## SUMMARY

The flow structures produced by the hydromedusae *Sarsia tubulosa*
and *Aequorea victoria* are examined using direct numerical simulation
and Lagrangian coherent structures (LCS). Body motion of each hydromedusa is
digitized and input to a CFD program. *Sarsia tubulosa* uses a jetting
type of propulsion, emitting a single, strong, fast-moving vortex ring during
each swimming cycle while a secondary vortex of opposite rotation remains
trapped within the subumbrellar region. The ejected vortex is highly energetic
and moves away from the hydromedusa very rapidly. Conversely, *A.
victoria*, a paddling type hydromedusa, is found to draw fluid from the
upper bell surface and eject this fluid in pairs of counter-rotating,
slow-moving vortices near the bell margins. Unlike *S. tubulosa*, both
vortices are ejected during the swimming cycle of *A. victoria* and
linger in the tentacle region. In fact, we find that *A. victoria* and
*S. tubulosa* swim with Strouhal numbers of 1.1 and 0.1, respectively.
This means that vortices produced by *A. victoria* remain in the
tentacle region roughly 10 times as long as those produced by *S.
tubulosa*, which presents an excellent feeding opportunity during swimming
for *A. victoria*. Finally, we examine the pressure on the interior
bell surface of both hydromedusae and the velocity profile in the wake. We
find that *S. tubulosa* produces very uniform pressure on the interior
of the bell as well as a very uniform jet velocity across the velar opening.
This type of swimming can be well approximated by a slug model, but *A.
victoria* creates more complicated pressure and velocity profiles. We are
also able to estimate the power output of *S. tubulosa* and find good
agreement with other hydromedusan power outputs. All results are based on
numerical simulations of the swimming jellyfish.

## INTRODUCTION

Two distinct types of hydromedusan propulsion are well known
(Colin and Costello, 2002).
Prolate species such as *Sarsia tubulosa* primarily use a jetting type
of propulsion with large jet velocities immediately behind the velar aperture.
Their swimming is characterized by quick accelerations during the contraction
phase of swimming followed by periods of gliding with relatively small
accelerations. However, the once widely accepted jetting model fails to
explain the swimming patterns seen in prolate species such as *Aequorea
victoria*. These hydromedusae use a paddling or rowing motion to swim and
produce more diffuse vortices shed from the bell margins during the
contraction phase.

Prolate, jetting hydromedusae retract their tentacles during swimming and feed by extending their tentacles while drifting (Madin, 1988). Swimming is used to escape predators or to ambush prey. Swimming and feeding are disparate activities since extending the tentacles during swimming could greatly decrease swimming performance. Conversely, oblate, paddling hydromedusae leave their tentacles extended during swimming and the vortices produced during swimming travel through the extended tentacles (Colin and Costello, 2002; Colin et al., 2003; Costello, 1992; Costello and Colin, 1994; Ford and Costello, 1997). Prey in the fluid near the bell region of paddling hydromedusae have even been observed to be carried into contact with the tentacles by the vortices formed during swimming (Costello and Colin, 1995). For these reasons, swimming complements feeding in paddling hydromedusae by helping to draw prey into the tentacles.

It has been shown that rowing propulsion is a necessary adaptation for larger hydromedusae due to morphological constraints and energy efficiency. For jetting propulsion, the force necessary for propulsion increases faster with size than the available muscle force to provide the jetting motion (Dabiri et al., 2007). Oblate hydromedusae make up for this by employing paddling type propulsion. Models show that the production of stopping vortices during the relaxation phase of paddling type propulsion allows large hydromedusae to swim effectively despite their morphological constraints (Dabiri et al., 2007). Specifically, the stopping vortex partially cancels the starting vortex, reducing the induced drag on the oblate hydromedusa and increasing swimming efficiency. The smaller, prolate species have lower drag due to their shape and further decrease drag by retracting their tentacles while swimming (Colin et al., 2003). These factors, combined with more rapid bell contractions, make jetting hydromedusae much more proficient swimmers than their oblate relatives (Daniel, 1983).

We are interested in hydromedusan propulsion as a basis for the design of new propulsion technologies for underwater vehicles. Recently, jet and vortex propulsion have become a focus in the areas of underwater maneuvering and locomotion of bio-engineered vehicles. A vortex thruster loosely mimicking hydromedusa propulsion was proposed by Mohseni (Mohseni, 2004; Mohseni, 2006). The current generation of these thrusters and their implementation on an underwater vehicle are discussed by Krieg and Mohseni (Krieg and Mohseni, 2008) and are capable of producing very strong vortices (formation time of up to 15, see Results for a discussion of formation numbers). Mohseni et al. (Mohseni et al., 2001) and Dabiri and Gharib (Dabiri and Gharib, 2005) also reported numerical and experimental results for jets formed from a nozzle with temporally varying exit velocity and diameter, respectively.

