Fish near-body flow dynamics

Several theories have been developed to explain fishswimming performance. Wu (1961, 1971a,b,c) and Lighthill(1960, 1975) have investigated the propulsion characteristicsof flexible two-dimensional and three-dimensional plates usinglinearized potential flow theory. While these seminal worksprovide insight into the basic swimming propulsivemechanisms, the details of the three-dimensional flow and thedynamics of the shed vorticity are still not well understood.Several comprehensive works on fish swimming thoroughlyanalyze the kinematics and the dynamics of straight-line andunsteady swimming motions for a variety of aquatic species(Videler, 1993; Blake, 1983) utilizing these linear theories toestimate the propulsive dynamics. However, such methodsoverpredict the thrust on more realistic fish-like forms(Videler, 1981) because basic assumptions about the linearslender body theory are violated.

The study of unsteady lifting surface motion for bothswimming and flying has provided insight into the details ofthe dynamics of fish propulsion methods. The aerodynamics ofhovering insect flight (Osborne, 1951; Ellington, 1984;Ellington et al., 1996) illustrates the importance of unsteadyflow mechanisms in achieving a large loading on the wing,including a delay in stall. The unsteady motions of hydrofoils

have been studied analytically and numerically by severalinvestigators (von Kármán and Burgess, 1935; Wu, 1961,1971a,b,c; Lighthill, 1960, 1975; Chopra, 1976; Chopraand Kambe, 1977; Lan, 1979), but these methods employlinearized body boundary conditions as well as assumed wakeshapes.

Most fish move with amplitudes and frequencies that exceedthe limit of linear lifting surface theories. Videler and Hess(1984) showed that two species of fish exhibited highpropulsive efficiency, close to the actuator disk limit, withlarge lateral motion amplitude. Triantafyllou et al. (1993)showed that a variety of fish and cetaceans swim with afrequency and amplitude of tail motion that are within a narrowrange of Strouhal numbers, minimizing energy lost in the wakefor a given momentum and increasing efficiency. This Strouhalnumber range corresponds to the regime of maximum stabilityof the vortex wake thrust jet.

Several investigators have studied the mechanisms ofvorticity control achieved by unsteady motion of a body in afluid (Taneda and Tomonari, 1974; ffowcs-Williams and Zhao,1989; Tokomaru and Dimotakis, 1991). Gopalkrishnan et al.(1994), Streitlien et al. (1996) and Anderson et al. (1998)utilized experimental and computational methods to

2303The Journal of Experimental Biology 202, 2303–2327 (1999)Printed in Great Britain © The Company of Biologists Limited 1999JEB1956

We consider the motions and associated flow patternsof a swimming giant danio (Danio malabaricus).Experimental flow-visualization techniques have beenemployed to obtain the unsteady two-dimensional velocityfields around the straight-line swimming motions and a 60 °turn of the fish in the centerline plane of the fish depth. Athree-dimensional numerical method is also employed topredict the total velocity field through simulation.Comparison of the experimental and numerical velocityand vorticity fields shows good agreement. The fishmorphology, with its narrow peduncle region, allows forsmooth flow into the articulated tail, which is able to sustaina large load for thrust generation. Streamlines of the flowdetail complex processes that enhance the efficiency of flow

actuation by the tail. The fish benefits from smooth near-body flow patterns and the generation of controlled body-bound vorticity, which is propagated towards the tail, shedprior to the peduncle region and then manipulated by thecaudal fin to form large-scale vortical structures withminimum wasted energy. This manipulation of body-generated vorticity and its interaction with the vorticitygenerated by the oscillating caudal fin are fundamental tothe propulsion and maneuvering capabilities of fish.

Key words: carangiform, swimming, hydrodynamics, unsteadypropulsion, manoeuvering, giant danio, Danio malabaricus, vorticitycontrol, digital particle image velocimetry, panel method.

Summary

Introduction

NEAR-BODY FLOW DYNAMICS IN SWIMMING FISH

M. J. WOLFGANG1, J. M. ANDERSON1,*, M. A. GROSENBAUGH2, D. K. P. YUE1 AND M. S. TRIANTAFYLLOU1,‡1Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA and

2Department of Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institution, Woods Hole,MA 02543, USA

*Present address: Charles Stark Draper Laboratory, Cambridge, MA 02139, USA‡Author for correspondence (e-mail: [email protected])

Accepted 27 May; published on WWW 9 August 1999

2304

demonstrate active control of a shear flow with an oscillatingfoil, revealing energy recovery flow control mechanisms thatcan increase or decrease the efficiency. Direct experimentaldynamic measurements of the energetic make-up andefficiency of live swimming fish are difficult to obtain,however, and may not be characteristic of natural behavior. Asa result, Barrett (1996) constructed a robotic underwaterflexible hull vehicle, in the shape of a tuna with a lunate tail,on which direct dynamic measurements and energetic analysescould be performed. Over small ranges of various swimmingvariables, the robot could achieve extremely high propulsiveefficiencies and lower drag than that of the rigid body hullthrough mechanisms of boundary layer relaminarization andvorticity control by the tail fin (Barrett et al., 1999).

Several researchers have studied, through visualization, thewake of a naturally swimming fish. Techniques that havebeen used include dye visualization (Rosen, 1959; Aleyev,1977) and visualization using a stratified layer (Rosen, 1959;McCutchen, 1976). Neither method has elucidated all thedetails of the flow near the body and in the wake. Stamhuisand Videler (1995) utilized experimental particle imagevelocimetry to capture the flow dynamics around several liveswimming organisms and to analyze the energetic make-upof the wake. Anderson (1996) used experimental digitalparticle image velocimetry (DPIV) to visualize the wakebehind a swimming giant danio (Danio malabaricus) andidentified the active manipulation of shed wake vorticity tocreate a reverse Kármán vortex street. Müller et al. (1997)analyzed the wake of a swimming mullet (Chelon labrosusRisso) using similar DPIV techniques and concluded that themanipulation of the wake structure resulted in highpropulsive efficiencies.

In the present study, we present quantitative, multipointmeasurements of the flow field around a small naturallyswimming fish using DPIV. The results include flowmeasurements for straight-line swimming and rapidmaneuvering of a giant danio (Danio malabaricus). We alsodevelop a numerical method to simulate the hydrodynamicsof an actively swimming fish-like body. The characteristicsof real fish swimming at large Reynolds numbers areapproximated by assuming that viscous effects are confined toa thin boundary layer and wake region. The formulation allowsfor the satisfaction of the exact body boundary conditions andfor the nonlinear evolution of the shed wakes. This inviscidapproach is computationally an order of magnitude more rapidthan the fully viscous numerical methods that are essential atlower Reynolds numbers, such as those developed by Liu etal. (1997) to study tadpole locomotion. Our numerical methodis then applied to analyze the wake dynamics and the near-body hydrodynamics of the motions of the giant danio, whichwere captured using DPIV, including both continuous straight-line swimming and a 60 ° turning maneuver. We present asample of the experimental and numerical results, whichdemonstrate that the controlled actuation of the caudal fin andrelease of bound vorticity contributes to the efficientproduction of a reverse von Kármán street thrust jet.

Materials and methodsExperimental methods and apparatus

We employ digital particle image velocimetry (DPIV), firstintroduced by Willert and Gharib (1991), for flowmeasurement and visualization. The DPIV algorithm is appliedto two images of the flow separated by small time dt, and smallsubsets of image data (interrogation windows) are compared atthe same location in each image by computing the spatial crosscorrelation. When the particle images match well, the cross-correlation function is sharply peaked, and the location of thepeak marks the displacement of the particle image pattern. Thealgorithm is then applied throughout the images to obtain theentire flow field.

For the DPIV algorithm to work well, there must beadequate particle seeding, randomly distributed over the entireimage plane. Willert and Gharib (1991) systematically studiedthe seeding requirement and found that 10–20 particles perwindow are sufficient to ensure good correlations. For acomplete discussion of general DPIV capabilities andlimitations, see Willert and Gharib (1991). Unique problemsarise in using DPIV to visualize flows around moving,deformable bodies. As the fish swims and boundaries move,interrogation windows cross boundaries and contain spuriousimage data. This biases the cross-correlation peak towards thedisplacement of the boundary, resulting in spuriousdisplacement data for the fluid near the boundary. A completediscussion of the near-body DPIV techniques we employ,including their capabilities, limitations and validation, can befound in Anderson (1996).

The windowing process in DPIV filters out wavenumbers(spatial frequencies) larger than the inverse interrogationwindow size. This is a problem when the flow field of interesthas fine-scale motions and flow reversals, such as in a vortexcore. To resolve such fine-scale features correctly, the imageview should be enlarged in the region of interest by movingthe camera or by reducing the window size. Willert and Gharib(1991) showed that there is approximately 70 % attenuation of32 pixel features when the data are processed with 16 pixel ×16 pixel interrogation windows, and one should always looselyinterpret displacement data for scales less than twice thewindow size.

We calibrated our hardware and processing techniques in thesame manner as Willert and Gharib (1991) using an artificiallygenerated test image displaced with a rotary table instead of alinear table. The algorithm is very robust up to the Nyquistlimit, despite curved particle trajectories. Our relativedisplacement error is generally less than 5 %, and usually lessthan 2 % for moderate (>1 pixel) displacement.

Fig. 1 summarizes the experimental apparatus for DPIVimplementation in these experiments. The measurementvolume, a 51 cm × 25 cm × 28 cm glass tank, was seeded withsmall (20–40 µm diameter), neutrally buoyant fluorescentpolymer spheres. A planar slice of the flow approximately3 mm thick was illuminated with a 6 W argon-ion laserconnected to a fiberoptic cable and light sheet optics (Dantec).

M. J. WOLFGANG AND OTHERS

2305Fish near-body flow dynamics

The particle flow was imaged using a high-resolution (768pixels × 480 pixels) black-and-white CCD video camera(Texas Instruments, Multicam MC-1134P) and recorded tovideo disc using a Sony CRVdisc videodisc recorder (LVR-5000A). Images were acquired in real time at the camera framerate of 30 Hz in dual-field mode, allowing full verticalresolution with no interlace. Asynchronous shuttering of thelaser light with a timing box (General Pixels) synchronized tothe video signal allows for higher effective frame rates.Although image pairs are collected at 15 Hz, asynchronousshuttering allows the difference between images in a pair to bevery small (approximately 5 ms).

As illustrated in Fig. 1, the fish was confined within a layerof fluid approximately 3 cm thick, between the free surface anda highly permeable screen consisting of a steel wire grid(2.5 cm openings) and a fine nylon mesh (approximately 3 mmopenings). The clearance between the fish and the screen andfree surface was estimated to be 3 mm. The screen was locatedapproximately 25 cm from the tank bottom. The screenconfined the fish yet allowed fluid to pass through it, and theplane in which the fish was allowed to swim was sufficientlywide to eliminate any wall effects. The flow in the horizontalsymmetry plane of the fish was revealed by orienting the laserlight sheet at the mid-depth of the confinement layer. Ingeneral, the light sheet illuminated the fish at the caudal fork;however, in some instances, slight bending of the caudal fincaused the tips of the tail to be illuminated. The free surfacedistortions caused by typical swimming and maneuveringmotions were observed visually to be negligible. This suggeststhat there was minimal interaction between the wake vorticityand the free surface.

