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2D-Numerical Simulation of Flexible Oscillating Fin Mimetic Bony Fishes Fin
Abstract
A controllable flexible biomimetic fin actuated by Shape Memory Alloy (SMA) was presented to imitate the flexible bending of fin rays in bony fishes. Based on the bending experiment of the biomimetic fin, the kinematics model was set up. Then a two-dimensional numerical simulation on the oscillatory biomimetic fin was performed by computational fluid dynamic to investigate the interaction with the surrounding fluid and the propulsive force production. By solving the 2D, incompressible, laminar, unsteady Navier-Stokes equations, the computational results show that the biomimetic fin generates a vortex ring after a complete period in still water which can generate a thrust force. The dynamics performance of the vortex ring is achieved and then the average propulsive force is calculated on different oscillatory frequencies and maximum bending angles to find the change character. At last, an important parameter, the Strouhal number was discussed in this paper. Different Wake pattern were observed depending on different velocities of the background flow, which decide the Strouhal number. When the 0.2<St<0.3, the wake shows evidence of an approximate momentum balance.
Biomimetic caudal fin propulsion systems are a topic of growing interest recently. However, the choice of a particular caudal fin shape for a given robotized underwater vehicle is not straightforward. In order to address this problem, numerical simulations are performed assuming laminar flow, considering the operational regimes of small underwater vehicles. The numerical models are validated against results from literature and then used to compare the propulsive performance of caudal fin shapes ranging from a rectangular fin to a highly forked tail with various combinations of geometric parameters. The optimal shape of the tail based on propulsive efficiency is shown to exist between these two extremes. The presence of leading edge vortices in caudal fins is also shown to have an important role in achieving propulsive performance. The underlying mechanisms and the cause for high thrust and power consumption are discussed in detail with more focus on relating such parameters directly with the geometrical features such as the amount of forking and fin leading edge angle. Researchers can use these insights to arrive at optimal designs for small robotic vehicles with flapping fin-like propulsion, as the relations between geometric features and swimming performance are clearly brought out.
Motivated by our interest in unsteady aerodynamics of insect flight, we devise a computational tool to solve the Navier Stokes equation around a two-dimensional moving wing, which mimics biological locomotion. The focus of the present work is frequency selection in forward flapping flight. We investigate the time scales associated with the shedding of the trailing- and leading-edge vortices, as well as the corresponding time-dependent forces. We present a generic mechanism of the frequency selection as a result of unsteady aerodynamics.
A technique is described whereby the vortex wake of birds in slow forward flight may be investigated with a view towards testing some of the assumptions and predictions of existing theoretical models of bird flight. Multiflash stereophotogrammetry was used to analyse the wake as a pigeon passed through a cloud of neutrally-buoyant helium bubbles.
All photographs obtained support the hypothesis that the wake is composed of a chain of discrete, small-cored vortex rings. This being the case, velocity profiles taken from sections through the wake allow us to estimate the momentum in the wake as represented by vortex rings.
The momentum in the wake appears to be approximately half that required for weight support in unaccelerated, level flight. The possible causes and consequences of this paradoxical result are discussed.
Thrust-producing harmonically oscillating foils are studied through
force and power
measurements, as well as visualization data, to classify the principal
characteristics of
the flow around and in the wake of the foil. Visualization data are obtained
using
digital particle image velocimetry at Reynolds number 1100, and force and
power
data are measured at Reynolds number 40 000. The experimental results are
compared
with theoretical predictions of linear and nonlinear inviscid theory and
it is found
that agreement between theory and experiment is good over a certain parametric
range, when the wake consists of an array of alternating vortices and either
very
weak or no leading-edge vortices form. High propulsive efficiency, as high
as 87%, is
measured experimentally under conditions of optimal wake formation. Visualization
results elucidate the basic mechanisms involved and show that conditions
of high
efficiency are associated with the formation on alternating sides of the
foil of a
moderately strong leading-edge vortex per half-cycle, which is convected
downstream
and interacts with trailing-edge vorticity, resulting eventually in the
formation of a
reverse Kármán street. The phase angle between transverse
oscillation and angular
motion is the critical parameter affecting the interaction of leading-edge
and
trailing-edge vorticity, as well as the efficiency of propulsion.
