Flow patterns generated by oblate medusan jellyfish: field measurements and laboratory analyses

Using computational and mechanical models to study animal locomotion

Recent advances in computational methods have made realistic large-scale simulations of animal locomotion possible. This has resulted in numerous mathematical and computational studies of animal movement through fluids and over substrates with the purpose of better understanding organisms' performance and improving the design of vehicles moving through air and water and on land. This work has also motivated the development of improved numerical methods and modeling techniques for animal locomotion that is characterized by the interactions of fluids, substrates, and structures. Despite the large body of recent work in this area, the application of mathematical and numerical methods to improve our understanding of organisms in the context of their environment and physiology has remained relatively unexplored. Nature has evolved a wide variety of fascinating mechanisms of locomotion that exploit the properties of complex materials and fluids, but only recently are the mathematical, computational, and robotic tools available to rigorously compare the relative advantages and disadvantages of different methods of locomotion in variable environments. Similarly, advances in computational physiology have only recently allowed investigators to explore how changes at the molecular, cellular, and tissue levels might lead to changes in performance at the organismal level. In this article, we highlight recent examples of how computational, mathematical, and experimental tools can be combined to ultimately answer the questions posed in one of the grand challenges in organismal biology: "Integrating living and physical systems."

Flow structure and transport characteristics of feeding and exchange currents generated by upside-down Cassiopea jellyfish

Quantifying the flows generated by the pulsations of jellyfish bells is crucial for understanding the mechanics and efficiency of their swimming and feeding. Recent experimental and theoretical work has focused on the dynamics of vortices in the wakes of swimming jellyfish with relatively simple oral arms and tentacles. The significance of bell pulsations for generating feeding currents through elaborate oral arms and the consequences for particle capture are not as well understood. To isolate the generation of feeding currents from swimming, the pulsing kinematics and fluid flow around the benthic jellyfish Cassiopea spp. were investigated using a combination of videography, digital particle image velocimetry and direct numerical simulation. During the rapid contraction phase of the bell, fluid is pulled into a starting vortex ring that translates through the oral arms with peak velocities that can be of the order of 10 cm s(-1). Strong shear flows are also generated across the top of the oral arms throughout the entire pulse cycle. A coherent train of vortex rings is not observed, unlike in the case of swimming oblate medusae such as Aurelia aurita. The phase-averaged flow generated by bell pulsations is similar to a vertical jet, with induced flow velocities averaged over the cycle of the order of 1-10 mm s(-1). This introduces a strong near-horizontal entrainment of the fluid along the substrate and towards the oral arms. Continual flow along the substrate towards the jellyfish is reproduced by numerical simulations that model the oral arms as a porous Brinkman layer of finite thickness. This two-dimensional numerical model does not, however, capture the far-field flow above the medusa, suggesting that either the three-dimensionality or the complex structure of the oral arms helps to direct flow towards the central axis and up and away from the animal.

Effects of shape and stroke parameters on the propulsion performance of an axisymmetric swimmer

In nature, there exists a special group of aquatic animals which have an axisymmetric body and whose primary swimming mechanism is to use periodic body contractions to generate vortex rings in the surrounding fluid. Using jellyfish medusae as an example, this study develops a mathematical model of body kinematics of an axisymmetric swimmer and uses a computational approach to investigate the induced vortex wakes. Wake characteristics are identified for swimmers using jet propulsion and rowing, two mechanisms identified in previous studies of medusan propulsion. The parameter space of body kinematics is explored through four quantities: a measure of body shape, stroke amplitude, the ratio between body contraction duration and extension duration, and the pulsing frequency. The effects of these parameters on thrust, input power requirement and circulation production are quantified. Two metrics, cruising speed and energy cost of locomotion, are used to evaluate the propulsion performance. The study finds that a more prolate-shaped swimmer with larger stroke amplitudes is able to swim faster, but its cost of locomotion is also higher. In contrast, a more oblate-shaped swimmer with smaller stroke amplitudes uses less energy for its locomotion, but swims more slowly. Compared with symmetric strokes with equal durations of contraction and extension, faster bell contractions increase the swimming speed whereas faster bell extensions decrease it, but both require a larger energy input. This study shows that besides the well-studied correlations between medusan body shape and locomotion, stroke variables also affect the propulsion performance. It provides a framework for comparing the propulsion performance of axisymmetric swimmers based on their body kinematics when it is difficult to measure and analyze their wakes empirically. The knowledge from this study is also useful for the design of robotic swimmers that use axisymmetric body contractions for propulsion.

