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Abstract Biological systems have often been sources of inspiration for engineering design. Over the past decade, advances in soft robotics have enabled the development of bioinspired technology across a wide range of sizes and applications. When paired with recent advances in miniaturization and manufacturing techniques, soft robotics can be used to investigate the locomotion and bio-hydrodynamics of millimeter-scale swimmers that operate at intermediate Reynolds numbers (100–103). However, it is important to understand the kinematics and dynamics of biological model systems in order to leverage the true potential of bioinspired robots/devices. Ctenophores (comb jellies) are gelatinous marine invertebrates with soft bodies and flexible appendages composed of bundles of millimeter-long cilia; they are the largest animals in the world to locomote using cilia, with each appendage operating at a Reynolds number of approximately 102. Their efficiency, maneuverability, and ubiquity in the global ocean make them a potentially attractive candidate for bioinspired design applications. Each ctenophore has eight rows of paddle-like ciliary bundles (ctenes) that beat metachronally, with a phase lag between neighboring appendages, producing a “metachronal wave” that propagates along the row. This strategy, known as metachronal coordination, is also used by many other organisms (including crustaceans, annelids, and insects) to facilitate feeding, respiration, and locomotion. In general, the performance of a metachronal system depends on a large number of geometrical and dynamical parameters (e.g. beat frequency, phase lag, appendage length, appendage spacing, et al). However, it is unclear how these parameters interact to affect the hydrodynamics of the system overall. We take advantage of natural variation between different species of ctenophores to explore the role of beating frequency, body size, and propulsor spacing in metachronal systems. Using Particle Shadow Velocimetry (PSV), we compare velocity and vorticity fields generated by actively beating ctene rows in three distinct ctenophore species, across a range of beating frequencies and body shapes. Our findings show that ctenophores with more densely packed ctenes (i.e., closer propulsor spacing) generate more coherent flow fields compared to those with higher propulsor spacing at similar Reynolds numbers. Our results highlight the importance of subtle geometric/kinematic differences in driving fluid flow by flexible appendages, and provide a foundation for further investigation of the role of appendage spacing in metachronal coordination for both biological and bioinspired systems. 

Abstract Ctenophores swim using flexible rows of appendages called ctenes that form the metachronal paddling. To generate propulsion, each appendage operates a power stroke that strokes backward, followed by a recovery stroke that allows the appendage to readjust its position. Notably, strokes of most metachronal swimmers are asymmetric, with faster power strokes while slower recovery strokes. Previously, the material properties are assumed as isotropic. So, the faster power stoke will lead to more pronounce deformation and the slower recovery stroke will lead to less deformation. However, this contradicts with the observations that power-stroking ctenes have the least deformation and recover deforms more, indicating an anisotropic material behavior. Such anisotropic material is hard to be manufactured, but the anisotropic behavior may be achieved by making the initial structural shape curved. The pre-curved ctene, that bending towards downstream, will be straighten in power stoke while easy to bend during recovery stroke. Our study aims to demonstrate the feasibility of using pre-curved shapes to achieve anisotropic material properties during metachronal swimming. Treating it as fluid-structure interaction (FSI) problem, we integrate our in-house computational fluid dynamics (CFD) solver with a finite element method (FEM) solver, utilizing strong coupling methods for convergence. By comparing the performance of pre-curved ctenes with straight ones, which represent isotropic material properties, we found that the curved ctenes exhibited 26.05% to 65.69% higher cycle-averaged thrust compared to the straight one as stiffness is lower. However, as stiffness increased, the pre-curved ctenes produced 3.92% to 30.58% less thrust than the straight ones. Similar trends were observed in propulsive efficiency, with the pre-curved ctenes demonstrating 46.97% better efficiency at the lowest stiffness but dropping to 34.02% less efficient as stiffness rise. Thus, while the pre-curved initial shape led to better performance at lower stiffness, exceeding a certain stiffness threshold resulted in worse performance compared to straight ctenes. The thrust enhancement from pre-curve shape is due to the drag reduction during recovery stroke, where the curved shape mitigate part of force to point more downward. 

