HYDRODYNAMICS OF FISHLIKE SWIMMING

Annu. Rev. Fluid Mech. 2000. 32:33–53Copyright ! 2000 by Annual Reviews. All rights reserved

0066–4189/00/0115–0033$12.00 33

HYDRODYNAMICS OF FISHLIKE SWIMMING

M. S. Triantafyllou1, G. S. Triantafyllou2,and D. K. P. Yue1

1Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139; e-mail: [email protected] Technical University of Athens, Athens, Greece

Key Words vorticity control, biomimesis

Abstract Interest in novel forms of marine propulsion and maneuvering hassparked a number of studies on unsteadily operating propulsors. We review recentexperimental and theoretical work identifying the principal mechanism for producingpropulsive and transient forces in oscillating flexible bodies and fins in water, theformation and control of large-scale vortices. Connection with studies on live fish ismade, explaining the observed outstanding fish agility.

INTRODUCTION

Marine propulsion and maneuvering has a long history of development and hasalready reached a level of maturity. This level is in some aspects satisfactory andin other aspects, particularly for transient motion, very limiting. Fast fish andcetaceans, on the other hand, move in water with great agility. They propel them-selves through rhythmic unsteady motions of their body, fins, and tail; they offera different paradigm of locomotion than that conventionally used in man-madevehicles. The view that the resulting unsteadiness in the flow is exploited by fishto their advantage is based on a series of studies demonstrating that they (a)generate large, short-duration forces efficiently, (b) coordinate rhythmic unsteadybody and tail motion to minimize the energy required for steady propulsion, and(c) coordinate transient motion of the body and tail to minimize the energy lostin the wake during maneuvering.Novel propulsion ideas based on unsteady flow control have emerged recently

as progress in robotics, new materials, and actuators have become available.Unsteady propulsors and vehicles that emulate the motion of fish have been devel-oped for technological application, using state-of-the-art technology and roboticsand an understanding of how fish swim. In turn, from the development and testingof new systems, gains in understanding the principles of fish-like swimming havebeen obtained as well.

34 TRIANTAFYLLOU ET AL

The basis for enhancing performance through unsteady flow control is theformation of large-scale vortices through body motion, the sensing and manipu-lation of these vortices as they move down the body, and the eventual reposition-ing through tail motion. These concepts constitute the essence of vorticity control.

VORTICITY CONTROL

Unsteady motion of a body and unsteady forcing in the flow can be used forefficient flow control (Tokomaru & Dimotakis 1991, Cortelezzi 1996, Koumout-sakos 1999). The mechanics of flow body interaction are reviewed in Rockwell(1998). Unsteady propulsion may offer certain advantages when compared withconventional propulsors. In Gursul & Ho (1992) it is shown that unsteady motionof airfoils can cause a very high lift coefficient; and in Hoppe (1989) and Ander-son et al (1998) foils are shown to produce propulsive thrust very efficiently.Also, an oscillating foil can be used to alter and reposition oncoming vorticity(Koochesfahani & Dimotakis 1988, Cortelezzi et al 1997) and recapture energycontained in the eddies of an oncoming flow using vorticity control (Gopalkrish-nan et al 1994, Streitlien 1994, Streitlien et al 1996, Anderson 1996).The interaction of oncoming vorticity with a fin is a basic problem in the study

of vorticity control mechanisms; its principles are of great importance to under-standing fish swimming and maneuvering. As identified in Gopalkrishnan et al(1994), Streitlien et al (1996), and Anderson (1996), a harmonically oscillatingfoil may interact with oncoming large-scale vortices, which have a typical coresize that is comparable with the foil chord, in three distinct ways to produce thrust.It is found that oncoming vortices

1. interact with foil-shed vorticity constructively, generating stronger vortices inthe wake, arranged in the form of a reverse Karman street

2. interact with foil-shed vorticity destructively, generating weak vortices in thewake, arranged in the form of a reverse Karman street

3. pair with foil-shed vorticity of the opposite sign, generating a wake of vortexpairs, which drift away from the centerline, thus creating a wide wake.

Foil vorticity may be generated at the trailing edge, the leading edge, or both.The process of repositioning the oncoming vorticity is a crucial part of the inter-action. As found theoretically (Streitlien 1994) and confirmed experimentally(Gopalkrishnan et al 1994), a foil may extract energy from the oncoming vorticeswhen destructive interference with foil-generated vorticity is observed, increasingits efficiency substantially. Figure 1 (Gopalkrishnan et al 1994) shows the strongvortices created by an upstream-located D-section cylinder when it interacts witha foil in the destructive mode associated with maximum energy extraction. Flowis from right to left; vortices downstream from the foil are weak and drasticallyrepositioned, compared with their original strength and location.