In the present study, we present the Lagrangian coherent structures (LCS)
seen in the results of numerical simulations of hydromedusae swimming as well
as several examples of particle motion in the resulting flow. The hydromedusae
examined are *Aequorea victoria* Murbach and Shearer 1902, a paddling
or rowing type of hydromedusa, and *Sarsia tubulosa* M. Sars 1835, a
jetting type of hydromedusa. We believe this to be the first numerical study
of this type. The actual motion of the hydromedusa, reproduced from digitized
videos of the swimming hydromedusae, is used to compute the surrounding
velocity field. A brief description of the numerical method for computing the
velocity field is included in Materials and methods. The use of computational
fluid dynamics (CFD) data instead of an empirical velocity field from digital
particle image velocimetry (DPIV), or similar, results in higher resolution of
the LCS as well as greater accuracy in subsequent calculations. Additionally,
there are significant difficulties in obtaining high-quality results from DPIV
for swimming hydromedusae. DPIV results are only available for the time during
which the hydromedusa is properly oriented within the field of view, perhaps
only a few swimming cycles depending on many factors. Additionally, the
resolution obtained from DPIV depends on the concentration of particles in a
given region. In general, the distribution of particles may be highly
non-uniform. The particles will be drawn toward certain flow structures, just
as dye is drawn into vortices in dye visualization experiments, but other
areas of the flow may be left with few particles. None of these difficulties
are present in our method. It is only necessary to capture a good swimming
cycle. The periodic swimming motion may then be determined up to the
resolution of the camera used and iterated for as many swimming cycles as
desired.

As expected, *S. tubulosa* produces strong vortices along the axis
of symmetry. These vortices move quickly away from the hydromedusa, providing
a high momentum transfer for rapid swimming while negating opportunities for
feeding while swimming. In fact, if *S. tubulosa* were to extend its
tentacles during swimming, they would create additional drag and could
negatively impact swimming performance. Conversely, the structures formed by
the paddling hydromedusa, *A. victoria*, transport fluid from the outer
bell surface to linger in the tentacle region, enhancing feeding opportunities
since the flow passes through the region where the tentacles drift.

We also note the presence of previously unobserved flow structures in the
subumbrellar region of *S. tubulosa*. As previously mentioned, *A.
victoria* produces a starting and stopping vortex during each swimming
cycle. Since these vortices are ejected together during the contraction phase,
they interact and influence each other. We find that *S. tubulosa* also
produces a stopping vortex during the relaxation phase. However, only the
starting vortex is ejected during contraction while the stopping vortex mostly
dissipates within the subumbrellar cavity. Finally, we examine the pressure on
the subumbrellar wall of both hydromedusae as well as the velocity profiles in
the wake and across the velar opening. *Sarsia tubulosa* produces a
nearly uniform jet through the velar opening and an equally uniform pressure
along the subumbrellar wall. We expect that this type of swimming could be
very well approximated by a slug model or a piston-cylinder arrangement.
Conversely, *A. victoria* produces a much more complicated wake and
pressure profile.

## MATERIALS AND METHODS

### Hydromedusa motion

The motion of each hydromedusa was determined from videos of physical specimens of the swimming hydromedusae. These videos were provided by Dr Sean Colin (Roger Williams University) and are further discussed in Sahin et al. (Sahin et al., 2009). The hydromedusae were placed in filtered seawater within a glass vessel of sufficient size to allow each hydromedusa to swim freely. The outline of the bell was illuminated using a planar laser directed through the central axis. Fluorescein dye was injected near the bell to enhance the illumination.

After recording, each frame of the video was analyzed, and the body motion
of the hydromedusa was determined. The geometry of the hydromedusa was
approximated using NURBS curves (Piegl,
1991), and Fourier-series interpolation in time was used to create
a numerical model of the periodic contraction of the swimming hydromedusae. In
contrast to the relatively short time frames allowed by using DPIV, this model
allows us to analyze many periods of swimming *via* numerical solution
for the flow around the hydromedusa.