We chose to experiment with a fast-swimming tropicalspecies, Danio malabaricus (Jerdon), the giant danio, whosemorphology is very similar to that of the pearl danio studied

by Rosen (1959). The dominant fins are the caudal fin, theflexible dorsal fin and the stiffer anal fin, which acts like a keel.Anterior to the anal fin, the ventral fins flare out duringmaneuvering but participate little in straight-line swimming.Similarly, the pectoral fins are used in maneuvering and aregenerally kept flush to the body during straight-line motion.

Several fish were used with body lengths between 5 and10 cm. The maximum span of the caudal fin was approximately23 % of the body length. Whenever possible, data will bepresented normalized with respect to fish body length L. Wefound the giant danio well suited to our experiments since theyare steady, fast swimmers and hardy with respect to handling.The laser light sheet was oriented horizontally in the midplaneso that it hit the fish near the vertical plane of symmetry. Thefish were allowed to swim freely in the confinement layer,during which time the video was recorded to laser disc. Thefish showed no adverse reaction to either the particles or thelaser light, and the kinematic behavior of the fish was identicalwhether the laser was on or off.

The DPIV images obtained of the fish swimming in astraight line or undergoing maneuvering motions are processedto produce velocity field images. Each arrow in a velocity fieldplot represents one cross-correlation calculation, and the lengthof each arrow indicates the relative velocity magnitude. Theimages obtained were processed using 32 pixel × 32 pixelinterrogation windows, shifted in eight-pixel steps, whichcorresponds to oversampling the available image data fourtimes. This interrogation window dimension is sufficientlysmall compared with the length of the fish (0.076L) such thatflow scales of size 0.15L and above can be measuredaccurately. The velocity data are then used to compute thecomponent of the vorticity perpendicular to the DPIV plane,so that large-scale vorticity components along the body and inthe wake may be observed and quantified.

Fig. 1. Schematic diagram of experimental apparatus. The fish is confined within a thin layer of fluid between the water surface and apermeable screen. The horizontal midplane of the fish is illuminated from the side and imaged from above.

2306

Numerical methodWe develop a computational tool for investigating the

swimming characteristics of a three-dimensional flexible bodywith arbitrary motions and geometry. We consider as acanonical problem the abrupt starting from rest to a constantvelocity U of a streamlined flexible body geometry undergoingprescribed undulations about its mean line, with arbitrarydistribution of sharp-trailing-edged fins. Thin shear layerwakes are shed continuously from the sharp trailing edges ofthe body as time proceeds. With the exception of the wake, thefluid is assumed to be inviscid and irrotational, as well asincompressible, allowing for the existence of a velocitypotential Φ(x→,t). We define two coordinate systems, an inertialglobal coordinate system O,X,Y,Z, fixed in space in the fluid,and a local coordinate system o,x,y,z, instantaneously fixed inthe flexible body and orthonormal to the stretched-straightmean line and body section plane. The mean periodicundulations of the body are described with reference to o,x,y,z,and the global translational and rotational motions of the bodyare described with respect to O,X,Y,Z. Fig. 2 details thecoordinate systems used in the problem. All time and lengthscales are chosen to be non-dimensional with respect to thebody length L=1.

We consider the computational domain to be enclosed bythree surfaces, the body surface Sb, an infinitesimally thin wakesheet Sw, and a far-field boundary at infinity S∞. Laplace’sequation for the velocity potential Φ(x→,t) governs theconservation of mass of the fluid, and the unsteady Bernoulliequation describes the pressure everywhere in the field. Thevelocity potential can be further described by the linearsuperposition of a body perturbation velocity potential Φb(x→,t)and a wake perturbation velocity potential Φw(x→,t), eachsatisfying the Laplace field equation. Although the strength of

the wake being shed at the body trailing edges at any time isunknown, the strengths of all previously shed wake surfacesare known, with the exception of the initial condition of theimpulsive start when no previously shed wakes are present.Thus, the wake perturbation velocity potential Φw(x→,t) isknown, and we formulate the boundary value problem for thebody perturbation velocity potential Φb(x→,t).

The prescribed body motion V→b(x→,t) gives the kinematicbody boundary condition in terms of the body perturbationpotential for a body normal vector n̂b:

where !→

is a gradient vector. The prescribed motion of thebody forces the length of the shed wake to increase, as anunsteady Kutta condition is applied to the sharp trailing edgesof the body to enforce smooth flow separation. The strength ofthis portion of the shed wake sheet is unknown. At any time,however, we can set the strength of the shed wake velocitypotential to the jump in the body perturbation potentialbetween the upper and lower surfaces near the trailing edge(TE) (Kinnas and Hsin, 1992):

Φw(x→,t)|TE =ΔΦb(x→,t)|TE . (2)

A radiation condition is imposed such that the influence of thevelocity potential Φ(x→,t) decays rapidly to zero in the far field.

Green’s formulation was considered to solve the problem forthe body perturbation velocity potential Φb(x→,t). The secondform of Green’s theorem is employed for the body perturbationpotential Φb(x→,t) and for the three-dimensional source potentialGreen’s function 1/|r|, both of which independently satisfy theLaplace equation. r is the distance from observation point x→ toany point ξ

→on the boundary, r=|x→−ξ

→|. This choice of Green’s

function is standard in the development of source-dipoleboundary integral methods for an ideal fluid in the absence ofa free surface. Thus, the body perturbation velocity potentialΦb(x→,t) at any point x→ can be described in terms of the integralaround the boundaries of the contributions to the perturbationvelocity potential. If this point x→ is on the boundary, Green’stheorem can be written:

where S is composed of all the bounding surfaces,S=Sb+Sw+S∞, ∂n is a derivative in the direction of the outwardsurface normal vector and ds is an elemental unit of surfacearea. The influences of the source and dipole singularities inthe integrands of equation decay as 1/r and 1/r2, respectively.Thus, the boundary at infinity S∞ can be shown to havenegligible influence.

The unsteady wake strength is a function of space and time,Φw(x→,t). The pressure across the wake shear layer iscontinuous, and the wake surface is a material surface and canthus be represented by a smoothly varying strength dipole

(3)ds− ds ,

2πΦb(x→,t)=

∂Φb(x→,t)∂n

∂(1/|r|)∂n

1|r|

⌠⌡SΦb(x→,t)

⌠⌡

⌠⌡S

⌠⌡

(1)= [V→b(x→,t)−!→Φw(x→,t)]· n̂b ,

∂Φb(x→,t)∂nb


xy

z

Sw

S b

U

X

Y

Z

Fig. 2. Coordinate system convention for the analysis of straight-lineswimming (top view). Local body motions are described with respectto the body-fixed o,x,y,z coordinate system, and the body trajectory isdescribed with respect to an inertial reference frame O,X,Y,Z. U,swimming speed; Sb, body surface in the computational domain; Sw,wake surface in the computational domain.


sheet. From Kelvin’s theorem, as the foil’s circulation changes,so does the strength of the shed wake dipole. At any time, thestrength of the entire previously shed wake is known, and thusthe strength of Φw(x→,t) is known, except for the recently shedshear layer which smoothly separates from predeterminedtrailing edges. The unknown strength of the newest portion ofthe wake sheet is addressed through the Kutta condition asdescribed by equation 2. Thus, the body perturbation velocitypotential Φb(x→,t) and the change in circulation of the variouswake-shedding lifting surfaces of the body can be written interms of surface integrals over the body surface Sb and thewake sheet surface Sw.

Representation of the time-dependent continuous potentialdistributions over the body and wake surfaces is developed indiscrete form. The body and the wake are each divided intoquadrilateral panels. The source-dipole distributionrepresenting the body is approximated as piecewise constantover each body panel; similarly, the dipole distributionrepresenting the thin shear layer wake is approximated aspiecewise constant over each wake panel. The collocationpoint for each panel is the geometric center of the panel. Thebody grid is generated with higher panel densities in regionsof presumed rapid potential variation and in regions withcomplex geometric considerations, such as areas of largecurvature or lifting surface tips, to reduce panel skew and toincrease accuracy with respect to the unsteady, continuousvelocity potential distribution.

At any time t, the body perturbation velocity potential ateach panel collocation point φb(x→,t) can then be found in termsof the perturbation potentials at all the other panels. A linearsystem of equations results for φb(x→,t), with the wakeperturbation velocity potential φw(x→,t) of the new wake panelsdescribed in terms of φb(x→,t) at the trailing edges. A discreteform of the kinematic boundary condition on the body surface(equation 1) gives the normal velocity at each panel. A simplesystem of linear equations for φb(x→,t) at any given time t isexpressed in the general form:

[Q̃] {φb} = [P̃] , (4)

where Q̃ is the matrix of influence coefficients for the source-dipole potential panel distribution, φb is a column vector of thebody perturbation velocity potentials, and P̃ is the columnvector of normal velocities for the source-dipole potentialpanel distribution. Details of the linear system development arefound in Appendix A. At each time step, this linear system ofequations is solved to find φb(x→,t) and φw(x→TE,t) through theuse of an iterative scheme. φw(x→TE,t) is representative of thechange in the total circulation of the body lifting surfaces.While the number of body panels k is constant, the totalnumber of wake panels nw increases with time, as a wake oftime-varying strength is continually shed from the trailingedges of the body. The numerical scheme is temporallyintegrated using a fourth-order Runge–Kutta scheme.

Thin fins and wakesIn general, we consider fish geometries which are composed

of a main body, with smoothly varying elliptical cross sections,and a tail fin, with rounded leading and sharp trailing edges.On occasion, we model additional fins to investigate the effectsof a more accurate geometrical representation. Modeling thesefins as finite-thickness lifting surfaces with rounded leadingand sharp trailing edges becomes impractical as the finsbecome smaller and thinner with respect to the main body. Asthe angle between adjacent panels at the leading edgeapproaches 2π, the resulting square-root singularitydestabilizes the influence coefficient matrix (Katz and Plotkin,1991).

The addition of thin fins to the main body and caudalfin arrangement is accomplished using vortex latticesurfaces, such as those investigated by Lan (1979), Chenget al. (1991) and mostly recently H. Kagemoto, D. Yue andM. Triantafyllou (unpublished data). A surface of zerothickness is discretized into a grid of planar quadrilateralelements, and the doublet jump strength at each panel φb(x→,t)is assumed to be constant. The source distribution strengthacross each panel on the thin surface is set identically to zero,and a zero-normal velocity boundary condition is againapplied at each thin panel’s collocation point x→f with normaln̂ ′.

The application of this no-flux boundary condition results in aset of equations for a geometry with both a typical body andthin fins for the doublet jump strength φb across a thin fin panelat x→f.

