On the role of copepod antennae in the production of … · using an anatomically realistic,...

3019

INTRODUCTIONCopepods are equipped with a wide range of appendages, which aredeployed selectively for various modes of locomotion and feeding(Fig.1A). Copepods use mainly their cephalic (head) appendages togenerate a feeding current toward their mandibular palps duringsteady swimming or cruising (Koehl and Strickler, 1981; Strickler,1975; van Duren et al., 2003) but use mainly strokes of their antennaeand legs when they need to rapidly hop (jump) (Buskey et al., 2002;Fields and Yen, 1997; Strickler, 1975; van Duren and Videler, 2003).The hopping may be in response to a predator threat (Trager et al.,1994), presence of prey (Yen and Strickler, 1996), or to attract amate (van Duren et al., 1998). A typical hop starts with the power(effective) stroke followed by the return (recovery) stroke of theappendages to the initial position. The power stroke starts with thebeating of the antenna, followed by multiple metachronal beatingsof the legs while other mouth appendages stay in the retracted position(Strickler, 1975). In the return stroke all the appendages move toinitial positions synchronously (Alcaraz and Strickler, 1988; Strickler,1975; van Duren and Videler, 2003).

The hydrodynamics of a feeding copepod is significantly differentfrom that of a hopping copepod due to drastic differences in theReynolds number – the ratio of the inertial to viscous forces definedas:

where U and L are characteristic velocity and length scales,respectively, and is the kinematic viscosity of the fluid (Panton,

1996) – of the flow in each case. The flow field created by thefeeding copepod is characterized by relatively large viscous forces,and thus involves a low Reynolds number, with typical values inthe range Re1–10, based on the copepod length and swimmingspeed (Koehl and Strickler, 1981; Yen, 2000). The Reynoldsnumber during hopping, by contrast, is of the order 100–1000(Strickler, 1975; van Duren and Videler, 2003; Yen and Strickler,1996). For such values, the inertial forces begin to dominate theflow dynamics, which is in the so-called transitional regime (vanDuren and Videler, 2003; Yen, 2000). It is well known that in thelimit of Stokes flow (Rer0), entirely symmetric (reversible)kinematics – i.e. the appendage motion during the power stroke isthe mirror image of that during the return stroke – produces no netforce. Since time can be reversed in Stokes flow, the thrust producedduring the power stroke is cancelled exactly by the drag producedduring the return stroke (Panton, 1996; Purcell, 1977). Consequently,propulsion in a low Re environment is only possible if the animalemploys asymmetric strokes to break the symmetry of the Stokesregime and generate net thrust during each swimming cycle. Blake(Blake, 2001) discusses three possible symmetry-breaking effectsin the context of ciliary propulsion that can be deployed by an animalto propel itself in a viscosity-dominated environment: (1) the speedeffect, which refers to appendage motion with higher velocity duringthe power stroke than during the return stroke; (2) the orientationeffect, which refers to an appendage that changes its orientationduring the swimming cycle such that its largest frontal area isoriented perpendicular to the direction of motion during the powerstroke (to maximize thrust) and tangential to the direction of motion

Re =

LU

ν , (1)

The Journal of Experimental Biology 213, 3019-3035© 2010. Published by The Company of Biologists Ltddoi:10.1242/jeb.043588

On the role of copepod antennae in the production of hydrodynamic force duringhopping

Iman Borazjani1, Fotis Sotiropoulos1,*, Edwin Malkiel2 and Joseph Katz2

1St Anthony Falls Laboratory, University of Minnesota, 2 Third Avenue SE, Minneapolis, MN 55414, USA and 2Department ofMechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA

*Author for correspondence ([email protected])

