Session Q03: Biological Fluid Dynamics: Locomotion (3:55pm - 4:40pm CST)

Efficient biomimetic propulsors with combined internal-external actuation

Bio-inspired robotic swimmers can be propelled by periodic oscillations of an elastic caudal fin. Conventional designs use an external actuation source to create heaving motion. Recently the emergence of smart materials enables bio-inspired fins to be actuated by internal source such as a bending piezoelectric moment. We use three-dimensional computer simulations to probe the effect of combining these two distinct types of fin actuation. The fin, represented by a rectangular elastic plate, is actuated at the root with a harmonic heaving motion and by a distributed internal bending moment. The two actuations share an equal frequency. We vary the magnitude and phase difference between the actuation methods to investigate the resulting hydrodynamic thrust and efficiency. We find that the hybrid actuation can outperform either of the actuation methods. We identify the parameter space in which the synergy of the two actuation methods results in the enhanced hydrodynamic performance.   

The roles of tail and body angles on metachronal swimming performance

Freely-swimming crustaceans range from only a few millimeters to over a meter in body length, and some species are known to migrate across long distances. They swim individually and in large schools, and can rapidly maneuver in all directions using a swimming technique called metachronal paddling, which involves the sequential, periodic motion of closely spaced limbs. A number of factors, including body morphology and limb kinematics can affect metachronal swimming performance, and various species have been observed to flex their abdomen and tail as a way to vector the thrust generated by the paddling motion. Using a robotic paddling model, we examine how variation of the body and tail angles impact swimming performance, as well as affecting the momentum and angle of the paddling wake. Increasing the angle between the tail and the longitudinal axis of the body resulted in slightly increasing the angle of the wake, as well as slightly decreasing the total momentum of the wake, while changing the body angle resulted in larger changes in wake angle and swimming speed.   

Field testing of biohybrid robotic jellyfish to demonstrate enhanced swimming speeds

Biohybrid robots incorporating self-contained microelectronic systems embedded into live animals can potentially leverage the animals' self-healing properties and offset robotic power constraints using their metabolism. Previous work has shown that a biohybrid robotic jellyfish can exhibit enhanced swimming speeds up to 2.8 times in laboratory environments, but it remains unknown whether these results also occur in natural, dynamic ocean environments. The present work demonstrates a proof of concept that biohybrid robotic jellyfish can be successfully implemented in the coastal waters of Massachusetts. We demonstrate comparable field results to prior laboratory work, with enhancement factors up to 2.3 times the baseline speed, or absolute swimming speeds up to 6.6 $\pm$ 0.3 cm s$^{-1}$. A theoretical model was developed to predict experimental swimming speeds with mean errors within 1 cm s$^{-1}$, using morphological and time-dependent input parameters from individual animals. With future work to increase maneuverability and incorporate sensors to track environmental changes, we can potentially use biohybrid robotic jellyfish as a ubiquitous and energy-efficient tool in ocean monitoring.   

BOSO-Micro: The Bank Of Swimming Organisms at the Micron Scale

Unicellular microscopic organisms living in aqueous environments outnumber all other creatures on Earth. A large proportion self-propel in fluids with a vast diversity of swimming gaits and motility patterns. Here we introduce the BOSO-Micro database (acronym for Bank Of Swimming Organisms - Microscopic), a survey of the available experimental data produced to date [August 2020] on the motile behaviour of four broad categories of unicellular microswimmers: bacteria (and archaea), flagellated eukaryotes, spermatozoa and ciliates. Whenever possible, we first gathered the following parameters: species, geometry and size of the organisms, swimming speeds, actuation frequencies, actuation amplitudes, number of flagella/cilia and properties of the surrounding fluid. We then analyse the data in the light of the established fluid mechanics principles for propulsion at low Reynolds numbers. We reproduce expected scalings for the locomotion of cells within the same taxonomic groups of organisms while demonstrating the variability for organisms of different species within a group. The material gathered in our work is a summary of the established knowledge in the domain, providing a convenient and practical reference point for future studies while highlighting uncharted territories.   

Scaling the Performance of Hydrofoils with Combined Pitching and Heaving Motion in Underwater Swimming

This study evaluates the scaling of the propulsive performance of an oscillating teardrop foil with combined heaving and pitching motions at a range of Reynolds numbers ($Re=1000-8000$), reduced frequencies ($f^*=0.16-0.64$), Strouhal numbers ($St=0.1-0.8$), and phase angles ($\phi = 0^\circ-270^\circ$). Over 150 simulations were completed using Overset Grid Assembly incorporated into Direct Numerical Simulations in OpenFOAM. The numerically verified results are validated against the experimental performance data of Van Buren et al. (2020). Using this large dataset, we developed new scaling relations that incorporate $Re$, $f^*$, $St$, and $\phi$. This study extends the scaling laws originally developed for a solely pitching or heaving foil by Floryan et al. (2017) at a fixed $Re$, and then extended by Van Buren et al. (2020) to combined heaving and pitching motion for fixed $Re$ and varying $f^*$. The preliminary scaling suggests that the performance data scales with $Re^{1/2}$, which is a laminar behaviour. Moreover, the effect of $\phi$ dominates the scaling at different $f^*$ and $St$. These results also suggest that combining heaving and pitching motion alters the impact of effective oscillation amplitude, and thus the wake behaviour compared to a pure pitching motion.   