### Numerical procedure

For completeness, a brief overview of the numerical procedure used in the computation of the velocity field induced by the swimming hydromedusae is included here. For complete details of the numerical procedure we have developed for this problem, including code validation, see Sahin and Mohseni (Sahin and Mohseni, 2008a; Sahin and Mohseni, 2008b). The flow field was computed based on the periodic swimming model derived from the videos of swimming hydromedusae. The surrounding velocity field was computed using a new arbitrary Lagrangian–Eulerian (ALE) (Hirt et al., 1974) method developed for this purpose. In this method, the mesh follows the moving boundary between the fluid and the hydromedusa body, and the cylindrically symmetric governing equations are solved on a moving, unstructured quadrilateral mesh. The pressure is solved on a staggered grid, eliminating the need for pressure boundary conditions since pressure is defined only at interior points. The mesh motion is determined by solving the linear elasticity equation at each time to avoid remeshing (Dwight, 2006; Johnson and Tezduyar, 1994), and the linear systems produced by the discretization are solved using the GMRES method (Saad and Schultz, 1986) combined with several preconditioners.

### Lagrangian coherent structures

Lagrangian coherent structures provide a method of analyzing a flow field from a dynamical systems perspective. LCS were introduced by Haller and Yuan (Haller and Yuan, 2000) and further defined by Shadden et al. (Shadden et al., 2005). LCS represent lines of negligible fluid flux in a flow and therefore govern transport and mixing in the flow. Due to the general framework provided by LCS, they have been applied to a wide range of different areas including transport of pollutants in the ocean (Lekien et al., 2005), two-dimensional turbulence (Haller and Yuan, 2000; Manikandan et al., 2007), vortex shedding behind an airfoil (Cardwell and Mohseni, 2008; Lipinski et al., 2008) and transport in empirical vortex rings as well as hydromedusa swimming (Shadden et al., 2006). LCS have proven to be an effective tool for identifying exact vortex boundaries and can even be used to divide a flow into lobes that govern transport, as is done in classical lobe dynamics analysis (Rom-Kedar and Wiggins, 1990). We follow the procedure for computing LCS outlined by Shadden et al. (Shadden et al., 2005) and we provide here a brief overview for those unfamiliar with this concept.

It is simplest to think of LCS as a post-processing technique to reveal
coherent structures in a given flow. In our case, the flow is determined by
numerical simulations of the swimming hydromedusae. LCS are based on the
finite time Lyapunov exponent (FTLE), which is analogous to the standard
Lyapunov exponent of classical dynamical systems theory. The FTLE is defined
as:
(1)
where *t*_{0} is the time being considered, *T* is the
`integration time', which will be further explained later, **x** is the
position vector, and λ_{max}(Δ) is the maximum eigenvalue
of the finite time deformation tensor, Δ. In practice, the domain of
interest is seeded with passive tracer particles, which are then advected from
time *t*_{0} to time *t*_{0}*+T* using
the known velocity field. The resulting particle positions are then used to
compute Δ and then the FTLE field. The flow considered here is a
two-dimensional, axisymmetric flow, and Δ was calculated appropriately
to take this into account. In swirl-free axisymmetric coordinates,
(*r*, θ, *x*), Δ becomes:
(2)
(3)
where *r*_{i}, *r*_{f}, *x*_{i}
and *x*_{f} are the initial and final radial and axial
coordinates of a particle in the flow, respectively.

The resulting FTLE field depends on the integration time, *T*, in
that larger values of *T* reveal more structures than smaller values of
*T*. Therefore, *T* may be chosen to reveal the desired level of
detail without worrying about influencing the major structures that are
revealed. Additionally, *T* may be positive or negative, representing
forward and backward particle advection, respectively. Therefore, there are
two types of FTLE fields: forward time and backward time.

Once the FTLE field has been calculated, LCS are defined as ridges in the FTLE field, following Shadden et al. (Shadden et al., 2005). In practice, LCS are usually visualized by looking at contour plots of the FTLE field. Conceptually, ridges in the forward FTLE field, called forward LCS, represent lines where particles diverge most quickly, and backward LCS represent lines where particles converge. For this region, dye visualization experiments reveal structures very similar to backward LCS. Additionally, for LCS which are sufficiently strong, the flux across the LCS is negligible, a property that makes LCS extremely useful for analyzing transport in flows. Finally, forward LCS are analogous to the stable manifolds of a dynamical system and act as repelling material lines while backward LCS are analogous to the unstable manifolds of a dynamical system and act as attracting material lines. The interaction of these LCS largely govern transport in a flow, and their intersections can be used to exactly define a vortex without the use of arbitrary thresholds of vorticity (Shadden et al., 2006).