Visual control of steering in the box jellyfish Tripedalia cystophora

Box jellyfish carry an elaborate visual system consisting of 24 eyes, which they use for driving a number of behaviours. However, it is not known how visual input controls the swimming behaviour. In this study we exposed the Caribbean box jellyfish Tripedalia cystophora to simple visual stimuli and recorded changes in their swimming behaviour. Animals were tethered in a small experimental chamber, where we could control lighting conditions. The behaviour of the animals was quantified by tracking the movements of the bell, using a high-speed camera. We found that the animals respond predictably to the darkening of one quadrant of the equatorial visual world by (1) increasing pulse frequency, (2) creating an asymmetry in the structure that constricts the outflow opening of the bell, the velarium, and (3) delaying contraction at one of the four sides of the bell. This causes the animals to orient their bell in such a way that, if not tethered, they would turn and swim away from the dark area. We conclude that the visual system of T. cystophora has a predictable effect on swimming behaviour.

Faking giants: the evolution of high prey clearance rates in jellyfishes

Jellyfishes have functionally replaced several overexploited commercial stocks of planktivorous fishes. This is paradoxical, because they use a primitive prey capture mechanism requiring direct contact with the prey, whereas fishes use more efficient visual detection. We have compiled published data to show that, in spite of their primitive life-style, jellyfishes exhibit similar instantaneous prey clearance and respiration rates as their fish competitors and similar potential for growth and reproduction. To achieve this production, they have evolved large, water-laden bodies that increase prey contact rates. Although larger bodies are less efficient for swimming, optimization analysis reveals that large collectors are advantageous if they move through the water sufficiently slowly.

A biomimetic robotic jellyfish (Robojelly) actuated by shape memory alloy composite actuators

An analysis is conducted on the design, fabrication and performance of an underwater vehicle mimicking the propulsion mechanism and physical appearance of a medusa (jellyfish). The robotic jellyfish called Robojelly mimics the morphology and kinematics of the Aurelia aurita species. Robojelly actuates using bio-inspired shape memory alloy composite actuators. A systematic fabrication technique was developed to replicate the essential structural features of A. aurita. Robojelly's body was fabricated from RTV silicone having a total mass of 242 g and bell diameter of 164 mm. Robojelly was able to generate enough thrust in static water conditions to propel itself and achieve a proficiency of 0.19 s(-1) while the A. aurita achieves a proficiency of around 0.25 s(-1). A thrust analysis based on empirical measurements for a natural jellyfish was used to compare the performance of the different robotic configurations. The configuration with best performance was a Robojelly with segmented bell and a passive flap structure. Robojelly was found to consume an average power on the order of 17 W with the actuators not having fully reached a thermal steady state.

Volumetric imaging of shark tail hydrodynamics reveals a three-dimensional dual-ring vortex wake structure

Understanding how moving organisms generate locomotor forces is fundamental to the analysis of aerodynamic and hydrodynamic flow patterns that are generated during body and appendage oscillation. In the past, this has been accomplished using two-dimensional planar techniques that require reconstruction of three-dimensional flow patterns. We have applied a new, fully three-dimensional, volumetric imaging technique that allows instantaneous capture of wake flow patterns, to a classic problem in functional vertebrate biology: the function of the asymmetrical (heterocercal) tail of swimming sharks to capture the vorticity field within the volume swept by the tail. These data were used to test a previous three-dimensional reconstruction of the shark vortex wake estimated from two-dimensional flow analyses, and show that the volumetric approach reveals a different vortex wake not previously reconstructed from two-dimensional slices. The hydrodynamic wake consists of one set of dual-linked vortex rings produced per half tail beat. In addition, we use a simple passive shark-tail model under robotic control to show that the three-dimensional wake flows of the robotic tail differ from the active tail motion of a live shark, suggesting that active control of kinematics and tail stiffness plays a substantial role in the production of wake vortical patterns.

