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First published online May 18, 2006
Journal of Experimental Biology 209, 2025-2033 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02242
Fast-swimming hydromedusae exploit velar kinematics to form an optimal vortex wake
1 Graduate Aeronautical Laboratories and Bioengineering, California
Institute of Technology, Mail Code 138-78, 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 27 March 2006
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Summary |
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Key words: locomotion, wake, vortices, jellyfish, Nemopsis bachei
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Introduction |
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;
Donaldson et al., 1980)]. In
cnidarians such as Aglantha, the jetting strategy is the sole means
of propulsion. By contrast, some higher organisms such as cephalopods make use
of pulsed jets primarily for high-speed swimming (e.g. squid in their
tail-first configuration), relying on lateral fins for locomotion at low
speeds (Bartol et al., 2001;
Anderson and Grosenbaugh,
2005).
The body plan of hydromedusan jellyfish is most commonly prolate, or
torpedo-shaped, consistent with the need to reduce resistive forces associated
with their rapid swimming accelerations. This acceleration reaction, the
effect of body added-mass (Daniel,
1983), is reduced for more prolate body shapes. In addition, these
animals possess a characteristic funnel-shaped velum at the exit of their oral
cavity. In a resting position, the velum is oriented in a plane perpendicular
to the body axis of radial symmetry (Fig.
1A). As water is ejected from the subumbrellar oral cavity, the
generated fluid pressure causes the velum to `outpocket', effectively creating
a funnel through which the jet flow emerges
(Fig. 1B). The velum has
therefore been assumed to function primarily as a flow accelerator, augmenting
thrust via an increase in jet flow velocity through the funnel.
However, improved thrust generation by this mechanism would be achieved at the
expense of increased costs of locomotion
(Gladfelter, 1972;
Daniel, 1983;
Daniel, 1985).
|
). The
energy cost of ejecting fluid in the form of a vortex ring with a trailing jet
behind it has been shown to be higher than fluid transport via an
isolated vortex ring with no trailing jet
(Krueger and Gharib, 2003).
Together, these results suggest that the efficiency of fluid transport in
pulsatile jets can be maximized by the formation of `optimal' vortex rings, in
the sense of the largest possible vortex during each jet pulse without the
formation of a trailing jet behind the vortex.
Given the broad role of pulsatile jet propulsion in a diverse range of
biological fluid transport functions, from aquatic locomotion to intracardiac
blood transport, it is useful to ask whether animal systems are capable of
optimizing vortex formation to take advantage of the aforementioned functional
benefits discovered in mechanical fluid transport systems. A key difference
between the mechanically generated jet flows previously studied and those
occurring in nature is the appearance of valve-like structures, such as the
hydromedusan velum described above, which vary the jet exit diameter during
the vortex formation process. It has recently been shown that a proper
extension of the optimal vortex formation concept to animal systems must
incorporate these time-dependent kinematics in the analysis
(Dabiri and Gharib, 2005a;
Dabiri and Gharib, 2005b).
Failure to do so can obscure the importance of optimal vortex formation to
biological fluid transport processes. In particular, time-dependent valve (or
similar structure) kinematics are capable of changing the optimal vortex
formation number T*lim from the specific value
T*lim
4(±0.5) found in
constant-diameter mechanically generated jets
(Dabiri and Gharib, 2005b).
Consequently, the existence of optimal vortex formation (or a lack thereof) in
a biological system cannot be proven based on the assumption of a particular
numerical value of the vortex formation number.
To better understand the role of optimal vortex formation in biological
pulsed jets, this paper investigates the possible occurrence of optimal vortex
formation in the fast-swimming jellyfish Nemopsis bachei Agassiz
1849. First, we use high-speed video imaging of free-swimming specimens to
measure the time-dependent velar kinematics during the entire swimming cycle.