## RESULTS

### Sarsia tubulosa

*Sarsia tubulosa* employs a jetting type of propulsion. The swimming
cycle consists of three phases: (1) a rapid contraction, (2) relaxation and
(3) a coasting phase. Each cycle results in the expulsion of a strong vortex
along the axis of symmetry, which rapidly propels the jellyfish forward as
seen in the LCS shown in Fig. 1
(also see Movie 1 in supplementary material).

The jetting nature of *S. tubulosa*'s propulsion, forming a single
vortex ring with each swimming pulse, is clearly reflected in the LCS seen in
Fig. 1. During the contraction
phase, a vortex which draws fluid in the positive *x*-direction along
the axis of symmetry is ejected from the hydromedusa. This will be referred to
as the starting vortex. One of the most striking details of this figure is the
presence of very complex flow structures within the subumbrellar region of the
jetting hydromedusa. This region is very difficult to view experimentally with
DPIV or other techniques. However, our numerical technique shows that during
the relaxation phase of swimming (Fig.
1C), a stopping vortex of opposite sense to the ejected vortex
forms within the subumbrellar cavity. In this hydromedusa, the presence of the
velum traps the stopping vortex within the subumbrellar cavity. Since the
stopping vortex is not ejected, it cannot interact with the ejected starting
vortex. This is the first time this stopping vortex has been observed in
jetting hydromedusae, likely due to the difficulty in imaging the subumbrellar
cavity of jetting hydromedusae during experiments. In fact, it has been
recently stated that no stopping vortex is formed in a jetting swimmer
(Weston et al., 2009), but we
suspect this vortex is present, even if difficult to detect, in all jetting as
well as paddling hydromedusae.

Our simulations for *S. tubulosa* used a hydromedusa with a maximum
bell radius of 0.63 cm, a minimum bell radius of 0.57 cm and a swimming cycle
length of 1 s with 100 time steps per cycle. In addition, the subumbrellar
volume had a maximum value of approximately 0.45 cm^{3} and a minimum
value of 0.26 cm^{3}.

*Sarsia tubulosa* is very efficient at producing a strong vortex. We
can quantify this ability by examining the formation time of the vortices
produced. Gharib et al. (Gharib et al.,
1998) defined the dimensionless formation time as:
(4)
for a jet of velocity *U*_{e} through a nozzle of constant
diameter *D*_{e} over a time *t* where the bar denotes a
running mean. However, for a non-constant exit diameter it is necessary to
consider an integral form of this equation. Calculations for a slug of fluid
ejected from an orifice with time varying diameter and velocity were presented
by Mohseni (Mohseni, 2000).
Using a similar approach, Dabiri and Gharib
(Dabiri and Gharib, 2005)
derived:
(5)
In the case of constant density flow, such as a hydromedusa swimming in water,
conservation of mass allows us to express this as:
(6)
where *V*_{c} is the volume of the cavity; in our case, the
subumbrellar volume.

A larger formation time indicates the formation of a more effective vortex,
where an impulsively started piston-cylinder arrangement produces a formation
time of about 4, after which additional fluid expulsion results in a trailing
jet behind the vortex (Gharib et al.,
1998). *Sarsia tubulosa* produces a vortex with a formation
time *T*^{*}≈7. This large formation number was expected, as
discussed by Dabiri et al. (Dabiri et al.,
2006), and is made possible largely by the decrease in diameter of
the velar opening during the contraction phase of swimming (see
Fig. 2). A trailing jet
represents a decreased efficiency of momentum and energy transfer
(Krueger and Gharib, 2003).
The vortex ring produced by *S. tubulosa* has no such trailing jet,
meaning the vortices are formed with maximal efficiency.

The axial velocity in the wake is plotted in
Fig. 3. This plot represents
one swimming cycle and clearly shows that the velocity disturbances are
concentrated near the axis of symmetry. In fact, beyond about one-half a
diameter away from the axis of symmetry, the disturbances quickly decay to
negligible levels. As a vortex is produced, a large axial velocity appears at
the velar opening of the hydromedusa, which then decays as the vortex moves
away from the hydromedusa. Also, the velocity in the wake is almost entirely
positive due to the jetting nature of the swimming. The strong jet can also be
seen in the plot of axial velocity across the velar opening as well as the
pressure on the subumbrellar wall (Fig.
4 and Fig. 5).
Notice, in particular, that the velocity across the velar opening forms a very
uniform jet, with only a small shear layer near the velum. This is in contrast
to the profile that we will see for *A. victoria*. Additionally, we can
divide the swimming cycle for *S. tubulosa* into three parts: a strong
contraction, a relaxation phase and then a brief coasting phase before the
next contraction. These three phases can be clearly seen as a decrease, then
increase, in the subumbrellar volume followed by a time of nearly constant
volume as seen in Fig. 2.