Figure 1 Oscillating foil manipulating oncoming vortices generated by an upstream located D-shaped cylinder. Strong vortices comingfrom right to left are repositioned and interact destructively with foil-generated vorticity, resulting in a weak reverse Karman street in thewake (Gopalkrishnan et al 1994). The propulsive efficiency of the foil increases significantly.

35


Periodic forcing of a flow may alter the flow features considerably. Roberts &Roshko (1985) show that the vortex structure and mixing can be influencedthrough pulsing the flow. Taneda (1978) showed that the wake width behind acylinder in an oncoming stream can be reduced through forced harmonic rotaryoscillations of the cylinder. Tokomaru & Dimotakis (1991) considered the sameproblem as Taneda and found that rotary oscillation exerts a maximum influenceon the flow when the frequency is close to the Strouhal frequency. Under properconditions the wake width and, hence, the drag can be reduced substantially.Tokomaru & Dimotakis (1991) found that in unforced wakes the separation pointsmove relatively little, and the natural instability of the flow redistributes the shedvorticity to create the large-scale patterns; whereas, in a forced oscillation, theseparation points move substantially. Additional vorticity forms at the cylinderand is then released in the wake. Taneda & Tomonari (1974) and Taneda (1977)considered the effect of imposing a traveling wave motion on the flow around aflexible plate placed in a stream of velocity U. They found that turbulence in theboundary layer of the plate is suppressed when the speed of propagation of thetraveling wave cp exceeds the free stream velocity.

OSCILLATING, THRUST-PRODUCING FOILS

A foil moving forward at steady speed and oscillating harmonically in a combi-nation of lateral motion (heave) and angular motion (pitch) may produce thrustunder proper parametric conditions. Thrust is produced when the time-averagedflow downstream of the foil has the form of a jet; however, if the flow is subjectto drag force, it has the form of a drag wake. In either case, unsteady large-scalevortical patterns form in the wake, which has a shape affected by the parametricchoices, as shown through visualization in Oshima & Oshima (1980), Oshima &Natsume (1980), Freymuth (1988), Koochesfahani (1989), Anderson et al (1998),and Kirby et al (1999). Two-dimensional flow studies show that the downstreamflow may be characterized by the formation of a sinuous wake; or a wake thathas either two large or four vortices per period. High propulsive efficiency isassociated with the formation of two vortices per cycle, forming a staggered arrayof vortices resembling a Karman street behind bluff bodies, but with the vorticesrotating in opposite directions (reverse Karman street).A high-aspect ratio flapping foil has the following prime parameters: (a) the

amplitude of heave motion compared with the chord length; (b) the featheringparameter (ratio of pitching angle compared with the maximum angle of attackinduced by the heave motion); (c) the phase angle between heave and pitch; (d)the reduced frequency; and (e) the relative position of the pitching axis. TheStrouhal number and the nominal angle of attack may be used instead of thereduced frequency and feathering parameter. The nominal angle of attack isthe maximum difference between the heave-induced angle and the pitch angle.

FISH HYDRODYNAMICS 37

The Strouhal number is a nondimensionalized frequency, defined in analogy withbluff body flows as

St " fA/U, (1)

where f is the frequency of oscillation, A is the width of the wake (often approx-imated by the lateral total excursion of the foil), and U is the velocity of motion.Although the reduced frequency provides a better measure of unsteadiness than

the Strouhal number by comparing spatial wavelength of the flow disturbancewith the chord length, the Strouhal number is a better parameter for characterizingthe dynamics of wake flow. Both parameters are often needed because of theimportance of both the unsteadiness in the foil and the dynamics of the wake(Luznik & Bose 1998). Ohmi et al (1990, 1991) studied the vortex formation ina translating and harmonically pitching foil at Reynolds numbers between 1,500and 10,000 with mean incidence angle of 15! or 30!. At large incidences theyfound that the patterns in the vortex wake depend on whether the translational orrotational motion dominates the flow, which is determined primarily by thereduced frequency. They also found that when the flow is dominated by therotational motion it is governed by a parameter proportional to the Strouhalnumber.Triantafyllou et al (1991, 1993) proposed that optimum efficiency is obtained

when the large-scale vortical pattern formation is compatible with the dynamicsof the wake or, more specifically, when the frequency of foil oscillation coincideswith the frequency of maximum amplification of disturbances, which is deter-mined from the detailed linear stability analysis of the wake. A discussion of therelation between the disturbances after they have been rolled up into vortices andthe produced thrust can be found in Streitlien & Triantafyllou (1998). Becausethe width of the wake is not a priori known, the lateral excursion of the foil isoften used to define the Strouhal number. The optimum range of Strouhalnumber—between 0.25 and 0.35—is found for certain specific profiles used inTriantafyllou et al (1993); in other cases, different values may be obtained.Fish swimming kinematic data showed that the nondimensional frequencies