Undulatory swimmers generate thrust by passing a transverse wave down their body. Thrust is generated not just at the tail, but also to a varying degree by the body, depending on the fish's morphology and swimming movements. To examine the mechanisms by which the body in particular contributes to thrust production, we chose eels, which have no pronounced tail fin and hence are thought to generate all their thrust with their body. We investigated the interaction between body movements and the flow around swimming eels using two-dimensional particle image velocimetry. Maximum flow velocities adjacent to the eel's body increase almost linearly from head to tail, suggesting that eels generate thrust continuously along their body. The wake behind eels swimming at 1.5 Ls(-1), where L is body length, consisted of a double row of double vortices with little backward momentum. The eel sheds two vortices per half tail-beat, which can be identified by their shedding dynamics as a start-stop vortex of the tail and a vortex shed when the body-generated flows reach the 'trailing edge' and cause separation. Two consecutively shed ipsilateral body and tail vortices combine to form a vortex pair that moves away from the mean path of motion. This wake shape resembles flow patterns described previously for a propulsive mode in which neither swimming efficiency nor thrust is maximised but sideways forces are high. This swimming mode is suited to high manoeuvrability. Earlier recordings show that eels also generate a wake reflective of maximum swimming efficiency. The combined findings suggest that eels can modify their body wave to generate wakes that reflect their propulsive mode.
Eels undulate a larger portion of their bodies while swimming than many other fishes, but the hydrodynamic consequences of this swimming mode are poorly understood. In this study, we examine in detail the hydrodynamics of American eels (Anguilla rostrata) swimming steadily at 1.4 L s(-1) and compare them with previous results from other fishes. We performed high-resolution particle image velocimetry (PIV) to quantify the wake structure, measure the swimming efficiency, and force and power output. The wake consists of jets of fluid that point almost directly laterally, separated by an unstable shear layer that rolls up into two or more vortices over time. Previously, the wake of swimming eels was hypothesized to consist of unlinked vortex rings, resulting from a phase offset between vorticity distributed along the body and vorticity shed at the tail. Our high-resolution flow data suggest that the body anterior to the tail tip produces relatively low vorticity, and instead the wake structure results from the instability of the shear layers separating the lateral jets, reflecting pulses of high vorticity shed at the tail tip. We compare the wake structure to large-amplitude elongated body theory and to a previous computational fluid dynamic model and note several discrepancies between the models and the measured values. The wake of steadily swimming eels differs substantially in structure from the wake of previously studied carangiform fishes in that it lacks any significant downstream flow, previously interpreted as signifying thrust. We infer that the lack of downstream flow results from a spatial and temporal balance of momentum removal (drag) and thrust generated along the body, due to the relatively uniform shape of eels. Carangiform swimmers typically have a narrow caudal peduncle, which probably allows them to separate thrust from drag both spatially and temporally. Eels seem to lack this separation, which may explain why they produce a wake with little downstream momentum while carangiform swimmers produce a wake with a clear thrust signature.
We employ a three-dimensional, nonlinear inviscid numerical method, in conjunction
with experimental data from live fish and from a fish-like robotic mechanism, to
establish the three-dimensional features of the flow around a fish-like body swimming
in a straight line, and to identify the principal mechanisms of vorticity control
employed in fish-like swimming. The computations contain no structural model for
the fish and hence no recoil correction. First, we show the near-body flow structure
produced by the travelling-wave undulations of the bodies of a tuna and a giant
danio. As revealed in cross-sectional planes, for tuna the flow contains dominant
features resembling the flow around a two-dimensional oscillating plate over most
of the length of the fish body. For the giant danio, on the other hand, a mixed
longitudinal-transverse structure appears along the hind part of the body. We also
investigate the interaction of the body-generated vortices with the oscillating caudal
fin and with tail-generated vorticity. Two distinct vorticity interaction modes are
identified: the first mode results in high thrust and is generated by constructive
pairing of body-generated vorticity with same-sign tail-generated vorticity, resulting
in the formation of a strong thrust wake; the second corresponds to high propulsive
efficiency and is generated by destructive pairing of body-generated vorticity with
opposite-sign tail-generated vorticity, resulting in the formation of a weak thrust
wake.
