How Non-Uniform Stiffness Affects the Propulsion Performance of a Biomimetic Robotic Fish

Live fish in nature exhibit various stiffness characteristics. The anguilliform swimmer, like eels, has a relatively flexible body, while the thunniform swimmer, like the swordfishes, has a much stiffer body. Correspondingly, in the design of biomimetic robotic fish, how to balance the non-uniform stiffness to achieve better propulsion performance is an essential question needed to be answered. In this paper, we conduct an experimental study on this question. First, a customized experimental platform is built, which eases the adjustment of the non-uniform stiffness ratio, the stiffness of the flexible part, the flapping frequency, and the flapping amplitude. Second, extensive experiments are carried out, finding that to maximize the propulsion performance of the biomimetic robotic fish, the non-uniform stiffness ratio is required to adapt to different locomotor parameters. Specifically, the non-uniform stiffness ratio needs to be reduced when the robotic fish works at low frequency, and it needs to be increased when the robotic fish works at high frequency. Finally, detailed discussions are given to further analyze the experimental results. Overall, this study can shed light on the design of a non-uniform biomimetic robotic fish, which helps to increase its propulsion performance.

A critical comparison of Lagrangian methods for coherent structure detection

We review and test twelve different approaches to the detection of finite-time coherent material structures in two-dimensional, temporally aperiodic flows. We consider both mathematical methods and diagnostic scalar fields, comparing their performance on three benchmark examples: the quasiperiodically forced Bickley jet, a two-dimensional turbulence simulation, and an observational wind velocity field from Jupiter's atmosphere. A close inspection of the results reveals that the various methods often produce very different predictions for coherent structures, once they are evaluated beyond heuristic visual assessment. As we find by passive advection of the coherent set candidates, false positives and negatives can be produced even by some of the mathematically justified methods due to the ineffectiveness of their underlying coherence principles in certain flow configurations. We summarize the inferred strengths and weaknesses of each method, and make general recommendations for minimal self-consistency requirements that any Lagrangian coherence detection technique should satisfy.

Effect of fluid inertia on the motion of a collinear swimmer

The swimming of a two-sphere system and of a three-sphere chain in an incompressible viscous fluid is studied on the basis of simplified equations of motion which take account of both Stokes friction and added mass effects. The analysis is based on an explicit expression for the asymptotic periodic swimming velocity and a corresponding evaluation of the mean rate of dissipation. The mean swimming velocity of the two-sphere system is found to be nonvanishing provided that the two spheres are not identical. The swimming of a comparable chain of three identical spheres is much more efficient.

Undulatory locomotion of flexible foils as biomimetic models for understanding fish propulsion

An undulatory pattern of body bending in which waves pass along the body from head to tail is a major means of creating thrust in many fish species during steady locomotion. Analyses of live fish swimming have provided the foundation of our current understanding of undulatory locomotion, but our inability to experimentally manipulate key variables such as body length, flexural stiffness, and tailbeat frequency in freely-swimming fish has limited our ability to investigate a number of important features of undulatory propulsion. In this paper we use a robotic apparatus to create an undulatory wave in swimming passive flexible foils by creating a heave motion at their leading edge, and compare this motion to body bending patterns of bluegill sunfish (Lepomis macrochirus) and clown knifefish (Notopterus chitala). We found similar swimming speeds, Reynolds and Strouhal numbers, and patterns of curvature and shape between these fish and foils suggesting that passive flexible foils provide a useful model for understanding fish undulatory locomotion. We swam foils with different lengths, stiffnesses, and heave frequencies while measuring forces, torques, and hydrodynamics. From measured forces and torques we calculated thrust and power coefficients, work, and cost of transport for each foil. We found that increasing frequency and stiffness produced faster swimming speeds and more thrust. Increasing length had minimal impact on swimming speed, but had a large impact on Strouhal number, cost of transport, and thrust coefficient. Foils that were both stiff and long had the lowest cost of transport (in mJ m-1 g-1) at low cycle frequencies, and the ability to reach the highest speed at high cycle frequencies.

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.

Integrating dynamic energy budget (DEB) theory with traditional bioenergetic models

Dynamic energy budget (DEB) theory offers a systematic, though abstract, way to describe how an organism acquires and uses energy and essential elements for physiological processes, in addition to how physiological performance is influenced by environmental variables such as food density and temperature. A ‘standard’ DEB model describes the performance (growth, development, reproduction, respiration, etc.) of all life stages of an animal (embryo to adult), and predicts both intraspecific and interspecific variation in physiological rates. This approach contrasts with a long tradition of more phenomenological and parameter-rich bioenergetic models that are used to make predictions from species-specific rate measurements. These less abstract models are widely used in fisheries studies; they are more readily interpretable than DEB models, but lack the generality of DEB models. We review the interconnections between the two approaches and present formulae relating the state variables and fluxes in the standard DEB model to measured bioenergetic rate processes. We illustrate this synthesis for two large fishes: Pacific bluefin tuna (Thunnus orientalis) and Pacific salmon (Oncorhynchus spp.). For each, we have a parameter-sparse, full-life-cycle DEB model that requires adding only a few species-specific features to the standard model. Both models allow powerful integration of knowledge derived from data restricted to certain life stages, processes and environments.