The vortex lattice solution algorithm requires thedifferentiation of equation 3 an additional time to apply theNeumann boundary condition when the panel source strengthis zero. The details of the development of this higher-orderalgorithm are analogous to those presented here for a source-dipole panel method and thus will not be discussed. However,it should be mentioned that some care must be taken in thegeometric modeling of these thin fins not to use a fine panelmesh density, because a hypersingularity present in the self-influence coefficients can destabilize the numerical algorithmfor very fine discretizations. The size of the thin fin panelsshould be of the order of the largest body panels to minimizethe matrix condition number. For constant potential jumpdistributions over a quadrilateral geometry, thehypersingularity requires no special treatment for the typicalgrid discretizations applied in these simulations, and theinfluence coefficients calculations can performed by standardvortex-lattice algorithm (Katz and Plotkin, 1991). Removalof the hypersingularity for more complex geometries andpotential jump distributions could be achieved throughvarious numerical and analytical techniques (Martin andRizzo, 1989; Guiggiani et al., 1992; Martin, 1996). Thesystem of equations is expressed as a coupled mixedboundary-value problem, consisting of a set of matrixequations for φb on the body geometries φbb and for the φbjump across the thin fin geometries φbf, as well as the strength

(5)= V→b · n̂ ′ .∂φb(x→f,t)∂n′

12

2308

of the unsteadily shed wake. In general form, the system isexpressed as:

where Q11, Q12, Q21 and Q22 are sub-matrices of the influencecoefficients defined in the list of symbols, and P1 and P2 arepartial column vectors of panel normal velocities defined in thelist of symbols. A wake equivalent to that described in theprevious section is shed from the thin fin trailing edge as timeprogresses, as prescribed by a Kutta condition. The Kuttacondition for the thin wing states that the change in circulationaround the lifting surface section must equal the change invorticity shed into the wake, in accordance with Kelvin’stheorem, which states that the change in total circulation forthe entire fluid domain must be zero:

where ! is the circulation strength. Thus, the strength of a newwake doublet panel is equal to the strength of the thin fintrailing edge panel doublet at the previous time step, t−dt. Thecoupling of these two types of surface representation allowsfor the simulation of complex bodies undergoing unsteadyarbitrary flexing motions.

When the solution of the boundary value problem has beenobtained at each time step, the wake panel endpoints areconvected by a velocity field described by a desingularizedversion of the Biot–Savart law. The desingularizationtechnique employed (Krasny, 1986) eliminates theinfinitesimal vortex sheet singularity and the associated ill-posedness. The inclusion of the desingularization is necessaryto prevent nonlinear energy transfer to the highest wavenumbermodes and simulation breakdown, in addition to non-physicalsolution growth caused by the numerical instabilities.

A wake vortex element of strength !v, consisting of the sideedge of one quadrilateral doublet panel, is assigned core radiusδw, such that the velocity v→ induced by this vortex element atξ→

can be expressed as:

where s→ is the tangent vector to the vortex element, r→ is thevector with magnitude r from the vortex element to the fieldpoint ξ

→, the path of integration is along the length of the vortex

element, and dl is an elemental unit of length along the vortexsegment. As ξ

→approaches the vortex, r approaches zero, and

the velocity field expressed by equation 8 approaches a finitelimit.

Similarly, the body and thin fin doublet panels aredesingularized by the same technique, with arbitrary bodydesingularization radius δb. This desingularization assists instabilizing the solution algorithm for coupled mixed boundaryvalue problems with bodies of fine mesh discretizations.Additionally, when upstream-shed vorticity impinges on

downstream body elements, non-physical free wakeacceleration and deformation are avoided. These impingingwake panels pass around the outside of the body surface, oftentangentially, convected by the body surface normal andtangential velocities. Without this body desingularization, freewake panels interacting with flexing body panels mightconvect inside the body.

Integrated quantitiesAt each time step, we find the velocity and force distribution

over the body panels. Once the unknown body perturbationvelocity potential φb is known for each panel, the tangentialsurface velocity on each panel can be found numerically byspatial differentiation of φb over the body using a finite-difference scheme. The pressure coefficient CP at each surfacepanel can then be found by employing the unsteady Bernoulliequation:

The partial derivative ∂φb/∂t is found by taking the materialderivative of φb at each panel using a temporal second-orderbackward-difference technique and subtracting the convectiveterm arising from the imposed motions of the unsteady flexingbody. The force Fb is found by integrating the pressure on eachpanel over the body:

where ρ is the density of the fluid. For thick bodies, thepressure is calculated on only one side of each panel, whereasfor thin fin surfaces, the total force realized is related directlyto the pressure jump between the upper and lower surfaces ofthe fin. The lower surface velocities are calculated for the thinfins, and the pressure jump is integrated to find the total forceon the thin fin Fb,thin. The method of Lan (1979) is employedto model the leading edge suction forces on any thin fins,provided that the sweep angle of the fin is not high. The caudalfin is modeled as a finite-thickness high-aspect-ratio foil, andthe leading edge suction acting on the caudal fin is resolvedthrough direct pressure integration over the body.

The power transmitted to the fluid by the motions of thebody Pb is the integral of the product of the local force and thelocal imposed body velocity. In general, for a geometry witha number of thick bodies and thin fin surfaces, the time recordsof the total force Fb and the total power delivered to the fluidPb may be quite complex, even for sinusoidal transversemotions. The total force is decomposed into a side force Lbperpendicular to the direction of straight-line swimming, whichshould have a zero temporal mean, and a longitudinal force Tb,which should have a non-zero mean value of thrust or drag fora mean swimming speed U in the absence of skin frictionaldrag modeling. The mean value of thrust generated over oneperiod for harmonic body undulations is denoted T–b. Similarly,the mean power transmitted to the fluid over a period of

(10)12

Fb = ρU2

Sb

CPn̂b ds ,!

(9)∂φb

∂t−1

U2 !→φb · !

→φb + 2CP = .

(8)v→(ξ→

) = dl ,Γv

4πs→× r→

r3 + δw3

⌠⌡

(7)= 0 ,DΓDt

(6)= ,P1

P2

Q11 Q12

Q21 Q22

φbb

φbf



harmonic body motion is denoted P–b. This allows us tomeasure the classical hydrodynamic propulsive efficiency ηfor swimming speed U as η=UT–b/P–b.

The unsteady code was validated by simulating theimpulsively started motion to a constant speed U of a finite-aspect ratio foil at a small angle of attack, in a fluid otherwiseat rest. Convergence of the numerical method was confirmedby systematically varying the time step size dt and the numberof panels k around the foil, in addition to the aspect ratio ARand the desingularization parameter δw for the wake, andcomparing the steady lift coefficient with the experimentalvalues illustrated in Fig. 17-5 of Hoerner (1985).

ResultsSteady swimming of a giant danio

DPIV experimental resultsIn the DPIV experiments, the fish exhibited varied behavior

with tail double amplitudes between 0.11L and 0.21L andfrequencies up to 6.2 Hz. Consequently, the Strouhal numberalso showed variability. Typical relative errors in the analysisof the fish and wake kinematics were 25 % for the tail doubleamplitude measurements (0.02–0.05L), 15 % for the frequencymeasurements (up to 0.9 Hz), 5 % for the speed measurement(up to 0.055 L s−1) and 30 % for the Strouhal numbercalculation. Much of the error is simply because the fish didnot always swim with a constant tail-beat amplitude orfrequency and does not actually represent measurement error.The values reported here for experiments are means. The wakeStrouhal numbers (St) clearly support the conclusions ofTriantafyllou et al. (1993) that fish tend to swim within anarrow range of Strouhal numbers in order to exploit thenatural instabilities of the thrust jet created. Although ourkinematic measurements are approximate, most of the Strouhalnumbers observed in our experiments are within the predictedoptimal range of 0.25<St<0.4.

Several swimming cycles were captured and studied usingDPIV; here, we present one particularly clear case where thefish swam directly away from the light source, thus revealingthe flow on both sides of the body and in the wake withoutshadows. For this case, the fluid ahead of the fish was nearlystationary, and all motion can be attributed to the motion of thefish. Velocities very near the body boundary are not shown inthe following results because the image of the fish itselfcontaminates the flow image data. Spurious data are directlyremoved, and adjacent data are treated specially to avoid theinclusion of contaminated data in the smoothing filter andcalculations of vorticity and circulation.

Fig. 3 shows the velocity field and vorticity contoursadjacent to and in the wake of the fish during steady straightswimming. The DPIV images reveal that the fish wasswimming at an approximately constant mean velocity of8.9 cm s−1 (1.1 L s−1), a normal observed swimming speed forthe giant danio, for the 0.23 s duration of the cycle captured.The fish traveled steadily from one end of the tank to the other,passing through the imaging window in the center of the tank.

Any acceleration or deceleration of the body occurred at eitherend of the tank, well before or after the images shown in Fig. 3in the center of the tank, so the flow features are likely to betypical of steady-state motion. Nearly an entire swimmingcycle is shown with the successive data sets separated byapproximately 0.22T, where T is the period of the tailoscillation. The figures show the fish body exactly as it appearsin the first image of each image pair except that it is coloreduniformly for clarity. Slight variations in the recorded bodyimage exist due to the fact that in some images the caudal forkis illuminated by the laser sheet while in others the caudal fintips are illuminated.

At the start of the swimming cycle (Fig. 3A), the fish tail isflexed maximally to the left. A clockwise vortex is formingwith its center near the tip of the tail as a result of shed body-generated vorticity associated with the body undulation. Thevortex is already well formed, although at this point the tail hasnot begun its rightward motion. Subsequent rightward tailmotion will contribute same-signed vorticity to this vortex.Immediately to the left of the fish and slightly anterior to thetail, the flow follows the rightward velocity of the body.Slightly upstream of this position at approximately the mid-body of the fish, longitudinal flow towards the tail marks theformation of the next bound counterclockwise vortex.

In Fig. 3B, the tail has moved to the right, and the newclockwise vortex is now clearly visible in the wake. The regionof longitudinal flow has moved rearward and to the right withthe tail motion. In Fig. 3C, the tail is maximally displaced tothe right ready to begin the return stroke. The flow to the rightof the fish follows the body motion to the left which definesthe circular flow that becomes the next counterclockwisevortex. Fig. 3D shows this new vortex in the wake as well asthe next clockwise vortex forming along the body. Theswimming cycle then concludes with the tail in the maximumdownward position and the next clockwise vortex positionedat the tail tip, ready to be shed on the upstroke.

The closed circular contours shown in Fig. 3 indicateconcentrated vortices arranged in a jet pattern. Body-boundvorticity cannot manifest itself, as it would require processingof the velocity field through the body of the fish. The wakeorganizes into a stationary reverse Kármán street: vorticesalign in a staggered pattern such that the net wake produces ajet flow. The maximum velocity along the centerline of the jetis approximately 90 % of the mean velocity of the fish. Theflow adjacent to the fish is strongly influenced by the bodyundulation, as demonstrated by the clockwise flow atapproximately three-quarters of the body length from the nose(Fig. 3D). This clockwise flow is the beginning of the nextvortex destined to be shed by the tail into the wake. The new‘vortex’ is formed well before encountering the tail. Theincreasing lateral motion of the posterior body region augmentsthe circulation and strengthens the vortex before it is shed intothe wake, just before and at the peduncle. The tail manipulatesthis vorticity through repositioning and shedding of additionalvorticity.

