Characteristics of vortex formation and thrust performance in drag-based paddling propulsion

The hydrodynamic performance of duck feet for submerged swimming resembles oars rather than delta-wings

Waterfowl use webbed feet to swim underwater. It has been suggested that the triangular shape of the webbed foot functions as a lift-generating delta wing rather than a drag-generating oar. To test this idea, we studied the hydrodynamic characteristics of a diving duck's (Aythya nyroca) foot. The foot's time varying angles-of-attack (AoAs) during paddling were extracted from movies of ducks diving vertically in a water tank. Lift and drag coefficients of 3D-printed duck-foot models were measured as a function of AoA in a wind-tunnel; and the near-wake flow dynamics behind the foot model was characterized using particle image velocimetry (PIV) in a flume. Drag provided forward thrust during the first 80% of the power phase, whereas lift dominated thrust production at the end of the power stroke. In steady flow, the transfer of momentum from foot to water peaked at 45° < AoA < 60°, due to an organized wake flow pattern (vortex street), whereas at AoAs > 60° the flow behind the foot was fully separated, generating high drag levels. The flow characteristics do not constitute the vortex lift typical of delta wings. Rather, duck feet seem to be an adaptation for propulsion at a wide range of AoAs, on and below the water surface.

Pleobot: a modular robotic solution for metachronal swimming

Metachronal propulsion is widespread in aquatic swarming organisms to achieve performance and maneuverability at intermediate Reynolds numbers. Studying only live organisms limits our understanding of the mechanisms driving these abilities. Thus, we present the design, manufacture, and validation of the Pleobot-a unique krill-inspired robotic swimming appendage constituting the first platform to study metachronal propulsion comprehensively. We combine a multi-link 3D printed mechanism with active and passive actuation of the joints to generate natural kinematics. Using force and fluid flow measurements in parallel with biological data, we show the link between the flow around the appendage and thrust. Further, we provide the first account of a leading-edge suction effect contributing to lift during the power stroke. The repeatability and modularity of the Pleobot enable the independent manipulation of particular motions and traits to test hypotheses central to understanding the relationship between form and function. Lastly, we outline future directions for the Pleobot, including adapting morphological features. We foresee a broad appeal to a wide array of scientific disciplines, from fundamental studies in ecology, biology, and engineering, to developing new bio-inspired platforms for studying oceans across the solar system.

Hydrodynamics of the fast-start caridoid escape response in Antarctic krill, Euphausia superba

Krill are shrimp-like crustaceans with a high degree of mobility and variety of documented swimming behaviors. The caridoid escape response, a fast-start mechanism unique to crustaceans, occurs when the animal performs a series of rapid abdominal flexions and tail flipping that results in powerful backward strokes. The current results quantify the animal kinematics and three-dimensional flow field around a free-swimming Euphausia superba as it performs the caridoid escape maneuver. The specimen performs a single abdominal flexion-tail flip combination that leads to an acceleration over a 42 ms interval allowing it to reach a maximum speed of 57.0 cm/s (17.3 body lengths/s). The krill's tail flipping during the abdominal closure is a significant contributor to the thrust generation during the maneuver. The krill sheds a complex chain of vortex rings in its wake due to the viscous flow effects while the organism accelerates. The vortex ring structure reveals a strong suction flow in the wake, which suggests that the pressure distribution and form drag play a role in the force balance for this maneuver. Antarctic krill typically swim in a low to intermediate Reynolds number (Re) regime where viscous forces are significant, but as shown by this analysis, its high maneuverability allows it to quickly change its body angle and swimming speed.

Trends in Stroke Kinematics, Reynolds Number, and Swimming Mode in Shrimp-Like Organisms

Metachronal propulsion is commonly seen in organisms with the caridoid facies body plan, i.e. shrimp-like organisms, as they beat their pleopods in an adlocomotory sequence. These organisms exist across length scales ranging several orders of Reynolds number magnitude, from 10 to 104, during locomotion. Further, by altering their stroke kinematics, these organisms achieve three distinct swimming modes. To better understand the relationship between Reynolds number, stroke kinematics, and resulting swimming mode, Euphausia pacifica stroke kinematics were quantified using high-speed digital recordings and compared to the results for the larger E. superba. Euphausia pacifica consistently operate with a greater beat frequency and smaller stroke amplitude than E. superba for each swimming mode, suggesting that length scale may affect the kinematics needed to achieve similar swimming modes. To expand on this observation, these euphausiid data are used in combination with previously-published stroke kinematics from mysids and stomatopods to identify broad trends across swimming mode and length scale in metachrony. Principal component analysis (PCA) reveals trends in stroke kinematics and Reynolds number as well as the variation among taxonomic order. Overall, larger beat frequencies, stroke amplitudes, between-cycle phase lags, and Reynolds numbers are more representative of the fast forward swimming mode compared to the slower hovering mode. Additionally, each species has a unique combination of kinematics that result in metachrony, indicating that there are other factors, perhaps morphological, which affect the overall metachronal characteristics of an organism. Finally, uniform phase lag, in which the timing between power strokes of all pleopods is equal, in 5-paddle systems is achieved at different Reynolds numbers for different swimming modes, highlighting the importance of taking into consideration stroke kinematics, length scale, and the resulting swimming mode.