were close to the value predicted by instability analysis (Triantafyllou et al 1993,Rohr et al 1998). Figure 2 from Rohr et al (1998) shows the Strouhal number asa function of the Reynolds number for numerous observations of trained dolphins,showing good overall agreement between theory and experiment.Dynamic-stall or leading-edge vortices form when the flow separates near the

leading edge. Maxworthy (1979), Ellington (1984), Freymuth (1990), Rayner(1995), and Liu et al (1998) studied the aerodynamics related to the flight ofhovering insects, and all groups concluded that unsteady flow mechanisms playa very important role. High values of lift coefficient were associated with theformation of a leading-edge vortex (dynamic-stall vortex), which for specificparametric combinations was subsequently amalgamated with trailing-edge vor-ticity. Reynolds & Carr (1985) discussed the basic mechanism that governs thegeneration of leading-edge vorticity also described by McCroskey (1982). Elling-


Figure 2 Strouhal number for swimming dolphins as a function of Reynolds’ number(Rohr et al 1998, with permission).

ton (1984) and Ellington et al (1996) also note the significant delay in stall causedby unsteady effects, as found earlier by Maresca et al (1979) for a foil at largeincidences in steady flow, that are undergoing axial oscillations. The Reynoldsnumber effect was found to be of secondary importance. Liu et al (1998) showclearly the three-dimensional formation and evolution of a substantial leading-edge vortex in a wing-simulating hawkmoth hovering.Although leading-edge vortices are used primarily to provide a large lift force

through the low pressure they induce, moderate formation of leading-edge vor-tices has also been associated with high propulsive efficiency, up to 87%, formoderately loaded foils in Anderson et al (1998). Figure 3 from Anderson et al(1998) shows the efficiency achieved in a flapping airfoil under various conditionsof oscillation. Conditions for high efficiency are (a) amplitude of heave motioncomparable to the chord length, (b) nominal angle of attack at about 20! and (c)Strouhal number compatible with the formation of a reverse Karman street.Three-dimensional effects in finite-aspect ratio foils become smaller for oscil-

lating foils as frequency increases, compared with those of steadily moving foils.This is because of the formation of alternating-sign tip vortices (Cheng &Murillo1984), which result in weaker induced velocities. The structure of vortices shedin the wake resembles vortex rings with adjoining backs, as first sketched in


Figure 3 Experimentally measured efficiency of a foil of chord (c) flapping with ampli-tude (ho), frequency ( f ), nominal angle of attack (!o), pitch to heave phase angle (w), asfunction of the Strouhal’s number St. Curve (‘‘case’’) 1: ho /c " 0.75, !o " 21!, w "75!; curve 2: ho /c " 0.75, !o " 17!, w " 105!; curve 3: ho /c " 0.25, !o " 15!, w "90!; curve 4: ho /c " 0.75, !o " 5!, w " 90!; curve 5: ho /c " 0.75, !o " 25!, w " 90!;curve 6: ho /c " 0.75, !o " 20!, w " 90!; curve 7: !o " 10!, w " 90!; curve 8: ho /c "0.75, !o " 30!, w " 90!.

Lighthill (1969). Visualization has shown a pattern compatible with this picturealthough the reconnection details of the rings among themselves and the foil arecomplex, as shown in the flow visualization of Figure 4 (see color insert) (Hartet al 1992), obtained through air injection at the tip of a foil oscillating near awall. The flow structure behind flapping foils in live fish is studied experimentallyin Drucker & Lauder (1999).


STEADILY SWIMMING FISH

Fish swim with great agility, and certain species can reach high speeds. Therelatively large-amplitude unsteady motion of their bodies is striking and intrigu-ing to the observer, hence many studies addressed the subject of fish propulsion,summarized in Hertel (1966), Gray (1968), Lighthill (1969), Aleyev (1977),Blake (1983), and Videler (1993). Interest in explaining the mechanisms of fishpropulsion peaked after Gray (1936) published a study that estimated that themuscular power of a dolphin is sevenfold less than the power needed to propel arigid-straight model of its body at a speed between 15 and 20 knots, which Graypresumed to be the maximum speed achievable by dolphins (Gray’s paradox).Subsequent studies tried to explain or dispute Gray’s conclusion (Kramer 1957,Lang & Daybell 1963, Hoyt 1975, Fish & Hui 1991, Fein 1998).Wu (1961, 1971a,b) and Lighthill (1960, 1970) investigated the propulsion