Ontogenetic changes in the bell morphology and kinematics and swimming behavior of rowing medusae: the special case of the limnomedusa Liriope tetraphylla

Swimming animals may experience significant changes in the Reynolds number (Re) of their surrounding fluid flows throughout ontogeny. Many medusae experience Re environments with significant viscous forces as small juveniles but inertially dominated Re environments as adults. These different environments may affect their propulsive strategies. In particular, rowing, a propulsive strategy with ecological advantages for large adults, may be constrained by viscosity for small juvenile medusae. We examined changes in the bell morphology and swimming kinematics of the limnomedusa Liriope tetraphylla at different stages of development. L. tetraphylla maintained an oblate bell (fineness ratio ≈ 0.5-0.6), large velar aperture ratio (R(v) ≈ 0.5-0.8), and rapid bell kinematics throughout development. These traits enabled it to use rowing propulsion at all stages except the very smallest sizes observed (diameter = 0.14 cm). During the juvenile stage, very rapid bell kinematics served to increase Re sufficiently for rowing propulsion. Other taxa that use rowing propulsion as adults, such as leptomedusae and scyphomedusae, typically utilize different propulsive strategies as small juveniles to function in low Re environments. We compared the performance values of the different propulsive modes observed among juvenile medusae.

A wake-based correlate of swimming performance and foraging behavior in seven co-occurring jellyfish species

It is generally accepted that animal–fluid interactions have shaped the evolution of animals that swim and fly. However, the functional ecological advantages associated with those adaptations are currently difficult to predict on the basis of measurements of the animal–fluid interactions. We report the identification of a robust, fluid dynamic correlate of distinct ecological functions in seven jellyfish species that represent a broad range of morphologies and foraging modes. Since the comparative study is based on properties of the vortex wake – specifically, a fluid dynamical concept called optimal vortex formation – and not on details of animal morphology or phylogeny, we propose that higher organisms can also be understood in terms of these fluid dynamic organizing principles. This enables a quantitative, physically based understanding of how alterations in the fluid dynamics of aquatic and aerial animals throughout their evolution can result in distinct ecological functions.

A Lagrangian approach to identifying vortex pinch-off

A criterion for identifying vortex ring pinch-off based on the Lagrangian coherent structures (LCSs) in the flow is proposed and demonstrated for a piston-cylinder arrangement with a piston stroke to diameter (L/D) ratio of approximately 12. It is found that the appearance of a new disconnected LCS and the termination of the original LCS are indicative of the initiation of vortex pinch-off. The subsequent growth of new LCSs, which tend to roll into spirals, indicates the formation of new vortex cores in the trailing shear layer. Using this criterion, the formation number is found to be 4.1+/-0.1, which is consistent with the predicted formation number of approximately 4 of Gharib et al. [Gharib et al. J. Fluid Mech. 360, 121 (1998)]. The results obtained using the proposed LCS criterion are compared with those obtained using the circulation criterion of Gharib et al. and are found to be in excellent agreement. The LCS approach is also compared against other metrics, both Lagrangian and Eulerian, and is found to yield insight into the pinch-off process that these do not. Furthermore, the LCS analysis reveals a consistent pattern of coalescing or "pairing" of adjacent vortices in the trailing shear layer, a process which has been extensively documented in circular jets. Given that LCSs are objective and insensitive to local errors in the velocity field, the proposed criterion has the potential to be a robust tool for pinch-off identification. In particular, it may prove useful in the study of unsteady and low Reynolds number flows, where conventional methods based on vorticity prove difficult to use.

Functional morphology and fluid interactions during early development of the scyphomedusa Aurelia aurita

Scyphomedusae undergo a predictable ontogenetic transition from a conserved, universal larval form to a diverse array of adult morphologies. This transition entails a change in bell morphology from a highly discontinuous ephyral form, with deep clefts separating eight discrete lappets, to a continuous solid umbrella-like adult form. We used a combination of kinematic, modeling, and flow visualization techniques to examine the function of the medusan bell throughout the developmental changes of the scyphomedusa Aurelia aurita. We found that flow around swimming ephyrae and their lappets was relatively viscous (1 < Re < 10) and, as a result, ephyral lappets were surrounded by thick, overlapping boundary layers that occluded flow through the gaps between lappets. As medusae grew, their fluid environment became increasingly influenced by inertial forces (10 < Re < 10,000) and, simultaneously, clefts between the lappets were replaced by organic tissue. Hence, although the bell undergoes a structural transition from discontinuous (lappets with gaps) to continuous (solid bell) surfaces during development, all developmental stages maintain functionally continuous paddling surfaces. This developmental pattern enables ephyrae to efficiently allocate tissue to bell diameter increase via lappet growth, while minimizing tissue allocation to inter-lappet spaces that maintain paddle function due to boundary layer overlap.