First published in 1967, Professor Batchelor's classic work is still one
of the foremost texts on fluid dynamics. His careful presentation of the
underlying theories of fluids is still timely and applicable, even in
these days of almost limitless computer power. This reissue ensures that
a new generation of graduate students experiences the elegance of
Professor Batchelor's writing.
The results of numerical experiments aimed at investigating the topology of the vortex structures shed by an oscillating foil of finite span are described. The motion of the foil and its geometry are chosen to mimic the tail of a fish using the carangiform swimming. The numerical results have been compared with the flow visualizations of Freymuth [J. Fluids Eng. 111, 217 (1989)] and those of von Ellenrieder et al. [J. Fluid Mech.490, 129 (2003)]. The results show that a vortex ring is shed by the oscillating foil every half a cycle. The dynamics of the vortex rings depends on the Strouhal number St. For relatively small values of St, the interaction between adjacent rings is weak and they are mainly convected downstream by the free stream. On the other hand, for relatively large values of St, a strong interaction among adjacent rings takes place and the present results suggest the existence of reconnection phenomena, which create pairs of longitudinal counter-rotating vortices.
The flow field generated by a foil during transient motions is investigated by means of numerical experiments. The numerical simulations have some advantages with respect to laboratory experiments. Indeed, having access to the velocity and pressure fields both in space and in time, it is possible to 'measure' quantities like vorticity, forces and torques which are quite difficult to obtain in laboratory. Moreover, data can be easily gained for different foil kinematics. The obtained results show that the time history of the propulsive force strongly depends on the details of the kinematics of the foil. Moreover, the numerical simulations have allowed to understand the main mechanisms employed by fish to propel themselves during fast starts and to identify the values of the parameters providing optimal propulsive performances.
The unsteady hydrodynamics of the tail flapping and head oscillation of a fish, and their phased interaction, are considered in a laboratory simulation. Two experiments are described where the motion of a pair of rigid flapping foils in the tail and the swaying of the forebody are simulated on a rigid cylinder. Two modes of tail flapping are considered: waving and clapping. Waving is similar to the motion of the caudal fin of a fish. The clapping motion of wings is a common mechanism for the production of lift and thrust in the insect world, particularly in butterflies and moths. Measurements carried out include dynamic forces and moments on the entire cylinder-control surface model, phase-matched laser Doppler velocimetry maps of vorticity-velocity vectors in the axial and cross-stream planes of the near-wake, as well as dye flow visualization. The mechanism of flapping foil propulsion and maneuvering is much richer than reported before. They can be classified as natural or forced. This work is of the latter type where discrete vortices are forced to form at the trailing edge of flapping foils via salient edge separation. The transverse wake vortices that are shed, follow a path that is wider than that given by the tangents to the flapping foils. The unsteady flap-tip axial vortex decays rapidly. Significant higher order effects appear when Strouhal number (St) of tail flapping foils is above 0.15. Efficiency, defined as the ratio of output power of the flapping foils to the power input to the actuators, reaches a peak below the St range of 0.25-0.35. Understanding of two-dimensional flapping foils and fish reaching their peak efficiency in that range is clarified. Strouhal number of tail flapping does emerge as an important parameter governing the production of net axial force and efficiency, although it is by no means the only one. The importance of another Strouhal number based on body length and its natural frequency is also indicated. The relationship between body length and tail flapping frequency is shown to be present in dolphin swimming data. The implication is that, for aquatic animals, the longitudinal structural modes of the body and the head/tail vortex shedding process are coupled. The phase variation of a simulated and minute head swaying, can modulate axial thrust produced by the tail motion, within a narrow range of ±5 percent. The general conclusion is that, the mechanism of discrete and deterministic vortex shedding from oscillating control surfaces has the property of large amplitude unsteady forcing and an exquisite phase dependence, which makes it inherently amenable to active control for precision maneuvering.