characteristics of flexible two- and three-dimensional plates by using potentialflow theory and linearized boundary conditions. Because of the elongated shapeof most fishes, so that geometrical changes along the longitudinal (x axis) aremuch slower than those in the cross yz plane (the y axis is presumed to be in thedirections of transverse motion, the z axis is in the nonflexing direction), theslender-body theory has dominated the theoretical modeling of fish hydrodynam-ics. In the slender-body theory it is assumed that the crossflow caused by thetransverse motion of the fish’s body is approximately two-dimensional alongplanes perpendicular to the axis of the fish (in the yz plane). By further assumingunseparated flow the following expression is obtained for the transverse (lift)force per unit length (L) (Lighthill, 1960):

" " "h "hL " # $ U a $ U , (2)! " # ! "$"t "x "t "x

where U is the speed of the fish, a is the local (two-dimensional) added mass perunit length, and h(x,t) is the vertical displacement of the backbone of the fish. Ifone assumes that shedding of vorticity occurs at sections along the contractingpart of the body, then the following modified expression is obtained (Wu 1971b,Newman & Wu 1973):

2" "L " #a $ U h. (3)! ""t "x

Recent flow visualization work provides support for using the second expressionfor the posterior part of the body. Lighthill (1971) extended the slender-bodytheory to large amplitude motions. These seminal works provided a firm theo-retical basis for studying fish swimming and gave insight into the basic swimmingpropulsive mechanisms.Flow visualization, however, shows that the flow may differ qualitatively from

the flow presumed in slender-body theory (Wolfgang et al 1999b). Depending on


the phase of oscillation, the flow may not be two-dimensional along yz (trans-verse) planes, and instead appears approximately two-dimensional along xy (lon-gitudinal) planes, which resembles the flow over a flapping two-dimensionalsheet, except near the body edges. Hence, quantitative predictions require numer-ical simulation, which provides more accurate results especially for the parametersassociated with fish motion (Cheng et al 1991, Barrett et al 1999).The details of the three-dimensional flow around fishlike bodies, especially

the form and evolution of the vortical structures providing propulsive and maneu-vering forces, were addressed only relatively recently. The potential flow theory(Lighthill 1970, Wu 1971b) assumes a frozen wake form, translating with thespeed of the flow. The linearly unstable dynamics of the wake are known, how-ever, to be very important in bluff body wakes in which they determine thefrequency and wavelength of the dominant structure. By analogy, and by usingthe methodology developed by Triantafyllou et al (1986), the jet that forms behindself-propelled, actively swimming fishlike bodies is found to have characteristicslargely dependent on the dynamics of the wake flow (Triantafyllou et al 1991,1993) as described in the previous section. In a similar unsteady problem, Gharibet al (1994, 1998) and Rosenfeld et al (1998) found that in the transient formationof a circular jet flow through a piston, the leading vortex ring has reached itsmaximum circulation when the nondimensional ‘‘formation time’’ is approxi-mately 4. The formation time is equal to Ut/D, where U is the mean speed of thepiston, t the time of formation, and D the diameter of the ring.The presence of vorticity in the wake of a self-propelled body in a viscous

fluid is a consequence of the need for a propulsive jet to counter the body drag.The dynamics of the free-wake vorticity play a prominent role in the selection ofthe frequency and form of swimming motion. Gray (1964, 1968) providessketches of the flow around nematodes and a flow visualization figure of a liveswimming eel, which demonstrates the importance of body-generated vorticityand the formation of a reverse Karman street in the wake. A matter of somecontroversy is the presence of separated vorticity around the body of the fish.Rosen (1959) visualized the flow around and in the wake of a naturally swimmingfish, showing large-scale vortices traveling near the body, well ahead of the pedun-cle region. Aleyev (1977) used dye visualization to show that such near-bodyvortices appear only when the fish swim in a shallow-depth tank. In deep waterthe flow remains attached; Aleyev noted the formation of a reverse Karman streetin the wake of fish.Stamhuis & Videler (1995) used experimental particle image velocimetry to

capture the flow dynamics around several live swimming organisms and to thenanalyze the energetic makeup of the wake. Anderson (1996) used experimentaldigital particle image velocimetry (DPIV) to visualize the wake behind a swim-ming Giant Danio (Danio malabaricus) and identified the active manipulation ofshed wake vorticity to create a reverse Karman vortex street (Triantafyllou et al1996). Muller et al (1997) analyzed the wake of a swimming mullet (Chelonlabrosus risso) using DPIV and concluded that the manipulation of the wake