Accepted 10 May 2010

SUMMARYWe integrate high-resolution experimental observations of a freely hopping copepod with three-dimensional numericalsimulations to investigate the role of the copepod antennae in production of hydrodynamic force during hopping. Theexperimental observations revealed a distinctive asymmetrical deformation of the antennae during the power and return strokes,which lead us to the hypothesis that the antennae are active contributors to the production of propulsive force with kinematicsselected in nature in order to maximize net thrust. To examine the validity of this hypothesis we carried out numerical experimentsusing an anatomically realistic, tethered, virtual copepod, by prescribing two sets of antenna kinematics. In the first set, eachantenna moves as a rigid, oar-like structure in a reversible manner, whereas in the second set, the antenna is made to moveasymmetrically as a deformable structure as revealed by the experiments. The computed results show that for both cases theantennae are major contributors to the net thrust force during hopping, and the results also clearly demonstrate the significanthydrodynamic benefit in terms of thrust enhancement and drag reduction derived from the biologically realistic, asymmetricantenna motion. This finding is not surprising given the low local Reynolds number environment within which the antennaoperates, and points to striking similarities between the copepod antenna motion and ciliary propulsion. Finally, the simulationsprovide the first glimpse into the complex, highly 3-D structure of copepod wakes.

Supplementary material available online at http://jeb.biologists.org/cgi/content/full/213/17/3019/DC1

Key words: copepod, hopping, hydrodynamic forces, antenna deformation, numerical simulation.

THE JOURNAL OF EXPERIMENTAL BIOLOGY

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during the recovery stroke to minimize drag; and (3) the wall effect,which refers to possible symmetry-breaking effects due to thepresence of a solid boundary in the vicinity of the animal appendages(Blake, 1974; Winet, 1973). For a copepod, all three effects couldpotentially become important. In our study, however, we onlyconsider a copepod hopping in an ambient flow extending to infinityin all directions and as such, there is no contribution from exteriorwall boundaries. Nevertheless, the copepods can benefit from thewall effect by moving the appendages near the body and hidingthem in the body’s wall boundary layer.

It has already been established in previous work that copepodappendages exhibit the speed effect during hopping or escape, i.e.the power stroke is faster than the return stroke (Alcaraz andStrickler, 1988; Lenz et al., 2004; van Duren and Videler, 2003).It has also been documented that the cephalic appendages exhibitasymmetry in the shape (orientation effect) of the power and returnstrokes during feeding (Gauld, 1966; Jiang and Osborn, 2004;Strickler, 1984). The motion of the copepod antennae duringhopping or escape, however, has received considerably less attention

in previous studies. For example, the only quantitative data availablein the literature on the antennae kinematics to date are in terms ofthe angle and the frequency of the antennae beat, while noinformation is available about possible changes in shape duringpower and return strokes (Lenz et al., 2004; Strickler, 1975; vanDuren and Videler, 2003; Yen and Strickler, 1996). Furthermore,there is a controversy in the literature regarding the role of theantennae, namely whether they are passive or active contributorsin the propulsion process. Many researchers, such as Storch (Storch,1929), have suggested that the copepod antennae motion is passiveand they just bend out of the way. However, Strickler (Strickler,1975) argues that the motion of the antennae is active since thecopepod accelerates as the antennae moves before the legs start theirpower strokes.

Conclusively resolving questions regarding the possible role ofthe antennae during hopping is very challenging experimentally,since it is very difficult if not impossible to measure forces exertedon the flow by each individual appendage. Few experimental studiesthat have succeeded in reporting force measurements during thepower and return strokes, have done so for a tethered copepod andonly reported the total hydrodynamic force exerted by the flow onthe tether. Alcaraz and Strickler (Alcaraz and Strickler, 1988), forinstance, examined the relationship of the tether force to appendagemovement during escape by measuring the spring force attached toa tethered copepod while simultaneously filming the appendagesmovement from the side. They found a thrust-type force during thepower stroke and a drag-type force during the return stroke. Morerecently, Lenz et al. (Lenz et al., 2004) also performed a similarexperimental study. They reported that the peaks in the force recordcorrespond to the power stroke of each leg. Malkiel et al. (Malkielet al., 2003) estimate the propulsive force that a feeding appendagegenerates for a free copepod while feeding based on the three-dimensional (3-D) flow around it. It is important to reiterate that inall above studies the force record was the total force produced bythe collective action of all appendages.