Swimming response of marine copepod species to small-scale turbulent-like eddies

The swimming kinematics of three species of marine copepods (\textit{Acartia tonsa}, \textit{Temora longicornis}, and \textit{Calanus finmarchicus}) were studied in response to small-scale turbulent-like eddies using a Burgers vortex model. A stable Burgers vortex structure is generated in the laboratory that simulated an individual turbulent eddy at dissipation rates between 0.002 and 0.25 cm$^{\mathrm{2}}$s$^{\mathrm{-3}}$, which copepods are likely to encounter in their environments. The vortex structure was generated in four intensities, plus stagnant flow conditions, to examine how copepods respond to the intensity and orientation of turbulent eddies in their habitat. All treatments were generated with horizontal and vertical orientations of the vortex axis (separately). Tomographic PIV was used to measure the three-dimensional flow field and to quantitatively assess that it matches the target parameters of natural turbulent structures. Three-dimensional trajectories were digitally reconstructed and overlaid on the vortex flow field to obtain swimming kinematics relative to the flow field. The results show behavioral responses that are species dependent. As the vortex intensity increases, the copepods are more likely to swim in circular trajectories around the vortex axis. Species dependent responses also include changes in relative swimming speed, path complexity, hop or escape frequency, and other measures that depend on their mechanosensory system and swimming mode.   

Microswimmers in anisotropic media

Biological microswimmers sometimes have to move through complex fibrous environments whose microstructures are anisotropic. Examples include the cytoplasm of eukaryotic cells, gels and polymer networks. In this work, we develop theoretical modelling of simple microswimmers interacting with slender fibres. We first use a combination of asymptotic calculations and numerical simulations based on the method of regularised stokeslets to address the fundamental case of a single force-free swimmer near an infinite slender fibre and then address the stochastic dynamics of an ensemble of swimmers. We finally show how to use our results in order to design useful anisotropic media.   

Rebound and scattering of motile Chlamydomonas al- gae in confined chambers

Recent experiments showed that motile algae, as opposed to dead cells, get trapped in draining aquatic foams. Motivated by these observations, we study the swimming behaviour of Chlamydomonas reinhardtii (CR) cells confined in two-dimensional microscopic chambers imitating the cross-sectional shape of a single foam internal channel. We first carry out experiments to track swimming cells and deduce the probability density function of the cells in the chambers, as well as their scattering dynamics along walls. The analysis of the phase space of trapped and escaping trajectories inside a simple three-circles billiard with constant bouncing angle shows that the experimentally observed accumulation of swimmers in the corners has a geometric origin. We then develop a more detailed model based on experimental data to quantitatively reproduce the distribution of swimmers in the chamber. We determine that the trapping of CR is controlled by a combination of the shape of the concave chambers, the finite size of the CR cells, and the angle distribution of the cells bouncing off the walls. We finally observe that the cells are significantly slowed down in the vicinity of walls and show, using numerical simulations, that this effect is of purely hydrodynamic origin.   

Hydrodynamics and bio-locomotion of fast start mechanism in krill: Caridoid escape response

Krill are shrimp-like crustaceans with a high degree of mobility and a variety of documented swimming behaviors. The caridoid escape response, a fast-start mechanism unique to crustaceans, occurs when the crustacean performs a series of rapid abdominal flexions and tail flipping that result in powerful backward strokes. For the first time, the propulsion behavior and flow disturbance of a caridoid response performed by an Antarctic krill (\textit{Euphausia superba}) has been quantified using a tomographic Particle Image Velocimetry (tomo-PIV) system. This system was used to quantify the three-dimensional flow field around and in the wake of a free-swimming \textit{E. superba} as it performs the maneuver. From kinematic analysis, it was determined that the animal performs an abdominal flexion and tail flip combination that leads to an acceleration over a 50 ms interval allowing it to reach a maximum speed of 57 cm/s. Counter-rotating vortices are shed in the wake of the krill located off of the tip of the antennae. Antarctic krill typically swim in a low-to-intermediate Reynolds number (\textit{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. The velocity and vorticity field data shed light on both the flow behavior in the intermediate \textit{Re} regime and the intricacies of bio-locomotion of zooplankton.   

High speed metachronal swimming by the peacock mantis shrimp.