structure resulted in high propulsive efficiencies. Figure 5 (see color insert) is aDPIV figure from Muller et al (1997) showing clearly the flow about the fishbody and the tail and the formation of a reverse Karman street in the wake.Wolfgang et al (1999a) compare experimental DPIV with numerical results, andthey find that an inviscid numerical formulation, which satisfies the exact bodyboundary conditions, allows for the nonlinear evolution of the shed vorticity, andprovides good description of the kinematics of the flow around the body and inthe wake. A similar methodology was developed and used by Liu & Bose (1997)to treat flexible unsteadily operating foils.Anderson (1996), Muller et al (1997), and Wolfgang et al (1999a) provide the

following process of vorticity control used by fish: the body motion has the formof a wave traveling along the fish. This motion generates a flow that is charac-terized by spatially traveling waves of body-bound vorticity, when viewed at thelongitudinal mid-height plane (intersecting at right angles the direction of trans-verse motion). Three-dimensionality of the flow (i.e. flow perpendicular to thislongitudinal plane of view) is strongest near the upper and lower edges of thefish (Wolfgang et al, 1999b); this is a deviation from the assumptions of slender-body theory. The flow is, for part of the oscillatory cycle, closer to the flow thatWu’s (1961) theory provides for a two-dimensional sheet undergoing travelingwaves. Figure 6 (see color insert) shows three transverse cuts (parallel to the yzaxes), demonstrating that the flow is not two-dimensional within the yz plane, butcontains strong features characteristic of two-dimensional flow in xy planes. Themix between longitudinal and transverse flow features varies with the phase ofoscillation.Figure 7 (see color insert) reinforces this view by showing two longitudinal

cuts along the xy axes, demonstrating the persistence of two-dimensional–likeflow from the mid-height plane (lower) to a plane at 20% of depth from the top(upper). This description is in agreement with the quantitative findings of Chenget al (1991), who show that the numerically predicted value for the transverseforce lies between the estimates of the two-dimensional theory and the slender-body theory. Figure 7 (Figure 7 upper; see color insert) shows clearly the sepa-ration from the body and the tail interacting with oncoming vorticity to form thereverse Karman street.As the amplitude of body motion increases from head to tail, the bound-vor-

ticity amplitude grows as well. As it reaches the point of maximum width, atabout mid-length, vorticity at the upper and lower edges is shed in the wake, veryweakly in the beginning and strongly as it reaches the peduncle region. Thepeduncle region is assumed for the purposes of this discussion to have a smalltransverse area. The details of shedding of vorticity depend strongly on the formof the body of the fish and the shape of the fins. Qualitatively, this flow separationprovides support for Wu’s modified expression for obtaining the lateral forcewithin the slender-body theory (Wu 1971b, Newman & Wu 1973).In the peduncle area, free (shed) three-dimensional vortical structures have

formed that interact next with the tail. The tail repositions vortices coming from


the fish’s left side to its right side, and vortices from the right to its left side. Theinteraction of body-generated vorticity with the tail resembles the control ofoncoming vorticity through a fin, described in Gopalkrishnan et al (1994). Alongitudinal planar cut of the resulting flow in the wake at the fish’s mid-heighthas the form of a reverse Karman street, consisting of two alternating sign vorticesper period, inducing a jet.The shedding of vorticity by the contracting part of the body can also be seen

in Liu et al (1997), who consider the computational fluid dynamics simulation ofthe swimming motion of a tadpole at Reynolds number 7200. Figure 8 (see colorinsert) from Liu et al (1997) shows the onset of separation from the edges of thebody, whereas the wake consists of antisymmetrically rotating and positionedthree-dimensional vortical structures. Their paper also describes the significanteffect of the snout motion on the flow in the head region.Ames’ (1998) flow visualization studies with live fish showed that shallow

water effects cause stronger shedding of body-generated vorticity furtherupstream from the body, but the qualitative mechanics of swimming remain thesame as for fish in deep water. Shallow-water fish visualization shows clearly thatthe vorticity control sequence applied by the fish through its tail manipulatesoncoming vorticity. This reconciles the observations of Rosen (1959) and Aleyev(1977) and provides a unified explanation of the details of fish swimming.