Forces on individual appendages can be obtained via numericalsimulations, provided, of course, that the virtual copepod thatgenerates these forces has sufficient realism for the results to bebiologically relevant. However, simulating the flow induced byanatomically realistic copepods poses a major challenge to even themost advanced numerical methods available today. This challengeis a result of the complex copepod body shape and the presence ofmultiple thin moving appendages whose detailed kinematics arecomplex and need to be prescribed from experimental observations.Consequently, only a handful of studies have been reported so farin the literature that have attempted to simulate copepod flows. Thefirst attempt to simulate copepod swimming numerically wasreported in a series of papers by Jiang et al. (Jiang et al., 1999; Jianget al., 2002a; Jiang et al., 2002b; Jiang et al., 2002c). In theirsimulations, however, the multiple swimming appendages wereneglected and their effect was collectively accounted for bydistributing body forces in the equations governing the fluid motion(the Navier–Stokes equations) at grid nodes adjacent and ventral tothe body. This work yielded important novel insights into thehydrodynamics of copepod swimming, but because of the simplifiedapproach adopted to model the copepod appendages, it could neitherquantify the hydrodynamic forces contributed by individualappendages nor illuminate the copepod wake structure. The firstsimulation of an anatomically realistic copepod with all of its majorswimming appendages was reported by Gilmanov and Sotiropoulos(Gilmanov and Sotiropoulos, 2005) who employed a sharp-interfaceCartesian method to model swimming of a tethered copepod.

I. Borazjani and others

TailAntenna

X2

X2

X3

X3

X3

X1

X1

θtail

θleg

θmax

θadd

θ

Legs

Maxilla

Maxillipeds

A

B

C

Fig.1. (A)The copepod with all the appendages modeled as rigid, andmeshed with triangular elements needed for the immersed boundarymethod. (B)The angle definitions for the tail and the legs of the copepod.(C)The angle definition for the antennae.


3021Copepod hydrodynamics

However, they carried out these simulations in order to illustratethe capabilities of their numerical method rather than study thehydrodynamics of copepod locomotion.

In this work we couple high resolution numerical simulationsand experiments to systematically investigate the hydrodynamicperformance of the copepod antennae during hopping. Morespecifically, we seek to: (1) examine whether the antenna is capableof producing net thrust; and (2) elucidate the role of the speed andorientation and/or wall effects in thrust production during theantennae motion. The approach we adopted was to construct andcompare the hydrodynamic performance during hopping of twoanatomically realistic, tethered virtual copepods: one with rigidantennae whose shape remains the same during power and returnstrokes (moving back and forth like an oar), i.e. only the speedeffect is present; and the other with deformable antennae whoseshape and speed change from the power to the recovery stroke, i.e.both speed and orientation and/or wall effects are present. The bodyshape and appendage kinematics of the former copepod model areidentical to those used by Gilmanov and Sotiropoulos (Gilmanovand Sotiropoulos, 2005). The model of Gilmanov and Sotiropoulos(Gilmanov and Sotiropoulos, 2005) is anatomically realistic andwas constructed using experimental data available at the time. Itincludes all major copepod appendages with kinematics prescribedto closely match what was reported in the literature (for details,see Gilmanov and Sotiropoulos, 2005). For the latter case, theGilmanov and Sotiropoulos (Gilmanov and Sotiropoulos, 2005)body shape and leg shape and kinematics were retained, but theantennae kinematics and shape deformation were obtained fromexperimental observations of freely hopping copepods. Theexperiments were carried out using a high resolution cinematic dualdigital holography (CDDH) technique. By comparing the forcesproduced by the rigid and flexible antennae models, we coulddiscern the importance of speed effect versus orientation and/orwall effect of a hopping copepod. By integrating the pressure andviscous forces directly on the various body parts of the copepod,we calculated the magnitude and direction of force on eachappendage, and their contributions to swimming. These resultsconclusively show that the antennae are significant contributors tohydrodynamic force during a hopping cycle. We also analyze theflow field produced by such a tethered hopping copepod, andcompare it with available experimental data. It is important to notethat our numerical simulations can be considered the virtual

equivalent of experiments with live tethered copepods, in whichthe copepods have been observed to respond with rapid strokes ofantennae and legs (similar to hopping) when a hydrodynamicdisturbance is applied (Fields and Yen, 1997; Kiørboe et al., 1999).In our case, of course, no stimulus is necessary since the appendagekinematics is prescribed.