Metachronal rowing is a common swimming technique among organisms with multiple swimming legs in which posterior legs stroke first and are sequentially followed by anterior neighbors. Metachronal rowing leg kinematics have been previously measured for a wide range of swimming speeds, but flow fields near the appendages of freely swimming animals have only been measured for hovering or slowly swimming animals. Here we present time-resolved 2D PIV measurements of the flow generated by a peacock mantis shrimp swimming at speeds of 0.2-1.9 m/s, and advance ratios of 1.1-2.6. Measurements are acquired in sagittal, near-frontal, and transverse planes on an animal with body and pleopod lengths of 114 mm and 15 mm, respectively. Measurements in the animal's sagittal plane show that each stroking pleopod pair creates a vortex which is advected backwards. At these high advance ratios, the vortex created by the anteriormost pleopod pair interacts with and is strengthened by the power stroke of the posteriormost pleopod pair. Flow measurements in the near-frontal plane show a jet of counter-rotating vortices resembling a reverse von Karman vortex street in the animal's wake. Counter-rotating vortex pairs are also seen in the animal's wake in the transverse plane.   

Effect of Reynolds number on the thrust generation of a pitching Prandtl-D root-ns foil

The Prandtl-D root-ns (NASA Preliminary Research Aerodynamic Design to Lower Drag) foil inspired from flight of birds, designed by NASA is intended to develop future low drag foils. The flow around a Prandtl-D root-ns foil pitching about its quarter chord point is studied numerically using Ansys-Fluent. The objective is to understand the effect of variation in Reynolds number on the thrust generation of a Prandtl-D root-ns foil. Numerical simulations are conducted over a range of Reynolds numbers, Re $=$ 200 - 2000, for~various Strouhal numbers at a pitching amplitude of $5^{o}$. In order to further test the thrust generation performance of Prandtl-D root foil, the obtained results are compared with NACA (National Advisory Committee for Aeronautics) series foil NACA0012. The Prandtl-D root-ns foil is observed to perform better compared to NACA0012 foil, in terms of thrust generation. The reduction in viscous component of drag for the Prandtl-D root-ns foil compared to NACA0012 foil is responsible for enhancement in thrust generation. The results of the present study will help to design high performance under water vehicles.   

Continuum Simulation of Hybrid Locomotion on Granular Terrain in SPH

Wheeled robots can move fast on flat surfaces but suffer from loss of traction and immobility on soft ground because of sinkage and slipping. Legged machines, however, have superior mobility over wheeled locomotion when they are in motion over flowable ground or a terrain with obstacles but can only move at relatively low speeds on flat surfaces. A plausible question is the following: If legged and wheeled locomotion are combined, can the resulting hybrid leg-wheel locomotion move fast in any terrain condition? Locomotive interactions are sensitive to the underlaying ground soil, which is a granular material. Continuum treatment is a good choice for dealing with dynamic interaction problems when the particle size of the ground is smaller than the intruder size. A Smooth particle hydrodynamics framework has been proposed for this problem. The mesh-free nature of SPH makes it easy to capture the large deformation and the post-failure state of the granular substrate. Great agreement is found amongst the obtained numerical results and theoretical as well as experimental results across a wide range of robot leg shapes. The results are expected to form a good basis for robot navigation and exploration in unknown and complex terrains.   

Hydrodynamics of elastic plates with external and internal actuation

Fish are able to achieve swimming efficiencies and burst speeds far exceeding any man-made device through passive (flow induced) and active (muscles) deformations of membranes. The emergence of smart materials has allowed new approaches for efficient design of robotic fish. Using numerical simulations based on a coupled lattice Boltzmann and finite differences model and experiments we investigate different actuation methods on the hydrodynamics of elastic plates oscillating at resonance. We compare the conventional heaving actuation with an internal actuation mimicking smart materials, such as macro-fiber composites (MFC). We identify linear and nonlinear regimes of plate oscillations for a wide range of Reynolds numbers. We find that the actuation method drastically impacts the propulsion thrust and efficiency. Heaving plates significantly outperform plates with internal actuation. This result is explained based on the bending pattern of the plate and emerging flow structures. We also find that the inertia coefficient is a strong function of the actuation method, amplitude, aspect ratio and Reynolds number. Our results point to the need to develop methods for improving the hydrodynamic efficiency of propulsors made of internally actuated smart materials.   

Enhanced swimming performance of tapered plates

In nature fish rely on different type of swimming motion to propel themselves. On the one hand anguilliform fish (eel-like) generate travelling waves while ostraciiform fish utilize standing waves for propulsion. Different type of fish swimming can be formally characterized by investigating the nature of the wave propagation in terms of the standing wave ratio (SWR). Although standing wave based propulsion yields higher swimming velocity, creating a travelling wave enabling efficient swimming is not trivial. The acoustic black hole (ABH) effect is a phenomenon arising in tapered structures that prevents wave reflection thereby leading to traveling waves. Through three-dimensional fully coupled fluid structure interaction simulations, we show that ABH is an efficient passive solution to maintain travelling waves in finite-sized structures that can be used for efficient underwater propulsion. We explore the effects of different tapering shapes on the SWR and demonstrate that the SWR is a critical metric relating to the hydrodynamic efficiency of tapered plates.   