Flow properties were quantified by averaging over one

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swimming cycle. The maximum wake velocity along the wakecenterline was computed by averaging the streamwise velocityat several locations over one wavelength downstream, at oneinstant in time. The maximum wake velocity was 0.88U or 0.97 L s−1. In the regions near the body where the undulationwave has the most effect, the maximum longitudinal(streamwise) velocity component was approximately 0.14U,and the maximum transverse velocity (towards or away fromthe body in the normal direction) was approximately 0.39U.The experimental circulation !e of the wake vortices wascalculated by integrating along circular contours around several

wake vortices using equation 11. The average circulation of thewake vortices was !–e/LU=0.3 with core radius Ro≈0.1L:

Numerical modelingIn the numerical calculations, we approximate the body to

be as close as possible to the giant danio body, including thecaudal fin, and the dorsal and anal fins. The tail generates thelargest wake, and a primary separation line is chosen to be the

(11)Γe = u→(X,Y) dL→ .!


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Fig. 3. (A–D) Velocity field (left) and vorticity contours (right) during steady, straight-line swimming viewed from above. Each arrowrepresents one cross-correlation calculation with the length indicating the relative magnitude of the velocity. The maximum velocity shown isapproximately 8 cm s−1. Vorticity contours range from −5.0 to −20.0 s−1 (dotted lines) and from +5.0 to +20.0 s−1 (solid lines) in ±1.67 s−1

increments. Time t is given in each view. Kinematic variables of the straight-line swimming motion: swimming speed U=1.1 L s−1; tail-beatfrequency f=3.3 Hz; tail-tip double amplitude A=0.16 L; backbone wavelength λ=1.1 L; Strouhal number St=0.45. L fish body length.


trailing edge of the tail. Although the giant danio possesses alarge dorsal fin, a large anal fin and smaller ventral and pectoralfins, these edges are treated differently in the following cases.It is obvious from the experimental evidence that the tailrealizes the greatest unsteady motion amplitudes, which wouldsustain the largest unsteady lifting forces and generate thestrongest vorticity structures; however, the influence ofvorticity shed upstream of the tail by secondary lifting surfaceson the flow around the tail, the total wake structure and theforce on the body is unclear.

Lighthill (1960, 1975) hypothesized that if there was a largegap between the dorsal and caudal fins, any vorticity shed fromdorsal or ventral lifting surfaces would behave very differentlyfrom that in the presence of a fin bridging the gap between thetwo fins. Specifically, for certain values of the phase betweenthe tail and the dorsal fin motions, recovery of upstream shedenergy may be achieved. Streitlien et al. (1996) showedcomputationally such energy recovery using an oscillating foilin a shear flow.

To clarify this point, a computational body geometry waschosen which includes the main portion of the body, the caudalfin and the smaller dorsal and anal fins (Fig. 4). The main bodysections are assumed to be elliptical with a major-to-minor axisratio of AR=2.2, where the major axis corresponds to the heightof the body. A curve-fitting technique is used to determine theprofile shape of the body with L=1.0, and is given simply by:

z(x) = p(x) ± 0.305tanh(6x + 1.8)for −0.3 " x " 0.1 , (12)

z(x) = p(x) ± [0.15−0.152tanh(6.3x− 3.08)]for 0.1 " x " 0.7 , (13)

p(x) = 0.195tanh[−(0.3 + x)/0.15] + 0.195 , (14)

where z(x) is the vertical coordinate along the length of thebody in the o,x,y,z coordinate system, and p(x) is the verticalcoordinate of the mean line along the length of the body in thatcoordinate system. The caudal fin is assumed to havechordwise sections of NACA 0016 shape, to allow for efficientresolution of leading-edge suction forces. The caudal finleading-edge and trailing-edge profiles for the semi-span arealso determined through a curve-fitting technique, and aregiven simply by:

x(z)LE = 39.543z3−3.685z2 + 0.636z , (15)

x(z)TE =−40.74z3 + 9.666z2−0.15z + 0.1075 , (16)

where x(z)LE and x(z)TE are the longitudinal coordinate of theleading and trailing edges, respectively, along the span of thefin, and where 0#z#0.15. The leading edge of the tail atmidspan intersects the posterior end of the body’s contractionregion, or the region of the body which narrows in cross-sectional area anterior to the caudal peduncle. The entire bodylength is then scaled to L=1. The smaller dorsal and anal finsare attached to the body along the midbody profile axes of theventral and dorsal contraction regions anterior to the caudalpenducle, and they are modeled as zero-thickness vortex-lattice

lifting surfaces. These fins remain attached to the body alongthe same axes but are allowed to flex with the prescribedmotion of the body. The trailing edge of the caudal fin isassumed to be the largest wake-producing edge on the fish inthe wake-shedding scheme. Additionally, in order to model theeffects of upstream shed vorticity on the flow into the caudalfin, the downstream wake structure and the total force on thefish, the thin dorsal fin and anal fin are assumed each to sheda vortex wake from the trailing edge. The separation lines ofthe wake sheets are shown in Fig. 4.

The numerical grid in the region of the caudal peduncle,where the body and tail intersect, is artificially smoothed. Thissimplification allows rapid repanelization of the structured gridand ease of relative panel motion. The differences in thepeduncle contraction region between the numerical form andthe actual giant danio are assumed to have a small effect onthe hydrodynamic similitude, as this region sustains a smallhydrodynamic load and serves to transmit force from theoscillating tail to the center of mass of the body (Lighthill,1960, 1975). Additional geometric differences between thenumerical body representation and the live giant danio includethe zero-thickness representation of the anal and dorsal fins andthe omission of the smaller ventral and pectoral fins. Thesmaller ventral and pectoral fins are assumed to have minimaleffects on propulsion and are neglected in this analysis. Theroles of the anal and dorsal fins in propulsion are investigatedby implementation of the wake-shedding scheme.

Fig. 5 presents a case of typical straight-line swimming,revealing the nonlinear wake sheet deformations andinteractions. Clean separation of the main wake from thetrailing edge of the caudal fin is shown, and the wake rolls upbehind the tail, forming a reverse Kármán vortex streetstructure and thrust jet, under the self-influence of the shedvorticity and also the body perturbation velocity influence. Thethrust jet is composed of connected vortex-ring-like structures,which are evident from the concentric contours of dipolestrength on the deforming wake surface. The wakes shed fromthe vortex-lattice lifting surfaces of the thin dorsal and anal finsare also evident, with strength which oscillates due to theunsteady fin undulations caused by the traveling backbonewave. While the thin fin trailing edges are not entirely vertical,they are close to the leading edge of the caudal fin and removedfrom the main body, which reduces the roll-up effects causedby the influence of oscillating body perturbation velocities and

Fig. 4. Computational form chosen for a giant danio. Wakeseparation lines are shown for the dorsal fin, anal fin and caudal finseparation scheme.

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the self-influence of the shed vorticity. Thus, the interaction ofthe vortex sheets shed from the thin fins with the caudal finwake occurs before the thin fin wakes have significantlytightened into vortex lines. These unrolled thin fin wakes maythen affect more of the flow around the leading edge of thecaudal fin than if they had an opportunity to deformsignificantly, increasing the likelihood of energy recapture bythe caudal fin through increased leading-edge suction.

Imposed motion descriptionA kinematic history of giant danio straight-line swimming

motions was obtained from experimental sequences ofimages taken over three swimming cycles using DPIV, suchas that shown in Fig. 3. From this information, we derivedan analytical representation of the swimming motions. Giantdanio swim using carangiform motion. Carangiformswimmers possess tapering tails which terminate in a well-defined caudal fin. Carangiform motion is characterized by asmooth, amplitude-modulated traveling wave moving alongthe length of the fish backbone with a phase speed cp=ω/kw,which is usually different from the swimming speed U. Inthis description, kw=2π/λ is the wavenumber, for awavelength of the backbone perturbation λ, and ω is thecircular frequency of oscillation. The transverse amplitude ofthe backbone motion a(x) increases gradually along itslength, such that the primary propulsive motions are confinedto the tail region.

On the basis of experimental measurements of the giantdanio using DPIV, discretized from Fig. 3, the analyticaldescription of the motion was curve-fitted to be purelysinusoidal and to consist of a smooth amplitude-modulatedtraveling wave along the body length with constant phasespeed cp=ω/kw for a constant swimming speed U. The imposedtransverse motion y(x,t), with x measured from the nose, hasthe form:

y(x, t) = a(x)sin(kwx−ωt) , (17)

where a(x) is the amplitude envelope, given the form of aquadratic function:

a(x) = c1x + c2x2 , (18)

where c1 and c2 are adjustable coefficients, as described byBarrett et al. (1999), and are chosen to achieve a specific shapeof the amplitude envelope a(x) along the length of the bodyand a specific value of the double amplitude of motion A at thetail. The proper frequency scaling of data is thus achieved bymaintenance of a constant non-dimensional Strouhal numberSt, based on the wake Strouhal law (Triantafyllou et al., 1991,1993) observed in live fish, where St is given by:

St = fA/U , (19)

where f is the frequency of oscillation and A is the total meanlateral excursion of the tail fin. These variables, obtained fromthe discretizations of the DPIV data of Fig. 3, are used todevelop the imposed motion description for the computationalinviscid model. We chose to impose the measured motionbecause the basic purpose is to corroborate numericalpredictions of the detailed flow against experimental results.The numerical results are then used to investigate details of theflow development that cannot be easily observed in theexperiment.

Flow profile comparisonOur computational geometry is employed to investigate the

straight-line swimming of the giant danio as obtained duringDPIV experiments. The kinematic variables that prescribe themotion are those calculated from experimental DPIV imagesfor the sequence shown in Fig. 3, specifically: swimming speedU=1.1 L s−1; tail-beat frequency f=3.3 Hz; tail-tip doubleamplitude A=0.16L; backbone wavelength λ=1.1L; Strouhalnumber St=0.45; phase angle between pitch and heave of thetail motion φ=95 °; and tail angle of attack α=6 °.

