms are unimportant. On the other hand,
maximum body accelerations at the ends of half-strokes would strongly
implicate such force-producing mechanisms as rotational circulation, wake
capture and the acceleration reaction (see
Sane, 2003
).
|
![]() |
Results |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
We calculated mean kinematic parameters from all ten sequences, but present wingtip positional data only for the eight sequences in which the digitized stroke was initiated on the ventral surface of the animal (i.e. the upstroke). Aspect ratio of the wings ranged from 1.66 to 4.47 (Table 1).
|
All data given subsequently are presented as the mean value ±1 S.E.M. (standard error of the mean).
Qualitative description of the wingbeat
A typical wingbeat of an ascending C. antarctica is presented in
Fig. 1, together with vectors
indicating instantaneous velocity and acceleration of the geometric center of
body area. The wingbeat exhibits distinctive propulsive and recovery phases in
each half-stroke, whereby ascent velocity during the each propulsive power
phase exceeds ascent velocity during the recovery phase. As wing rotation
begins at the end of the down- and upstrokes, wings are curved around the body
with the wingtips nearly touching or overlapping
(Fig. 1a,e). As the wings begin
to straighten and translate posteriorly, they simultaneously twist along their
length, initiating the power stroke (Fig.
1b,g). By mid-stroke, the wing has completely straightened out,
and the wingtip has reached its maximum positional angle within the stroke
(Fig. 1c,h). At this point in
time, the body reaches its maximum translational velocity. Wings are
distinctly cambered positively relative to the direction of wing motion. At
the completion of the power phase of the half-stroke, the wings start to curl
around the body (Fig. 1d,i).
The wingtip then translates anteriorly along the body axis to recover for
subsequent strokes (Fig.
1df).
When the body of the pteropod is viewed laterally, the path of the wingtip relative to the wing base resembles a distorted figure-of-eight that is wrapped around the anteriorposterior axis of the body (Fig. 2). The wingtip follows a counterclockwise path around the ventral loop of this figure-of-eight, and a clockwise path around the dorsal loop. Due to the slightly ventral location of the wing base, the wingtip path is not symmetrical about the body axis, and the paths traced out by the upstroke and downstroke intersect at a point approximately 10% of the wing length ventral to the wing base. For two individuals, we digitized two consecutive wingbeats, and observed no major differences in wing motion (see Fig. 2). Overall, the visual impression of wingbeat motions is one of very high repeatability among consecutive wingbeats during ascent.
Quantitative wingbeat kinematics
We define =0 as the beginning of the power phase of the
upstroke, with the maximum wingtip velocity occurring between
=0.08 and
=0.14
(Fig. 3). The power phase ends
at approximately
=0.23. At
=0.29, the wingtip
typically rests against the dorsal surface of the animal and slowly moves
anteriorly in preparation for the next half-stroke. At
=0.36,
the wing continues to move anteriorly, but spanwise curvature is progressively
reduced. The wingtip reaches its most anterior point at
=0.50,
ending the half-stroke. The downstroke is largely symmetrical with respect to
the upstroke. The mean wingbeat frequency, n, was 1.36±0.25
Hz. The ratio of the duration of the upstroke that of the downstroke,
US:DS, averaged 0.98±0.05. Neither n nor the stroke
area was significantly correlated with body translational velocity
(P>0.1). The Re for the mean wing chord ranged from
15123 (Table 1). However, Re of the mean wing chord during the power phase of the
upstroke was nearly twice the mean value for the half-stroke, ranging from 26
to 223. The advance ratio J (calculated as the ratio of the mean
ascent velocity to the mean wingtip velocity) ranged from 0.06 to 0.18,
indicating that unsteady effects may be important, particularly in the larger
animals (see Table 1).
|
The stroke plane angle, ß, is largely orthogonal to the longitudinal body axis (ß=1.98±1.65°), but the power stroke plane angle, ßps, ranges from 18.5° to 31.7°, with a mean of 24.8±1.7°. At low velocities, ßps is fairly shallow, but increases at higher translational velocities of the body (Fig. 4).