TURNING AND FAST-STARTING FISH

Fast-starting and -maneuvering fish exhibit outstanding agility. Domenici &Blake(1997) review the literature on observed fast-starting kinematics of live fish,reporting for fast fish, such as the pike (Esox lucius), maximum accelerations inexcess of 150 m/s2 (Frith & Blake 1995, Harper & Blake 1991). The body of thefish is bent into either a C shape or an S shape, and then it is rapidly unwoundin a traveling wave fashion. Weihs (1972, 1973) extended Lighthill’s large-ampli-tude, slender-body theory to estimate the axial and transverse forces acting on amaneuvering fish using its body and fins.Flow visualization studies have shown that, in fast-starting and maneuvering

as in propulsion, the role of formation and control of vorticity is very important.Ahlborn et al (1991, 1997) show flow visualization caused by a fin forced tomove in a double flip simulating the caudal fin motion of a fast-starting fish (i.e.motion to one side, shedding a vortex, followed by a reverse-direction motion,shedding a vortex of the opposite sign). This results in the formation of a pair ofthrust-producing vortices. These experiments show that a time lag between flipsenhances thrust production because it allows for the proper growth of the thrust-producing eddies.Anderson (1996), Triantafyllou et al (1996), and Wolfgang et al (1999a) show

experimental and simulation results of the flow around a maneuvering GiantDanio. Figure 9 (see color insert) shows DPIV flow visualization in a live Giant


Figure 10 Body-bound and wake vorticity control during fish turning maneuver. CircledL indicates a region of low pressure that is manipulated by the body and tail to reinforcethe formation of the turning thrust jet (Wolfgang et al 1999b).

Danio performing a C maneuver, compared with numerical prediction. The fieldof view is planar, intersecting the fish body at about its mid-height. We see theflow organized initially into two circular-like features, one centered near the tailand one near the head, as the tightening of the body into a C shape nears com-pletion. These two features can be described as two body-bound vortices of oppo-site sign.Subsequently, the tail begins to move to the fish’s left, and the counterclock-

wise vortex moves toward the tail region and is shed before the peduncle region.The shedding of vorticity before the peduncle region happens in the same wayas described for a straight-swimming fish, but is more intense. The shed vorticityreaches the tail and is repositioned by the caudal fin, which interacts with vorticityshed at the trailing edge of the tail, ultimately moving in the wake. During thesubsequent stroke of the tail, which is now moving in the opposite transversedirection (to the fish’s right), the clockwise vortex, initially seen near the headregion, moves posteriorly, and sheds before the peduncle region. Then it is alsomanipulated by the tail and finally moves in the wake pairing with the previouslyshed counterclockwise vortex.The result of the C-shape maneuver is a vortex pair, forming a local jet flow

directed slightly downward and to the reader’s right, as shown in Figure 10, whichsummarizes the principal steps in the turning of the fish. The vortices in this pairare packets of counterrotating large-scale wake vorticity. The vortices comprisingthe jet have average nondimensional circulation 42% greater than the typical wakevortex strength for steady swimming of the Giant Danio (Wolfgang et al 1999b).The core radius of the jet vortices is more than double that of those produced instraight swimming. The vortex pair supplies the force needed to change themomentum of the fish.The vortex pair that forms in rapidly turning fish is generated to a great extent

by the body. The body-generated vorticity is repositioned by the tail and interacts


Figure 11 Robotic laboratory robot in the shape of a bluefin tuna, with a length of 1.25m, to measure the power needed to propel an actively swimming body (left). Autonomousflexible hull robot in the shape of a pike, with a length of 0.81 m, implementing vorticitycontrol for fast maneuvering. (Photographs by S. Ogden)

with tail-shed vorticity to form two precisely controlled vortices. The timing offormation, shedding, and position of the vortices is crucial to an effective maneu-ver. In fact, flow visualization shows the absence of any other (parasitic) shedvorticity, hence the absence of separation drag. A rigid streamlined body under-going a similar maneuver would be subjected to considerable flow separation andresulting drag force. As in the case of a straight-swimming fish, the flow has atwo-dimensional appearance except near the edges of the fish. Vorticity controlby the tail may be described qualitatively in terms of two-dimensional vortexmanipulation concepts, although the detailed flow picture and force estimationrequire a three-dimensional procedure.The agility of fish is explained by the rapid generation of a vortex pair through

body flexing and tail manipulation and by the absence of separation drag.