This paper is organized as follows. First, in the materials andmethods section we describe briefly the experimental and numericalmethods and the copepod model parameters. Then, we discuss theresults of the simulations and the new kinematics for the antennaeobtained by analyzing the experimental data. In the discussionsection, we analyze the major findings from the simulations, andrelate them to previous experimental observations. At the end, wediscuss the limitations of the current work, and lay out theexperimental data needed to overcome these limitations.

MATERIALS AND METHODSExperimental techniques

The swimming behavior of the copepods (Eurytemora affinis Poppe1880) was recorded using a high-speed, dual-view, in-line, digitalholography system, illustrated in Fig.2. The principles of thistechnology, including a theoretical background and analysisprocedures are described by Malkiel et al. (Malkiel et al., 2003) andSheng et al. (Sheng et al., 2003; Sheng et al., 2006). Twoperpendicular digital holograms were recorded simultaneously inorder to maintain the same spatial resolution in all directions. Theholograms were acquired at 2000 frames per second using a pairof 1K�1K pixels, CMOS cameras at a resolution, aftermagnification, of 6.8mpixel–1. The light source was a pulsed (Q-switched), diode-pumped Nd:YLF laser, whose beam was expanded,collimated and split before illuminating the sample volume. We useda red light, 660nm, since the copepods were less sensitive in thiswavelength range, unlike green light that immediately triggered aresponse.

The region of interest was the central 7mm�7mm�7mm of glasscells whose total dimensions were 25mm�25mm�25mm and10mm�10mm�10mm. The calanoid copepods, Eurytemoraaffinis, were collected at Chesapeake Bay, USA, a couple of hoursbefore the experiments. They were brought to the laboratory, andkept in the same bay water during the experiments. We did not useany means to trigger their motion, did not add any food, and forthe sample shown in this paper, did not add seed particles.

Photron Ultima APX, 1Kx1K CMOS, 2000 fps camera with onboard RAM for storing 6144 frames = 3 s

Schneider Optics Macro Componon 50 mm lens operating at 3� in reverse

Crystalaser Nd-Ylf (660 nm) laser w/2 cm coherence length diode pumped, Q-switched , 80 mJ, 10 ns pulses at 2 KHz

7 mm�7 mm�7 mm region ofinterest in 25 mm�25 mm�25 mmand 10 mm�10 mm�50 mm cell

Fig.2. Optical setup of the dual-view, high-speed, in-linedigital holography system.


3022

The digital holograms were reconstructed numerically in severalplanes. In-house-developed software was then used to detect theplanes of focus for each organism, and for generating movies of in-focus images of the swimming copepods. Thus, the depth of eachframe was adjusted to maintain focus. Further improvements inimage quality were achieved by subtracting the time-averaged imagefrom each frame.

Numerical techniquesThe numerical method is identical to that used in our previouswork on fish-like swimming (Borazjani and Sotiropoulos, 2008;Borazjani and Sotiropoulos, 2009a), in which the complex flexiblemoving bodies are handled with the sharp-interface, immersedboundary method; the readers are referred to the above-mentionedpapers and other papers from our group (Borazjani et al., 2008;Ge and Sotiropoulos, 2007; Gilmanov and Sotiropoulos, 2005)for more details. In summary, we solve the unsteady, three-dimensional, incompressible Navier–Stokes equations in a domainthat contains the copepod with all the moving appendages usingthe hybrid Cartesian/immersed boundary (HCIB) methodologydeveloped by our group (Gilmanov and Sotiropoulos, 2005). TheHCIB method employs an unstructured, triangular mesh to trackthe position of a complex, moving immersed solid surface. Theimmersed surfaces are treated as sharp interfaces by reconstructingboundary conditions for the velocity field at grid nodes only inthe immediate vicinity of the moving boundary, the so-calledimmersed boundary (IB) nodes, by interpolating along the localnormal to the boundary. No explicit boundary conditions arerequired for the pressure field at the IB nodes because of the hybridstaggered–non-staggered mesh formulation (Gilmanov andSotiropoulos, 2005). The reconstruction method has been shownto be second-order accurate on Cartesian grids with movingimmersed boundaries (Borazjani and Sotiropoulos, 2008;Gilmanov and Sotiropoulos, 2005). The IB nodes at each time stepare identified using an efficient ray-tracing algorithm describedby Borazjani et al. (Borazjani et al., 2008).