The numerical comparison of flow patterns and propulsive performances for the hydromedusae Sarsia tubulosa and Aequorea victoria

The thrust-generating mechanism of a prolate hydromedusa Sarsia tubulosa and an oblate hydromedusa Aequorea victoria was investigated by solving the incompressible Navier–Stokes equations in the swirl-free cylindrical coordinates. The calculations clearly show the vortex dynamics related to the thrust-generating mechanism, which is very important for understanding the underlying propulsion mechanism. The calculations for the prolate jetting hydromedusa S. tubulosa indicate the formation of a single starting vortex ring for each pulse cycle with a relatively high vortex formation number. However, the calculations for the oblate jet-paddling hydromedusa A. victoria indicate shedding of the opposite-signed vortex rings very close to each other and the formation of large induced velocities along the line of interaction as the vortices move away from the hydromedusa in the wake. In addition to this jet propulsion mechanism, the hydromedusa's bell margin acts like a paddle and the highly flexible bell margin deforms in such a way that the low pressure leeward side of the bell margin has a projected area in the direction of motion. This thrust is particularly important during refilling of the subumbrella cavity where the stopping vortex causes significant pressure drag. The swimming performances based on our numerical simulations, such as swimming velocity,thrust, power requirement and efficiency, were computed and support the idea that jet propulsion is very effective for rapid body movement but is energetically costly and less efficient compared with the jet-paddling propulsion mechanism.

Lagrangian coherent structures in low Reynolds number swimming

This work explores the utility of the finite-time Lyapunov exponent (FTLE) field for revealing flow structures in low Reynolds number biological locomotion. Previous studies of high Reynolds number unsteady flows have demonstrated that ridges of the FTLE field coincide with transport barriers within the flow, which are not shown by a more classical quantity such as vorticity. In low Reynolds number locomotion (O(1)-O(100)), in which viscous diffusion rapidly smears the vorticity in the wake, the FTLE field has the potential to add new insight to locomotion mechanics. The target of study is an articulated two-dimensional model for jellyfish-like locomotion, with swimming Reynolds number of order 1. The self-propulsion of the model is numerically simulated with a viscous vortex particle method, using kinematics adapted from previous experimental measurements on a live medusan swimmer. The roles of the ridges of the computed forward- and backward-time FTLE fields are clarified by tracking clusters of particles both backward and forward in time. It is shown that a series of ridges in front of the jellyfish in the forward-time FTLE field transport slender fingers of fluid toward the lip of the bell orifice, which are pulled once per contraction cycle into the wake of the jellyfish, where the fluid remains partitioned. A strong ridge in the backward-time FTLE field reveals a persistent barrier between fluid inside and outside the subumbrellar cavity. The system is also analyzed in a body-fixed frame subject to a steady free stream, and the FTLE field is used to highlight differences in these frames of reference.

The hydrodynamics of ribbon-fin propulsion during impulsive motion

Weakly electric fish are extraordinarily maneuverable swimmers, able to swim as easily forward as backward and rapidly switch swim direction, among other maneuvers. The primary propulsor of gymnotid electric fish is an elongated ribbon-like anal fin. To understand the mechanical basis of their maneuverability, we examine the hydrodynamics of a non-translating ribbon fin in stationary water using computational fluid dynamics and digital particle image velocimetry (DPIV) of the flow fields around a robotic ribbon fin. Computed forces are compared with drag measurements from towing a cast of the fish and with thrust estimates for measured swim-direction reversals. We idealize the movement of the fin as a traveling sinusoidal wave, and derive scaling relationships for how thrust varies with the wavelength, frequency,amplitude of the traveling wave and fin height. We compare these scaling relationships with prior theoretical work. The primary mechanism of thrust production is the generation of a streamwise central jet and the associated attached vortex rings. Under certain traveling wave regimes, the ribbon fin also generates a heave force, which pushes the body up in the body-fixed frame. In one such regime, we show that as the number of waves along the fin decreases to approximately two-thirds, the heave force surpasses the surge force. This switch from undulatory parallel thrust to oscillatory normal thrust may be important in understanding how the orientation of median fins may vary with fin length and number of waves along them. Our results will be useful for understanding the neural basis of control in the weakly electric knifefish as well as for engineering bio-inspired vehicles with undulatory thrusters.

