Flow patterns generated by oblate medusan jellyfish: field measurements and laboratory analyses
1 Bioengineering and Graduate Aeronautical Laboratories, California
Institute of Technology, Mail Code 301-46, Pasadena, CA 91125, USA
2 Biology and Marine Biology, Roger Williams University, MNS 241, Bristol,
RI 02809, USA
3 Biology, Providence College, Providence, RI 02918, USA
* Author for correspondence (e-mail: jodabiri{at}caltech.edu)
Accepted 31 January 2005
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Summary |
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Key words: jellyfish, Aurelia aurita, flow pattern, flow patterns, vortex rings, jet propulsion
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Introduction |
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Any vortex formation that may occur during the recovery stroke of the
propulsive cycle is neglected in existing models of the swimming process (e.g.
Daniel, 1983;
Colin and Costello, 2002
;
McHenry and Jed, 2003
).
Therefore interactions between adjacent vortex rings in the wake have been
previously examined with the assumption of a uniform train of starting vortex
rings in the animal wake, each with identical rotational sense.
Weihs (1977) used a
quasi-steady model of such a vortex ring train to conclude that substantial
thrust augmentation can occur up to 150% relative to a steady jet
if the vortex rings are spaced sufficiently close together (i.e.
C/(B+D) is small; see
Fig. 1). The thrust benefit
arises from the induced downstream velocity of the vortex ring train on each
of its members. Interestingly, the flow pattern depicted by Ford et al. (see
fig. 5 in Ford et al., 1997
)
for the oblate jellyfish Chrysaora quinquecirrha shows a train of
closely spaced vortex rings.
The proximity of the wake vortex rings to each other and to the swimming
animal as measured by Ford et al.
(1997) is unexpected given the
tendency of individual vortex rings to rapidly propagate away from the flow
source due to self-induced velocity (Lamb,
1932
) and convection by any jet-like flow that is present behind
each ring. An estimate by Daniel
(1983
) of the medusan wake
vortex ring train using these assumptions predicts a spacing of 10 vortex ring
radii between adjacent rings. This result agrees with a physical intuition for
the formation of successive vortex rings with identical rotational sense, but
is incompatible with the experimental measurements of Ford et al.
(1997
).
Given difficulties in quantitatively measuring vortex ring properties using
particle tracking methods, one cannot make a priori preference for
either the measurements of Ford et al.
(1997) or the model estimate
of Daniel (1983
). An
experimental re-examination of both the measured flow patterns of Ford et al.
(1997
), and the assumptions
leading to the model of Daniel
(1983
) is necessary to clarify
this issue. The importance of this effort is underscored by the heavy reliance
on both the model of Daniel
(1983
) and observations of
power stroke vortex ring formation to develop generalized dynamical models of
medusan swimming (e.g. Colin and Costello,
2002
; McHenry and Jed,
2003
).
Of further concern is the fact that existing models of medusan behavior
make little association between swimming and the prey capture mechanisms
required for feeding. Yet, a clear relationship between swimming and feeding
has been documented for a variety of medusae, notably larger oblate forms
(Costello and Colin 1994,
1995
).
Owing to the inherent physical coupling between the locomotor and feeding structures in medusae, it is important to investigate how the generated flow patterns can efficiently meet the demands of both systems. Fluid motions must be heavily utilized in the process of capturing prey and transporting it to feeding surfaces, while maintaining a capability to generate thrust for swimming. The physical mechanisms whereby the observed vortex ring structures accomplish both tasks are currently unknown.
The objective of this paper is to elucidate the nature and utility of flow structures generated by oblate swimming medusae. In order to understand how flow contributes to both thrust production and feeding mechanisms, we utilized flow visualization to qualitatively and quantitatively examine flow patterns surrounding the oblate scyphomedusa Aurelia aurita L. swimming in its natural marine habitat.
Specifically, we observed the motion of fluorescent dye markers injected at several locations around the bells of swimming A. aurita medusae. These field observations eliminate the need for an imposed flow current, as is necessary in typical laboratory tanks that house medusae (i.e. kreisel facilities). The swimming kinematics and dynamics measured by this method should be more relevant to animal behavior in the marine water column than those achieved by any type of laboratory tank. Further, the use of a continuous dye marker instead of discrete prey for particle tracking simplifies the identification of large-scale fluid structures in the flow, such as vortex rings. In addition, the three dimensional nature of the flow (in the absence of significant vortex stretching) does not limit the effectiveness of the dye marker technique, as is the case with particle tracking. By observing vortex dynamics in the wake and fluidstructure interactions near the medusa bell, the role of the observed flow structures is clarified.