A robotic fish caudal fin: effects of stiffness and motor program on locomotor performance

We designed a robotic fish caudal fin with six individually moveable fin rays based on the tail of the bluegill sunfish, Lepomis macrochirus. Previous fish robotic tail designs have loosely resembled the caudal fin of fishes, but have not incorporated key biomechanical components such as fin rays that can be controlled to generate complex tail conformations and motion programs similar to those seen in the locomotor repertoire of live fishes. We used this robotic caudal fin to test for the effects of fin ray stiffness, frequency and motion program on the generation of thrust and lift forces. Five different sets of fin rays were constructed to be from 150 to 2000 times the stiffness of biological fin rays, appropriately scaled for the robotic caudal fin, which had linear dimensions approximately four times larger than those of adult bluegill sunfish. Five caudal fin motion programs were identified as kinematic features of swimming behaviors in live bluegill sunfish, and were used to program the kinematic repertoire: flat movement of the entire fin, cupping of the fin, W-shaped fin motion, fin undulation and rolling movements. The robotic fin was flapped at frequencies ranging from 0.5 to 2.4 Hz. All fin motions produced force in the thrust direction, and the cupping motion produced the most thrust in almost all cases. Only the undulatory motion produced lift force of similar magnitude to the thrust force. More compliant fin rays produced lower peak magnitude forces than the stiffer fin rays at the same frequency. Thrust and lift forces increased with increasing flapping frequency; thrust was maximized by the 500× stiffness fin rays and lift was maximized by the 1000× stiffness fin rays.

A numerical study of the effects of bell pulsation dynamics and oral arms on the exchange currents generated by the upside-down jellyfish Cassiopea xamachana

Mathematical and experimental studies of the flows generated by jellyfish have focused primarily on mechanisms of swimming. More recent work has also considered the fluid dynamics of feeding from currents generated during swimming. Here we capitalize on the benthic lifestyle of the upside-down jellyfish (Cassiopea xamachana) to explore the fluid dynamics of feeding uncoupled from swimming. A two-dimensional mathematical model is developed to capture the fundamental characteristics of the motion of the unique concave bell shape. Given the prominence of the oral arms, this structure is included and modeled as a porous layer that perturbs the flow generated by bell contractions. The immersed boundary method is used to solve the fluid–structure interaction problem. Velocity fields obtained from live organisms using digital particle image velocimetry were used to validate the numerical simulations. Parameter sweeps were used to numerically explore the effects of changes in pulse dynamics and the properties of the oral arms independently. Numerical experiments allow the opportunity to examine physical effects and limits within and beyond the biologically relevant range to develop a better understanding of the system. The presence of the prominent oral arm structures in the field of flow increased the flux of new fluid from along the substrate to the bell. The numerical simulations also showed that the presence of pauses between bell expansion and the next contraction alters the flow of the fluid over the bell and through the oral arms.

Fluid transport and coherent structures of translating and flapping wings

The Lagrangian coherent structures (LCSs) of simple wing cross sections in various low Reynolds number motions are extracted from high-fidelity numerical simulation data and examined in detail. The entrainment process in the wake of a translating ellipse is revealed by studying the relationship between attracting structures in the wake and upstream repelling structures, with the help of blocks of tracer particles. It is shown that a series of slender lobes in the repelling LCS project upstream from the front of the ellipse and "pull" fluid into the wake. Each lobe is paired with a corresponding wake vortex, into which the constituent fluid particles are folded. Flexible and rigid foils in flapping motion are studied, and the resulting differences in coherent structures are used to elucidate their differences in force generation. The clarity with which these flow structures are revealed, compared to the vorticity or velocity fields, provides new insight into the vortex shedding mechanisms that play an important role in unsteady aerodynamics.