By noting that *S. tubulosa* is a jetting swimmer, propelling itself
*via* a jet created by pressurizing the subumbrellar cavity, we can
calculate the power output similarly to a biological (such as a heart) or
mechanical pump, as has been done for squids
(O'Dor, 1988). The power
output is simply given by
*P*_{out}=*pQ̇*, where
*p* is the subumbrellar pressure and
*Q̇* is the jet flow rate. Since the
pressure and jet velocity are very uniform in space (see Figs
4 and
5), we can use the mean value
at each time step without losing much accuracy.
*Q̇, p* and *P*_{out} are
plotted in Fig. 6.

The mean power output is found to be about 16 g cm^{2}
s^{–3}. To account for body size, power is divided by mass to
the 5/3 power (see Daniel,
1983), where this *S. tubulosa* has a mass of ∼0.65 g
as calculated from the volume of the body and the assumption of neutral
buoyancy (1 g cm^{–3}). This results in a mean power output of∼
0.33 W kg^{–5/3}, which is within the reported range
(0.2–0.75 W kg^{–5/3}) of the experimentally measured
power outputs for the hydromedusae *Gonionemus vertens* and
*Stomotoca atra* (Daniel,
1985).

Since there is only negligible flux across LCS, they largely govern
transport in a given flow (Shadden et al.,
2005). In fact, the intersections of forward and backward LCS
divide the flow into lobes that have distinct mixing characteristics. It is
interesting to ask where the particles in a vortex come from. Figs
7,
8,
9 show the motion of passive
tracers for the swimming *S. tubulosa* (also see Movie 2 in
supplementary material). In this jetting hydromedusa, the LCS are complicated
and evolve very quickly during the contraction phase so that the groups of
tracers that end up in a vortex are not clearly separated upstream of the
hydromedusa (Fig. 7A). As the
hydromedusa swims, the green group of tracers is pulled into the subumbrellar
cavity while the blue and red groups collect outside the bell (see
Fig. 8B). Then, just before the
contraction phase, the blue group is pulled into the subumbrellar cavity while
the red group remains outside (Fig.
9A). The red and blue groups of tracers merge and are ejected with
the vortex ring, at which point the dark red group of tracers begins to be
wrapped into the vortex ring as well (Fig.
9B,C). The ejected drifters travel away from the hydromedusa very
quickly as they are carried with the traveling vortex ring.

The best way to quantify this is to define the Strouhal number as
*St*=*fL*/*v*, where *f* is frequency, *L*
is the mean hydromedusa radius and *v* is the rate of vortex separation
from the hydromedusa. *Sarsia tubulosa* has a Strouhal number of about
0.10, meaning that the vortices separate from the hydromedusa by about 10
radii per swimming cycle. As we will discuss in more detail later, this
presents little opportunity for feeding during swimming.

### Aequorea victoria

The paddling propulsion of *A. victoria* is very different from the
jetting propulsion of *S. tubulosa*. This paddling or rowing propulsion
produces two vortices during each swimming cycle, which are ejected together
during the contraction phase. This results in more energy-efficient swimming
but cannot provide the fast accelerations and rapid swimming seen in jetting
hydromedusa. The LCS produced by *A. victoria* are markedly different
as well. The forward and backward LCS can be seen in
Fig. 10 (also see Movie 3 in
supplementary material).

The *A. victoria* hydromedusa used in our simulations has a maximum
bell radius of 2.3 cm and a minimum bell radius of 1.9 cm during contraction
and is therefore much larger than *S. tubulosa*. The swimming cycle
took 1.17 s, and 100 time steps per cycle were used for the CFD runs. Since
*A. victoria* does not use a jetting motion to swim, the concept of a
vortex formation time does not make much sense. However, if we naively define
a vortex formation time by considering the volume in the subumbrellar region,
this results in a formation number of *T*^{*}≈0.07. A very
small formation time is characteristic of a thin vortex ring located away from
the axis of symmetry, which is precisely what we see in *A. victoria*.
Clearly, *A. victoria* has not optimized its swimming to produce the
most powerful vortices since even nozzles of constant diameter are capable of
producing vortices with *T*^{*}≈4
(Gharib et al., 1998).