Optimal movement in the prey strikes of weakly electric fish: a case study of the interplay of body plan and movement capability

Animal behaviour arises through a complex mixture of biomechanical, neuronal, sensory and control constraints. By focusing on a simple, stereotyped movement, the prey capture strike of a weakly electric fish, we show that the trajectory of a strike is one which minimizes effort. Specifically, we model the fish as a rigid ellipsoid moving through a fluid with no viscosity, governed by Kirchhoff's equations. This formulation allows us to exploit methods of discrete mechanics and optimal control to compute idealized fish trajectories that minimize a cost function. We compare these with the measured prey capture strikes of weakly electric fish from a previous study. The fish has certain movement limitations that are not incorporated in the mathematical model, such as not being able to move sideways. Nonetheless, we show quantitatively that the computed least-cost trajectories are remarkably similar to the measured trajectories. Since, in this simplified model, the basic geometry of the idealized fish determines the favourable modes of movement, this suggests a high degree of influence between body shape and movement capability. Simplified minimal models and optimization methods can give significant insight into how body morphology and movement capability are closely attuned in fish locomotion.

Transitions in morphology, nematocyst distribution, fluid motions, and prey capture during development of the scyphomedusa Cyanea capillata

Like that of most scyphozoans, the ontogeny of Cyanea capillata medusae involves substantive alterations in feeding structures and mechanics. We used video and optical microscopy approaches to quantify these ontogenetic changes in morphology, flow, and feeding of C. capillata medusae. We found that alterations in gross morphology and nematocyst distributions coincided with a shift from prey capture on the manubrium or lappets of ephyrae (bell diameter 0.2-0.4 cm) to capture primarily on the tentacles in adult medusae (diameter >1.0 cm). These changes occurred within a hydrodynamic framework that itself changed due to medusan growth. Viscous forces were important in flows around small ephyrae (maximum Re <10(1)), whereas viscosity was less influential in the inertially dominated flows around adult medusae (Re > 10(2)). The relative timing of these events indicates that ontogenetic processes are closely synchronized with alterations in the hydrodynamic environment within which C. capillata medusae develop.

An overview of a Lagrangian method for analysis of animal wake dynamics

The fluid dynamic analysis of animal wakes is becoming increasingly popular in studies of animal swimming and flying, due in part to the development of quantitative flow visualization techniques such as digital particle imaging velocimetry (DPIV). In most studies, quasi-steady flow is assumed and the flow analysis is based on velocity and/or vorticity fields measured at a single time instant during the stroke cycle. The assumption of quasi-steady flow leads to neglect of unsteady (time-dependent) wake vortex added-mass effects,which can contribute significantly to the instantaneous locomotive forces. In this paper we review a Lagrangian approach recently introduced to determine unsteady wake vortex structure by tracking the trajectories of individual fluid particles in the flow, rather than by analyzing the velocity/vorticity fields at fixed locations and single instants in time as in the Eulerian perspective. Once the momentum of the wake vortex and its added mass are determined, the corresponding unsteady locomotive forces can be quantified. Unlike previous studies that estimated the time-averaged forces over the stroke cycle, this approach enables study of how instantaneous locomotive forces evolve over time. The utility of this method for analyses of DPIV velocity measurements is explored, with the goal of demonstrating its applicability to data that are typically available to investigators studying animal swimming and flying. The methods are equally applicable to computational fluid dynamics studies where velocity field calculations are available.

The flow generated by an active olfactory system of the red swamp crayfish

Crayfish are nocturnal animals that mainly rely on their chemoreceptors to locate food. On a crayfish scale, chemical stimuli received from a distant source are dispersed by an ambient flow rather than molecular diffusion. When the flow is weak or absent, food searching can be facilitated by currents generated by the animal itself. Crayfish employ their anterior fan organs to produce a variety of flow patterns. Here we study the flow generated by Procambarus clarkii in response to odour stimulation. We found that while searching for food the crayfish generates one or two outward jets. These jets induce an inflow that draws odour to the crayfish's anterior chemoreceptors. We quantified velocity fields in the inflow region using Particle Image Velocimetry. The results show that the inflow velocity decreases proportionally to the inverse distance from the animal so that it takes about 100 s for an odour plume to reach the animal's chemoreceptors from a distance of 10 cm. We compare the inflow generated by live crayfish with that produced by a mechanical model. The model consists of two nozzles and an inlet and provides two jets and a sink so that the overall mass flux is zero. Use of the model enabled us to analyze the inflow at various jet parameters. We show that variation of directions and relative intensities of the jets allows the direction of odour attraction to be changed. These results provide a rationale for biomimetic robot design. We discuss sensitivity and efficiency of such a robot.