|
Body kinematics
C. antarctica ascends at absolute velocities ranging from 1 to 7
mm s1, corresponding to non-dimensional velocities from 0.35
to 1.87, respectively, and to Re between 6 and 49, respectively (see
Table 1). The mean angle of
ascent, , is largely coincident with the sum of mean body angle
and
ß. However, the trajectory of an ascending pteropod is punctuated by
cyclic oscillations due to large asymmetrical forces produced by the wings
during each half-stroke. Body motions follow a saw-toothed path, and at the
end of each half-stroke the body slows to a halt before the subsequent
half-stroke is initiated (Fig.
5). The mean angle of ascent during the power stroke was
64.7±2.2°, and increased with non-dimensional body velocity (see
Fig. 4). This angle of ascent
was largely perpendicular to the wing surface. The three individuals with
Rebody<25, exhibited downwards body motion (i.e.
negative translational velocities) during the recovery phase. Of the other
seven individuals, only individual no. 10, with an unusually long recovery
stroke, exhibited net downwards motion.
|
|
|
In addition to inducing oscillations of the center of mass, forces produced
by the flapping wings also act on the moment arm between the wing bases and
the center of body mass, causing the body angle to oscillate during a
wingbeat. During a half-stroke, Clione oscillates through an average
angle of 11.0±1.0° (Fig.
3). These oscillations exhibit a significant decrease in amplitude
with body mass (r2=0.8, P=0.0005). On average,
maximum and minimum values of occur at
=0.44 and
=0.9, respectively, during the recovery phase at the end of each
half-stroke (Fig. 3). Maximum
angular accelerations occur at the beginning of the power phase of each
half-stroke (i.e.
=0.07 for the upstroke, and
=0.55 for the downstroke), and maximum decelerations occur at
the beginning of each recovery phase (
=0.26 for the upstroke,
and
=0.75 for the downstroke;
Fig. 8). Dorsoventral
accelerations of the wingtip precede these body accelerations, whereas wingtip
velocities peak after maximum body accelerations
(Fig. 9).
|
|
![]() |
Discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Kinematics of the wingtip and body provide further support for the
assertion that C. antarctica is using its wing primarily as a paddle
and not as an hydrofoil. First, the angles of attack were high, ranging from
60° in the slowest swimmers to 80° in the fastest ones. At Re
of 160, the lift coefficient of a model insect wing declines substantially as
the angle of attack increases from 60° to 80°, and the lift-to-drag
ratio is maximal at much lower angles of attack, namely between 20° and
40° (Sane and Dickinson,
2001). Although circulation certainly develops around the flapping
pteropod wing, lift produced during wing translation is unlikely to be an
important component in the overall force balance during flapping ascent.
Indeed, the stroke plane angle of the power phase as well as the wing's angle
of attack increase with swimming velocity, which would serve to orient drag
forces more vertically and any potential lift forces more dorsoventrally. In
addition, the fact that the wing surface is nearly perpendicular to the
pteropod's ascent trajectory during the power phase provides further support
for the inference of drag-based locomotion. By ruling out lift as an important
propulsive mechanism for C. antarctica, these results buttress a
general conclusion that lift-based flapping is ineffective at
Re<100 (Bennett,
1972
; Daniel and Webb,
1987
; Horridge,
1956
; Webb and Weihs,
1986
). The tiny wasp Encarsia formosa oscillates its
wings at an Re of 17 (Weis-Fogh,
1973
), but it is unclear whether it and other minute flying
insects employ lift-based locomotion or are simply rowing through the air
(Bennett, 1972
;
Horridge, 1956
;
Thompson, 1942
).
The potential role of unconventional mechanisms of force generation in the
rowing locomotion of C. antarctica, and of other organisms operating
at Re between 10 and 100, merits further investigation. In C.
antarctica, large forces must be produced at stroke initiation because of
the rapid accelerations estimated for the center of body mass. For a model
insect wing, Sane and Dickinson
(2001,
2002
) demonstrated that force
transients at stroke initiation are primarily due to wake recapture, although
smaller peaks during impulsive starts at stroke transitions are caused by the
acceleration reaction as they precede vortex formation
(Birch and Dickinson, 2003
).