Second, we combine these results with dye flow visualization of the wake
created by N. bachei and a model for the associated swimming dynamics
to determine whether the animal achieves optimal vortex formation, in a manner
similar to that observed in recent mechanically generated jet flow experiments
(Dabiri and Gharib, 2005b). In
the process, generally applicable aspects of the optimal vortex formation
concept are identified.
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Materials and methods |
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5 mm) were
placed in a Petri dish filled with saltwater and observed under a light
microscope. Swimming motions were elicited using tactile stimulation from an
external needle brought in contact with the tentacles. The results were
recorded digitally from a 720 x480 pixel CCD camera to a PC at 250
frames s-1. Motion of the velum was observed from camera views into
the oral cavity (from an oblique angle) and from the side during separate
swimming cycles, to ensure that the velum was properly identified. In
addition, a larger-field view was used during separate swimming cycles to
observe the wake created by the animals. This wake flow was made visible by
injecting a milk solution into the oral cavity prior to the introduction of
the tactile stimulation that elicited swimming motions. Since not all of the
injected dye was ejected during each contraction phase, residual dye in the
subumbrellar region facilitated visualizations of the subsequent refilling
phase as well.
Image analysis (Dabiri and Gharib,
2003) was used to reconstruct the animal morphology and kinematics
based on the manually selected locations of the apex of the subumbrellar oral
cavity, the edges of the subumbrellar margin, and the velar tips, as observed
in each frame of the video measurements
(Fig. 1C). The algorithm then
created a best-fit curve connecting the selected control points. The fidelity
of each morphological reconstruction was confirmed based on comparison with
local image intensity and contrast signatures in the original image frame.
Given the approximation that the animals are radially symmetrical [reasonable
for the present analysis (Gladfelter,
1972)], the elliptical reconstruction (e.g.
Fig. 1C) was sufficient to
compute the volume of the oral cavity for each video frame captured during the
swimming cycles. Volume measurements possess a maximum uncertainty of
±6%, stemming from determination of the oral cavity boundary (i.e.
broken curve in Fig. 1C).
Due to inherent challenges in capturing the rapid swimming motions in a manner such that they could be subsequently analyzed quantitatively (e.g. motion in a plane parallel to the camera image plane and away from the Petri dish bottom, straight swimming trajectories that are contained in the field of view, etc.), the number of specimens that could be examined quantitatively was limited (N=2). These animals exhibited behavior that was qualitatively and quantitatively similar when investigated in depth. However, the generality of the quantitative conclusions that can be drawn from this study is necessarily tempered by the limited sample size. In the subsequent presentation of measurements, two distinct swimming cycles are presented as an indication of the repeatability of the phenomena observed here.
Vortex formation parameter
The vortex `formation time', the dimensionless parameter which is used to
describe the vortex formation process, can be generalized
(Dabiri and Gharib, 2005a;
Dabiri and Gharib, 2005b) to
the case of a time-varying jet velocity U(t) and diameter
D(t) by expressing the parameter as an integral:
 | (1) |
where time t=0 corresponds to the start of fluid ejection (e.g.
the beginning of medusa bell contraction). A complementary modification
(Krueger et al., 2004) can be incorporated in Eqn 1 to account for flow past
the animal (irrespective of the animal shape) that occurs due to forward
motion of the animal relative to the surrounding fluid:
 | (2) |
where 
(t) is the
instantaneous forward velocity of the animal. The vortex formation time
parameter in Eqn 2 was computed for the N. bachei specimens based on
the measured forward velocity
(t) as well as the measured
oral cavity volume VOC(t) and velar diameter
DV(t):
 | (3) |
Note that the first term in the numerator of Eqn 3 is the jet velocity during fluid ejection and is positive since the oral cavity volume is decreasing during fluid ejection.
Fluid dynamic model
The measured body kinematics, VOC(t) and
DV(t), of swimming N. bachei were input
to a recently developed force estimation model
(Dabiri, 2005) to discern the
effect of velar kinematics on the swimming performance predicted by the model.