The axial velocity in the wake is plotted in
Fig. 11. Near the axis of
symmetry, the flow moves toward the jellyfish, but near the bell margins the
axial flow velocity oscillates with the swimming strokes. Beyond the bell
margins, the flow disturbances quickly decay, leaving the flow far from the
axis of symmetry largely undisturbed as well. The axial velocity across the
velar opening and the pressure on the subumbrellar surface of the hydromedusa
are also presented in Fig. 12
and Fig. 13. Note that, again,
the large changes in velocity across the velar opening as well as pressure on
the hydromedusa occur near the bell margins. Additionally, the swimming cycle
for *A. victoria* is equally split into a contraction phase and a
relaxation phase as it uses a rowing motion to propel itself forward.

These results are all significantly different from those seen for *S.
tubulosa*, which produced velocity disturbances only near the axis of
symmetry and displayed a very uniform pressure profile and velocity across the
velar opening. Additionally, there is no coasting phase for *A.
victoria*. Each swimming cycle is a continuous transition from contraction
to relaxation and back to contraction. Due to these fundamental differences in
locomotion, it is not possible to calculate the power output of *A.
victoria* in the same way we have done for *S. tubulosa*. Although
this is still possible, it requires a more complicated analysis of the
surrounding flow.

The resulting forward and backward LCS are shown in
Fig. 10. The backward LCS show
attracting material manifolds and reveal vortical structures that look very
similar to the results of previous dye visualization experiments
(Dabiri et al., 2007;
Costello et al., 2008). As
discussed by Dabiri et al., the paddling type of hydromedusa creates two
vortices of opposite rotation during each swimming cycle
(Dabiri et al., 2007). During
the relaxation phase (Fig.
10D), a stopping vortex forms in the subumbrellar region with a
rotation that draws fluid towards the hydromedusa along the axis. Then, during
the contraction phase (Fig.
10B), a starting vortex of opposite sense is formed very near the
first, and these two vortices are ejected from the hydromedusa together. Once
the vortices have been ejected, the weaker stopping vortex acts to cancel out
some of the vorticity from the starting vortex. This improves the swimming
efficiency of the hydromedusa (Dabiri et
al., 2007). This swimming motion is repeated periodically,
generating a series of vortices and propelling the hydromedusa forward. Unlike
*S. tubulosa*, where only the starting vortex has been previously
noted, both of these have been observed before and are known to play a key
role in the swimming of *A. victoria*.

Fig. 14 and Fig. 15 show the motion of passive tracers placed in the flow that end up in the ejected vortices (also see Movie 4 in supplementary material). Note that, in Fig. 10B, during the contraction phase, the forward LCS have formed a loop along the outer surface of the bell, which is labeled A in the figure, as well as a loop in the subumbrellar region (labeled B). Tracers that end up in lobe A begin upstream of the hydromedusa in one coherent group and are swept around the tip of the bell in one cycle (see Fig. 14F to Fig. 15B). On the other hand, the tracers in lobe B are more dispersed until they group together in Fig. 14E. As a contraction takes place, lobe A is swept into the subumbrellar region, along with the tracers contained therein, and combines with lobe B, merging the two groups of tracers. From here, the tracers are immediately ejected with the next vortex pair.

Since particles (such as food) are collected at the core of the vortex, the
ejected vortices and the particles contained therein remain at about the same
radius as the bell margin and the hydromedusa's tentacles as the hydromedusa
moves upstream. In fact, *A. victoria* swims with *St*≈1.1,
meaning that the particles in a vortex separate downstream from the
hydromedusa at a rate of only 1.1 radius per swimming cycle, in contrast to
the 10 radius separation seen for *S. tubulosa*. This provides an
excellent opportunity for the hydromedusa to feed and offers a plausible
explanation for why *A. victoria* swims with its tentacles
extended.

## DISCUSSION

We have seen that *S. tubulosa* and *A. victoria* fall into
two different categories based on their method of swimming and that feeding
and swimming are coupled activities. For hydromedusae, effective feeding is
completely dependent on bringing prey into contact with the tentacles or oral
arms, where it may be captured. These two groups have addressed this problem
in very different ways.

The jetting hydromedusa, *S. tubulosa*, retracts its tentacles while
swimming and feeds primarily by ambushing its prey. Swimming is used primarily
to escape predators or relocate to a new feeding location. To this end,
jetting hydromedusae have optimized their swimming to move quickly, despite
the extra energy costs.