We employ a three-dimensional, nonlinear inviscid numerical method, in conjunction
with experimental data from live fish and from a fish-like robotic mechanism, to
establish the three-dimensional features of the flow around a fish-like body swimming
in a straight line, and to identify the principal mechanisms of vorticity control
employed in fish-like swimming. The computations contain no structural model for
the fish and hence no recoil correction. First, we show the near-body flow structure
produced by the travelling-wave undulations of the bodies of a tuna and a giant
danio. As revealed in cross-sectional planes, for tuna the flow contains dominant
features resembling the flow around a two-dimensional oscillating plate over most
of the length of the fish body. For the giant danio, on the other hand, a mixed
longitudinal–transverse structure appears along the hind part of the body. We also
investigate the interaction of the body-generated vortices with the oscillating caudal
fin and with tail-generated vorticity. Two distinct vorticity interaction modes are
identified: the first mode results in high thrust and is generated by constructive
pairing of body-generated vorticity with same-sign tail-generated vorticity, resulting
in the formation of a strong thrust wake; the second corresponds to high propulsive
efficiency and is generated by destructive pairing of body-generated vorticity with
opposite-sign tail-generated vorticity, resulting in the formation of a weak thrust
wake.
Over the past 20 years, experimental analyses of the biomechanics of locomotion in fishes have generated a number of key findings
that are relevant to the construction of biomimetic fish robots. In this paper, we present 16 results from recent experimental
research on the mechanics, kinematics, fluid dynamics, and control of fish locomotion that summarize recent work on fish biomechanics.
The findings and principles that have emerged from biomechanical studies of fish locomotion provide important insights into
the functional design of fishes and suggest specific design features relevant to construction of robotic fish-inspired vehicles
that underlie the high locomotor performance exhibited by fishes.
A numerical study of undulatory locomotion is presented. Unsteady hydrodynamics around an undulatory swimming body is solved using a time-accurate solution of the three-dimensional, incompressible, laminar Navier–Stokes equations. A realistic tadpole-shaped body is modeled, which “swims” by sending a laterally compressing, sinusoidal wave down the tail tip. The method is validated by an extensive numerical study of the thrust generation of an oscillating airfoil, involving comparisons with reliable experimental results. For a three-dimensional tadpole model that undergoes undulatory swimming, the hydrodynamics and mechanism of the undulatory swimming were then analyzed and compared with conventional hydrodynamic theories, which provide a general understanding of the relationship between the dynamic vortex flow and the jet-stream propulsion associated with undulatory locomotion of vertebrates.
Quantifying the locomotor forces experienced by swimming fishes represents a significant challenge because direct measurements of force applied to the aquatic medium are not feasible. However, using the technique of digital particle image velocimetry (DPIV), it is possible to quantify the effect of fish fins on water movement and hence to estimate momentum transfer from the animal to the fluid. We used DPIV to visualize water flow in the wake of the pectoral fins of bluegill sunfish (Lepomis macrochirus) swimming at speeds of 0.5-1.5 L s(-)(1), where L is total body length. Velocity fields quantified in three perpendicular planes in the wake of the fins allowed three-dimensional reconstruction of downstream vortex structures. At low swimming speed (0.5 L s(-)(1)), vorticity is shed by each fin during the downstroke and stroke reversal to generate discrete, roughly symmetrical, vortex rings of near-uniform circulation with a central jet of high-velocity flow. At and above the maximum sustainable labriform swimming speed of 1.0 L s(-)(1), additional vorticity appears on the upstroke, indicating the production of linked pairs of rings by each fin. Fluid velocity measured in the vicinity of the fin indicates that substantial spanwise flow during the downstroke may occur as vortex rings are formed. The forces exerted by the fins on the water in three dimensions were calculated from vortex ring orientation and momentum. Mean wake-derived thrust (11.1 mN) and lift (3.2 mN) forces produced by both fins per stride at 0.5 L s(-)(1) were found to match closely empirically determined counter-forces of body drag and weight. Medially directed reaction forces were unexpectedly large, averaging 125 % of the thrust force for each fin. Such large inward forces and a deep body that isolates left- and right-side vortex rings are predicted to aid maneuverability. The observed force balance indicates that DPIV can be used to measure accurately large-scale vorticity in the wake of swimming fishes and is therefore a valuable means of studying unsteady flows produced by animals moving through fluids.