BIOMIMETIC ROBOTS FOR HYDRODYNAMICEXPERIMENTATION

Biomimetics is an emerging field, using principles from living organisms to deriveman-made mechanisms and vehicles that are capable of emulating the perfor-mance of animals. The field also has an impact on our understanding of therelevant mechanics of propulsion. In hydrodynamics highly accurate robotic


Figure 12 Reduction in power needed by the actively swimming laboratory robot aspercent of power needed to tow the robot rigid-straight, as function of Strouhal numberSt " fA/U. Body flexes in traveling wave with maximum double amplitude at the tailequal to A, frequency f, moving at speed U " 0.7 m/s. The wavelength is chosen to beequal to 1.22 m, the nominal angle of attack at the tail is 15!, the phase between linearand angular motion at the tail is 90! (Barrett et al 1999).

mechanisms are used to measure the power needed and forces generated, as wellas visualize the flow around their external skin structure (Hover & Triantafyllou1998).Several theoretical and experimental studies have explored the possibility of

applying harmonically oscillating propulsors to surface and underwater vehicles.Paidoussis (1976) studied an ingenuous ichthyoid propulsion device. Tsuhara &Kimura (1987) studied propulsive foils that use the Weis-Fogh mechanism togenerate thrust, and propulsive efficiency was measured to be #50%. Lai et al(1993), Yamamoto et al (1995), Yamaguchi & Bose (1994), Belibassakis et al(1997), Politis et al (1999), and Czarnowski et al (1997) studied the applicationof flapping foils to ship propulsion. Kato & Inaba (1997), Bandyopadhyay et al(1997a,b), Kato (1998) and Read (1999) studied the use of fins in small under-water vehicles for hovering and low-speed maneuvering.


Figure 13 Transverse averaged longitudinal velocity profiles at a given location, markeda, and at a particular phase of a plate, undergoing a wavy motion (see inset). The maximumamplitude a " 0.032L (L is the plate length), the wavelength is k " 0.4L and the phasespeeds of the motion are c/U " 0.0 (circle), c/U " 0.4 (square) and 1.2 (triangle). U isthe speed of the incoming turbulent stream. The Reynolds number based on L is 6000.The instantaneous (transversely averaged) friction velocity u∗/U are, for c/U" 0.0: 0.055;for c/U " 0.4: 0.049; and for c/U " 1.2: 0.041.


Robotic mechanisms actuating flexible hulls are used as laboratory vehicles toobtain precise power measurements (Barrett 1996, Triantafyllou & Triantafyllou1995, Barrett et al 1999) and to achieve the same outstanding agility in fast-starting and maneuvering observed in live fish, for technological use (Kumph &Triantafyllou 1998, Anderson 1998). The development of new materials and actu-ation mechanisms, such as shape memory alloys and conducting polymers,enhances the capability to emulate the performance of live organisms; they useactuators similar to the muscles and tendons used by the animals. Figure 11 (left)shows a laboratory robot, 1.2 m long, with an external shape in the form of abluefin tuna (Barrett 1996), consisting of eight links. Figure 11 (right) shows afree-swimming autonomous robot 0.81 m in length, consisting of three indepen-dently controlled links, in the shape of a pike (Kumph & Triantafyllou 1998).Barrett et al (1999) used a laboratory robot to study the effect of a fishlike

body motion on the axial and transverse forces and hence the power required forswimming. The development of this high-precision, eight-link, robotic mecha-nism, which can emulate very closely the swimming of the tuna (Barrett 1996,Triantafyllou & Triantafyllou 1995, Wolfgang et al 1998), overcomes the diffi-culties associated with working with live fish because it allows the acquisition ofdetailed measurements of the forces on an actively controlled flexible body.Repeated precise measurements on the flexible robot have shown that, within aparametric range, the power needed to propel an actively swimming body at aReynolds number of #106 can be reduced by $50% compared with the powerneeded to tow the same vehicle in a rigid-straight configuration. Minimal requiredpower is found for nondimensional parametric conditions that are very similar tothose observed in live tuna (Dewar 1993, Dewar & Graham 1994). Figure 12shows the reduction in required power in the actively swimming robot, as apercent of its required towing power and as a function of the Strouhal number,based on the tail fin excursion. There are two distinct peaks, one at a low Strouhalnumber, #0.13, and another at a value of #0.25. The shape of the curve isremarkably similar to the efficiency of a flapping foil as a function of the Strouhalnumber, shown in Figure 3 above.In Barrett et al (1999) it is also found that if the body is subdivided longitu-

dinally into three parts of roughly equal lengths, the front part absorbs 15%, themiddle part 46%, and the rear part 39% of the total input power. This is inagreement with direct measurement of the power produced by the muscle of livefish along its length (Rome et al 1993). Rome et al concluded that the middle andrear parts of the fish produce most of the power and share it almost equally. Thisfurther confirms the important role played by the body in the generation of thepropulsive force, as also revealed through flow visualization.The robotic fish study was conducted at near-transitional Reynolds numbers,

in which flow can be fully relaminarized through traveling-wave motion as shownin Taneda & Tomonari (1974) and Taneda (1977). Zhang (1999) studied, throughdirect numerical simulation, the three-dimensional flow around a two-dimensionalplate undergoing traveling-wave–like oscillations at Re " 6,000 in a steady