Overall, forces produced by the acceleration reaction are approximately equal
in magnitude to those produced via wing rotation, but are on average
small in comparison with translational aerodynamic forces
(Sane and Dickinson, 2002
).
The acceleration reaction clearly plays a role in propulsion at much higher
Re (>1000), as exemplified by such diverse locomotor modes as the
rowing of dytiscid beetles (Nachtigall,
1980
), frogs (Gal and Blake,
1988a
,b
),
the jet propulsion of medusae (Daniel,
1983
), and the tail-flick of carridean shrimp
(Daniel and Meyhöfer,
1989
). However, by combining kinematic data, force measurements
and blade-element modeling, McHenry et al.
(2003
) demonstrated that the
acceleration reaction is negligible in the undulatory swimming of ascidian
larvae (1<Re<100). Similarly, Williams
(1994a
,b
)
used kinematic results, physical models and theoretical analysis to discount
the importance of the acceleration reaction in rowing Artemia (for
which 1<Re<10). Downwards acceleration of the mass of the
muscular wing itself may also contribute to vertical force production in
pteropods. Body acceleration is more tightly phase-coupled to wing
acceleration than to peak wing foreaft velocity (see
Fig. 6), a finding consistent
with accelerational effects of either the wing mass or the wing added mass. To
conclusively evaluate the relative importance of wing inertia, rotational
circulation, the acceleration reaction and wake recapture in rowing locomotion
at these Re, we suggest similar integrative approaches in future
studies of pteropod locomotion, in particular quantification of flows.
Rowing with flexible paddles
Wingbody interactions and wing flexibility may also confound direct
comparison of pteropod hydrodynamics with those derived empirically from the
mechanical flapping of rigid model wings. Whereas pterygote insects possess
wing veins and volant vertebrates use bones to support their wings, pteropod
wings are a modification of the generalized molluscan foot and lack any rigid
support. As pteropods row, they not only change the position of the wingtip in
space but also alter the shape of the wing itself. At mid-stroke, the wing is
nearly straight, but at the end of each half-stroke it is adpressed to the
body, forming a tight arc. Such flexibility may allow the wing to remain
within the boundary layer surrounding the body during the recovery phase of
each half-stroke. One of the major problems with drag-based propulsion at low
Re is that, during the recovery stroke, the drag on an appendage
oriented parallel to flow may be only slightly less than that when
perpendicular to flow (Walker,
2002). Pteropods may partially mitigate this problem by removing
their appendages from the free-stream velocity during recovery. During the
beginning and end of each half-stroke, C. antarctica may also utilize
`peel-like behavior' reminiscent of `clap and fling' and squeeze mechanisms
previously described in other taxa (Daniel
and Meyhöfer, 1989
;
Ellington, 1984b
;
Weis-Fogh, 1973
), and proposed
for pteropods (Satterlie et al.,
1985
). Such body interactions are explicitly addressed elsewhere
in a hydrodynamic model of pteropod swimming
(Childress and Dudley, 2004
).
Overall, the distinctive kinematic features of the pteropod wingbeat exploit
largely drag-based force production during the power phase of each
half-stroke, together with a recovery phase enabled by low aspect ratio and
flexible wings pressed close to the body. This mode of locomotion therefore
uniquely combines flapping and rowing motions, and is distinct from other
drag-based stroke regimes that rely on asymmetry between half-strokes to
effect recovery to the starting position of the stroke.
