Fast-swimming hydromedusae exploit velar kinematics to form an optimal vortex wake

Pulsatile jet propulsion is one of the earliest known forms of aquatic locomotion exhibited by animals. Using a simple action-reaction principle,jet-propelled animals move forward by periodically ejecting jets of water in a direction opposite to the desired trajectory. Hydromedusan jellyfish have been especially successful in implementing this strategy. For example, measurements of Aglantha digitale medusae suggest that swimming speeds above 13 body lengths (BL) s^-1 are achieved during fast-swims,outpacing all but the very fastest predatory bony fishes [albeit for short bursts (Mackie, 1980; 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 flow created by pulsatile jets is dominated by radially symmetric,rotating currents called vortex rings. Laboratory studies using mechanical pulsatile jet generators have shown that a physical limit exists on the maximum size of any vortex ring created by a pulsatile jet. Once this limit-called the vortex `formation number' (denoted T<sup>*</sup><sub>lim</sub>, where the asterisk indicates that the parameter is dimensionless) - is reached, additional fluid ejected during the same pulse forms a jet of fluid that trails behind the vortex ring(Gharib et al., 1998). 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.

Specimens of Nemopsis bachei Agassiz (BL ∼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 ×480 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.

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.

The measured body kinematics, V_OC(t) and D_V(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 F_L 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.

where ρ is the density of the fluid (water), 1 is the identity matrix (a matrix with each diagonal element equal to 1), C_AMis the wake vortex added-mass tensor (a 3 ×3 matrix in the present case), φ is the velocity potential, and the normal vector n is directed out of the vortex wake. The integral is evaluated on the surface S_V 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^-1_AM in Eqn 4 can be evaluated in an element-wise fashion.

The assigned 30% entrainment is compatible with laboratory studies of vortex formation from pulsed jets (Dabiri and Gharib, 2004).

where m is the mass of the animal, estimated from the body volume and by assuming that the animal is neutrally buoyant (i.e.ρ _body=ρ_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.

Fig. 2 shows that the velar diameter changes in phase with the subumbrellar cavity volume, indicating that contraction of the velum during bell contraction dictates velar kinematics to a greater degree than the out-of-phase outpocketing effect as the velum swings open. At the end of jet ejection, the velum has decreased to one-fifth of its resting diameter, or 4% of its resting area. The superimposed data from a single swimming contraction of each animal(Fig. 2) suggests the level of inter-animal repeatability of the observed velar kinematics.

Calculation of the vortex formation time of the ejected jet based on the measured N. bachei kinematics (i.e. Eqn 3) evaluated over the duration of the contraction phase) indicates that swimming function of the N. bachei sample achieves a total vortex formation time T^*_max≈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^*_lim≈4(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.

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).

Small hydromedusae swim in order to escape predation and to reposition themselves to feed (Colin et al.,2003). 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^*_max≈8 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^*_max≈8 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).

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 S_V 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 C_AM will have non-zero elements only along the diagonal. Hence, the inverse can be computed in an element-wise fashion.

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.

List of symbols
 
• 1

  identity matrix

 
• A

  wake vortex shape constant

 
• \({\bar{c}}_{\mathrm{ii}}\)

  two-dimensional wake vortex added-mass coefficient

 
• c_ii

  wake vortex added-mass coefficient

 
• C_AM

  wake vortex added-mass tensor

 
• D

  jet diameter

 
• D_V

  velar diameter

 
• e₁

  unit vector

 
• F_L

  locomotive force

 
• φ

  velocity potential

 
• Γ

  vortex circulation

 
• I

  total impulse

 
• I_v

  vortex impulse

 
• m

  body mass

 
• n

  normal vector

 
• R

  vortex radius

 
• ρ

  fluid density

 
• Re

  Reynold's number

 
• S

  wake vortex width

 
• S_V

  vortex surface

 
• t,τ

  dimensional time variables

 
• T^*

  instantaneous vortex formation time (dimensionless)

 
• T^*_lim

  vortex formation number (dimensionless)

 
• T^*_max

  total vortex formation time (dimensionless)

 
• U(t)

  jet velocity

 
• U_SI

  self-induced wake vortex velocity

 
• U_U

  unsteady wake vortex velocity

 
• U_Ui

  unsteady wake vortex speed in i-direction

 
• U_V

  total wake vortex velocity

 
• U_Vi

  total wake vortex speed in i-direction

 
• V_OC

  oral cavity volume

 
• V_V

  vortex volume

 
• Wa

  wake vortex ratio

 
• ω

  fluid vorticity

 
• X

  body position

 
• ẋ

  body velocity

 
• X

  body acceleration

This research is supported in part by National Science Foundation Grants OCE-0116236 and OCE-0350834 (to J.H.C.) and OCE-0351398 (to S.P.C.). The authors thank the anonymous reviewers of the manuscript for their thorough and insightful comments.