On the role of phase lag in multi-appendage metachronal swimming of euphausiids

Metachronal paddling is a common method of drag-based aquatic propulsion, in which a series of swimming appendages are oscillated, with the motion of each appendage phase-shifted relative to the neighboring appendages. Ecologically and economically important euphausiid species such as Antarctic krill (Euphausia superba) swim constantly by stroking their paddling appendages (pleopods), with locomotion accounting for the bulk of their metabolic expenditure. They tailor their swimming gaits for behavioral and energetic needs by changing pleopod kinematics. The functional importance of inter-pleopod phase lag (ϕ) to metachronal swimming performance and wake structure is unknown. To examine this relation, we developed a geometrically and dynamically scaled robot ('krillbot') capable of self-propulsion. Krillbot pleopods were prescribed to mimic published kinematics of fast-forward swimming (FFW) and hovering (HOV) gaits ofE. superba, and the Reynolds number and Strouhal number of the krillbot matched well with those calculated for freely-swimmingE. superba. In addition to examining published kinematics with unevenϕbetween pleopod pairs, we modifiedE. superbakinematics to uniformly varyϕfrom 0% to 50% of the cycle. Swimming speed and thrust were largest for FFW withϕbetween 15%-25%, coincident withϕrange observed in FFW gait ofE. superba. In contrast to synchronous rowing (ϕ= 0%) where distances between hinged joints of adjacent pleopods were nearly constant throughout the cycle, metachronal rowing (ϕ> 0%) brought adjacent pleopods closer together and moved them farther apart. This factor minimized body position fluctuation and augmented metachronal swimming speed. Though swimming speed was lowest for HOV, a ventrally angled downward jet was generated that can assist with weight support during feeding. In summary, our findings show that inter-appendage phase lag can drastically alter both metachronal swimming speed and the large-scale wake structure.

Dual Phase-Shifted Ipsilateral Metachrony in Americamysis bahia

Previously documented metachrony in euphausiids focused on one, 5-paddle metachronal stroke, where contralateral pleopod pairs on the same abdominal segment beat in tandem with each other, propelling the animal forward. In contrast, the mysid shrimp Americamysis bahia's pleopods on the same abdominal segment beat independently of each other, resulting in two, 5-paddle metachronal cycles running ipsilateral along the length of the body, 180° out of phase. The morphology, kinematics, and nondimensional measurements of efficiency are compared primarily to the one-cycle Euphausia superba to determine how the two-cycle approach alters the design and kinematics of metachrony. Pleopodal swimming in A. bahia results in only fast-forward swimming, with speeds greater than 2BL/s (body lengths per second), and can reach speeds up to 12BL/s, through a combination of increasing stroke amplitude, beat frequency, and changing their inter-limb phase lag. Trends with Strouhal number and advance ratio suggest that the kinematics of metachrony in A. bahia are favored to achieve large normalized swimming speeds.

Metachronal motion across scales: current challenges and future directions

Metachronal motion is used across a wide range of organisms for a diverse set of functions. However, despite its ubiquity, analysis of this behavior has been difficult to generalize across systems. Here we provide an overview of known commonalities and differences between systems that use metachrony to generate fluid flow. We also discuss strategies for standardizing terminology and defining future investigative directions that are analogous to other established subfields of biomechanics. Lastly, we outline key challenges that are common to many metachronal systems, opportunities that have arisen due to the advent of new technology (both experimental and computational), and next steps for community development and collaboration across the nascent network of metachronal researchers.

Hydrodynamics of metachronal paddling: effects of varying Reynolds number and phase lag

Negatively buoyant freely swimming crustaceans such as krill must generate downward momentum in order to maintain their position in the water column. These animals use a drag-based propulsion strategy, where pairs of closely spaced swimming limbs are oscillated rhythmically from the tail to head. Each pair is oscillated with a phase delay relative to the neighbouring pair, resulting in a metachronal wave travelling in the direction of animal motion. It remains unclear how oscillations of limbs in the horizontal plane can generate vertical momentum. Using particle image velocimetry measurements on a robotic model, we observed that metachronal paddling with non-zero phase lag created geometries of adjacent paddles that promote the formation of counter-rotating vortices. The interaction of these vortices resulted in generating large-scale angled downward jets. Increasing phase lag resulted in more vertical orientation of the jet, and phase lags in the range used by Antarctic krill produced the most total momentum. Synchronous paddling produced lower total momentum when compared with metachronal paddling. Lowering Reynolds number by an order of magnitude below the range of adult krill (250-1000) showed diminished downward propagation of the jet and lower vertical momentum. Our findings show that metachronal paddling is capable of producing flows that can generate both lift (vertical) and thrust (horizontal) forces needed for fast forward swimming and hovering.