Focusing on the midbody plane flow dynamics, Figs 6 and7 highlight many prominent in-plane flow features which


Z

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-50 -30 -10 10 30 50 s-1

Fig. 5. Effect of upstream-vorticity shedding on the wake structure behind the computational giant danio geometry during straight-lineswimming simulation. Wake sheets are represented by deforming, ungridded surfaces, with superimposed contour lines of wake dipole strength(range −50 s−1 to 50 s−1) on the deforming wake surfaces.


develop during the course of a straight-line swimming period.Fig. 6 illustrates the midbody plane velocity vectors andstreamlines at eight discrete intervals over the swimming cycle.Velocity vectors densely grid the plane, scaled in size by themagnitude of the local velocity, so that far from the fish theymay appear as points. In-plane streamlines are superimposedon the velocity vector grid to clarify the direction and thestructure of the flow perturbations. Although the streamlines inthis representation are continuous, the magnitude of the fluidvelocity decreases greatly in this reference frame as thedistance from the fish body increases. While the velocityvectors in this plane, in general, may have three-dimensionalvertical components, careful analysis reveals that the flow isstrongly two-dimensional in most regions along the body atthis depth for all separation schemes and body geometries.While the flow around the top and bottom of the fish body

shows strong three-dimensional behavior, sectional planeprofiles along the body length show that the flow in themidplane is entirely longitudinal and has no verticalcomponents. The complexity of the flow at other depths is thesubject of continuing investigation.

The color contours of the velocity vectors are correlated tothe sign and magnitude of the vertical vorticity component ωz.The red vorticity indicates a clockwise or positive circulation,and the blue vorticity indicates a counterclockwise or negativecirculation. The green areas are irrotational. These vorticitycontours compare favorably with the DPIV experimentalresults shown in Fig. 3, with similar wake strength anddeveloping wake dynamics. The formation of a steady thrustjet is evident through the unsteady dynamics of the oscillatingcaudal fin, continuously shedding and manipulating wakevorticity of oscillating strength. Streamline trajectories are

A

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Fig. 6. Midplane depth (z=0.5H, where H is fish body maximum total depth) flow profiles for one period of the fish swimming motion viewedfrom above. In-plane velocity streamlines (black) are superimposed on a dense velocity vector grid. Velocity vectors are scaled in size byvelocity magnitude and scaled in color by vertical vorticity (ωz) contours (range −10 s−1 to 10 s−1). Red vorticity indicates clockwise rotation,blue vorticity indicates counterclockwise rotation, green regions are irrotational. The sequence is shown in A to H at intervals of T/8, where T isthe swimming cycle period.

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determined using a fourth-order Runge–Kutta spatialintegration with linear interpolation between grid points. Theabsence of small-scale turbulent fluctuations in the numericalrepresentation of the fluid dynamics obviates the need forhigher-order integration and interpolation schemes. Therectangular grid relevant length is dL=0.01 based on the bodylength L=1.

Fig. 7 illustrates the thrust jet region and the near-body flowdynamics by showing the dynamic pressure contours atmidbody depth, at regular intervals over the course of theswimming cycle. The dynamic pressure contours reveal low-pressure regions formed upstream, along the midsection regionand contraction region of the fish body length, which are thenmanipulated by the oscillation of the afterbody. Specifically,the controlled actuation of the caudal fin intercepts these low-pressure regions as they progress posteriorly in the local bodyframe of reference and pass into the region on the inside of the

maximum lateral excursion of the caudal fin. As the caudal finis swept to the other side through the low-pressure region, theseparation from the sharp trailing edge of the caudal fincontributes to the formation of a reverse Kármán vortex street.This vorticity control and production interaction mechanismcontributes to the efficient production of the thrust jetcomprising the wake of the straight-line swimming motions,as is evident from the contours of dynamic pressure ormomentum in the fluid (Fig. 7).

Turning of a giant danioFish are known to have outstanding capabilities for fast-

starting and maneuvering. Fish can turn through 180 ° on aradius considerably less than their body length, whereas man-made underwater vehicles require several body lengths toexecute a similar turn. Experimental measurements of live fishmaneuvering and fast-starting can be found in Weihs (1972),


A

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E

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Fig. 7. Midplane depth (z=0.5H, where H is fish body maximum total depth) flow profiles for one period of the fish swimming motion viewedfrom above. Dynamic pressure coefficient contours are shown (range 0 to 2). Blue regions are high pressure, and red regions are low pressure,clearly revealing the wake thrust jet. Numerical desingularization of the rolled-up wake structures allows the vortex cores bounding the thrustjet to be easily identified. The sequence is shown in A to H at intervals of T/8, where T is the swimming cycle period.


Blake (1983), Harper and Blake (1990) and Domenici andBlake (1997). A thorough kinematic analysis of unsteadyturning and maneuvering motions can be found in Videler(1993). In the present study, we employ our numerical methodto simulate transient fish-like motions, such as turning andmaneuvering, and then compare the simulation results withexperimentally observed fish turning motions. By examiningthe near-body flow and the wake produced by the turningmotions of the fish, we extend the concepts of vorticityshedding and manipulation by the tail to explain fishmaneuvering performance.

Experimental resultsThe flow patterns around a swimming giant danio

performing a 60 ° turn were captured on video. Some of thevelocity-field data were missing because of fish shadows. InFig. 8, horizontal slices of the flow are shown with the fish

viewed from above. The laser is located at the top of eachfigure. The fish of length L=8.32 cm begins the maneuvercoasting in a straight line at a speed U=1.50 L s−1, havingrecently completed another turn; then it rapidly contorts itsbody into a tight ‘C’-curve and subsequently recoils to resumestraight-line swimming at U=1.58 L s−1 in a direction 60 °towards the right as viewed by the fish from the startingdirection. A total of 25 images were taken during the turn, at0.0333 s intervals, for a total of 0.8 s. The backbone mean linepositions of the fish were obtained from 13 of these imagegraphically, at 0.0667 s intervals, by manual discretization ofthe backbone at each time into 20 equal arc length segments.The backbone locations were later used to analyze thetrajectory of the fish for numerical simulation. The fishintersected the laser plane at mid-body depth for the durationof the turn.

The DPIV results for this maneuver are shown in Fig. 8. The

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Fig. 8. Experimental velocity fields capturedusing digital particle image velocimetry in thewake of a turning giant danio. Six images areshown during the course of the maneuver atdifferent times t, detailing the formation of a ‘C’-shape through local body contortion and thegeneration of a strong pair of counter-rotatingvortices which provide the large transient forceneeded for turning.

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region immediately to the left of the fish was in its shadow, sothese data were removed. Shortly preceding the position shownin Fig. 8A, the fish executed a turn originating in the upper left-hand corner of Fig. 8A and resumed straight-line swimming.Residual vorticity from these movements remains above andto the left of the fish. In Fig. 8B,C, the fish begins to bend,moving its head towards the new swimming direction whilethe tail moves in the opposite sense, forming a characteristic‘C’-shape. The fluid motion follows that of the contortingbody, moving directly towards the fish on the concave side ofthe body and directly away from the fish on the convex side ofthe body, at the points of maximum curvature.

In Fig. 8D, we see the flow organize into two circular-likeflows, one centered at the tail and one near the head, as thetightening of the body into a ‘C’-shape nears completion. InFig. 8E, the tail has begun to move towards the right of theviewed image, shedding in the wake the previously formedcounterclockwise vortex. At this instant, the effect of boundvorticity near the head is obscured by reflection of light from thebody. The next several frames are not presented because thesebody reflections obscured a large portion of the flow. However,analysis of the video record shows that during the stroke of thetail, sweeping downwards in the viewed image as the fish bodyheads to the left, the counterclockwise vortex sheds into thewake. During the next stroke of the tail, which sweeps backupwards in the viewed image as the fish continues moving to theleft, the clockwise vortex initially associated with the frontregion of the body moves posteriorly and sheds in the wake.

The net result is shown in Fig. 8F: a strong vortex pair forms

which provides a jet directed slightly downwards and to theright. The vortices in this pair are packets of counter-rotatinglarge-scale wake vorticity, and Fig. 9 shows the vorticitycontours at the same instant. The asymmetric shapes of thecontours of the two vortices that make up the turning jet are dueto the unsteady, non-periodic body motions and velocity.Additional vorticity shown in this figure, but not labeled withcontour strength values, was created by the fish before theinitiation of the turning maneuver. The entire turning sequence,from a straight-line coasting trajectory following a previous turnto a steady swimming trajectory 60 ° to the right as viewed bythe fish from the starting direction, takes slightly more than 0.5 s.

The vortices comprising the jet have average non-dimensional circulation !e*=!e/LU=0.43, which is 42 %greater than the typical wake vortex strength for steadyswimming at similar speeds. The core radius of the jet vorticesis more than double that of those produced in straightswimming. The jet is 0.34L wide, with maximum jet velocityof uj≈U oriented approximately 60 ° to the left of the initialstarting position as seen by the fish. Interestingly, the fishvelocity after the turn is roughly the same as the jet velocity.Also, the direction of the jet is roughly in the direction requiredby horizontal momentum balance. Three-dimensional effectsare significant, and the vigorous motion of the fish somewhatdisrupts our two-dimensional planar results. After completingthe turn, the fish began to swim steadily at U=1.5 L s−1.

Numerical modelingWe impose on the fish model the body motions observed in

live fish, including the movements of the tail as well as thetrajectory of the fish. The shape and position of the fish wereonly known at 13 time steps during the maneuver; an accuratewake picture could not be produced if these 13 positions werethe sole simulation input with such a large time differentialbetween body positions. An interpolative scheme was thereforenecessary to calculate the location of the fish at intermediatetime steps, from which the fish turn could be simulated.

Fig. 10 details the coordinate systems used. The O,X,Y,Zcoordinate system is global and fixed. The o,x,y,z coordinatesystem is a coordinate system whose origin is fixed at the noseof the fish. The x-axis always passes through the tail of the fish.Thus, the flexing motion of the fish can be described in a localcoordinate system, while its trajectory can be described in theglobal coordinate system. The z coordinates of the fish bodydepth were assumed to be invariant as the plane of the lasersheet intersected the fish at midbody depth for the duration ofthe turning motion.

The 13 discretized fish backbone images were then analyzedas follows. First, the position of the nose in the globalcoordinate system was recorded as a function of time, servingas the origin of the o,x,y,z coordinate system. The angle θ thatthe o,x,y,z coordinate system made with the global coordinatesystem was also recorded as a function of time, by constructingthe x-axis through the fish tail at each time interval. Theexperimental data of fish head and tail locations during theturning maneuver can be seen in Fig. 11A.


Fig. 9. Vorticity contours in the wake of a turning fish at the instantshown in Fig. 8F, time t=0.534 s. Solid lines denote positivevorticity, broken lines negative vorticity. Contour values are asfollows: 6.7 s−1 to 16.9 s−1 and −6.7 s−1 to −20.0 s−1 in 1.03 s−1

increments. Unlabeled vorticity contours are remnants of priorturning motions and the cessation of straight-line swimming motions.

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A periodic motion was assumed for the local flappingmotion of the fish. As the fish had a fairly uncambered meanline shape at both the first time step and the last time step ofthe turn, it was assumed that the local shape of the fish meanline at the extrapolated time of 0.8667 s would be equal to theoriginal mean line shape. In this way, the local x and ycoordinates of the mean line, as well as the thicknessdistribution h, could be expanded as Fourier sine series asfunctions of time. Additionally, as the local y coordinate andthe thickness h are always zero at the nose and tail, these couldbe expanded as Fourier sine series as functions of backbonearc length, while the local x coordinate could be expanded asa linear function plus a Fourier sine series as a function ofbackbone arc length. Thus, the local behavior of the backbonecan be found at any time during the simulation.