In the model (Dabiri, 2005),
the magnitude of the swimming force FL generated by N.
bachei during the contraction phase is dictated by the transfer of
momentum from the animal to the wake in the form of fluid vorticity (rotation
and shear) and wake vortex added-mass (cf.
Dabiri, 2006), an unsteady
contribution from the acceleration of the fluid vorticity.
It is important to note that the governing equation for force estimation in
Dabiri's model (Dabiri, 2005)
contains an error that is corrected here [see Appendix and published
Corrigendum (Dabiri, 2005)].
The correct equation for the locomotive force, in its most general form, is:
 | (4) |
where 
is the density of the fluid (water), 1 is the identity
matrix (a matrix with each diagonal element equal to 1), CAM
is the wake vortex added-mass tensor (a 3 x3 matrix in the present
case), 
is the velocity potential, and the normal vector n is
directed out of the vortex wake. The integral is evaluated on the surface
SV of the vortex wake. The added-mass tensor of a vortex
will typically have non-zero elements only on the diagonal (see Appendix).
Hence, the matrix inverse C-1AM in Eqn 4
can be evaluated in an element-wise fashion.
The integral of the velocity potential in Eqn 4 can be expressed in terms
of the volume VV(t) and velocity
UV(t) of the vortex wake (see Appendix):
 | (5) |
The volume VV(t) of the wake vortex created by
N. bachei during each swimming cycle was calculated by assuming that
all of the fluid ejected during a pulse is contained in the generated wake
vortex, and that an additional volume of fluid equal to 30% of the vortex
volume was also entrained into the wake vortex from the ambient fluid:
 | (6) |
or, using the present measurement data,
 | (7) |
The assigned 30% entrainment is compatible with laboratory studies of
vortex formation from pulsed jets (Dabiri
and Gharib, 2004).
The vortex velocity UV(t) was approximated as
the ratio of the vortex ring circulation 
(t) to the vortex
ring radius R(t). The circulation was estimated using the
`slug model' (Didden, 1979),
which assumes that fluid is initially ejected as a straight, spatially uniform
jet. With this assumption, the model gives:
 | (8) |
or, using the present measurement data,
 | (9) |
The second term in Eqn 9 reflects the fact that flow past the animal
reduces the relative strength (i.e. circulation content) of the ejected jet by
reducing its velocity relative to the ambient flow
(Krueger et al., 2003;
Krueger et al., 2006). The
vortex radius R(t) was derived from the volume calculation
in Eqn 7 as:
 | (10) |
Finally, dye visualizations (e.g. Fig.
4) indicated that that the wake vortex ring created by N.
bachei is closely approximated by an ellipsoid with added-mass
coefficient cii=0.6 in the axial direction. This component
of the full added-mass tensor is the only one of relevance since the net
locomotive force is aligned with the axial direction. Combining Eqn 5, Eqn 7,
Eqn 9 and Eqn 10, the complete equation for the locomotive force
FL deduced from the measurements of the N. bachei
wake is:
 | (11) |
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(t), the equations
must be solved iteratively. To do so, the animal was assumed to begin its
motion from rest [(0)=0] and the
swimming velocity was modelled based on the instantaneous swimming
acceleration 
(t) derived from
Newton's law:
 | (12) |
where m is the mass of the animal, estimated from the body volume
and by assuming that the animal is neutrally buoyant (i.e.
body=fluid). The swimming trajectory
X(t) was computed by integrating the forward swimming
velocity in Eqn 12 over the duration of the swimming cycle.
To complete the model of swimming dynamics, the refilling phase was assumed
to consist of a spatially uniform inflow
(Daniel, 1983). This
assumption is consistent with dye visualizations of the refilling phase (see
Results), which did not indicate the prominent vortex formation that has been
seen in a recent study of the oblate schyphozoan refilling phase
(Dabiri et al., 2005).