Abstract Many organisms use flexible appendages for locomotion, feeding, and other functional behaviors. The efficacy of these behaviors is determined in large part by the fluid dynamics of the appendage interacting with its environment. For oscillating appendages at low Reynolds numbers, viscosity dominates over inertia, and appendage motion must be spatially asymmetric to generate net flow. At high Reynolds numbers, viscous forces are negligible and appendage motion is often also temporally asymmetric, with a fast power stroke and a slow recovery stroke; such temporal asymmetry does not affect the produced flow at low Reynolds numbers. At intermediate Reynolds numbers, both viscous and inertial forces play non-trivial roles—correspondingly, both spatial and temporal asymmetry can strongly affect overall propulsion. Here we perform experiments on three robotic paddles with different material flexibilities and geometries, allowing us to explore the effects of motion asymmetry (both spatial and temporal) on force production. We show how a flexible paddle’s time-varying shape throughout the beat cycle can reorient the direction of the produced force, generating both thrust and lift. We also evaluate the propulsive performance of the paddle by introducing a new quantity, which we term ‘integrated efficiency’. This new definition of propulsive efficiency can be used to directly evaluate an appendage’s performance independently from full-body swimming dynamics. Use of the integrated efficiency allows for accurate performance assessment, generalization, and comparison of oscillating appendages in both robotic devices and behaving organisms. Finally, we show that a curved flexible paddle generates thrust more efficiently than a straight paddle, and produces spatially asymmetric motion—thereby improving performance—without the need for complex actuation and controls, opening new avenues for bioinspired technology development. 

Abstract Ctenophores employ flexible rows of appendages called ctenes that form the metachronal beating pattern. A complete cycle of such paddling consists of a power stroke that strokes backward to produce propulsion and a recovery stroke that allows the appendage to recover its initial position. Effective locomotion in these creatures relies on maximizing propulsion during the power stroke while minimizing drag in the recovery stroke. Unlike rigid oars, the ctenes are flexible during both the power stroke and the recovery stroke, and notably, their strokes are asymmetric, with faster movement during the power stroke. As previous research assumed uniform material properties. This assumption will eventually make the ctene deform more intensively in the power stroke than the recovery stroke due to the asymmetrical hydrodynamic forces. However, observations contradict these assumptions. One explanation posits that ctenes stiffen during the power stroke, enhancing their propulsive force, and become more flexible in the recovery stroke, reducing drag by minimizing the water-countering area. This study focusses on the influence of asymmetric stiffness on their propulsion mechanism. Inspired by nature, we conducted three-dimensional fluid-structure interaction (FSI) using an in-house immersed-boundary-method-based flow solver integrated with a nonlinear finite-element solid-mechanics solver. This integrated solver uses a two-way coupling that ensures a higher accuracy regarding the complexity due to the involvement of the multiple ctenes in a ctene row. The preliminary results show that the anisotropic stiffness of the ctene have better accuracy of deformation as compared to the deformation recorded by the high-speed camera. The asymmetric properties of the ctene material allow both the spatial and temporal asymmetry of the ctene beating pattern. Our investigation suggests that while symmetrical beating can only generate negative net thrust, a slightly asymmetrical beating can make the thrust positive. We find that power stroke period that cost 30% whole period can generates the highest thrust. As multiple ctenes involves, the interaction among ctenes can amplified the effects of the asymmetrical beating, so that the thrust generation is enhanced by 9 to 13 times because of it. 

Abstract Metachronal motion is a unique swimming strategy widely adopted by many small animals on the scale of microns up to several centimeters (e.g., ctenophores, copepods, krill, and shrimp). During propulsion, each evenly spaced appendage performs a propulsive stroke sequentially with a constant phaselag from its neighbor, forming a metachronal wave. To produce net thrust in the low-to-intermediate Reynolds number regime, where viscous forces are dominant, the beat cycle of a metachronal appendage must present significant spatial asymmetry between the power and recovery stroke. As the Reynolds number increases, the beat cycle is observed to change from high spatial asymmetry to lower spatial asymmetry. However, it is still unclear how the magnitude of spatial asymmetry can modify the shear layers near the tip of appendages and thus affect its associated hydrodynamic performance. In this study, ctenophores are used to investigate the hydrodynamics of multiple appendages performing a metachronal wave. Ctenophores swim using paddle-like ciliary structures (i.e., ctenes), which beat metachronally in rows circumscribing an ovoid body. Based on high-speed video recordings, we reconstruct the metachronal wave of ctenes for both a lower spatial asymmetry case and a higher spatial asymmetry case. An in-house immersed-boundary-method-based computational fluid dynamics solver is used to simulate the flow field and associated hydrodynamic performance. Our simulation results aim to provide fundamental fluid dynamic principles for guiding the design of bio-inspired miniaturized flexible robots swimming in the low-to-intermediate Reynolds number regime. 