The orientation of the local coordinate system within theglobal frame of reference is similarly assessed. Expansion ofthe global X, Y and θ coordinates in terms of Fourier series intime yields the trajectory and inclination of the local coordinatesystem within the global reference frame for any time duringthe simulation. Details of the complete analytical descriptionof the unsteady motion can be found in Appendix B. From thisanalytical description of the motion, we can produce asimulation with an arbitrary number of time steps, and thedifficulties in accurately reproducing the body motion and theresulting wake are alleviated by having a sufficient number oftime steps in the simulation. As an example, Fig. 11B showsthe backbone trajectory and flexing behavior of the giant danioover the entire simulation employing 50 intermediate timesteps of information.

The viscous effects of the body are again assumed to beconfined to a thin boundary layer near the body and in a small

wake described by shear layers. The Reynolds numbers Re ofthe giant danio before and after the turning motion areapproximately Re=10 400 and Re=11 000, respectively. Atthese values, we would expect the boundary layer to be laminaror within a transition region, as illustrated by experimental datafor the drag coefficients of axisymmetric streamlined bodieswith varying L/d ratios (where d is the diameter of thestreamlined body) found in Fig. 6.22 of Hoerner (1965). It isreasonable to assume that the near-body viscous effects on thebody are small in comparison with the effects of unsteady

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Fig. 10. Coordinate system conventions for the analysis ofmaneuvering (top view). Local unsteady body motions are describedwith respect to the body-fixed o,x,y,z coordinate system, and thebody trajectory is described with respect to an inertial referenceframe O,X,Y,Z. U, swimming speed or instantaneous translationalvelocity; Sb, body surface in the computational domain; Sw, wakesurface in the computational domain; θ and θ·, angular orientationand rotational velocity, respectively, of the local coordinate systemo,x,y,z about the z axis with respect to O,X,Y,Z.

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Fig. 11. Trajectory of a maneuvering fish. (A) Experimental data:circles and crosses indicate head and tail positions, respectively; thecurves represent a few of the body backbone positions. The timeinterval between sequential plotted head/tail positions is dt=0.0333 s.The starting position is vertical with the head pointing downwards inthe view shown. The fish contorts its body into a tight ‘C’-shape,then recoils to resume swimming to its right. (B) Simulation input:the fish mean backbone trajectory is shown at intervals of dt=0.02 s,corresponding to every other time step in the numerical simulation.Backbone lines are shown in bold every 0.1 s for clarity.

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vorticity generated by separation, such as that occurring at thesharp trailing edges of the body which generate wakes. Again,the tail is assumed to generate the largest wake, and a primaryseparation line is chosen to be the trailing edge of the tail. Thelarge dorsal and anal fins are also included in the course of ourinvestigation, to examine the influence of upstream-generatedvorticity on the flow around the tail, the overall wake structureand the unsteady body forces generated.

The body shape can be described at each time step, and theimposed normal velocities at each panel midpoint are known;hence, the boundary integral equation can be solved to find thesolution of the unsteady boundary value problem at each timestep, as described above.

Numerical results and comparisonsThe turning motion of the giant danio is simulated using the

method described above. It was observed in the DPIVexperiments that the near-tail flow dynamics are affected bythe motions of the dorsal and anal fins during the maneuver.These interactions are less apparent during straight-lineswimming, in which the dorsal and anal fin lateral motions aresmaller. As a result, the accurate modeling of the fish geometryduring the turning maneuver was deemed crucial to resolvingthe flow dynamics around the tail and around the dorsal andanal fins with their large excursions.

The simulation time step size dt=0.01 s was chosen to besmall enough to describe the details of the motion completelyand to resolve wake-body dynamics and wake self-interactions.Fourier expansions of the mean line shapes in the localcoordinate system reference frame employ 15 temporal modesand 10 spatial modes. Similar expansions of the localcoordinate system trajectories within the global coordinatesystem employ 10 temporal modes. Numericaldesingularization radii for the wake and body were chosen tobe δw=0.02 and δb=0.025, respectively, based on a body lengthof L=1.0, selected through a convergence process ensuringoptimal stability of the numerical solution.

To eliminate the influence of any starting vorticity shed bythe caudal, dorsal and anal fins during the impulsive start ofthe numerical simulation, a ramping-up time interval wasadded to the initial portion of the simulation. The initialconditions of the simulation are such that the fish mean line islocally perturbed to assume the shape of the giant danio at thestart of the turning maneuver. The global position of thegeometry is such that it translates at a constant velocity, equalto the velocity at the start of the turning maneuver. At theconclusion of the ramping-up time interval, the position,velocity and shape of the geometry are identical to those of thegiant danio at the initial frame of the experimental DPIV data.The total simulation length of 160 time steps thus modeled the0.800 s turning maneuver, with 0.800 s of initial coastingmotion before the turn was initiated to eliminate unwantedstarting vorticity dynamics.

The results of the simulation are shown in Figs 12 and 13.These are taken at 0.100 s intervals, shown from t=−0.100 sbefore the start of the turning maneuver until the turn is nearly

completed at t=0.600 s. These frames are representative of timesteps ts=70 to ts=140 at 10-time-step intervals.


A

B

C

D

E

F

G

H

Fig. 12. Midplane depth (z=0.5H, where H is fish body maximumtotal depth) flow profiles during the unsteady turning motion, viewedfrom above. In-plane velocity streamlines (black) are superimposedon a dense velocity vector grid. Velocity vectors are scaled in size byvelocity magnitude and scaled in color by vertical vorticity (ωz)contours (range −20 s−1 to 20 s−1). Red vorticity indicates clockwiserotation, blue vorticity counterclockwise rotation, green regions areirrotational. The sequence is shown in A to H at intervals dt=0.100 s,corresponding to t=−0.100 s to t=0.600 s of the original experimentalrecording.


The sequence of images shown in Fig. 12 details the two-dimensional flow patterns around the turning fish in a plane atmidbody depth on the fish. In the initial plots of the sequence,at the extreme top of the fluid plane there is residual vorticityresulting from the starting motions in the simulation. This is

caused by the translation of the giant danio geometry from thetop of the frame into its starting position. The turning maneuverproduces additional vorticity which is largely unaffected by thedecaying starting vorticity. The sequence of images shown inFig. 13 illustrates the evolution of dynamic pressure contoursduring the turn.

From both Figs 12 and 13, it can be seen that as the fishcontorts itself into a tight ‘C’-shape, the flow is organized intothree circular patterns, around the head, the midbody and thetail fin, respectively. As the fish begins to sweep its tail towardsits left side, the tail fin sheds positive vorticity, as negativebound vorticity moves from the contraction region towards thetail. A large low-pressure region coming from the right of thebody crosses over to the left, passing over the tail andeventually pairing with the previous vortex to form a strongjet, which turns the fish. The negative vorticity in thecontraction region moves posteriorly to the tail during thestroke of the tail to the left, and a new low-pressure region isformed below the fish. This negative vorticity is released intothe wake during the subsequent return stroke to the right, asthe second low-pressure region passes over the leading edge ofthe tail fin. As the tail concludes its stroke to the right andpasses through the second region of high fluid momentum, itrecovers energy from the jet through the enhanced separationfrom the trailing edge of the tail, reinforcing the negativevorticity that is being shed into the wake and resulting inadditional loading on the tail, which propels the body forward.

The dynamics of the maneuver are shown in Fig. 14. Theforces on the fish body during the turn are depicted as afunction of time, with the total force decomposed into its X andY components of the global coordinate system (Fig. 14A). Inaddition, the fluid force vectors acting on the fish center ofmass at each time step of the simulation are shown, with themean line curves of the fish backbone superimposed on the plotevery four time steps (Fig. 14B). The forces are moderatelylarge at the outset of the maneuver, as the fish halts theoscillatory motions of straight-line swimming and releasesresidual body-bound vorticity into the wake, then increasesharply as the fish contorts into a tight curve and sweeps itscaudal fin to the left. The oscillatory forces gradually subsideas the turn is completed and straight-line swimmingcommences again. It is evident from the vector time historythat the forces on the body allow it to turn in a short distancerelative to its length.

DiscussionSteady straight-line swimming

Comparison between the experimental data and thenumerical simulation results shows good quantitative andqualitative agreement. Wake vortex strength and placementcompare well, with a thrust jet width of approximately 1.3times the maximum body width. Maximum computednumerical vorticity strengths are within 20 % of experimentalvalues. Simulation results reveal that the side forces aredistributed evenly between the tail and the main body, while

A

B

C

D

E

F

G

H

Fig. 13. Midplane depth (z=0.5H, where H is fish body maximumtotal depth) flow profiles during the unsteady turning motion viewedfrom above. Dynamic pressure coefficient contours are shown (range0 to 3). Blue regions are high pressure, and red regions are lowpressure, revealing the formation of a turning-assisting jet. Thesequence is shown in A to H at intervals dt=0.100 s, corresponding tot=−0.100 s to t=0.600 s of the original experimental recording.

2320

the tail sustains over 90 % of the longitudinal force. Thedetailed patterns shown in Fig. 15 elucidate the complexprocesses of bound vorticity release and thrust wake formationin the experimental results and those of the simulation. Thenear-body and wake streamline patterns near the trailing-edgeseparation line of the caudal fin show qualitatively similarfeatures.

Fig. 16 summarizes the development and control ofvorticity by the flexible body undulation during theswimming cycle. This model simplifies the complex three-dimensional nature of the near-body flow, but accuratelyportrays the dominant features of the mid-depth plane flowand the principal mechanisms involved in manipulation of the

large-scale two-dimensional vorticity features in the wake. InFig. 16A, clockwise (negative) vorticity is shed into thewake. Along the contraction region of the body,counterclockwise (positive) vorticity is fully developed, dueto the local undulation of the body and the coordinatedupward stroke of the caudal peduncle as seen by the reader.Fig. 16B reveals positive bound vorticity moving towards the


0 0.2 0.4 0.6-30

-20

-10

0

10

20

Y forceX force

Time (s)

Forc

e (N

)

5 10 15

4

6

8

10

12

14

X (cm)

Y (c

m)

Fig. 14. Force versus time history for the turning giant daniosimulation. (A) X (solid line) and Y (broken line) components of theforce on the fish body center of mass in a global reference frame as afunction of time. (B) Force vectors acting through the center of massshown as a function of position at every time step dt=0.01 s duringthe maneuver. Body mean line locations are superimposed on thevector plot every four time-step intervals.