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Materials and methods |
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Several hours of videotape were recorded in which medusae were observed swimming both in large groups and in isolation. The swimming kinematics were qualitatively consistent throughout the observations using the dye technique. In order to achieve more quantitative results, a subset of the full video database was analyzed in greater detail.
Kinematic analyses
Select recordings of animal swimming in the field were collected for
further laboratory analysis. For kinematic analyses, sequences of recorded
swimming motions of two medusae were chosen according to visibility of the
medusa subumbrellar surface, motion in planes parallel to the camera,
separation from other medusae, and sufficient dye marker in the flow. In
addition, the upper and lower limits of animal size were sought, to facilitate
comparisons of fluid dynamic and geometric scaling. The larger medusa in the
kinematic analyses is an order of magnitude larger than the small medusa (10.2
cm maximum bell diameter vs 3.6 cm maximum bell diameter, or 22 times
larger by volume).
Video recordings of up to five successive swimming contractions of each
animal were analyzed using the algorithm of Dabiri and Gharib
(2003). Between 10 and 20
control points along the subumbrellar surface of each medusa were tracked
during swimming motions to generate a computational reconstruction of the
medusae kinematics. The entire subumbrellar surface was clearly visible
through the transparent mesoglea in each of the selected frames, and any
fluorescent dye carried inside the medusa bell further improved the resolution
of the interface location.
In addition to measuring the shape of the bell, the properties of the formed vortex rings were analyzed. Specifically, the volume of each vortex ring and the inter-ring spacing of the vortex ring train in the animal wake were measured. The physical extent of each vortex ring was measured based on the distribution of the dye marker in the flow.
The kinematic analyses include only frames in which the animals are swimming parallel to the image plane of the camera. Hence, the orientation of the vortex rings can be assumed to be such that the ring axis is also parallel to the image plane. To calculate the volume of the visible toroidal vortex rings, we therefore require only measurements of the diameter across the toroid (B+D in Fig. 1) and the diameter of the vortex core (B in Fig. 1).
Important caveats associated with this measurement technique are discussed
in the following section. Each set of measurements is presented
conservatively, with an uncertainty calculated as the maximum difference
between any individual measurement and the average of the set. This metric is
used in lieu of the standard deviation (S.D.), since the
S.D. tends to underestimate the data spread for relatively small
sample sizes such as those studied here
(Freedman et al., 1998).
Strategies for dye marker interpretation
Despite the convenience of a passive dye marker for qualitative flow
visualization, there are important limitations to the technique. First,
although all of the dye labels fluid, not all of the fluid is labeled by dye.
Therefore if one desires to track the evolution of a fluid structure such as a
vortex ring using the passive dye marker, the visible labeled structure may be
smaller than the actual fluid structure. When identifying vortex rings in the
flow, the dye-labeled ring represents a lower bound on the size of the
vortex.
Secondly, the diffusion coefficient of dye markers such as that used in
these experiments (fluorescene) is substantially less than the diffusion
coefficient of fluid vorticity (i.e. rotation and shear), as measured by the
kinematic viscosity of water (102 cm2
s1). Hence, regions of compact vorticity such as the wake
vortex rings will tend to spread by diffusion at a rate faster than can be
observed in the dye. Again, the result is that the visibly labeled structure
may be smaller than the actual fluid structure. Consequently, all of our
estimates of vortex ring volume are conservative values.
Our strategy was to exploit these limitations by assuming that the measurements represent a lower bound on the size of the vortex rings. The effects of a departure from the lower bound case can be inferred from the available data.
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Results |
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During the power stroke the bell contracted and initiated the formation of a starting vortex ring. The starting vortex ring induced a motion of fluid originating from regions both inside the subumbrellar volume and outside the bell via entrainment of ambient fluid. The induced motion was oriented at an angle away from the bell margin and toward both the central axis of the bell and downstream (Fig. 2). At maximum contraction, the starting vortex was fully developed and traveling away from the medusa.
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As the bell relaxed, a stopping vortex was formed inside the subumbrellar cavity. The stopping vortex induced a motion of fluid originating from outside the bell and toward the subumbrellar cavity. Consequently, as the subumbrellar volume increased with bell relaxation, the stopping vortex served to refill the subumbrellar volume with fluid from outside the bell.