Aquatic organisms often employ maneuverable and agile swimming behavior to escape from predators, find prey, or navigate through complex environments. Many of these organisms use metachronally coordinated appendages to execute complex maneuvers. However, though metachrony is used across body sizes ranging from microns to tens of centimeters, it is understudied compared to the swimming of fish, cetaceans, and other groups. In particular, metachronal coordination and control of multiple appendages for three-dimensional maneuvering is not fully understood. To explore the maneuvering capabilities of metachronal swimming, we combine 3D high-speed videography of freely swimming ctenophores (Bolinopsis vitrea) with reduced-order mathematical modeling. Experimental results show that ctenophores can quickly reorient, and perform tight turns while maintaining forward swimming speeds close to 70% of their observed maximum—performance comparable to or exceeding that of many vertebrates with more complex locomotor systems. We use a reduced-order model to investigate turning performance across a range of beat frequencies and appendage control strategies, and reveal that ctenophores are capable of near-omnidirectional turning. Based on both recorded and modeled swimming trajectories, we conclude that the ctenophore body plan enables a high degree of maneuverability and agility, and may be a useful starting point for future bioinspired aquatic vehicles. 

Metachronal rowing is a biological propulsion mechanism employed by many swimming invertebrates (e.g. copepods, ctenophores, krill and shrimp). Animals that swim using this mechanism feature rows of appendages that oscillate in a coordinated wave. In this study, we used observations of a swimming ctenophore (comb jelly) to examine the hydrodynamic performance and vortex dynamics associated with metachronal rowing. We first reconstructed the beating kinematics of ctenophore appendages based on a high-speed video of a metachronally coordinated row. Following the reconstruction, two numerical models were developed and simulated using an in-house immersed-boundary-method-based computational fluid dynamics solver. The two models included the original geometry (16 appendages in a row) and a sparse geometry (8 appendages, formed by removing every other appendage along the row). We found that appendage tip vortex interactions contribute to hydrodynamic performance via a vortex-weakening mechanism. Through this mechanism, appendage tip vortices are significantly weakened during the drag-producing recovery stroke. As a result, the swimming ctenophore produces less overall drag, and its thrust-to-power ratio is significantly improved (up to 55.0 % compared with the sparse model). Our parametric study indicated that such a propulsion enhancement mechanism is less effective at higher Reynolds numbers. Simulations were also used to investigate the effects of substrate curvature on the unsteady hydrodynamics. Our results illustrated that, compared with a flat substrate, arranging appendages on a curved substrate can boost the overall thrust generation by up to 29.5 %. These findings provide new insights into the fluid dynamic principles of propulsion enhancement underlying metachronal rowing. 

The hydrodynamics of swimming at the millimeter-to-centimeter scale often present the challenge of having both viscous and inertial effects playing nontrivial roles. Inertial forces arise from the momentum of a moving fluid, while viscous forces come from friction within the flow. The non-dimensional Reynolds number (Re) compares the magnitudes of the inertial and viscous forces within a flow. At low Re (≪ 1), viscous forces dominate; at higher Re (≫ 1), inertial forces are more important. Efforts to understand the hydrodynamics of swimming have mainly focused on the extremes of fully viscous-dominated (Re ≪ 1) or inertia-dominated flow (Re ≫ 1). However, many animals swim in an intermediate regime, where inertia and viscosity are both significant. As an impactful and generalizable case study, we focus on ctenophores (comb jellies), a type of marine zooplankton. Ctenophores swim via the coordinated rowing of numerous highly flexible appendages (ctenes), with Reynolds numbers on the order of 10-100. Their locomotory dynamics present a unique opportunity to study the scaling of rowing (drag-based propulsion) across the low to intermediate Reynolds number range. With a combination of animal experiments, reduced-order analytical modeling, and physical-robotic modeling, we investigate how the kinematic and geometric variables of beating ctenes vary across Re, and how they affect swimming (including force production, speed, and maneuverability). Using animal experiments, we quantify the spatiotemporal asymmetry of beating ctenes across a wide range of animal sizes and Re. With our reduced-order model—the first to incorporate adequate formulations for the viscous-inertial nature of this regime—we explore the maneuverability and agility displayed by ctenophores, and show that by controlling the kinematics of their distributed appendages, ctenophores are capable of nearly omnidirectional swimming. Finally, we use a compliant robotic model that mimics ctenophore rowing kinematics to study rowing performance with direct calculation of thrust and lift forces distributed along the propulsor. These experiments shed new light on the relationship between motion asymmetries and thrust and lift production. This combination of animal experiments, analytical modeling, and physical modeling is the most detailed study of low to intermediate Re rowing to date, and provides a foundation for future applications in bio-inspired design.