Comparative Biomechanics: The Jellyfish Paradox Resolved

Studying the mechanics of swirling water has solved a mystery about the evolution of body shape and size in jellyfish.

Morphological diversity of medusan lineages constrained by animal–fluid interactions

Cnidarian medusae, commonly known as jellyfish, represent the earliest known animal taxa to achieve locomotion using muscle power. Propulsion by medusae requires the force of bell contraction to generate forward thrust. However, thrust production is limited in medusae by the primitive structure of their epitheliomuscular cells. This paper demonstrates that constraints in available locomotor muscular force result in a trade-off between high-thrust swimming via jet propulsion and high-efficiency swimming viaa combined jet-paddling propulsion. This trade-off is reflected in the morphological diversity of medusae, which exhibit a range of fineness ratios(i.e. the ratio between bell height and diameter) and small body size in the high-thrust regime, and low fineness ratios and large body size in the high-efficiency regime. A quantitative model of the animal–fluid interactions that dictate this trade-off is developed and validated by comparison with morphological data collected from 660 extant medusan species ranging in size from 300 μm to over 2 m. These results demonstrate a biomechanical basis linking fluid dynamics and the evolution of medusan bell morphology. We believe these to be the organising principles for muscle-driven motility in Cnidaria.

Non-invasive measurement of instantaneous forces during aquatic locomotion: a case study of the bluegill sunfish pectoral fin

Swimming and flying animals generate unsteady locomotive forces by delivering net momentum into the fluid wake. Hence, swimming and flying forces can be quantified by measuring the momentum of animal wakes. A recently developed model provides an approach to empirically deduce swimming and flying forces based on the measurement of velocity and vortex added-mass in the animal wake. The model is contingent on the identification of the vortex boundary in the wake. This paper demonstrates the application of that method to a case study quantifying the instantaneous locomotive forces generated by the pectoral fins of the bluegill sunfish (Lepomis macrochirusRafinesque), measured using digital particle image velocimetry (DPIV). The finite-time Lyapunov exponent (FTLE) field calculated from the DPIV data was used to determine the wake vortex boundary, according to recently developed fluid dynamics theory. Momentum of the vortex wake and its added-mass were determined and the corresponding instantaneous locomotive forces were quantified at discrete time points during the fin stroke. The instantaneous forces estimated in this study agree in magnitude with the time-averaged forces quantified for the pectoral fin of the same species swimming in similar conditions and are consistent with the observed global motion of the animals. A key result of this study is its suggestion that the dynamical effect of the vortex wake on locomotion is to replace the real animal fin with an `effective appendage', whose geometry is dictated by the FTLE field and whose interaction with the surrounding fluid is wholly dictated by inviscid concepts from potential flow theory. Benefits and limitations of this new framework for non-invasive instantaneous force measurement are discussed, and its application to comparative biomechanics and engineering studies is suggested.

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

Fast-swimming hydromedusan jellyfish possess a characteristic funnel-shaped velum at the exit of their oral cavity that interacts with the pulsed jets of water ejected during swimming motions. It has been previously assumed that the velum primarily serves to augment swimming thrust by constricting the ejected flow in order to produce higher jet velocities. This paper presents high-speed video and dye-flow visualizations of free-swimming Nemopsis bachei hydromedusae, which instead indicate that the time-dependent velar kinematics observed during the swimming cycle primarily serve to optimize vortices formed by the ejected water rather than to affect the speed of the ejected flow. Optimal vortex formation is favorable in fast-swimming jellyfish because,unlike the jet funnelling mechanism, it allows for the minimization of energy costs while maximizing thrust forces. However, the vortex `formation number'corresponding to optimality in N. bachei is substantially greater than the value of 4 found in previous engineering studies of pulsed jets from rigid tubes. The increased optimal vortex formation number is attributable to the transient velar kinematics exhibited by the animals. A recently developed model for instantaneous forces generated during swimming motions is implemented to demonstrate that transient velar kinematics are required in order to achieve the measured swimming trajectories. The presence of velar structures in fast-swimming jellyfish and the occurrence of similar jet-regulating mechanisms in other jet-propelled swimmers (e.g. the funnel of squid) appear to be a primary factor contributing to success of fast-swimming jetters, despite their primitive body plans.