oncoming turbulent flow of velocity U. Turbulence intensity was 5%, whereasthe speed of traveling waves c could be varied arbitrarily. Zhang found that tur-bulence is reduced significantly when c/U$ 1. Figure 13 shows the instantaneousshape of the plate (lower), with the amplitude quadratically increasing in thedirection of flow (left to right). Three average-velocity profiles in the boundarylayer are shown in location a (defined in the inset). The profile for c/U " 0 hasthe characteristic shape of a turbulent profile; for c/U " 0.4, it has undergone asmall change, but at c/U " 1.2 it has largely laminarized. As also found in Barrettet al (1999), Zhang found that the power needed to propel the foil is minimumat c/U " 1.2. Similar conclusions are reached in the experimental work of Techet(1999) and Anderson et al (1999), who performed DPIV flow visualization in theboundary layer of the laboratory tunalike robot, shown in Figure 11, at Re " 7% 105; as well as in the boundary layer of live fish.

ACKNOWLEDGMENTS

The authors received financial support from the Office of Naval Research contractN00014-96-1-1141 (monitored by P. Purtell, T. McMullen, and J. Fein) and theSea Grant Program under Grant NA46RG0434.

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TRIANTAFYLLOU ET AL C-1

Figure 4 Photograph of tip vortex formation downstream ofan oscillating hydrofoil (right) visualized by injecting air atthe tip of the hydrofoil near the trailing edge. Injected air isentrained into the low-pressure core of the tip vortex tracingthe complex structure of the vortex formation. The tip vortexcan be seen connecting the shed spanwise vortices resultingin the formation of an intricate vortex chain (Re = 1.5 × 105,angle of attack 5°, oscillation amplitude 15°, reduced fre-quency πfc/U = 5.2) (Hart et al 1992). Photograph from Hartet al (1992), provided by Prof. Hart.

C-2 TRIANTAFYLLOU ET AL

Figure 5 DPIV flow visualization in the vicinity of a straight-swimming mul-let. Arrows represent velocity in meters per second (Müller et al 1997). FromMüller et al (1997) with permission from the Company of Biologists Ltd.


Figure 6 (below and following page) Flow visualization along planes perpendicular to thelong axis of a straight-line–swimming Giant Danio. Longitudinal locations of the planesare shown on the outline of the Giant Danio (top). Streamlines (black) are superimposedon a dense velocity vector grid. Velocity vectors scaled in size by velocity magnitude andscaled in color by longitudinal vorticity ωx contours (range: –0.1 s-1, 0.1 s-1). Red vortici-ty indicates clockwise rotation; blue vorticity indicates counterclockwise rotation; greenregions denote irrotational flow. A primarily longitudinal pattern is visible at the head (cutAA, below). The midbody downstroke produces primarily a transverse flow (cut BB, top,next page). As the contraction region reverses undulation direction, longitudinal stream-lines to the left of the body and fins combine with a weakly transverse flow from boundvorticity generated upstream (cut CC, bottom, next page). From Wolfgang (1999).



Figure 7 Flow visualization along xy planes, containing the long axis anddirection of transverse oscillation of a straight-line–swimming Giant Danio ofdepth (D). Top, z/D = 0.2 measured from the top; bottom, z/D = 0.5. Streamlines(black) are superimposed on a contour plot of the vertical vorticity ωz compo-nent in the plane (range: –0.5 s-1, 0.5 s-1). Red vorticity indicates clockwisefluid rotation; blue vorticity indicates counterclockwise rotation; green regionsare irrotational (Wolfgang et al 1999b).


Figure 8 Views of a virtual tadpole at Re = 7200.Isovorticity contours are illustrated with whitelines at the base of the tail and at the mid-tail;wake vortices are shown in green. (A) The tail ina C-curve posture. (B) The tail in an S-curve pos-ture (Liu et al 1997). From Liu et al (1997) withpermission from the Company of Biologists Ltd.


Figure 9 Flow visualizationduring the 60° turning maneuverof a Giant Danio. Top, DPIVmeasurements of live fish; bot-tom, simulation results at themid-depth plane correspondingto the same time instant. In-plane velocity streamlines(black) are superimposed on adense velocity vector grid.Velocity vectors scaled in sizeby velocity magnitude andscaled in color by vertical vortic-ity ωz contours (range: –20 s-1,20 s-1). Red vorticity indicatesclockwise rotation, blue vortici-ty counterclockwise rotation,and green regions are irrotation-al (Wolfgang 1999).

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HYDRODYNAMICS OF FISHLIKE SWIMMING