Basal ray-finned fishes possess a heterocercal tail in which the dorsal lobe containing the extension of the vertebral column is longer than the ventral lobe. Clarifying the function of the heterocercal tail has proved elusive because of the difficulty of measuring the direction of force produced relative to body position in the aquatic medium. We measured the direction of force produced by the heterocercal tail of the white sturgeon (Acipenser transmontanus) by visualizing flow in the wake of the tail using digital particle image velocimetry (DPIV) while simultaneously recording body position and motion using high-speed video. To quantify tail function, we measured the vertical body velocity, the body angle and the path angle of the body from video recordings and the vortex ring axis angle and vortex jet angle from DPIV recordings of the wake downstream from the tail. These variables were measured for sturgeon exhibiting three swimming behaviors at 1.2 L s(-)(1), where L is total body length: rising through the water column, holding vertical position, and sinking through the water column. For vertical body velocity, body angle and path angle values, all behaviors were significantly different from one another. For vortex ring axis angle and vortex jet angle, rising and holding behavior were not significantly different from each other, but both were significantly different from sinking behavior. During steady horizontal swimming, the sturgeon tail generates a lift force relative to the path of motion but no rotational moment because the reaction force passes through the center of mass. For a rising sturgeon, the tail does not produce a lift force but causes the tail to rotate ventrally in relation to the head since the reaction force passes ventral to the center of mass. While sinking, the direction of the fluid jet produced by the tail relative to the path of motion causes a lift force to be created and causes the tail to rotate dorsally in relation to the head since the reaction force passes dorsal to the center of mass. These data provide evidence that sturgeon can actively control the direction of force produced by their tail while maneuvering through the water column because the relationship between vortex jet angle and body angle is not constant.
A key evolutionary transformation of the locomotor system of ray-finned fishes is the morphological elaboration of the dorsal fin. Within Teleostei, the dorsal fin primitively is a single midline structure supported by soft, flexible fin rays. In its derived condition, the fin is made up of two anatomically distinct portions: an anterior section supported by spines, and a posterior section that is soft-rayed. We have a very limited understanding of the functional significance of this evolutionary variation in dorsal fin design. To initiate empirical hydrodynamic study of dorsal fin function in teleost fishes, we analyzed the wake created by the soft dorsal fin of bluegill sunfish (Lepomis macrochirus) during both steady swimming and unsteady turning maneuvers. Digital particle image velocimetry was used to visualize wake structures and to calculate in vivo locomotor forces. Study of the vortices generated simultaneously by the soft dorsal and caudal fins during locomotion allowed experimental characterization of median-fin wake interactions.
During high-speed swimming (i.e. above the gait transition from pectoral- to median-fin locomotion), the soft dorsal fin undergoes regular oscillatory motion which, in comparison with analogous movement by the tail, is phase-advanced (by 30% of the cycle period) and of lower sweep amplitude (by 1.0cm). Undulations of the soft dorsal fin during steady swimming at 1.1bodylengths−1 generate a reverse von Kármán vortex street wake that contributes 12% of total thrust. During low-speed turns, the soft dorsal fin produces discrete pairs of counterrotating vortices with a central region of high-velocity jet flow. This vortex wake, generated in the latter stage of the turn and posterior to the center of mass of the body, counteracts torque generated earlier in the turn by the anteriorly positioned pectoral fins and thereby corrects the heading of the fish as it begins to translate forward away from the turning stimulus. One-third of the laterally directed fluid force measured during turning is developed by the soft dorsal fin. For steady swimming, we present empirical evidence that vortex structures generated by the soft dorsal fin upstream can constructively interact with those produced by the caudal fin downstream. Reinforcement of circulation around the tail through interception of the dorsal fin’s vortices is proposed as a mechanism for augmenting wake energy and enhancing thrust.
Swimming in fishes involves the partitioning of locomotor force among several independent fin systems. Coordinated use of the pectoral fins, caudal fin and soft dorsal fin to increase wake momentum, as documented for L. macrochirus, highlights the ability of teleost fishes to employ multiple propulsors simultaneously for controlling complex swimming behaviors.
As members of the derived teleost fish clade Scombridae, mackerel exhibit high-performance aquatic locomotion via oscillation of the homocercal forked caudal fin. We present the first quantitative flow visualization of the wake of a scombrid fish, chub mackerel Scomber japonicus (20-26cm fork length, FL), swimming steadily in a recirculating flow tank at cruising speeds of 1.2 and 2.2FLs-1. Thrust was calculated from wake measurements made separately in the horizontal (frontal) plane and vertical (parasagittal) planes using digital particle image velocimetry (DPIV)and compared with drag measurements obtained by towing the same specimens of S. japonicus post mortem.