Non-planar flagellar beating enables helical navigation by chiral microswimmers

Helical alignment towards gradients in space and time is a ubiquitous mechanism by which ciliated and flagellated organisms achieve deterministic reorientation in space. Here, we consider the biflagellate green alga Chlamydomonas reinhardtii which rotates steadily at 1-2Hz in the absence of stimuli while swimming on helical trajectories. To date, the mechanism underlying this behaviour has not been confirmed. We prove for the first time that rolling motion derives from a consistent, non-planar beat pattern. We demonstrate beat non-planarity by high-speed imaging and micromanipulation of live cells. To relate the observed flagellar beat patterns directly to the free-swimming dynamics, we construct a fully 3D theoretical model of Chlamydomonas. Incorporating geometrical parameters from the experimental data, we reproduce the sense and magnitude of the axial rotation observed in real cells. In particular, we deduce that helical swimming arises from an asymmetric flagellar driving. Cells are able to reorient towards or away from directional stimuli by actively modulating this bilateral asymmetry. We conjecture that molecular and physiological differences between cis and trans flagella underlie intracellular control over biflagellar dominance, which is critical for steering or taxis.   

Computational Analysis of Thrust Generation in a Foil with Travelling Wave on Its Surface

We present an analysis of propulsive performance in a 2D foil with travelling wave on its surface using the in-house level-set based immersed boundary method solver. Four governing parameters: Reynolds number , velocity ratio ($V_R = c/U_\infty=[1-10]$), amplitude ($A_m=[0.1-0.5]$) and wavenumber ($k=[5-20]$) of the travelling wave, have been varied to study its effect on the total drag. The propulsive efficiency and thrust coefficient have been used as output data for propulsive performance analysis. In the travelling wave motion, the generation of thrust force is because of the high-pressure and high-velocity zone at the trough of the wave on the foil surface. This high-pressure and high-velocity zone produces the localized thrust jet, which ultimately makes the body to move forward. The time-averaged value of total drag has been found to become negative (i.e. net thrust) for $V_R$ $\geq 1.2$ at $Re=5000$ with $A_{m}=0.3$ and $k=10$ of travelling wave. The propulsive efficiency increases with increasing $V_R$ and reaches to its maximum and starts to decrease after a further increase in $V_R$. The variation of the phase angle between the upper and lower surfaces of the foil has also been studied. At $\phi = 180^o$, $$ is maximum while $C_{L_{rms}}$ is minimum.   

Geometric and Dynamic Scaling of Marine Snail Swimming

Pteropods are holoplanktonic marine snails which swim by flapping their highly flexible pair of wings. Shelled pteropods are highly negatively buoyant as compared to shell-less pteropods. Swimming is essential for pteropod survival as these animals must escape from predators and perform diel vertical migration. Different pteropod species have different shell shapes and sizes, are distributed in tropical, temperate, and polar oceans around the world, and have significant ecological and biogeochemical impact. However, the effects of pteropod shell presence, shape, and size on swimming are not well studied. Here we acquire high speed stereophotogrammetry measurements of the swimming of seven shelled and one shell-less subtropical pteropod species. Using these data and previously measured data for temperate and polar species, we investigate the relationship among parameters such as size, flapping frequency, and seawater viscosity (which corresponds to temperature), translational Re, flapping Re, and Strouhal number. For four different pteropod species with coiled shells operating across seawater viscosities which differ by a factor of almost 2, beat frequency is inversely related to animal size and to water viscosity and the Strouhal number is found to lie between 0.2 and 0.4.   

Cambered Undulating Fin for Heading Control.

Some aquatic organisms use undulating membranes for forward swimming as well as to generate remarkable directional maneuvers. However, the control of the fin to produce similar swimming characteristics in underwater robotics remains a challenge. In this work, we explore the maneuver control of a robotic underwater vessel in different flow conditions with an arbitrary fin kinematics producing different wave shapes. The propulsion of the robotic vessel consists of a single undulating fin running along the length of the robot, which controls both forward motion and directional maneuvers. We tested the physical model in a water channel with different incoming flow. In addition, the robotic system was tested in pool to follow a predefined trajectory. During a trajectory following the robot is controlled to follow the path by the actuation of two states, i.e. the forward speed and the heading. These states are controlled by the adjustment of the amplitude of oscillation of the undulating fin, and the deflection of the totality of the rays in a curved way, as opposed to the linear offset rotation of only a subset of the rays of the fin. In contrast with this thrust vectoring approach, we found that adjusting the camber of the undulating fin produces a net yawing torque capable of changing the heading of the robot during motion. Preliminary results also show that the control scheme is robust for different flow conditions.   