It is important to note that the swimming model (Eqn 12) does not include the effects of any resistive fluid dynamic forces. This omission stems from the fact that the required measurements of the oral cavity and velar kinematics could not be obtained from a view that contained the full exumbrellar surface. This outer surface profile is necessary in order to estimate both viscous drag and the acceleration reaction (added-mass) of the animal body. Although this deficiency limits comparison between the measurements and models to a qualitative analysis, it will be shown that qualitative comparison alone is sufficient to demonstrate the importance of velar kinematics for the observed swimming performance.
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8, greater than the vortex formation
number that was observed in studies of rigid tube pulsed jets to coincide with
the limit of vortex growth,
T*lim4(Fig.
3). If vortex growth had ceased at any value of
T* prior to end of the ejection phase [i.e. any
T*lim<T*max),
a pronounced trailing jet of fluid would be observed directly behind the
vortex [e.g. see fig. 3C
(Gharib et al., 1998)].
|
However, visualization of the wake of the N. bachei (Fig. 4) conclusively demonstrates that the animals do in fact create only a single vortex without a trailing jet during each jet ejection phase, despite the fact that the total vortex formation time T*max is much greater than 4. This result is examined further in the Discussion.
The importance of temporal variations in the velar diameter becomes
apparent when one examines their effect on the forward trajectory of the
animal. Fig. 5 plots the
measured trajectory of a N. bachei specimen during a single swimming
cycle (solid black line) and compares this with the aforementioned model
(Dabiri, 2005) of the
trajectory (Eqn 11, Eqn 12) based on either the measured time-varying velar
kinematics (broken black line) or a hypothetical constant velar diameter
(equal to the velar diameter at rest; dotted grey line). The results in
Fig. 5 are presented such that
the measured trajectory and the models each have the same maximum forward
motion. This presentation provides an objective comparison despite the fact
that the effect of resistive fluid forces is not included in the models, as
mentioned previously. The swimming model based on transient velar kinematics
agrees qualitatively with the trend observed empirically, while the model
assuming a constant velar diameter does not. Specifically, a constant velar
aperture would result in a significant backward motion of the animal during
the refilling phase. In contrast, thrust generated during the ejection phase
of the animals (real and modelled) with time-varying velar kinematics is
sufficient to fully compensate for the retarding effect of the refilling
phase. Hence, this qualitative comparison suggests that the velar motion
exhibited by N. bachei is integral to the observed swimming
performance of the animals.
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An important component of the dynamic model for the refilling phase was the
assumption that vortex ring formation does not affect this portion of the
swimming cycle in hydromedusae, and therefore the refilling flow may be
treated as a uniform inflow jet (Daniel,
1983). Dye visualizations support this assumption from two
perspectives. First, the vortex ring formed during the contraction phase has
moved sufficiently far downstream at the end of the contraction phase that the
flow field it induces at the subumbrellar margin is negligible
(Fig. 4). The magnitude of the
flow induced by the vortex is inversely proportional to the distance from the
vortex (Lamb, 1932). Second,
there is no visual evidence of a pronounced stopping vortex during the
refilling phase (see movie in supplementary material), as has been observed in
scyphomedusae (Dabiri et al.,
2005).
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). Effective swimming for these small hydromedusae means that
they are capable of accelerating fast enough to avoid predation and
efficiently enough to avoid wasting valuable energy resources. We have
demonstrated that N. bachei is able to manipulate its velar diameter
during swimming in a manner that enables the medusa to produce a jet that
maximizes thrust while minimizing energy costs. Specifically, contracting the
velum during bell contraction resulted in N. bachei being able to
achieve a total formation time T*max8 while
producing only a single vortex (i.e. without a thick trailing jet directly
behind it).