The Navier–Stokes equations are solved on Cartesian grids usingan efficient, fractional step method (Ge and Sotiropoulos, 2007).The Poisson equation is solved with the flexible generalized minimalresidual method (FGMRES) method (Saad, 2003) and a multigridas a preconditioner using parallel computing libraries of portable,extensible toolkit for scientific computation (PETSc) (Balay et al.,2004). The numerical method has been validated extensively byGilmanov and Sotiropoulos (Gilmanov and Sotiropoulos, 2005) andBorazjani and Sotiropoulos (Borazjani and Sotiropoulos, 2008) forflows with moving boundaries, and successfully applied to studyvortex-induced vibrations (Borazjani and Sotiropoulos, 2009b),hydrodynamics of fish-like swimming (Borazjani and Sotiropoulos,2008; Borazjani and Sotiropoulos, 2009a), and the flow field arounda tethered copepod with moving appendages (Gilmanov andSotiropoulos, 2005).

Copepod model and the kinematics of the appendagesThe copepod body was modeled with all the appendages and meshedwith triangular elements needed for the sharp-interface immersedboundary method (see Fig.1A). Two sets of simulations have beenperformed to study the importance of speed vesus orientation and/orwall effect: one set with rigid and another set with deformableantennae.

For the rigid-antennae model, all appendages are considered asrigid bodies that can bend at their respective hinge. The appendagemovements used in this case are similar to those used in Gilmanov

and Sotiropoulos (Gilmanov and Sotiropoulos, 2005), which asexplained in that paper were based on experimental observationsby Jeannette Yen. The power stroke starts with the stroke of theantennae posteriorly and the flap of the tail (urosome) anteriorly,in opposite directions. This is followed by the posterior-directedreturn stroke of the urosome to its initial position and metachronalstrokes of the legs in the same direction. Next, in the return strokethe antennae and the legs move anteriorly to return to their initialposition. Note that all the legs return synchronously in the returnstroke. The position of each appendage at each time instant duringone cycle is shown in Fig.3 in terms of the angles defined in Fig.1.It can be observed from Fig.3 that the appendages move faster inthe power stroke, and return more slowly in the return stroke, i.e.they produce thrust using the speed effect.

The second set of simulations was performed by modeling allappendages, except the antennae, in exactly the same manner as inthe rigid-antennae model but incorporating the flexibility of theantennae to model its asymmetric motion, as observed in theexperiments. The approximate shape and dynamic deformation ofthe two antennae during the power and return strokes were obtainedby analyzing the experimental holographic movies, frame-by-frame,to develop the biologically-inspired model of antennae motiondescribed in detail in the Results section.

It is important to point out that in the experiments, only the motionof hopping copepods was recorded and analyzed, and no attemptwas made to obtain simultaneous measurements of the flow fieldgenerated by the hopping animals. In other words, the role of theexperiments in this work was to reveal the motion of the copepodantennae at a resolution sufficiently high to extract biologicallyrealistic kinematics. As such, and since no quantitative comparisonsbetween experiments and simulations are possible, we made noattempt to develop a virtual copepod model that replicated thespecific copepod species used in the experiments. Instead, weemployed the generic copepod anatomy described by Gilmanov andSotiropoulos (Gilmanov and Sotiropoulos, 2005).

Non-dimensional parameters for the tethered copepod modelWe considered a virtual copepod of length L that is tethered in aninitially stagnant fluid of kinematic viscosity . At t0, the virtualcopepod starts moving its appendages according to the prescribedkinematics, and the time it takes to complete both the power and


t /T

θ (d

eg)

0 0.2 0.4 0.6 0.8 1–20

0

20

40

60

80

100AntennaTailLeg 12Leg 34Leg 56Leg 78

Return strokePower stroke

Fig.3. The position of different appendages during one cycle in terms ofangles defined in Fig.1. The power stroke starts with the stroke of theantennae posteriorly and the flap of the tail (urosome) anteriorly in oppositedirections. This is followed by the posteriorly-directed return stroke of theurosome to its initial position and metachronal strokes of the legs in thesame direction. Next, in the return stroke the antennae and the legs moveanteriorly to return to their initiating position. Note that all the legs returnsynchronously in the return stroke.



return strokes once (i.e. the period of the appendage motion) is T.We calculate the total force on the tether and the force producedby each appendage for the first period, which represents a singlehop.