Intracellular coupling modulates biflagellar synchrony

Beating cilia and flagella exhibit diverse synchronization modes. This has long been attributed to hydrodynamic coupling between the flagella. However, recent work using different flagellated algae has indicated that a mechanism internal to the cell, acting through the contractile fibers connecting the flagellar basal bodies, must be at play to actively modulate flagellar synchrony. Exactly how basal coupling mediates flagellar coordination remain unclear. Here, we examine the role of basal coupling in the synchronization of the model biflagellate Chlamydomonas reinhardtii using a series of mathematical models of decreasing complexity. We report that basal coupling is sufficient to achieve inphase, antiphase, and bistable synchrony, even in the absence of hydrodynamic coupling and flagellar compliance. These modes can be reached by modulating the activity level of the individual flagella or the strength of the basal coupling. We observe a ‘slip’ mode when allowing for differential flagellar activity, just as in experiments with live cells. Lastly, we introduce a dimensionless ratio of flagellar activity to basal coupling, which is predictive of synchronization mode. This allows us to query biological parameters which are currently not accessible experimentally   

Transient eddies over ridged substrates enhance trapping and settlement of coral larvae

The low settlement success of planktonic larvae is an important problem that can inhibit the recovery of reefs from environmental damage. Tiny coral larvae ($<$ 1mm) must navigate the water column to find a suitable surface for permanent settlement, a process influenced by diverse chemical, biological, and physical mechanisms acting over multiple length scales. Using a custom-built oscillatory flume tank, we investigate coral larval settlement on substrates of different roughness scales to understand the effect of hydromechanical forces on larvae due to boundary layer flows. Dynamic flow fields, characterized by particle image velocimetry, contain regions of recirculation near millimeter-scale substrate features that correlate strongly with larvae settlement positions, despite the mean flow speed in the flume tank being an order of magnitude higher than the larval swimming speed. Using simulations, we explore how the recirculation regions influence the settlement of active swimmers over substrates of varying surface topography.   

Swimming of a Subtropical Soft-bodied Sea Angel at Intermediate Reynolds Number

Gymnosomatous pteropods, also known as sea angels, are shell-less planktonic marine snails which swim by flapping a pair of low aspect ratio, highly flexible wing-like parapodia. Locomotion is important for sea angels as these animals must swim to prey on shelled pteropods and to maintain their position in the water column. However, little is known about the fluid dynamics of their swimming as no flow measurements have previously been acquired. Here we present high speed stereophotogrammetry and novel time-resolved 2D micro-PIV measurements of a subtropical sea angel, Pneumoderma atlantica, acquired in Bermuda. The collected animals have body lengths up to 13.1 mm, wing spans up to 5.2 mm, beat their wings at frequencies up to 4.3 Hz, and swim upwards in sawtooth trajectories at speeds up to 34 mm/s, thus placing them at an intermediate Reynolds numbers of approximately 400. Three-dimensional wing and body kinematics show that this species performs a version of the cylindrical overlap-and-fling maneuver by pulling its wings close to its body at the end of each power and recovery stroke. We further compare the swimming of this warm water species with the sea angels Clione limacina and Clione antarctica which live in temperate and polar climates, respectively.   

Robokrill: understanding vortex generation during drag-based metachronal swimming

Metachronal, drag-based swimming in krill (sp. Euphausia superba) has been studied both to assess its ecological relevance as well as finding solutions for underwater locomotion at intermediate Reynolds numbers. While the use of submersible robots has proved useful to understand the benefits of metachrony and drag modulation as a means to propel forward, prominent questions regarding thrust generation remain unanswered. In this talk, we focus on fluid-structure interactions on the inner sections of the appendages of E. superba. We designed and constructed a scaled-up robotic model with geometric and kinematic similarity, reproducing the swimming kinematics of the appendages of free-swimming krill. Our robotic design allows the analysis of fine-scale kinematics and vortex generation in the vicinity of interior limb segments. We present Particle Image Velocimetry (PIV) measurements and flow dye visualizations using different limb shapes to investigate vortex formation mechanisms during drag-based propulsion. These findings feature important characteristics of metachronal propulsion that can be used in the development of underwater robots, especially in highly complex environments.   

Scaling law of ribbon fin propulsion

Some aquatic organisms swim by one or multiple elongated fins. In bony fish, this fin consists of an elastic membrane with hundreds of bony rays that allow them to stretch and manipulate the fin and to perform extraordinary swimming maneuvers. One of the most common kinematics of this elongated fin is transferring undulations in the form of traveling waves one end to the other, generating thrust in the opposite direction, though fishes adopt hybrid techniques for other maneuvering like rapid-reverse, station-keeping etc. Even though analytical formulation for this propulsion mode exists for few decades, very limited experimental work has focused to understand the law of scaling in undulating fin propulsion. The present work study how thrust produced by undulating fin can be scaled with respect to key variables. We used a bio-inspired robotic vessel that propels with an undulating ribbon fin, programmed to create sinusoidal motion. The vessel was tested for both static and free-swimming condition. First, we measured the dynamics of the vessel under free swimming condition, over a range of fin kinematics. Next, the drag force of the vessel was measured at a range of Reynold numbers. Finally, we measured the propulsive force and propulsive performance in a recirculating flume for different incoming flow speed (up to Re $=$ 6 x10$^{\mathrm{4}})$. We found that the propulsive force scales to the square of the relative velocity between the fin wave velocity and flow velocity. We will present the swimming performance and its implication to fish swimming.   