The stopping vortex remained in the subumbrellar cavity during the beginning of the contraction phase of the next swimming cycle. As the bell contracted, a part of the stopping vortex ring was ejected from the subumbrellar cavity and interacted with the starting vortex of the new cycle. In the interaction, a portion of the fluid from the stopping vortex co-joined with the starting vortex ring, completing formation of the downstream lateral vortex superstructure. The kinematics of the starting, stopping and co-joined lateral vortex structure are illustrated in Fig. 3. Although the schematic in Fig. 3B suggests a clear distinction between power and recovery stroke vortices in the wake, the two components actually become highly amalgamated because of fluid mixing. This process makes the individual vortices increasingly difficult to distinguish as the wake develops.
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There are several important observations to be made from the qualitative
analysis of the formation and interaction of the starting and stopping vortex
rings during swimming cycles. First, throughout the swim cycle, from the onset
of the contraction phase through the relaxation phase, there is a continual
flow of water that originates external to the bell and passes adjacent to the
bell margin before entering the bell. During the contraction, the flow
contributes to the fluid in the starting vortex ring but during bell
relaxation it contributes to the fluid in the stopping vortex ring. Second,
during a sequence of propulsive cycles we observed that the stopping vortex
ring from the preceding lateral vortex structure persists in the bell and
contributes to the formation of the subsequent starting vortex
(Fig. 2). A complex interaction
occurs as the starting vortex ring grows and the previous stopping vortex is
convected downstream away from the bell margin. The interaction appears to
increase the volume but decrease the velocity of the starting vortex ring,
via cancellation of starting vortex vorticity by the preceding
stopping vortex ring of opposite rotational sense (cf.
Lim and Nickels, 1995). As a
result of this interaction, the vortex ring volume and vortex ring spacing of
swimming oblate medusae do not relate to thrust in the manner predicted for a
unidirectional pulsed jet of fluid through an orifice (i.e.
Table 1). Existing models for
the time-dependent thrust neglect recovery stroke vortex formation altogether
and therefore do not apply to these medusae.
|
The interaction between starting and stopping vortices described above
tends to increase the total momentum of each wake vortex. This is because the
mass of each wake vortex increases (i.e. the starting and stopping vortex
masses combine in the wake) to a greater degree than the wake vortex velocity
decreases via vorticity cancellation. The effect of vorticity
cancellation is limited because it is a viscosity-dependent process that
occurs only at the interface between the starting and stopping vortices
(Shariff and Leonard,
1992).
Dye injected into the middle of the subumbrellar volume is not ejected directly outwards during bell contraction but instead spreads laterally along the subumbrellar surface of the bell. All of the dye leaves the subumbrellar cavity at the bell margin. However, it takes the medusa several swim cycles to eject all of the dye from the subumbrellar region. No fluid is ever directly ejected from the central subumbrellar region. In fact, there is always a net flow into the subumbrellar cavity in the central portion trailing to the bell. This is due to the direction of rotation of the stopping vortex ring in this region (Fig. 3).
Tentacle positioning
Throughout the pulsation cycle the tentacles of A. aurita were
primarily located in vortex rings (Fig.
4). At the beginning of the contraction phase they were in the
subumbrellar cavity in the stopping vortex ring. As the bell contracted the
tentacles become entrained in the starting vortex ring, which oriented the
tentacles in a trailing position. At the end of the contraction phase and the
onset of the relaxation phase, the tentacles became entrained in the stopping
vortex ring which drew them back up into the subumbrellar cavity.
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Vortex ring kinematic measurements
Several important aspects of the fully formed wake vortices were
investigated quantitatively. Wake vortices visualized with dye markers allow
for quantitative evaluation of thrust production mechanisms. In this regard,
an important variable is the volume of each wake vortex ring relative to the
volume of fluid ejected by the medusa. Although the results of the preceding
section suggest a more complex relationship between swimming thrust and vortex
kinematics than has been previously appreciated, we can anticipate that, for a
fixed swimming frequency (and corresponding characteristic flow velocity),
swimming thrust will maintain a direct relationship with vortex volume (cf.
Daniel, 1983). We cannot
neglect the fact that the characteristic flow velocity is reduced by vorticity
cancellation between the starting and stopping vortices. However, as mentioned
previously, this effect is limited in the present case. Therefore volume
measurements provide the most useful index of swimming thrust in the present
study.