6

5

4

3

2

1

Y(c

m)

6 7 8 9 10 11 12X (cm)

A

B

Fig. 15. Close-up of the separated and near-body flow at themidplane depth near the tail of the giant danio during straight-lineswimming. Magnification of near-body streamlines illustrates theeffect of body-bound and free vorticity on the adjacent flow. (A) Digital particle image velocimetry experimental data showingthe in-plane velocity streamlines near the trailing edge, revealing thelarge wake vortex and the body-bound vorticity near the trailingedge. (B) Numerical simulation results for the same instantsuperimpose streamlines (black) on a dense in-plane velocity vectorgrid. Velocity vectors are scaled in size by velocity magnitude and incolor by vertical vorticity contours (ωz) (range −10 s−1 to 10 s−1). Redvorticity indicates clockwise rotation, blue vorticity indicatescounterclockwise rotation, green regions are irrotational.


tail tip during the upward stroke of the tail. New negativebound vorticity develops in the midbody region of the fishlength, due to the downward midbody motion. Fig. 16Cillustrates the shedding of the positive bound vorticity intothe wake, while negative vorticity is fully developed in thecontraction region, owing to the local body contortion andcoordinated downward caudal peduncle motion. In Fig. 16D,the cycle is complete and recommences. The alternating-signfree vortices in the wake form a reverse Kármán street,resulting in the development of a thrust jet. Optimalpropulsion is achieved through proper selection of thefrequency and spacing of the wake vortices. The process ofthe formation of wake vortices through the release of body-generated vorticity and subsequent tail manipulation asrevealed in Figs 6, 7, 15 and 16 is an important result of the

present paper, extending the analysis and conclusions ofTriantafyllou et al. (1993).

TurningFig. 17 summarizes the development and control of

vorticity by the flexible body deformation during the turningmaneuver. This model simplifies the complex three-dimensional nature of the near-body flow, but accuratelyportrays the dominant features of the mid-depth plane flowand illustrates the principal mechanisms involved inmanipulation of the large-scale two-dimensional vorticity. InFig. 17A, the fish ceases its straight-line swimmingundulations and releases body-bound vorticity associatedwith straight-line swimming motions into the wake (notshown). In Fig. 17B, the fish initiates its backbone curve. Inthe center of the body, a pair of oppositely signed boundvortices develop, owing to the midbody translation to the leftand the opposite motion of the head and tail. In Fig. 17C, thelocal contortion of the backbone into a ‘C’-shape is complete,and the pair of body-generated, oppositely signed boundvortices separate as one of the vortices moves closer to thetail. Straightening of the body in a wave-like motioncommences in Fig. 17D, which releases counterclockwisevorticity into the wake through manipulation by the caudalfin. As straightening continues, the clockwise bound vorticitytravels down the length of the body towards the tail and isreleased into the wake. Initiation of straight-line swimmingmotions completes the release of clockwise vorticity into thewake, as shown in Fig. 17E, which then pairs with thecounterclockwise vorticity to form a thrust jet in the wake.The rapid formation of this thrust jet accomplishes the turningof the fish. It is remarkable that no uncontrolled vorticityappears to be shed during the turning process.

Thus, control of bound vorticity and wake formation by thebody and the tail fin for thrust vectoring to perform turningmaneuvers is achieved through large localized backbone andtail actuation. This mechanism of vorticity control allowsfor the efficient manipulation of near-body flows andempowers the generation of large, short-durationmaneuvering forces.

ConclusionsWe have presented detailed experimental measurements of

the flow around a live, naturally swimming fish and performednumerical simulations on a fish model performing the samemotions. Using DPIV, we have clarified the principal

D

A

B

C

Fig. 16. A model of the development and release of body-bound andtail-generated vorticity during one swimming cycle of steady,straight-line swimming. A top view of the swimming fish is shown.

EA B C D

Fig. 17. A model summarizing vorticity controlmechanisms in a giant danio executing a 60 ° turn toits right. A top view of the midplane of the swimmingfish is shown.

2322

propulsive mechanisms used by the fish. Three-dimensionalunsteady flexible-body numerical simulations are corroboratedagainst the experimental observations and then used toelucidate further details of the formation of the thrust jet.Specifically, fish control body-generated vorticity throughbody flexure and active manipulation by the caudal fin. Ourresults indicate that the structure of the jet wake is obtainedthrough the regulated release of body-generated vorticity andits constructive interaction with vorticity formed and shed bythe oscillating caudal fin.

We also presented detailed results for the near-body flowdynamics around a live fish executing an unsteady, rapidturning motion. Through precise body actuation, the fishregulates the formation and controlled release of body-generated vorticity, resulting in the production of a pair ofcounter-rotating vortices and, hence, a thrust jet. Thus, the fishis able to redirect its generated thrust rapidly and achieve itsdesired trajectory.

It has long been known that fish swimming in a straight lineproduce an unsteady wake in the form of a reverse Kármánstreet. However, it has generally been assumed that the wakevortices originate from the action of the tail as a lifting surface.Our results indicate that the tail fin has a secondary role in thegeneration of wake vortices. Vorticity is generated wellupstream of the tail by the undulations of the body. Uponarriving at the tail, body-generated vorticity is favorablyaffected by the motion of the tail: it is reinforced by the tail,while vorticity shed at the trailing edge of the tail merges withand amplifies each body-generated vortex. The undulation ofthe fish body and the caudal fin motion are synchronized suchthat vorticity formation and evolution are actively controlledfor both straight-line swimming and maneuvering.

Appendix A. Unsteady panel method solution algorithmFrom the discretized version of Green’s theorem at any time

t, the body perturbation velocity potential at each panelcollocation point φb(x→,t) can be found in terms of theperturbation potentials at all of the other panels:

where i is an index of observation panel number and j is anindex of field panel number in a discrete representation ofboundary integral formulation, k is the total number of bodypanels, and where:

where Pij is the source distribution influence of panel j on paneli, Qij is the dipole distribution influence of panel j on panel iand rij is the magnitude of the distance between the collocationpoint x→i of panel i and the collocation point x→j of panel j.

A linear system of equations results for φb(x→,t), the bodyperturbation velocity potential over each panel. However, thestrengths of the most recently shed wake panels φw(x→TE,t)adjacent to the trailing edge are unknown at this point, and theunique solution of the newly shed wake strength cannot bedetermined using the above system of equations 20–23 alone.This unknown can be absorbed into the system of equationsfor φb(x→,t) by the introduction of a discrete form of the Kuttaboundary condition (equation 2):

φw(x→TE,t) =φb(x→TE,t) | top panel−φb(x→TE,t) | bottom panel , (24)

where φb(x→TE,t)|top panel and φb(x→TE,t)|bottom panel are the bodyperturbation velocity potentials on trailing-edge panels at x→TEon the top and the bottom of the lifting surface, respectively.Therefore, we can describe the perturbation velocity potentialof the new wake panels in terms of the body perturbationvelocity potential φb(x→,t). The normal velocity on each bodypanel is composed of contributions from the imposed velocityof the body V→b and from the velocities induced by the discretewake panels, each with piecewise constant perturbationvelocity potentials φw(x→,t). We can write this boundarycondition in its discrete form:

where nw is the total number of wake panels of knownstrength, and p is an index of these wake panels. We thenrewrite the linear system (equation 20) for φb(x→,t):

where the influence coefficients Qiw are integrals over the newwake panels S′w of unknown strength adjacent to the trailingedges with collocation point x→TEw:

where riw is the magnitude of the distance between thecollocation point x→i of panel i and the collocation point x→TEwof the wake panel adjacent to the trailing edge of unknownstrength, and ∂nw is the partial derivative in the direction of thewake surface normal vector.

(27)ds ,Qiw =

(28)|x→i−xx→ΤΕw| ,riw =

∂(1/riw)∂nw

⌠⌡S′w

⌠⌡

(26)Pij ,

−2πφb(x→i,t) + φb(x→j,t)Qij + φw(x→TE,t)Qiw =

∂φb(x→j,t)∂nb

"k

j=1j!i

"k

j=1

(25)· n̂b ,∂φb(x→j,t)∂nb "

nw

p=1

= V→b(x→j,t)− !→φwp(x→,t)

(22)ds ,Qij =

(23)|x→i−xx→j| ,rij =

∂(1/rij)∂nbj

⌠⌡j

⌠⌡

(21)ds ,Pij =1rij

⌠⌡j

⌠⌡

(20)Pij ,−2πφb(x→i,t) + φb(x→j,t)Qij =∂φb(x→j,t)∂nb"

k

j=1j!i

"k

j=1



The discrete forms of the Kutta condition (equation 24) andthe kinematic body boundary condition (equation 25) can thenbe employed to arrive at a simple system of linear equationsfor φb(x→,t) at any given time t, expressed as equation 4.

Appendix B. Kinematic modeling of unsteadymaneuvering

Given the backbone shape and body thickness distributionat a discrete number of time steps for arbitrary fish swimmingmotions, we can determine the details of the local backboneundulations and the global body trajectory as functions oftime using a series expansion technique. This allows thegeneration of the smooth trajectory and body shape neededfor simulation.

At each time step, the local x and y coordinates of thebackbone mean line and the body thickness distribution h canbe expanded as Fourier sine series in backbone arc length s,where tp is the total number of arc length segments:

where Lo is the original arc length of the backbone, and An, Bnand Cn are the Fourier spatial expansion coefficients of lateralmotion, longitudinal motion and unsteady body thicknessdistribution, respectively. The linear function added to the x(s)expansion is not necessary for the y(s) and h(s) expansions,which are zero-valued at each end of the backbone in the localcoordinate system. In this way, the complex local shapes of themean line are always accommodated, whereas a functionaldescription such as y=f(x) for the expansion may suffer frommulti-valued violations at discrete times.

Given these spatial expansions at each time step, we wish tolink them through an expansion in time, where tt is the totalnumber of original time steps. For each mode in the spatialexpansion of x(s), y(s) and h(s), a periodic shape in time existsif we extrapolate to one more time step where all variablesreturn to their original values. Thus, the coefficients can beexpanded as functions of time:

where Anm is a Fourier coefficient of the lateral unsteadymotion spatial and temporal expansion. The equations thatdescribe the flapping motion of the fish in the local coordinatesystem as functions of backbone arc length and time thenbecome:

where N and M are the number of Fourier sine modes in spaceand time, respectively, Bnm and Cnm are the Fouriercoefficients of the spatial and temporal expansions of thelongitudinal unsteady motion and the unsteady body thicknessdistribution, respectively, and Lm is the Fourier coefficient ofthe temporal expansion of the backbone arc length.