Squirming motion of a sphere in a micropolar fluid

Micropolar fluid continuum equations involve both the velocity and internal spin vector fields resulting in antisymmetric and couple stresses. In such micro-structured fluid continua the spin plays a kinematical role comparable to that played by the velocity in classical Newtonian problems. In this investigation, the problem of swimming in micropolar fluids via a spherical squirmer model is analyzed. The idealized configuration allows analytical solutions for the velocity and spin fields surrounding the squirmer via Stokes stream function formulation. The propulsion speed is calculated using the force-free condition which is, surprisingly, the same as that of the spherical microorganism swimming in Newtonian fluids. The power dissipation and swimming efficiency results derived using non-zero spin boundary conditions on the squirmer surface, however, reveal the micro-rotational effects. The analytical solutions are also utilized to inspect the structure of flow fields surrounding the spherical squirmer. The results may be of interest in understanding microorganisms swimming mechanisms in fluids that exhibit angular momentum due to internal micro-rotation.   

Slender Phoretic Theory of chemically active filaments

Fuel-based autophoretic microbots drive propulsive slip flows via differential reaction rates of solute fuel at their surface. Typically, they are rigid particles, partially-coated in catalyst (Janus particles), but slender phoretic rods have become an increasingly prevalent design. Hitherto, asymptotic theories for slender phoretic rods have been restricted to straight rods with axisymmetric patterning. However, modern manufacturing methods will soon allow fabrication of slender phoretic filaments with complex three-dimensional shape. Thus, we have developed a fully-3D Slender Phoretic Theory (SPT) for the self-diffusiophoretic filaments of arbitrary 3D shape and patterning that reduces the solute concentration problem to a matter of evaluating a line integral. We demonstrate that, unlike other slender body theories, first-order azimuthal variations arising from curvature and confinement can have a leading order contribution to the swimming kinematics. This slender body theory could be used to rationally design phoretic microswimmers as filaments with complex 3D centrelines and chemical patterning, enabling exciting new dynamics.   

On the optimal flexibility condition of an impulsive starting jet nozzle for thrust generation

Looking at the nature, the propelling organs of underwater mollusks and the respiratory organs of fish and shellfish are commonly in the shape of a flexible nozzle, with an exceptional performance. In this study, the effect of the highly deformable nozzle on the jet thrust is studied. For the experiment, in-house cylindrical silicone nozzles (D $=$ 15mm) were manufactured while varying their elasticity (0.1-0.4 MPa), length (1.5-3D), and thickness (50-250 $\mu $m). A starting water jet was pushed using a piston with its exit velocity of 0.2-0.8 m/s, and acceleration time of 0.1-0.3 sec. Particle image velocimetry has been used to measure the velocity field of the jet, and image processing to quantify the nozzle deformation. We found that a larger thrust is generated than the rigid nozzle due to the nozzle deformation when the elasticity of the nozzle increases. In order to predict the interaction between the nozzle and the starting jet, a theoretical model has been constructed by combining the shell theory and tube law. From this model the governing dimensionless number, including the factors from nozzle geometry (aspect ratio and elasticity) and fluid flow (jet velocity and acceleration) are derived, which is proven to determine the optimal nozzle design for maximizing the thrust.   

Undulatory and peristaltic motion of California blackworms in water saturated sediments

The physical mechanisms by which organisms burrow in the loose sedimented Benthic layer at the bottom of lakes and oceans are not understood because of the difficulty in observing their motion in the opaque subsurface. We show that the California blackworms and earthworms use a combination of transverse undulatory strokes, and elongation-contraction peristaltic strokes, in water-saturated sediment beds. Using transparent granular hydrogels which refractive index-match with water, we dynamically track the shape of the worm, and its head and tail in real time inside the medium [1]. We show that the worm in fact moves faster in the sediments compared to moving in water exploiting the greater drag and drag anisotropy it experiences in the sediments compared with water while performing similar body motions. We also discuss the probing head motion of the worm as it makes decisions on the direction of motion and its impact on observed speed. [1]: "Burrowing dynamics of aquatic worms in soft sediments," Arshad Kudrolli and Bernny Ramirez, PNAS 116 25569-25574 (2019).   