Previous analyses (e.g. Gharib et al.,
1998) have demonstrated that it is not possible to form a single
vortex from a jet ejected with a total vortex formation time
T*max8 if the velar diameter is constant
(given that the corresponding vortex formation number
T*lim for a pulsed jet ejected from a
constant-diameter orifice or nozzle is equal to 4). Based on studies with
constant aperture area jets, total vortex formation times
T*max>4 indicate that jet ejection continued
beyond the formation number (i.e. the formation time at which vortex growth
ceases) and that they are not functioning in an energetically efficient
manner. However, mechanically generated jet flows exhibiting a temporally
variable aperture area (Dabiri and Gharib,
2005b) suggest that the rate of velar diameter decrease achieved
by N. bachei (approximately 20% of the average jet velocity) is
sufficient to extend vortex growth beyond T*4. In the
mechanically generated jet flows, aperture diameter contraction rates between
15% and 30% of the average jet velocity were sufficient to prolong vortex
growth. Although the velar geometry and kinematics in those experiments were
not identical to those of N. bachei (e.g. the velar motion was
prescribed a priori), the vortex formation process in both cases
shares similar physics.
It is also observed that at the end of the ejection phase of swimming
motions, the vortex formation time exhibited by N. bachei
(T*8) coincides very closely with the time at which
vortex growth is expected to terminate, based on those previous mechanical
experiments (Fig. 3). Together,
these results suggest that N. bachei creates a single vortex during
the contraction phase of each swimming cycle and that the vortex is the
largest that can be achieved with its given velar kinematics, since the
duration of jet ejection coincides with the duration of vortex growth. Hence,
optimal vortex formation is achieved via the velar motions, in the
sense of the largest possible vortex without the formation of a trailing jet
directly behind the vortex. This results in a maximization of swimming thrust
while avoiding the penalty of increased energetic costs that is associated
with the formation of a trailing jet.
Our results demonstrate that the velum functions to tune fluid ejection,
achieving efficient swimming by exploiting the concept of optimal vortex
formation. Important differences may potentially exist between the
manifestation of optimal vortex formation in simplified laboratory experiments
and the real animals, especially in the value of the vortex formation number
parameter. These differences must be appreciated in comparative biological
analyses, lest spurious conclusions be reached regarding the
structure-function relationships exhibited by these animals. The
time-dependent velar motion has been shown to be integral to the observed
swimming performance. The model prediction of backward swimming motion for a
constant velar diameter is consistent with previous observations of reduced
swimming performance in hydromedusae with the velum excised
(Gladfelter, 1972).
It is important to note the possibility of even further improved swimming
performance in other aquatic jet-propelled swimmers. While in the present case
vortex growth was extended until T*8, recent
theoretical studies suggest that vortex formation can, in principle, be
extended by another 35%, approaching T*11
(Kaplanski and Rudi, 2005).
This room for improvement may already be occupied by jetters whose performance
is known to surpass that of Nemopsis bachei, such as Aglantha
digitale (Colin and Costello,
2002).
The discussion of swimming energetics has thus far focused on the vortex
wake created by N. bachei. It is useful to consider also whether the
shape of the subumbrellar region and the velum may inherently pose energetic
benefits for locomotion outside of the aforementioned wake dynamics.
Engineering studies of flow exiting through orifices and nozzles have
demonstrated that fluid energy losses are a direct function of the exit
boundary shape (Smits, 2000).
As a rigid structure, the resting orientation of the N. bachei velum
in a plane normal to the jet flow would promote flow separation upstream of
the velar exit, resulting in energy losses during the process of fluid
ejection. By contrast, the outpocketed nozzle configuration of the velum
during the contraction phase minimizes upstream flow separation. We therefore
hypothesize that the flow-induced deformation of the velum from its resting
configuration to the outpocketed nozzle shape is a passive mechanism that
functions to minimize shape-related energy losses during the swimming cycle.