The volume contained by the bell at full contraction and relaxation was
measured using the subumbrellar surface profiles obtained from the video
analysis. The difference between these two values gives the volume of fluid
ejected. The ratio of wake vortex ring volume w to ejected
fluid volume
b was relatively insensitive to animal size:
3.09±0.35 for the smaller animal, and 3.57±0.40 for the larger.
Two primary effects contribute to the large volume of the wake vortex rings
relative to the volume of fluid ejected. First of these is the presence of the
stopping vortex ring, which is created from fluid external to the bell. This
stopping vortex is complexed with the starting vortex ring of the subsequent
lateral vortex to create each wake vortex ring. Secondly, both starting and
stopping vortex ring formation processes involve substantial entrainment of
ambient fluid from outside the bell.
Given the vortex ring kinematic measurements, we can also determine the inter-ring spacing in the wake vortex ring train. In both medusae, the wake vortex rings are very closely spaced. The inter-ring spacing is 1.36±0.08 ring radii in the smaller medusa and 1.06±0.06 ring radii in the large medusa.
Table 2 presents the measurements of vortex ring kinematics in dimensional form.
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Discussion |
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The observation of stopping vortex ring formation is not entirely new. It
is well known from experiments on vortex ring formation by mechanical
apparatus that such structures appear at the end of fluid ejection
hence their title (e.g. Didden,
1979). The source of vorticity for these structures is the
interaction between flow external to the vortex ring generator and its outer
surface. In a swimming medusa, this corresponds to the interaction of external
fluid with the expanding bell margin. The strength of the stopping vortex ring
will be directly proportional to the velocity of external fluid past the bell
(Didden, 1979
), and is
therefore coupled to the swimming speed of the animal and the local kinematics
of the bell margin.
The presence of such a pronounced stopping vortex ring has profound
consequences on thrust production and estimates of thrust. As mentioned, it
substantially increases the volume of fluid in each wake vortex ring,
w, relative to the volume of fluid from inside the bell
b. For laboratory-generated vortex rings in the absence of
background flow, Dabiri and Gharib
(2004
) found that approximately
one third of the vortex ring volume originates from ambient fluid external to
the jet flow. If we assume that the medusa stopping vortex ring is
approximately the same size as the starting vortex ring (a reasonable
assumption based on the dye visualizations), then a one-third entrained fluid
fraction in each starting and stopping vortex ring will lead to a ratio of
wake vortex ring volume to ejected fluid volume equal to 3, i.e.
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Comparing this value with the present measurements, it appears the two
effects that we have identified as contributing to the relatively large wake
vortex ring volume are sufficient to explain the observations. It is prudent
to note that the starting and stopping vortex volumes are indeed additive in
equation 1, despite the fact that
the vortices possess opposite rotational sense. This is because the volume of
each individual vortex is conserved despite the dynamical influences of
vorticity cancellation and induced velocity effects
(Lim and Nickels, 1995).
Since the swimming thrust increases directly with wake vortex volume (for a fixed swimming frequency and characteristic flow velocity), the presence of a stopping vortex ring has the potential to greatly increase swimming thrust. Based on the present measurements, the swimming thrust from the co-joined wake vortex ring motif would be larger than a steady jet of bell fluid by more than a factor of three. However, the estimated thrust advantage by this mechanism would probably be mitigated in practice by a lower convective velocity for the wake vortex rings relative to a steady jet flow, because of vorticity cancellation effects presented earlier.
The starting-stopping vortex interactions could also potentially increase
swimming thrust via the velocity field that the opposite-sign
vortices induce on one another, in a manner similar to the effect experienced
by vortices near solid surfaces (i.e. ground effect;
Rayner and Thomas, 1991;
Shariff and Leonard, 1992
).
Quantitative visualization techniques will be necessary to validate the
existence of such an effect in these animals.
The swimming efficiency is difficult to define for these animals because
the flow is highly unsteady, and there is no clear protocol for including the
relaxation phase in such a calculation. However, we can anticipate an increase
in swimming efficiency, given that a Froude-type calculation predicts higher
efficiencies for cases in which fluid is transported in high volume and at low
velocity, as is observed here (Vogel,
1994).
The above findings demonstrate that the effect of the complex vortex ring
wake structure must be considered in any realistic model of oblate medusan
swimming. Specifically, it is insufficient to estimate thrust production by
swimming oblate medusae using measurements of bell kinematics (e.g.