Optimal transport of a drop I -- internally actuated case

The Monge-Kantorovich problem of optimal mass transport is an old one, with deep connections to optimization theory and inviscid hydrodynamics and a range of applications to image analysis, machine learning etc. But can one use it or its variants to also construct policies to optimally transport real matter that obey complex physical dynamics? As a first example, we consider the motion of a drop of an active suspension by dynamically controlling the spatial profile of its internal active stress. Within the lubrication approximation, we use optimal control theory to pose and solve the problem of transporting such a drop with minimal expenditure of mechanical work. By parametrizing the position, size and shape of the drop, we uncover a general trade-off that bounds the maximum achievable displacement of the drop by its size, along with bistability in the optimal policies, determined using Pontryagin's Maximum Principle. Our analysis marries hydrodynamics and optimal control in a tractable and interpretable framework, paving the way forward for the spatio-temporal manipulation of active media.   

Magnetoelastic Filament Robot Mimics Anguilliform Swimming in Soft Sediments

We study the locomotion of a swimmer with a magnet head and elastic tail in fluid-saturated granular media driven by an oscillating magnetic field. Our studies are motivated by soft robot designs based on biomimetic principles which organisms exploit in response to the surrounding environment and stimuli. The applied oscillating field, magnetic field strength, and filament tail length, along with the sediment volume fraction and depth are control variables, in our study. Exploiting refractive index matching of the grains, we measure the shape and speed of the robot over time. We will discuss the nonlinear increase of swimming speed and oscillation amplitude of the tail observed with driving frequency, and a minimal model which captures the overall behavior.   

The Effects of Internal Damping on Locomotion in Frictional Environments

The periodic motion or gaits of undulating animals arise as the result of a complex interaction of their central nervous system, muscle and connective tissue, bone, and their environment. Previous studies have assumed that sufficient internal force necessary to produce observed kinematics are always achievable, thus not focusing on an understanding of the connection between force production in the crawling animals and their locomotion performance. For soft robotic applications, internal damping is a parameter in the designer's control, the effect of which is not well understood. We study how the internal damping affects the locomotion performance of a crawler with a continuous, visco-elastic, nonlinear beam model. Crawler muscle actuation is modeled as a traveling wave of bending moment that propagates posteriorly along the body. Consistent with the friction properties of the scales of snakes and limbless lizards, environmental forces are modeled using anisotropic Coulomb friction. We find that by varying the crawler body's internal damping, the performance of the crawler can be altered and distinct gaits emerge. Indeed, we find that crawling direction can be changed by appropriate control of internal damping. Further, we identify the parameters that produce optimal gaits.   

Metachronal versus hybrid stroke kinematics on multi-appendage swimming performance

Aquatic invertebrates have developed distinct locomotion strategies suited to their morphology and swimming behaviors. Many of these invertebrates, particularly crustaceans, move by paddling multiple limbs in sequence. The gaits used by different species can vary, with a purely metachronal (phase-delayed) stroke (MM) commonly found in continuously swimming species such as krill, and a metachronal power stroke followed by a near-synchronous recovery stroke (MS) commonly seen during rapid maneuvering behaviors in benthic and planktonic species such as mantis shrimp and copepods. Using a robotic model, we examine the effect of changing between these two swimming gaits on swimming speed and wake structure. Regardless of phase delay, the main benefit to using MS as opposed to MM is that it allows for a larger stroke amplitude. This allows MS to achieve faster swimming, even though MS is slower than MM for the same stroke amplitude. Additionally, the wake jet generated by MS is more dispersed, while MM generates a narrow, downward angled jet which may be useful for hydrodynamic signaling in schooling groups.   

Locomotive Performance of a Two Degree-of-Freedom Fish Model

A two degree-of-freedom fish model was used to investigate the phenomenological relationship between simplified fish body kinematics and locomotive performance. Its design, construction, and actuation provide control of maximum trailing edge excursion; heave-to-pitch ratio; phase offset between the tail and caudal fin; and oscillation frequency. The model was mounted to a carriage on linear air bearings aligned with the freestream and attached to a load cell that measured thrust. The input power was measured using torque sensors combined with angular velocity. The phase-averaged power input and thrust output were measured for a parameter space spanning all parameters except oscillation frequency that was fixed at 1Hz. Quasi-propulsive efficiency is calculated using the time-averaged thrust and power. The data will be used to determine optimal kinematics for maximum thrust and efficiency. This will include trends for each parameter as well as general trends in performance. These trends will be compared with the literature on pitching and/or heaving airfoils and panels to determine the applicability of extending their findings to full three-dimensional systems.   

Optimal transport of a drop II -- externally actuated case

The Monge-Kantorovich problem of optimal mass transport is an old one, with deep connections to optimization theory and inviscid hydrodynamics and a range of applications to image analysis, machine learning etc. But can one use it or its variants to also construct policies to optimally transport real matter that obey complex physical dynamics? As a second example, we consider the motion of a thin drop by dynamically controlling the spatial profile of an external driving stress such as gravity or capillarity. Within the lubrication approximation, we use optimal control theory pose and solve the problem of optimally transporting such a drop subject to some constraints. Using a minimal parametrization in terms of the position, size and shape of the drop allows us to recast the problem using Pontryagin's Maximum Principle and uncover the limits of controllability of the drop. Our analysis marries hydrodynamics and optimal control in a tractable and interpretable framework, paving the way forward for the design of strategies for the spatio-temporal manipulation of thin drops and films.   