Although the same benefit to the contraction phase could be achieved by a
rigid velum in the outpocketed configuration, the flow during the refilling
phase would be made much less efficient. The resting position of the velum and
its flexibility together enable flow-induced forces to create an
energy-preserving nozzle passage during both fluid ejection and refilling,
whereas a rigid outpocketed velum could not. The energy required to deform the
elastic velum in the process is likely negligible.
Finally, the concept of optimal vortex formation is not limited to the
formation of single vortices. Some species of jetting squid are known to form
a thick trailing jet flow directly behind the vortex, indicating that jet
ejection is continued after vortex growth has ceased
(Anderson and Grosenbaugh,
2005). However, temporal variations in the aperture area can still
be coordinated with the vortex formation time T* for
alternative behavioural aims, such as the maximization of absolute thrust
magnitude, irrespective of the required energy input
(Dabiri and Gharib, 2005a).
Such a behaviour would be appropriate in life-threatening circumstances, such
as during escape from predators. In each case, the velum or analogous
structure appears to be a primary factor contributing to success of
fast-swimming jetters, despite their primitive body plans. The appearance of
optimal vortex formation as a successful strategy in response to selective
pressures at the relatively low Reynolds numbers observed here
(Re<200) supports that the notion that optimal vortex formation
may exist even more broadly in aquatic locomotion and biological propulsion.
This suggestion of a broader role for the concept of optimal vortex formation
in swimming and flying has been buoyed by recent comparative biomechanics
studies at higher Reynolds numbers (Linden
and Turner, 2004; Dabiri and
Gharib, 2005a; Milano and
Gharib, 2005).
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). Traditional
analyses make use of the vortex impulse IV, which can be
computed given the distribution of vorticity in the (assumed
three-dimensional) wake:
 | (A1) |
The vortex impulse accounts for the total linear momentum carried by a
vortex if and only if that vortex is moving at its self-induced velocity
USI (i.e. the velocity induced by the distribution of
vorticity on itself). This can be seen more explicitly by rewriting the
integral in Eqn A1 using the theorem of Burgatti
(Saffman, 1992):
 | (A2) |
The first integral on the right-hand side of Eqn A2 is by definition equal
to the linear momentum of the fluid inside the wake vortex volume
VV. If the center of mass of the wake vortex moves at
USI (e.g. the aforementioned case of self-induced motion),
we have:
 | (A3) |
The second integral on the right-hand side of Eqn A2 - negative sign
included - is the contribution from wake vortex added-mass. By definition, it
is equal to the linear momentum of fluid surrounding the wake vortex volume
VV, and is computed by integrating the velocity potential
on the surface SV of the wake vortex. The negative
sign in Eqn A2 arises from the convention that the unit normal n is
directed out of the wake vortex into the surrounding fluid. This second
integral can be expressed in terms of the wake vortex volume
VV and the self-induced vortex velocity
USI using the wake vortex added-mass tensor
CAM:
 | (A4) |
Substituting Eqn A2-A4 into Eqn A1 and taking the time-derivative, we can
deduce that the locomotive force FL arising due to the
creation of a steady vortex propagating at its self-induced velocity
USI is given by:
 | (A5) |
where 1 is the identity matrix
(Lamb, 1932). It is worth
re-emphasizing that only in the case of a steady flow can the locomotive force
FL be deduced solely from the vortex impulse Eqn A1. In
general, flow unsteadiness caused by ambient flow conditions (relative to the
animal) and any interactions between the wake and the animal body/appendages
(e.g. during vortex formation, wake capture, etc.) will cause the vortex wake
to propagate at a velocity different from its self-induced velocity
USI. In these cases, the actual vortex motion
UV replaces the self-induced velocity USI
in Eqn A3 and Eqn A4. However, their sum,
 | (A6) |
is no longer equivalent to the integral in Eqn A1. Dabiri's equation 4
(Dabiri, 2005) incorrectly
accounts for the additional unsteady contribution by adding the added-mass
integral in Eqn A4 to the vortex impulse in Eqn A1. The proper solution is to
use Eqn A6 in its current form, as done in the present paper. Alternatively,
if an integral form is preferred (e.g. for numerical simulations), one can
rewrite the identity matrix in Eqn A6 using 1=C
-1AM CAM, factor out the added mass
tensor, and use Eqn A4 to arrive at the relation:
 | (A7) |
The locomotive force is then given by Eqn 4. The validity of Eqn A7 is dependent on the wake vortex added-mass tensor being non-singular, so that the matrix inverse can be performed. This does not appear to be a limitation in practice since the presence of fluid viscosity suggests that the wake vortex boundary SV will typically be geometrically regular (The interested reader is encouraged to pursue a rigorous proof of this statement). When the wake vortex possesses two orthogonal planes of symmetry, the added-mass tensor CAM will have non-zero elements only along the diagonal. Hence, the inverse can be computed in an element-wise fashion.