*Sarsia tubulosa* takes advantage of its jetting motion to produce
large accelerations to escape predators and reposition itself for feeding.
*Aequorea victoria*, however, experiences much lower accelerations and
uses swimming as an extension of its feeding mechanism. In fact, the velocity
profile in the wake of *A. victoria*
(Fig. 11) shows significant
negative velocities in the wake near the axis of symmetry. This indicates a
large added mass force that inhibits acceleration of the hydromedusa.

*Aequeria victoria* feeds by swimming with its tentacles extended.
In fact, oblate hydromedusae spend almost all their time swimming with
tentacles extended (Colin et al.,
2003). This is effective because each swimming stroke acts to
transport fluid that may contain food into the region of the tentacles in a
way that enables prey capture (see Fig.
15C).

The relatively high Strouhal number seen for *A. Victoria*
(*St*≈1.1) indicates that the produced vortices separate at a rate
of about 1.1 hydromedusa radii per swimming cycle (see
Fig. 10). During this time,
the particles entrained in the vortices are transported through the tentacles,
presenting an excellent opportunity for prey capture. Furthermore, as the
starting and stopping vortices interact, the vortices are stretched (see
Fig. 15D) and partially cancel
each other due to viscous effects and vorticity diffusion, decreasing the
rotation rate. This creates a relatively slowly rotating and translating
vortex, further enhancing the chance for prey capture.

*Sarsia tubulosa* swims with a much lower Strouhal number
(*St*≈0.10) than *A. victoria*. This means that the produced
vortices move about 10 radii away from the jetting hydromedusa during each
swimming cycle. This rapid transport of fluid away from the hydromedusa's body
offers little opportunity for prey capture during swimming since any prey in
the surrounding flow is quickly transported out of range of the tentacles.

*Aequorea victoria* primarily feed on small, soft-bodied zooplankton
(Costello and Colin, 2002),
which, to a good approximation, may be expected to largely drift along with
the surrounding flow. Haller and Sapsis have recently examined transport of
inertial particles in a general flow
(Haller and Sapsis, 2008) and
Peng and Dabiri have recently completed a study of the transport of inertial
particles in the flow around an oblate hydromedusa, *Aurelia aurita*,
using particle LCS (Peng and Dabiri,
2008) and they find regions of transport that are very similar to
those seen for passive tracers. Furthermore, inertial and finite size effects
decrease with prey size. If the fluid inside lobes A and B seen in
Fig. 10B contains potential
prey, at least a large portion of the prey will be drawn through *A.
victoria*'s tentacles during swimming. In particular, lobe A is
responsible for most of the transport from the upstream region into the
subumbrellar region and past the tentacles. Peng and Dabiri found that
64–91% (depending on the parameters used) of the volume of capture
regions for passive tracers (analogous to lobe A) was still a capture region
for inertial particles (Peng and Dabiri,
2008).

The wake structures produced by these two hydromedusae also help to explain some additional features of the anatomy of oblate and prolate hydromedusae. Oblate hydromedusae commonly have well-developed and prominent oral arms extending from the center of the bell while these structures are absent in prolate species (Costello et al., 2008). In fact, the presence of well-developed oral arms or tentacles extended in the flow in a jetting species could add drag and decrease swimming performance. Conversely, paddling hydromedusae take advantage of feeding structures, such as tentacles and oral arms, drifting freely in the flow by feeding while swimming.

By using a numerical model as the basis of our study, we can easily gain
additional information that would be difficult or impossible to get from
experiments. For example, for *S. tubulosa*, the velocity across the
velar opening is nearly uniform in space, exhibiting only a small shear layer
near the velum (Fig. 4). This
indicates that the fluid expelled by the hydromedusa can be well approximated
as a slug of uniform velocity, perhaps with a correction for the boundary
layer effects. Also, *A. victoria* swims by continuously paddling with
equal times dedicated to contraction and relaxation while *S. tubulosa*
follows a rapid contraction and relaxation with a brief period of coasting
before the next contraction. For *S. tubulosa*, the pressure seen on
the interior of the bell is also extremely uniform. This is very useful for
building engineered systems to imitate the propulsion of a jetting type of
hydromedusa. For example, knowing the pressure on the bell interior, combined
with the velar opening diameter, would allow for the design of a
piston-cylinder arrangement to mimic this behavior using current technologies.
Experimental studies have little hope of obtaining these types of information
due to the difficulty of directly measuring pressure on the surface of a
living organism as well as the challenges of obtaining a high-resolution
velocity profile.