Colin and Costello, 2002;
McHenry and Jed, 2003
) without
a wake vortex ring analysis, since the stopping vortex is fully formed from
the beginning of the bell motion. Also, the fluid inside the bell is not at
rest (with respect to the medusa) when bell contraction is initiated, as must
be assumed in a paddle model of the bell kinematics (e.g.
McHenry and Jed, 2003
).
In addition to the relationship between individual vortex kinematics and
thrust, the relation between inter-vortex ring spacing and thrust may also
differ from established models of jet propulsion. Similar to the observations
of Ford et al. (1997), we
observed reduced downstream propagation of wake vortices and, hence, a very
close spacing between adjacent wake vortex rings. The flow visualization
methods employed here enable us to resolve the apparent conflict between
measurements and physical intuition for the dynamics of individual vortex
rings. Reduced downstream propagation of the wake vortices is the result of
vorticity cancellation. For rapid self-induced motion of vortex rings from a
flow source it is fundamental that the vortex rings possess a single sign of
rotation (Lamb, 1932
). In the
event that opposite-sign rotation is present, vorticity cancellation will
occur in the vortex ring structure, resulting in a vortex ring with weaker
self-induced motion. Oblate medusae encounter vorticity cancellation by two
means as the flow pattern is being generated. First, the motion of each
starting vortex toward the bell axis of symmetry leads to vorticity
cancellation with mirror-image portions of the vortex on the opposite side of
the bell margin. This effect is enhanced by motion of the bell margin toward
the axis of symmetry during bell contraction. This type of vorticity
cancellation commences almost immediately after bell contraction is
initiated.
Secondly, as previously mentioned, the stopping vortex of the preceding lateral vortex structure interacts with each newly forming starting vortex ring. This additional vorticity cancellation occurs in proportion to the strength of the stopping vortex. Vorticity cancellation by this means can occur almost immediately after bell contraction because of the close proximity of the vortex structures carrying opposite-signed vorticity. The combined result of these processes is reduced downstream propagation of wake vortices and, hence, a very close spacing between adjacent wake vortex rings.
Although it is tempting to further suggest that the thrust benefits from
close inter-ring spacing predicted by Weihs
(1977) may be applicable to
the dynamics of these medusae, one must remember that the derivation of Weihs
(1977
) assumes a train of
single-sign vortex rings. Whether or not a similar benefit could be derived
for the co-joined wake vortex rings studied here is a question still to be
resolved.
Implications for medusan feeding behavior
Vortex formation and the use of accompanying induced flows are
fundamentally important to feeding by swimming A. aurita. Medusae
that feed as cruising predators, such as A. aurita, are highly
dependent on locally generated flow currents to capture prey (Costello and
Colin, 1994,
1995
;
Sullivan et al., 1994
;
Ford et al., 1997
). For these
animals, it is beneficial to generate flow regimes that increase encounter
rates with prey. The stopping and starting vortex rings and the lateral vortex
superstructure generated by A. aurita during swimming serve this
role. First, both the starting and the stopping vortices entrain fluid from
outside the bell throughout the pulsation cycle. This fluid is drawn past the
bell margin toward the tentacles that are positioned in the starting vortex
during bell contraction and the stopping vortex during bell relaxation
(Fig. 4). Since the starting
and stopping vortex rings entrain fluid during both the bell contraction and
relaxation, respectively, A. aurita is able to use the full pulsation
cycle for prey capture. Second, as discussed, the two vortex rings interact,
enabling the starting vortex ring to entrain a greater volume of fluid than
could be entrained by simplified jet flow. Therefore, medusae that generate
starting and stopping vortices during swimming are able to process a large
volume of water with each pulse. Finally, the prey entrained by the swimming
medusae propagate away from the bell at a reduced rate because of the
rotational flow of the wake vortex rings.
These observed structures differ considerably from the jet-propulsion flow model (Fig. 1) in which the flow structures take the form of a uniform slug with minimal fluid entrainment. Consequently, the flow structures of swimming oblate medusae, such as A. aurita, have the potential to greatly increase encounter rates with prey relative to the flow structures of jet-propulsion.
The reduction in wake vortex ring propagation away from the bell should be
amplified at higher swimming speeds, because the strength of the stopping
vortex (used for vorticity cancellation) possesses direct proportionality to
the fluid velocity past the bell (cf.