The flow field that results from the beating of the appendages ofsuch a virtual copepod swimmer is governed by the Reynoldsnumber of the flow equation (Eqn1). For the present tetheredcopepod model, the selection of an appropriate velocity scale U fordefining Re is not readily apparent. This is because there is noimposed far field velocity, as the initial flow is stagnant, and theflow field is generated by the beating of the copepod appendages.One obvious, albeit arbitrary, choice of a characteristic velocity scalefor the flow is an estimate of the average appendage velocity. Notethat since this velocity is transmitted to the fluid in the immediatevicinity of the copepod, a characteristic appendage velocity doesprovide an appropriate scale for the flow. Assuming that duringhopping a representative appendage covers a distance of about one-third of the body length, we can define a characteristic appendagevelocity as UL/(3T). Using this velocity scale in the definition forRe in Eqn1, we obtain:

In this work we carried out our simulations for Re300. Toappreciate what this value of the Reynolds number means in termsof dimensional quantities, we provide the following discussion.The variation of copepod size spans one order of magnitude, withL varying with species and age in the range of L0.5–5mm(Boxshall and Halsey, 2004). The appendage beating frequencyf1/T of copepods also varies greatly with age, e.g. for an adultfemale of E. rimana, f30Hz, whereas for a juvenile E. rimanaf100Hz (Yen and Strickler, 1996). Moreover, the period ofappendage movement of a single copepod can change significantlyfrom cycle to cycle, e.g. the jump duration for an adult malecopepod Acartia tonsa was found to vary between 9 and 44ms(Buskey et al., 2002). We selected values for L and T to berepresentative of corresponding values found in nature, by settingL3mm and T10ms (or f100Hz). For water with 10–6m2s–1,one obtains the aforementioned value of Re300. Given thedefinition of the Reynolds number in Eqn2, and assuming thatthe copepod size remains fixed, and that the kinematic viscosityof the fluid is constant, increasing the Reynolds number isequivalent to decreasing the appendage period, or increasing f,and vice versa. Alternatively, one can keep T constant and increaseRe by increasing the copepod size or changing the kinematicviscosity (fluid).

Computational grid and other detailsThe copepod is embedded in a background Cartesian grid, arectangular box with dimensions 6L�4L�6L. A fine uniform andisotropic mesh is used within a box with dimensions of0.2L�0.3L�L that covers the copepod body at all time. Outsidethis box, which is used to enhance numerical resolution in thevicinity of the body, the mesh is stretched along all three directionstoward the outer boundaries of the Cartesian domain via a hyperbolictangent stretching function. The far field boundary conditions areapplied at all outer boundaries. To examine the sensitivity of thecomputed solutions to mesh refinement, we carried our simulationson a coarse grid with 137�137�257 (4.7million) nodes and a finegrid with 153�201�293 (9million) nodes. The grid spacing in theinterior, uniform mesh domain that surrounds the copepod ish0.01L for the coarse grid and h0.005L for the fine grid. The

Re=

L2

3Tν . (2)

swimming period T is divided into 300 and 500 time steps for thecoarse and fine meshes simulations, respectively. The sensitivity ofthe results to mesh refinement is discussed in the Appendix A, whereit is shown that the coarse mesh is adequate for resolving all essentialfeature of the flow. The deformable antennae simulation has beenperformed on the fine mesh for Re300 as well.

Calculating the fluid forces on the body and appendagesThe forces on the copepod body and each appendage can becalculated by integrating the pressure and the viscous forces overthe respective immersed boundary surface:

where p is pressure, ij is the viscous stress tensor and nj is the normaldirection to the immersed boundary. The numerical details forcomputing the integral in Eqn3 in the context of complex immersedboundaries can be found in Borazjani (Borazjani, 2008).

A non-dimensional force coefficient along the ith direction canbe defined as follows:

The average value of the force coefficient is obtained by timeaveraging the instantaneous values over one swimming cycle (t0 isthe instant when the cycle starts):

In the present simulations the copepod main axis connecting headto tail is oriented in the X3 direction (i3; see Fig.1). Therefore, ifthe fluid force on the copepod is negative it would tend to propelit forward, i.e. the force would be of thrust type. In the oppositecase, the force would be of drag type.