Dabiri introduced the wake vortex ratio (Wa) as a measure of the importance
of flow unsteadiness to the total locomotive force
(Dabiri, 2005). The preceding
developments enable us to refine both the definition and quantitative
interpretation of the wake vortex ratio. Let us begin with the parameter Wa as
defined by Dabiri (Dabiri,
2005):
 | (A8) |
where 
is the
two-dimensional approximation for the wake vortex added-mass coefficient
corresponding to unidirectional motion in the i-direction, S
is the wake vortex width, and UVi is the total wake vortex
speed in the i-direction. The ratio in Eqn A8 compares the linear
momentum of the wake vortex added-mass (propagating with the wake vortex at
velocity UV) to the steady vortex momentum (i.e. wake vortex
propagation at the self-induced velocity USI instead of its
actual velocity UV); for details of the derivation, see
(Dabiri, 2005). Eqn A4-A6
indicate that the wake vortex ratio can be written more generally as:
 | (A9) |
where the L2-norm || || denotes the magnitude of the
enclosed vector, and UU is the unsteady component of the
wake vortex velocity, i.e.
UV-UU=USI. For
unidirectional vortex motion in the i-direction, straightforward
algebraic manipulation reduces Eqn A9 to:
 | (A10) |
Inspection of Eqn A10 suggests a simple, objective (i.e. independent of
measurement precision) criterion to determine the importance of unsteady wake
dynamics in locomotive forces: flow unsteadiness should be considered if:
 | (A11) |
Eqn A10 indicates that the left- and right-hand sides of Eqn A11 are equal
in the limit of a steady flow (UU=0), in which case Eqn A5
also holds. Alternatively, where Eqn A11 holds, Eqn A5 is no longer valid, and
unsteady wake vortex dynamics must be considered. In general, the wake vortex
added-mass tensor will be time-dependent, such that the relative importance of
flow unsteadiness varies during the propulsive cycle. For example, in the
period during the propulsive cycle when the vortex wake created by N.
bachei exhibits an added-mass coefficient cii=0.6,
the present criterion suggests that flow unsteadiness contributes to the
locomotive force if
 | (A12) |
Whether the unsteady wake vortex dynamics augment the locomotive force or reduce it can only be determined upon further investigation of the fluid dynamics, e.g. using Eqn A6.
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Footnotes |
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References |
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(2001). Swimming mechanics and behavior of the shallow-water
brief squid Lolliguncula brevis. J. Exp. Biol.
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Colin, S. P. and Costello, J. H. (2002).
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hydromedusae. J. Exp. Biol.
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Colin, S. P., Costello, J. H. and Klos, E. (2003). In situ swimming and feeding behavior of eight co-occurring hydromedusae. Mar. Ecol. Prog. Ser. 253,305 -309.
Dabiri, J. O. (2005). On the estimation of
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Dabiri, J. O. (2006). Note on the induced Lagrangian drift and added-mass of a vortex. J. Fluid Mech. 547,105 -113.[CrossRef]
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