Didden, 1979). The combined
effect is a feeding mechanism that uses both bell contraction and relaxation
productively for prey capture while being passively tuned for a range of
swimming speeds. These flows provide potentially powerful mechanisms for prey
encounter with the medusae.
The present dye visualization methods are insufficient to conclusively
validate the existence of the passive fluid dynamic tuning mechanisms
hypothesized here. However, future work using methods of quantitative
velocimetry (e.g. digital particle image velocimetry;
Willert and Gharib, 1991) will
enable the direct measurements of starting and stopping vortex strength that
are necessary to solidify these conclusions.
Implications for prolate medusae
Colin and Costello (2002)
suggested that oblate and prolate medusae differ in their dependence on jet
propulsion for thrust generation. Specifically, they concluded that prolate
medusae display a strong jet flow component in the wake, whereas oblate
species are dominated by vortex ring formation. Hence the results of the
present study are primarily applicable to oblate forms. In several respects,
however, our results may be relevant to prolate medusae.
The most important effect to recognize is that any accelerated jet with
flow separation from a circular orifice and sufficiently large Reynolds number
can form a vortex ring. Cantwell
(1986) computed that the
minimum Reynolds number required for vortex ring formation is approximately
six. This is well below the swimming regime of medusan swimmers, both prolate
and oblate (Gladfelter, 1973
;
Colin and Costello, 2002
). We
can therefore expect that prolate medusae will also experience vortex ring
formation.
A primary difference between the wake of a prolate swimmer and that of an
oblate form, however, lies in the duration of fluid ejection, as measured by
the formation time (A/D; see
Fig. 1). In laboratory
experiments, Gharib et al.
(1998) demonstrated that when
the formation time exceeds a value of four, the leading vortex ring at the
front of the fluid discharge stops growing and pinches off from the remaining
fluid discharge behind it. In the swimming medusae, several additional
parameters become important, such as temporal variation in the bell diameter
and swimming speed (Krueger et al.,
2003
; Dabiri and Gharib,
2005
). These factors will probably affect the ratio, changing it
from the nominal value of four. Nonetheless, we can expect the physical
principles to remain unchanged. Oblate medusae tend to eject fluid with small
A/D ratios, thereby avoiding both pinch-off of the leading
vortex ring and formation of a trailing jet flow. This leads to the observed
dominance of vortex ring structures in the wake of oblate swimmers. By
contrast, prolate medusae will tend to eject fluid with larger
A/D ratios, leading to a substantial presence of jet flow
behind the leading vortex rings in the flow. The difference in these ratios
appears to be consistent with the different foraging strategies of the two
forms. The low formation numbers of oblate forms suggest that their mode of
propulsion may be more efficient, which is consistent with their cruising
strategy (Table 1). Conversely,
larger A/D ratios in prolate forms are consistent with their
roles as ambush predators that periodically require large thrust generation,
perhaps at the expense of swimming efficiency.
Colin and Costello (2002)
show that the dominant presence of jet flow in prolate medusae makes them more
amenable to models based on jet flow such as that of Daniel
(1983
). However, one cannot
neglect to also examine the vortex ring formation that will occur early during
bell contraction in prolate medusae. Some form of the co-joined wake vortex
ring structures observed in the present study may also appear in the wakes of
a variety of medusae, particularly those with intermediate bell morphologies
between prolate and oblate. Further examination of wake structure and thrust
generation will be necessary to properly model the dynamic swimming behavior
of these animals.
A note on fluid dynamic and geometric scaling
One aim of this work has been to demonstrate the consistency of the
observed flow patterns over a wide range of medusae sizes. If the fluid
dynamical effects are insensitive to geometric scaling as we have seen
here then the kinematics of bell motion that dictate the fluid
dynamics should also be scale invariant. We demonstrate this effect
straightaway, by plotting the shape profile for the smaller and larger medusae
at full bell contraction and relaxation in coordinates normalized by the cube
root of the ejected bell fluid volume, b1/3
(Fig. 5). The cube root of the
volume is used rather than the volume itself so that the normalized
coordinates are dimensionless, thereby facilitating comparison across the full
range of animal sizes. Consistent with the arguments presented in this paper
and the findings of McHenry and Jed
(2003
), the bell kinematics
are very similar, despite the order of magnitude difference in the volumes of
the subjects. A similar examination should be made for other species of
medusae, to determine the generality of the fluid dynamic and geometric
scaling sensitivities observed here.
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Acknowledgments |
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