Effects of fish caudal fin sweep angle and kinematics on thrust production during low-speed thunniform swimming

Scombrid fish lunate caudal fins are characterized by a wide range of sweep angles. Scombrid that have small sweep-angle caudal fins move at higher swimming speeds, suggesting that smaller angles produce more thrust. Furthermore, scombrids occasionally use high angles of attack (AoA) suggesting this also has some thrust benefit. This work examined the hypothesis that a smaller sweep angle and higher AoA improved thrust in swimmers by experimentally analyzing a robophysical model. The robophysical model was tested in a water tunnel at speeds between 0.35 and 0.7 body lengths per second. Three swept caudal fins were analyzed at three different AoA, three different freestream velocities, and four different Strouhal numbers, for a total of 108 cases. Results demonstrated that the fin with the largest sweep angle of 50° resulted in lower thrust production than the 40° and 30° fins, especially at higher Strouhal numbers. Larger AoA up to 25° increased thrust production at the higher Strouhal numbers, but at lower Strouhal numbers, produced less thrust. Differences in thrust production due to fin sweep angle and AoA were attributed to the variation in spanwise flow and leading edge vortex dynamics.

Summary: The study examines the impact of scombrid fish caudal fin sweep angle and angle of attack on thrust production across a range of Strouhal numbers using a robophysical model.

On the role of tip curvature on flapping plates

During the flapping motion of a fish's tail, the caudal fin exhibits antero-posterior bending and dorso-ventral bending, the latter of which is referred to as chord-wise bending herein. The impact of chord-wise tip curvature on the hydrodynamic forces for flapping plates is investigated to explore potential mechanisms to improve the maneuverability or the performance of autonomous underwater vehicles. First, actuated chord-wise tip curvature is explored. Comparison of rigid curved geometries to a rigid flat plate as a baseline suggests that an increased curvature decreases the generated forces. An actuated plate with a dynamic tip curvature is created to illustrate a modulation of this decrease in forces. Second, the impact of curvature is isolated using curved plates with an identical planform area. Comparison of rigid curved geometries as a baseline corroborates the result that an increased curvature decreases the generated forces, with the exception that presenting a concave geometry into the flow increases the thrust and the efficiency. A passively-actuated plate is designed to capitalize on this effect by presenting a concave geometry into the flow throughout the cycle. The dynamically and passively actuated plates show potential to improve the maneuverability and the efficiency of autonomous underwater vehicles, respectively.

Self-propelled swimming simulations of bio-inspired smart structures

This paper presents self-propelled swimming simulations of a foldable structure, whose folded configuration is a box. For self-locomotion through water the structure unfolds and undulates. To guide the design of the structure and understand how it should undulate to achieve either highest speed or maximize efficiency during locomotion, several kinematic parameters were systematically varied in the simulations: the wave type (standing wave versus traveling wave), the smoothness of undulations (smooth undulations versus undulations of rigid links), the mode of undulations (carangiform: mackerel-like versus anguilliform: eel-like undulations), and the maximum amplitude of undulations. We show that the swimmers with standing wave are slow and inefficient because they are not able to produce thrust using the added-mass mechanism. Among the tested types of undulation at low Reynolds number (Re) regime of [Formula: see text] (Strouhal number of about 1.0), structures that employ carangiform undulations can swim faster, whereas anguilliform swimmers are more economic, i.e., using less power they can swim a longer distance. Another finding of our simulations is that structures which are made of rigid links are typically less efficient (lower propulsive and power efficiencies and also lower swimming speed) compared with smoothly undulating ones because a higher added-mass force is generated by smooth undulations. The wake of all the swimmers bifurcated at the low Re regime because of the higher lateral relative to the axial velocity (high Strouhal number) that advects the vortices laterally creating a double row of vortices in the wake. In addition, we show that the wake cannot be used to predict the performance of the swimmers because the net force in each cycle is zero for self-propelled bodies and the pressure term is not negligible compared to the other terms.

Passive energy recapture in jellyfish contributes to propulsive advantage over other metazoans

Gelatinous zooplankton populations are well known for their ability to take over perturbed ecosystems. The ability of these animals to outcompete and functionally replace fish that exhibit an effective visual predatory mode is counterintuitive because jellyfish are described as inefficient swimmers that must rely on direct contact with prey to feed. We show that jellyfish exhibit a unique mechanism of passive energy recapture, which is exploited to allow them to travel 30% further each swimming cycle, thereby reducing metabolic energy demand by swimming muscles. By accounting for large interspecific differences in net metabolic rates, we demonstrate, contrary to prevailing views, that the jellyfish (Aurelia aurita) is one of the most energetically efficient propulsors on the planet, exhibiting a cost of transport (joules per kilogram per meter) lower than other metazoans. We estimate that reduced metabolic demand by passive energy recapture improves the cost of transport by 48%, allowing jellyfish to achieve the large sizes required for sufficient prey encounters. Pressure calculations, using both computational fluid dynamics and a newly developed method from empirical velocity field measurements, demonstrate that this extra thrust results from positive pressure created by a vortex ring underneath the bell during the refilling phase of swimming. These results demonstrate a physical basis for the ecological success of medusan swimmers despite their simple body plan. Results from this study also have implications for bioinspired design, where low-energy propulsion is required.