The hydrodynamic implications of curvature of pteropod at midstroke are
unclear. The wings of insects, bats and numerous other flying and swimming
animals deform under inertial loads (Ennos,
1988,
1989
;
Swartz et al., 1996
;
Wootton, 1981
), and
fluid-dynamic loading is particularly important for wings of aquatic swimmers
(Combes and Daniel, 2003
;
Daniel and Combes, 2002
). Wing
deformation in some flies may advantageously alter lift production
(Ennos, 1988
). In pteropods,
positive camber likely derives from internal hydrostatic pressure and also
from contraction of radial and concentric muscle fibers in the wing. Active
wing flexion can contribute significantly to force production and enhanced
efficiency in flapping flight, but this phenomenon may be restricted to higher
Re (Combes and Daniel,
2001
). Rotation of the pteropod wing at the ends of half-strokes,
albeit as constrained anatomically by the relatively large wing base, must
also contribute to the patterns of deformation seen during the wingbeat.
Neuromuscular control of swimming speed
The ability to alter swimming speed is critical during predatorprey
interactions during which changes in translational and rotational velocities
are essential. Clione limacina, a pteropod found in the temperate
zone, exhibits two types of escape behaviors: a withdrawal response and an
escape swimming response (Norekian and
Satterlie, 2001). This pteropod species normally exhibits a slow
swimming behavior characterized by a flapping frequency of 12 Hz;
upwards propulsion barely exceeds negative buoyancy
(Satterlie et al., 1990
). When
disturbed, however, these pteropods increase flapping frequency substantially
to a maximum of 10 Hz, and ascent velocity approaches 100 mm
s1, approximately seven body lengths per second. This
response is mediated by the release of serotonin, which increases the cycle
frequency of the central pattern generator
(Satterlie and Norekian,
1996
). By contrast, C. antarctica failed to show an
escape swimming response during our experimental trials with over 400
individuals, and always showed a withdrawal response in reaction to even the
slightest disturbance. This withdrawal response involved apparent inhibition
of the swimming motoneurons, retraction of the wings, conglobation and
sinking. In C. limacina, such withdrawal behavior occurs only when
the animal is confronted with the most extreme stimuli
(Norekian and Satterlie,
1996
). We suspect that the increased expression of the withdrawal
response in C. antarctica is due, at least in part, to the apparent
inability to substantially increase flapping frequency necessary for an escape
swimming response.
Although C. antarctica does not exhibit a two-gear swimming
behavior, ascent velocity is positively correlated with flapping frequency
even within the small range of 12 Hz (R.D., B.J.B. and J.A.G.,
unpublished data). In pteropods, stroke amplitude is near maximal during
ascent swimming at all velocities, being geometrically constrained by the
presence of the body. Our present results suggest that additional aspects of
wingbeat kinematics may be important in the control of swimming speed. In
particular, changes in the inclination angle of the power phase of each
half-stroke will alter the effective angle of attack of the wing: steeper
power phases are associated with reduced body oscillations and a greater
ascent velocity. This situation is analogous to that of in flying insects, for
which increased stroke plane angles are strongly associated with increasing
forward velocity (see Dudley,
2000). This shift in stroke plane angle tilts the net aerodynamic
force vector forwards, yielding additional thrust but also maintaining
vertical force production. It is likely that individual pteropods similarly
modulate swimming speed via such a mechanism, although a challenge
for future studies will be to determine how the amplitude and timing of muscle
activation within the wingbeat cycle contributes to the complex geometry of
the wingtip path, and ultimately to the production of variable hydrodynamic
forces at different locomotor speeds.
![]() |
List of symbols |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
![]() |
Acknowledgments |
---|
![]() |
Footnotes |
---|
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Bellwood, D. R. and Wainwright, P. C. (2001). Locomotion in labrid fishes: Implications for habitat use and cross-shelf biogeography on the Great Barrier Reef. Coral Reefs 20,139 -150.[CrossRef]
Bennett, L. (1972). Effectiveness and flight of small insects. Ann. Entomol. Soc. Am. 66,1187 -1190.
Birch, J. M. and Dickinson, M. H. (2003). The
influence of wingwake interactions on the production of aerodynamic
forces in flapping flight. J. Exp. Biol.
206,2257
-2272.
Birch, J. M., Dickson, W. B. and Dickinson, M. H.
(2004). Force production and flow structure of the leading edge
vortex on flapping wings at high and low Reynolds numbers. J. Exp.
Biol. 207,1063
-1072.