Given the motions of the fish in the local coordinate system,the orientation of the local coordinate system within the globalframe of reference is found. Although not periodic, the locationand inclination of the local coordinate system origin in theglobal coordinate frame are determined by a related method.The global coordinates X, Y and θ are expanded in terms of aFourier sine series with an arbitrary number P of temporalmodes. A linear term is subtracted from each expansion toforce the coordinates to a value of zero at either end of thesimulation period. Superposition of the linear term and theFourier sine series expansion yields the trajectory andinclination of the local coordinate system within the globalreference frame:

where the coefficients Cθp, CXp and CYp are found in the usualmanner using the discretized mean line global location andinclination information. Simulations of the turning motiondescribed in the present study of turning were found to resultin optimal reproduction of the experimentally observedmotions when at least 10 Fourier sine modes were employedfor both temporal and spatial expansion of the local mean line

(37),+ Cθpsinθ(t) =θ|t=0− θ|t=0−θ|t=ttpπttt

ttt

"

P

p=0

(38),+ CXpsinX(t) = X|t=0− X|t=0−X|t=ttpπttt

ttt

"

P

p=0

(39),+ CYpsinY(t) = Y|t=0− Y|t=0−Y|t=tpπttt

ttt

"

P

p=0

"N

n=1"

M

m=1

(33),Bn|t=0 +

Lo(t) +x(s,t) =

(34)y(s,t) =

nπstp

,

,

mπttt

"N

n=1

Anmsin

Bnmsin sin

(35)h(s,t) ="N

n=1

Cnmsin

An|t=0 +

Cn|t=0 +

stp

"M

m=1

"M

m=1

nπstp

mπttt

sin

nπstp

mπttt

sin

, (36)Lo(t) = LmsinLo|t=0 + "M

m=1

mπttt

"$

m=1

(32),AnmsinAn(t) = An|t=0 +mπt

tt

stp "

$

n=1

(29),BnsinLo +x(s) =

(30)y(s) =

nπstp

"$

n=1

,Ansinnπstp

(31)h(s) ="$

n=1

,Cnsinnπstp

2324

behavior and for the temporal expansion of the local coordinatesystem global trajectory.

Given the position of the mean line (backbone) at a largenumber of intermediate time steps, the full three-dimensionalbody can be described at each time step. As the backbonedeforms, the sectional planes are assumed to remain orthogonalto the backbone, and it is also assumed that the body volumeis fixed. Therefore, an additional boundary condition must beapplied to this numerical method, that the volume of the bodymust be constant. Thus, the body is scaled at each time so thatthe integral of the total normal velocity around the body mustbe zero:

List of symbolsA total lateral excursion of the tail at the caudal

peduncleAn Fourier coefficients of lateral unsteady motion

spatial expansionAnm Fourier coefficients of lateral unsteady motion

spatial and temporal expansionAR aspect ratio of three-dimensional lifting surfacea(x) amplitude of the backbone motion along the length

of the body x in the o,x,y,z coordinate systemBn Fourier coefficients of longitudinal unsteady motion

spatial expansionBnm Fourier coefficients of longitudinal unsteady motion

spatial and temporal expansionCn Fourier coefficients of unsteady body thickness

distribution spatial expansionCnm Fourier coefficients of unsteady body thickness

distribution spatial and temporal expansionCP pressure coefficientCXp Fourier coefficients of the X-coordinate temporal

expansion for unsteady body motionsCYp Fourier coefficients of the Y-coordinate temporal

expansion for unsteady body motionsCθp Fourier coefficients of the θ-coordinate temporal

expansion for unsteady body motionscp phase speed of propagating backbone wavec1 linear backbone wave amplitude coefficientc2 quadratic backbone wave amplitude coefficientd diameter of streamlined body from Hoerner (1965)dl elemental unit of length along a vortex segment for

Biot–Savart integrationdL velocity grid length spacing based on body length LdL→

tangent vector for circulation contour integrationds elemental unit of surface areadt experimental DPIV image interval or computational

time stepFb force on the thick body surfacesFb,thin force on the vortex-lattice thin body surfacesf frequency of tail oscillation (Hz)

H fish body maximum total depthh fish body thickness distribution at midbody depth

along the length of the body in o,x,y,zcoordinate system

i index of observation panel number in discreterepresentation of boundary integral formulation

j index of field panel number in discreterepresentation of boundary integral formulation

k number of body panelskw wavenumber of the backbone waveL fish body lengthLm Fourier coefficients of the temporal expansion of

the backbone arc lengthLo original arc length of the backbone in unsteady

motion Fourier expansionsLb component of total body force Fb in the lateral

direction, perpendicular to the direction ofswimming

M number of temporal modes in the Fourierexpansions for unsteady body motion localtrajectories

m index of unsteady motion temporal Fourierexpansion

N number of spatial modes in the Fourierexpansions for unsteady body motion localtrajectories

n index of unsteady motion spatial Fourierexpansion

∂n derivative in the direction of the surface outwardnormal vector

n̂b body outward normal vector∂nb derivative in the direction of the body outward

normal vectorn̂ ′ vortex-lattice thin body surface normal vector∂n′ derivative in the direction of the thin fin outward

normal vectornw number of wake panels∂nw derivative in the direction of the wake surface

normal vectorO,X,Y,Z inertial reference frameo,x,y,z local body-fixed coordinate systemP number of temporal modes in the Fourier

expansions for unsteady body motion globaltrajectories

P̃ column vector of panel normal velocities forsource-dipole potential panel distribution

Pb power transmitted to the fluid by the bodyP–b temporal mean of Pb over an integral number of

steady cyclesPij source distribution influence of panel j on panel iP1 partial column vector of panel normal velocities

for thick body panel distributionP2 partial column vector of panel normal velocities

for vortex-lattice thin fin panel distributionp(x) vertical coordinate of the mean line along length

of body in o,x,y,z coordinate system

!→φb · n̂b = 0 . (40)

Sb!



p index of wake panels for panel normal velocitycalculation

Q̃ matrix of influence coefficients for source-dipolepotential panel distribution

Qij dipole distribution influence of panel j on panel iQiw dipole distribution influence of new wake panelQ11 sub-matrix of influence coefficients for source-

dipole potential panel distribution; thick bodypanels’ influence on thick body panels

Q12 sub-matrix of influence coefficients for source-dipole potential panel distribution; vortex-latticesurface panels’ influence on thick body panels

Q21 sub-matrix of influence coefficients for vortex-lattice potential panel distribution; thick bodypanels’ influence on vortex-lattice surface panels

Q22 sub-matrix of influence coefficients for vortex-lattice potential panel distribution; vortex-latticesurface panels’ influence on vortex-lattice surfacepanels

Re Reynolds numberRo core radius of observed wake vorticesr magnitude of distance between two pointsr→ vector of magnitude r from a vortex element to the

field point ξ→

for Biot–Savart integrationrij magnitude of distance between collocation points of

panel i and panel jriw magnitude of distance between collocation points of

panel i and new wake panel wS all bounding surfaces of computational domainSb all body surfaces in computational domain, subset

of SSw all wake surfaces in computational domain, subset

of SS′w new wake panels shed adjacent to trailing edgeS∞ far-field surface at x→=∞ in computational domain,

subset of SSt Strouhal numbers backbone arc length coordinate for unsteady motion

parametizations→ tangent vector to vortex element for Biot–Savart

integrationT period of one cycle of straight-line swimming

motionsTb thrust component of total body force Fb in the

direction of swimmingT–b temporal mean of Tb over an integral number of

steady cyclesTE trailing edge of lifting surfacest timetp total number of arc length segments for unsteady

motion parametizationts time step numbertt total number of time steps of experimental data for

unsteady motion parametizationU swimming speeduj velocity of fluid in thrust jet

u→(X,Y) velocity field in the X,Y plane of the O,X,Y,Zcoordinate system for circulation contourintegration.

V→b(x→,t) velocity of the body at point x→ on Sb or a panelat time t

v→b(ξ→

) velocity induced by vortex element at a point x→with respect to O,X,Y,Z coordinate system

X longitudinal coordinate of o,x,y,z origin withrespect to O,X,Y,Z

x longitudinal coordinate along length of body ino,x,y,z coordinate system

x(z)LE longitudinal coordinate of the leading edge ofthe tail as a function of the height z in o,x,y,zcoordinate system

x(z)TE longitudinal coordinate of the trailing edge ofthe tail as a function of the height z in o,x,y,zcoordinate system

x→ three-dimensional coordinate with respect toO,X,Y,Z

x→f three-dimensional coordinate with respect toO,X,Y,Z on a vortex-lattice thin body surface

x→i three-dimensional coordinate of the collocationpoint of the observation panel with respect toO,X,Y,Z

x→j three-dimensional coordinate of the collocationpoint of the field panel with respect to O,X,Y,Z

x→TE three-dimensional coordinate with respect toO,X,Y,Z at the trailing edge of lifting surfaces

x→TEw three-dimensional coordinate with respect toO,X,Y,Z of the collocation point of the wakepanel adjacent to the trailing edge of unknownstrength.

Y lateral coordinate of o,x,y,z origin with respectto O,X,Y,Z

y(x,t) transverse motion of the fish backbone as afunction of time t and length along the body xin the o,x,y,z coordinate system

z(x) vertical coordinate along length of body in theo,x,y,z coordinate system

α angle of attack of the tailΓ circulationΓe circulation of observed wake vorticesΓe* non-dimensional circulation of observed wake

vorticesΓv wake vortex element circulation strengthΦ(x→,t) total velocity potential field at point x at time tΦb(x→,t) body perturbation velocity potential field at

point x at time tΦw(x→,t) wake perturbation velocity potential field at

point x at time tΦw(x→,t)|TE wake perturbation potential field at the trailing

edge of lifting surfacesφ phase angle between the pitch and heave motion

of the tailφb column vector of body perturbation velocity

potentials

2326

φbb partial column vector of thick bodyperturbation velocity potentials

φbf partial column vector of vortex-latticethin body perturbation velocitypotentials

φb(x→,t) body perturbation velocity potential ona panel at x→ at time t

φb(x→TE,t)|top panel body perturbation velocity potential ona trailing edge panel at x→TE on thetop of the lifting surface at time t

φb(x→TE,t)|bottom panel body perturbation velocity potential ona trailing edge panel at x→TE on thebottom of the lifting surface at time t

φw(x→,t) wake perturbation velocity potentialfield on a panel at x→ at time t

φw(x→TE,t) wake perturbation velocity potential ata panel adjacent to the trailing edgeof a lifting surface

!→

gradient operator!→· divergence operatorΔΦb(x→,t)|TE jump in body perturbation potential

field at the trailing edge of liftingsurfaces

δb body panel desingularization radiusδw wake panel desingularization radiusλ wavelength of the backbone waveη propulsive efficiencyρ density of the fluidθ angular rotation of the local coordinate

system o,x,y,z about z axis withrespect to O,X,Y,Z

θ· angular rotational velocity of the local

coordinate system o,x,y,z about z axiswith respect to O,X,Y,Z

ω circular frequency of oscillation of thetail

ωz vertical component of vorticityξ→

three-dimensional coordinate withrespect to O,X,Y,Z

The financial support of the Office of Naval Research undercontract N00014-96-1-1141 monitored by P. Purtell, T.McMullen and J. Fein, the Office of Naval Research undergrants N00014-89-J-3186 and N00014-93-1-0774, theAdvanced Research Project Agency under contracts N00014-92-1726 and N00014-94-1-0735 and the Sea Grant Programunder Grant Number NA46RG0434 is gratefullyacknowledged.

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Fish near-body flow dynamics

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