Patterns of flow indicated that the wake consisted of a series of linked elliptical vortex rings, each with central jet flow. The length of the minor axis (height) of the vortex rings was approximately equal to caudal fin span;the length of the major ring axis was dependent on swimming speed and was up to twice the magnitude of ring height. Profiles of wake velocity components were similar to theoretical profiles of vortex rings.
Lift, thrust and lateral forces were calculated from DPIV measurements. At 1.2FLs-1, lift forces measured relative to the Xaxis were low in magnitude (-1±1mN, mean ± S.D., N=20)but oriented at a mean angle of 6° to the body axis. Reaction forces tend to rotate the fish about its center of mass, tipping the head down. Thus, the homocercal caudal fin of S. japonicus functions asymmetrically in the vertical plane. Pitching moments may be balanced anteriorly via lift generation by the pectoral fins. Thrust estimates for the two smallest fish based on DPIV analysis were not significantly different from drag measurements made by towing those same animals. At a speed of 1.2FLs-1,thrust magnitude was 11±6mN (mean ± S.D, N=40). Lateral force magnitudes were approximately double thrust magnitudes (22±6mN,mean ± S.D., N=20), resulting in a mean mechanical performance ratio (thrust/total force) of 0.32 at 1.2FLs-1. An increase in speed by a factor of 1.8 resulted in a mean increase in thrust by a factor of 4.4, a mean increase in lateral forces by a factor of 3, no change in the magnitude of lift produced and an increase in mean mechanical performance to 0.42. The relatively high lateral forces generated during swimming may be a necessary consequence of force production viapropagated waves of bending.
Although considerable progress has been made within the last decade in experimental hydrodynamic analyses of aquatic locomotion using two-dimensional digital particle image velocimetry (two-dimensional DPIV), data have been limited to simultaneous calculation of two out of the three flow velocity variables: downstream (U), vertical (V) and lateral(W). Here, we present the first biological application of stereo-DPIV, an engineering technique that allows simultaneous calculation of U, V and W velocity vectors. We quantified the wakes of rainbow trout (Oncorhynchus mykiss, 16.5-21.5 cm total body length, BL), swimming steadily in a recirculating flow tank at a slow cruising speed of 1.2 BL s-1. These data extend the comparative basis of current hydromechanical data on the wakes of free-swimming fishes to the salmoniforms and are used to test current hypotheses of fin function by calculations of mechanical performance and Froude efficiency.
Stereo-DPIV wake images showed three-dimensional views of oscillating jet flows high in velocity relative to free-stream values. These jet flows are consistent with the central momentum jet flows through the cores of shed vortex rings that have been previously viewed for caudal fin swimmers using two-dimensional DPIV. The magnitude and direction of U, V and W flows in these jets were determined over a time series of 6-8 consecutive strokes by each of four fish.
Although the fish swam at the same relative speed, the absolute magnitudes of U, V and W were dependent on individual because of body size variation. Vertical flows were small in magnitude (<1 cm s-1) and variable in direction, indicating limited and variable vertical force production during slow, steady, forward swimming. Thus, in contrast to previous data from sunfish (Lepomis macrochirus) and mackerel (Scomber japonicus), the trout homocercal caudal fin does not appear to generate consistent vertical forces during steady swimming. U flows were of the order of 3-6 cm s-1; lateral flows were typically strongest, with W magnitudes of 4-6 cm s-1. Such strong lateral flows have also been shown for more derived euteleosts with homocercal caudal fins.
The ratios of the magnitudes of wake flow, U/(U+V+W), which is a flow equivalent to mechanical performance, were also dependent on individual and ranged from 0.32 to 0.45, a range similar to the range of mechanical performance values previously determined using standard two-dimensional DPIV methods for caudal fin locomotion by more derived euteleosts. Strong lateral jet flow appears to be a general feature of caudal fin locomotion by teleosts and may reflect the nature of undulatory propulsion as a posteriorly propagated wave of bending. Froude efficiency (ηp) was independent of individual; meanη p was 0.74, which is similar to previous findings for trout.