Childress, S. and Dudley, R. (2004). Transition
from ciliary to flapping mode in a swimming mollusc: flapping flight as a
bifurcation in Re. J. Fluid
Mech. 498,257
-288.[CrossRef]
Combes, S. A. and Daniel, T. L. (2001). Shape, flapping and flexion: wing and fin design for forward flight. J. Exp. Biol. 204,2073 -2085.[Medline]
Combes, S. A. and Daniel, T. L. (2003). Into
thin air: contributions of aerodynamic and inertial-elastic forces to wing
bending in the hawkmoth Manduca sexta. J. Exp. Biol.
206,2999
-3006.
Daniel, T. L. (1983). Mechanics and energetics of medusan jet propulsion. Can. J. Zool. 61,1406 -1420.
Daniel, T. L. (1984). Unsteady aspects of aquatic locomotion. Am. Zool. 24,121 -134.
Daniel, T. L. and Combes, S. A. (2002). Flexible wings and fins: bending by inertial or fluid-dynamic forces. Integr. Comp. Biol. 42,1044 -1049.
Daniel, T. L. and Meyhöfer, E. (1989). Size limits in escape locomotion of carridean shrimp. J. Exp. Biol. 143,245 -266.
Daniel, T. L. and Webb, P. W. (1987). Physical determinants of locomotion. In Comparative Physiology: Life In Water and On Land (ed. P. Dejours, L. Bolis, C. R. Taylor and E. R. Weibel), pp. 343-369. New York: Liviana Press.
Daniel, T. L., Jordan, C. and Grunbaum, D. (1992). Hydromechanics of swimming. In Mechanics of Animal Locomotion (ed. R. M. Alexander), pp.17 -49. Berlin: Springer-Verlag.
Dudley, R. (2000). The Biomechanics of Insect Flight: Form, Function, and Evolution. Princeton: Princeton University Press.
Ellington, C. P. (1984a). The aerodynamics of hovering insect flight. III. Kinematics. Proc. R. Soc. Lond. B 305,41 -78.
Ellington, C. P. (1984b). The aerodynamics of hovering insect flight. IV. Aerodynamic mechanisms. Proc. R. Soc. Lond. B 305,79 -113.
Ellington, C. P. (1984c). The aerodynamics of hovering insect flight. V. A vortex theory. Proc. R. Soc. Lond. B 305,115 -144.
Ennos, R. (1988). The importance of torsion in the design of insect wings. J. Exp. Biol. 140,137 -160.
Ennos, R. (1989). Inertial and aerodynamic torques on the wings of Diptera in flight. J. Exp. Biol. 142,87 -95.
Fish, F. E. (1996). Transitions from drag-based to lift-based propulsion in mammalian swimming. Am. Zool. 36,628 -641.
Fuiman, L. A. and Batty, R. S. (1997). What a
drag it is getting cold: partitioning the physical and physiological effects
of temperature on fish swimming. J. Exp. Biol.
200,1745
-1755.
Gal, J. M. and Blake, R. W. (1988a). Biomechanics of frog swimming I. Estimation of the propulsive force generated by Hymenochirus boettgeri. J. Exp. Biol. 138,399 -411.
Gal, J. M. and Blake, R. W. (1988b). Biomechanics of frog swimming II. Mechanics of the limb-beat cycle in Hymenochirus boettgeri. J. Exp. Biol. 138,413 -429.
Horridge, G. A. (1956). The flight of very small insects. Nature 178,1334 -1335.
McHenry, M. J., Azizi, E. and Strother, J. A.
(2003). The hydrodynamics of locomotion at intermediate Reynolds
numbers: undulatory swimming in ascidian larvae (Botrylloides sp.).
J. Exp. Biol. 206,327
-343.
Morton, J. E. (1954). The biology of Limacina retroversa. J. Mar. Biol. Assn. UK 33,297 -312.
Morton, J. E. (1958). Observations on the gymnostomatous pteropod Clione limacina (Phipps). J. Mar. Biol. Assoc. UK 37,287 -297.