Sensitivity of Lagrangian coherent structure identification to flow field resolution and random errors

The effect of spatial and temporal resolutions and random errors on identification of Lagrangian coherent structures (LCSs) from Eulerian velocity fields is evaluated using two canonical flows: a two-dimensional vortex pair and a vortex ring formed by transient ejection of a jet from a tube. The flow field for the vortex pair case was steady and obtained analytically while the transient vortex ring flow was simulated using computational fluid dynamics. To evaluate resolution and random error effects, the flow fields were degraded by locally smoothing the flow and sampling it on a sparser grid to reduce spatial resolution, adding Gaussian distributed random noise to provide random errors, and/or subsampling the time series of vector fields to reduce the temporal resolution (the latter applying only for the vortex ring case). The degradation methods were meant to emulate distortions and errors introduced in common flow measurement methods such as digital particle image velocimetry. Comparing the LCS corresponding to the vortex boundary (separatrix) obtained from the degraded velocity fields with the true separatrix (obtained analytically for the vortex pair case or from high resolution, noise-free velocity fields for the vortex ring case) showed that noise levels as low as 5%-10% of the vortex velocity can cause the separatrix to significantly deviate from its true location in a random fashion, but the "mean" location still remained close to the true location. Temporal and spatial resolution degradations were found to primarily affect transient portions of the flow with strong spatial gradients. Significant deviations in the location of the separatrix were observed even for spatial resolutions as high as 2% of the jet diameter for the vortex ring case.

Using Lagrangian coherent structures to analyze fluid mixing by cilia

Motivated by the desire to understand the fluid flow within the airway surface liquid of the lung, we consider the flow generated by a computational model of a motile, internally actuated cilium. The cilium, along with a mucus layer modeled by linear elastic elements, is coupled to a viscous, incompressible fluid. The evolution of this coupled system is captured using an immersed boundary method. The Eulerian velocity field computed on a grid is used to compute finite-time Lyapunov exponent fields, whose maximal ridges identify Lagrangian coherent structures (LCSs). The computed LCS uncovers a barrier that separates a recirculation region of fluid that remains near the beating cilium from fluid that is advected downstream. Moreover, periodic stretching and folding of this region gives rise to complex mixing. Flow structures around a cilium propelling a mucus layer are compared to flow structures around a cilium with no mucus load.

Beyond robins: aerodynamic analyses of animal flight

Recent progress in studies of animal flight mechanics is reviewed. A range of birds, and now bats, has been studied in wind tunnel facilities, revealing an array of wake patterns caused by the beating wings and also by the drag on the body. Nevertheless, the quantitative analysis of these complex wake structures shows a degree of similarity among all the different wake patterns and a close agreement with standard quasi-steady aerodynamic models and predictions. At the same time, new data on the flow over a bat wing in mid-downstroke show that, at least in this case, such simplifications cannot be useful in describing in detail either the wing properties or control prospects. The reasons for these apparently divergent results are discussed and prospects for future advances are considered.

The `upstream wake' of swimming and flying animals and its correlation with propulsive efficiency

The interaction between swimming and flying animals and their fluid environments generates downstream wake structures such as vortices. In most studies, the upstream flow in front of the animal is neglected. In this study,we demonstrate the existence of upstream fluid structures even though the upstream flow is quiescent or possesses a uniform incoming velocity. Using a computational model, the flow generated by a swimmer (an oscillating flexible plate) is simulated and a new fluid mechanical analysis is applied to the flow to identify the upstream fluid structures. These upstream structures show the exact portion of fluid that is going to interact with the swimmer. A mass flow rate is then defined based on the upstream structures, and a metric for propulsive efficiency is established using the mass flow rate and the kinematics of the swimmer. We propose that the unsteady mass flow rate defined by the upstream fluid structures can be used as a metric to measure and objectively compare the efficiency of locomotion in water and air.

Vortex wake and flight kinematics of a swift in cruising flight in a wind tunnel

In this paper we describe the flight characteristics of a swift (Apus apus) in cruising flight at three different flight speeds (8.0, 8.4 and 9.2 m s(-1)) in a low turbulence wind tunnel. The wingbeat kinematics were recorded by high-speed filming and the wake of the bird was visualized by digital particle image velocimetry (DPIV). Certain flight characteristics of the swift differ from those of previously studied species. As the flight speed increases, the angular velocity of the wingbeat remains constant, and so as the wingbeat amplitude increases, the frequency decreases accordingly, as though the flight muscles were contracting at a fixed rate. The wings are also comparatively inflexible and are flexed or retracted rather little during the upstroke. The upstroke is always aerodynamically active and this is reflected in the wake, where shedding of spanwise vorticity occurs throughout the wingbeat. Although the wake superficially resembles those of other birds in cruising flight, with a pair of trailing wingtip vortices connected by spanwise vortices, the continuous shedding of first positive vorticity during the downstroke and then negative vorticity during the upstroke suggests a wing whose circulation is gradually increasing and then decreasing during the wingbeat cycle. The wake (and implied wing aerodynamics) are not well described by discrete vortex loop models, but a new wake-based model, where incremental spanwise and streamwise variations of the wake impulse are integrated over the wingbeat, shows good agreement of the vertical momentum flux with the required weight support. The total drag was also estimated from the wake alone, and the calculated lift:drag ratio of approximately 13 for flapping flight is the highest measured yet for birds.