Design optimization of a drag-based vibratory swimmer using averaging

Many aquatic animals and organisms use periodic motions of their body or limbs to generate forward motion. Using averaging techniques and experiments, we investigate the dynamics and design optimization of a class of vibratory swimmers which use asymmetric drag to achieve aquatic locomotion. The equation of motion of the system is a time-periodic, piecewise-smooth differential equation. Dynamic analysis of this class of systems shows that the maximum mean forward speed is achieved by minimizing the ratio of the drag coefficients in the forward and backward phases of motion. Though the drag can be a quadratic, linear, or other function of velocity, our results show that the optimum design is independent of this function. The analysis is also expanded to include the effects of the fluid added mass on the dynamics and design optimization of the class of vibratory systems. Using a centimeter-scale surface vessel with an oscillatory mass inside and an asymmetric rigid fin immersed in fluid, we verified the theoretical optimization results experimentally. The analysis presented here can aid in the design and optimization of bio-inspired and biomimetic robotic swimmers..   

Droving, Driving, and Mustering: Phases of Optimal Herding

While flocking behavior---of cells, animals, robots etc. ---is an area of growing interest, little is understood about how a few shepherds are able to control large groups of swarms, flocks, or herds. Here, we investigate how a shepherd (such as dogs, humans, or robots) should move in order to effectively herd and guide a flock towards a target. Using agent-based, ODE, and PDE models, we find that three distinct phases of control algorithms emerge as potential solutions---despite no specific control algorithm being prescribed---as a result of optimizing herd cohesion, distance to a target, and line of sight. Transitions between the phases are dependent on just two parameters: the scaled herd size and the scaled herd speed. Two of these phases---mustering and droving---show agreement with the behavior of sheepdogs in nature. The third, driving, is a novel phase that suggests an efficient control algorithm for the transport of a very large group of animals by a single agent. Several potential applications of driving can be seen in swarm robots.   

Measurements of performance and kinematics for a steady sinusoidal swimming gait of a 3 degree of freedom fish model

A scale model of a streamlined tuna was used to approximate a steady sinusoidal swimming gait. The model consisted of a stationary head and a tail with three passive joints. The tail was composed of a midbody piece, three ribs, and a trapezoidal acrylic caudal fin that were connected by a flexible spine. The midbody and ribs were able to pitch relative to one another with a finite but small amount of stiffness. The model was actuated at the midbody piece while the characteristics of the surrounding fluid determine the motion of the three ribs and caudal fin. Input torque, thrust, and tail kinematics were measured and compared for nine combinations of Strouhal number and tail flexibility (shim material). Future work will include flow field measurements, and variations of the prescribed actuation waveforms.   

A self-propelled flexible plate with a keel-like structure

The caudal keels, a pair of lateral keel-like structures along the caudal peduncle, are a remarkable specialization in tunas. Although various hypotheses about the function of caudal keels have been proposed, our understanding of their underlying hydrodynamic mechanism is still limited. The immersed boundary method is used to explore the self-propelled flexible plate with the keel-like structure. Vortical structures and pressure distributions are analyzed here to determine the mechanisms of thunniform propulsion. By comparing models with and without keels, caudal keels generate streamwise vortices that result in negative pressure and enhance the average cruising speed and thrust. The propulsion mechanisms are analyzed in detail in terms of phase of stroke. The average cruising speed and the swimming efficiency are increased by more than 3.6{\%} and 3.8{\%} with keels, respectively. The vortical structures are visualized to characterize the mechanism with keels qualitatively. A systematic study of the effects of variations in the keel shape vertically and horizontally is also presented.   

Cephalopod-inspired Jet Propulsion via Cyclic Deformation of an Axisymmetric Propeller

Inspired by the jet propulsion mechanism of cephalopods such as squids, we numerically investigate the propulsion performance of a deformable propeller with a pressure chamber and a nozzle by using an axisymmetric immersed-boundary model. In this system the propulsion is achieved via intermittent jetting enforced by cyclic inflation and deflation of the body The fluid dynamic force on the body includes the jet-related thrust, the added-mass-related thrust, and the viscous drag. The jet-related thrust consists of three parts, the momentum flux through the nozzle, the excessive pressure at the nozzle, and the jet acceleration. The performance of the system is determined by several parameters, including, e.g. the amount of fluid discharged within each deflation (represented by the `stroke ratio' or the `formation number') and the frequency of shape oscillation (the Strouhal number). Systematic simulations have been conducted to examine the force generation, flow characteristics, and energetics of the device at different conditions.