Nachtigall, W. (1980). Mechanics of swimming water-beetles. In Aspects of Animal Movement (ed. H. Y. Elder and E. R. Trueman), pp. 107-124. Cambridge: Cambridge University Press.
Norekian, T. and Satterlie, R. A. (1996).
Whole-body withdrawal system and its involvement in the behavioral hierarchy
of the mollusc Clione limacina. J.
Neurophysiol. 75,529
-537.
Norekian, T. and Satterlie, R. A. (2001). Serotonergic neural system not only activates swimming but also inhibits competing neural centers in a pteropod mollusc. Am. Zool. 41,993 -1000.
Podolsky, R. D. and Emlet, R. B. (1993).
Separating the effects of temperature and viscosity on swimming and water
movement by sand dollar larvae (Dendraster excentricus).
J. Exp. Biol. 176,207
-221.
Sane, S. P. (2003). The aerodynamics of insect
flight. J. Exp. Biol.
206,4191
-4208.
Sane, S. P. and Dickinson, M. H. (2001). The
control of flight force by a flapping wing: lift and drag production.
J. Exp. Biol. 204,2607
-2626.
Sane, S. P. and Dickinson, M. H. (2002). The
aerodynamic effects of wing rotation and a revised quasi-steady model of
flapping flight. J. Exp. Biol.
205,1087
-1096.
Satterlie, R. and Norekian, T. (1996).
Serotonergic modulation of swimming speed in the pteropod mollusc Clione
limacina. III. Cerebral neurons. J. Exp. Biol.
198,917
-930.
Satterlie, R. A., Labarbera, M. and Spence, A. N. (1985). Swimming in the pteropod mollusc, Clione limacina. I. Behavior and morphology. J. Exp. Biol. 116,189 -204.
Satterlie, R. A., Goslow, G. E. and Reyes, A. (1990). Two types of striated muscle suggest two-geared swimming in the pteropod mollusc, Clione limacina. J. Exp. Zool. 255,131 -140.[CrossRef]
Swartz, S. M., Groves, M. S., Kim, H. D. and Walsh, W. R. (1996). Mechanical properties of bat wing membrane skin. J. Zool. 239,357 -378.
Thompson, D. A. W. (1942). On Growth and Form. Cambridge: Cambridge University Press.
Usherwood, J. R. and Ellington, C. P. (2002).
The aerodynamics of revolving wings. II. Propeller force coefficients from
mayfly to quail. J. Exp. Biol.
205,1565
-1576.
Vogel, S. (1994). Life in Moving Fluids. Princeton: Princeton University Press.
Walker, J. (2002). Functional morphology and virtual models: physical constraints on the design of oscillating wings, fins, legs, and feet at intermediate Reynolds numbers. Integr. Comp. Biol. 42,232 -242.
Walker, J. A. (1998). Estimating velocities and
accelerations of animal locomotion: a simulation experiment comparing
numerical differentiation algorithms. J. Exp. Biol.
201,981
-995.
Walker, J. A. and Westneat, M. N. (1997).
Labriform propulsion in fish kinematics of flapping aquatic flight in the bird
wrasse Gomphosus varius (Labridae). J. Exp.
Biol. 200,1549
-1569.
Walker, J. A. and Westneat, M. W. (2000). Mechanical performance of aquatic rowing and flapping. Proc. R. Soc. Lond. B 267,1875 -1881.[CrossRef][Medline]
Webb, P. W. and Weihs, D. (1986). Functional locomotor morphology of early life history stages of fishes. Trans. Am. Fish. Soc. 115,115 -127.[CrossRef]
Weis-Fogh, T. (1973). Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol. 59,169 -230.
Williams, T. A. (1994a). Locomotion in
developing Artemia larvae: mechanical analysis of antennal propulsors
based on large-scale physical models. Biol. Bull.
187,156
-163.
Williams, T. A. (1994b). A model of rowing
propulsion and the ontogeny of locomotion in Artemia larvae.
Biol. Bull. 187,164
-173.
Wootton, R. J. (1981). Support and deformability in insect wings. J. Zool. 193,447 -468.