It is important to note that these advantages do not come at the expense of
relevance to biological experiments. For example, using a very simple
calculation, we find that *S. tubulosa* produces a power output of
about 0.33 W kg^{–5/3}, which matches very well with
experimental results presented by Daniel
(Daniel, 1985). Additionally,
even small details of the hydromedusan swimming are well captured by our
method. As briefly mentioned by Daniel
(Daniel, 1985), the
hydromedusa is an elastic, viscously damped system. During contraction,
muscles decrease the subumbrellar volume, but there are no muscles to
re-expand the bell. Elastic strain energy stored during the contraction
expands the bell during the relaxation phase to its resting state. Due to this
elastic expansion, during the coasting phase of swimming for *S.
tubulosa*, we are able to observe oscillations in both the oral cavity
volume and the velar opening diameter (see
Fig. 2). We attribute these
oscillations to elastic effects.

Another new and interesting feature discovered in this study is the complex
structure of the fluid in the subumbrellar region of *S. tubulosa* (see
Fig. 1). The subumbrellar
region even contains a stopping vortex that forms during the relaxation phase
of swimming. However, unlike in paddling hydromedusae, this stopping vortex
remains inside the subumbrellar cavity and therefore cannot influence the
development of the ejected vortex.

Fig. 16 shows the
development and dissipation of the stopping vortex for *S. tubulosa*.
In this figure, we display only the backward LCS so that activity in the
subumbrellar cavity is clear. As starting vortex A is ejected during
contraction, the previous stopping vortex, B, sits deep within the
subumbrellar cavity (Fig.
16A). During relaxation, a new stopping vortex, C is formed and
begins to push B along the walls of the cavity
(Fig. 16B) until C resides
deep within the cavity and B has been pushed near the velar opening
(Fig. 16C). At this point, B
is no longer recognizable as a well-defined vortex. In fact, as B is pushed
along the wall, C interacts with the wall to create secondary vorticity of an
opposite sense to vortex C. This secondary vorticity overwhelms vortex B so
that region D in Fig. 16D
actually has vorticity of opposite sense to the stopping vortex. Finally, a
new contraction begins and the fluid in region D
(Fig. 16D) is ejected as part
of a new starting vortex.

*Aequorea victoria* ejects both vortices together during the
contraction phase while *S. tubulosa* ejects only the starting vortex
while the stopping vortex remains within the bell. The different morphologies
of the two hydromedusae produce this distinction. In *A. victoria*,
each vortex is formed by the shear layer being shed off the tip of the bell.
However, the velum of *S. tubulosa* alters the way the vortex is formed
so that the stopping vortex moves deep within the velar cavity, instead of
remaining near the velar opening, so it is not ejected during contraction.
This stopping vortex in *S. tubulosa* has not been observed in DPIV or
dye visualization experiments, most likely due to its confinement within the
bell and the difficulty of imaging this area in experiments. Its effect on
energy requirements and swimming efficiency remains to be seen.

**LIST OF ABBREVIATIONS**

- Δ
- finite time deformation tensor
- λmax
- maximum eigenvalue of Δ
- ALE
- arbitrary Lagrangian–Eulerian
- CFD
- computational fluid dynamics
*D*_{e}- diameter of nozzle
- DPIV
- digital particle image velocimetry
- FTLE
- finite time Lyapunov exponent
- LCS
- Lagrangian coherent structure(s)
*p*- pressure
*P*_{out}- power output
*Q̇*- flow rate = –(d
*V*/d*t*) *r*_{f}- final radius
*r*_{i}- initial radius
*St*- Strouhal number
*T*- integration time
*T*^{*}- dimensionless formation time
*U*_{e}- velocity of jet
*V*- volume
*V*_{c}- volume of cavity (subumbrellar volume)
- x
- position vector
*x*_{f}- final axial position
*x*_{i}- initial axial position

## FOOTNOTES

Supplementary material available online at http://jeb.biologists.org/cgi/content/full/212/15/2436/DC1

The authors would like to acknowledge support from the National Science Foundation and Air Force Office of Scientific Research. The authors would also like to thank Dr Sean P. Colin of Rogers William University for providing videos used in the CFD calculations [as described in Sahin and Mohseni (Sahin and Mohseni, 2008a; Sahin and Mohseni, 2008b), which were used in the LCS calculations here.