The accuracy of our numerical procedure for calculating thepressure and viscous forces for moving boundary problems has beendemonstrated in Borazjani and Sotiropoulos (Borazjani andSotiropoulos, 2008). They simulated the flow induced by an axiallyvibrating cylinder and compared the results of their simulations withbenchmark computational data (Dütsch et al., 1998). Excellentagreement was reported both for the total force and its twocomponents.

Flow-field analysis and visualization toolsOur numerical simulations provide a wealth of information, thecomplete three-dimensional flow field generated by the tetheredhopping copepod in terms of instantaneous velocity componentsand pressure, which needs to be analyzed and interpreted. Tofacilitate such analysis, we employed fluid mechanics quantities thatare typically used to quantify the kinematics of a fluid element,which in general can be expressed as the superimposition of a rigidbody translation, a rigid body rotation, and a deformation. Thesethree motion components are quantified by the fluid velocity,vorticity and deformation (strain rate) fields, respectively (Kiørboeand Visser, 1999; Panton, 1996).

The vorticity vector i is defined as the curl of the velocity, whichis related to the antisymmetric part of the velocity gradient tensorij as follows:

1 = 223, 2 = 231, 3 = 212 . (7)

ci t( ) =Fi t( )

ρU 2 L2 . (4)

ci =

1

Tci (t) dt

t0

t0 +T

∫ . (5)

Fi t = − pni dA + τ ijnj dA∫∫ , (3)( )

Ωij =1

2

∂ui

∂xj

−∂uj

∂xi

⎛

⎝⎜⎞

⎠⎟ , (6)


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Fluid element deformation is quantified in terms of the strain ratetensor Sij, which is defined as the symmetric part of the velocitygradient tensor:

Velocity gradients can be perceived by the copepods through theirmechanosensory ability (Kiørboe and Visser, 1999). Both strain rateand vorticity depend, of course, on the velocity gradients, but onlythe strain rate has been shown to correlate well with the copepodresponse (Kiørboe et al., 1999). Vorticity has been regularly usedto visualize eddies in a specific plane of the flow field inexperimental studies, and will also be used for visualization purposesin this work, but there is no direct evidence that copepods respondto vorticity (Catton et al., 2007; Kiørboe et al., 1999). We use thesecond invariant of the strain rate tensor ||S|| as measure of the strainrate at a given spatial location, where || || is the Euclidean matrixnorm defined as follows (repeated indices imply summation):

||S|| √SijSij . (9)

As discussed above, a major advantage of numerical simulationsis that they can provide the complete description of the entire 3-Dflow field. We are thus able to demonstrate the complexity ofcopepod flows by visualizing, for the first time, the 3-D structureof the various vortices generated by the beating appendages – inprevious experiments vortical structures have been visualized so faronly in a two-dimensional slice through the flow field (Catton etal., 2007; van Duren et al., 1998; van Duren and Videler, 2003;Yen and Strickler, 1996). To accomplish this we visualize the 3-Dwake structure using an iso-surface of the variable q (Hunt et al.,1988). It is defined as, qG(||||2–||S||2), where S and denote thesymmetric and antisymmetic parts of the velocity gradient tensor,respectively, and || || is the Euclidean matrix norm. According toHunt et al. (Hunt et al., 1988), regions where q>0, i.e. regions wherethe rotation rate dominates the strain rate, are occupied by vorticalstructures.

RESULTSExperimentally observed antennae kinematics

By analyzing the high-resolution reconstructed holographic movies,frame-by-frame, it was observed that during the return stroke thecopepod antenna undergoes a distinctive deformation sequence,which is shown in Fig.4. First, at the end of the power stroke thecopepod is seen to fold its antennae backward, toward the posteriorof the body, such that they approach and ultimately come in contactand align with the body, as shown in Fig.4A. We can reasonablyhypothesize that this final (see Fig.4A), folded position of theantennae is hydrodynamically optimal as it reduces the drag throughthe orientation and wall (close to body) effects, and provides a morestreamlined body shape for as long as the antennae remains folded.Obviously, this is the first major difference between the real lifeantennae kinematics and the kinematics assumed in the rigid-

Sij =1

2

∂ui

∂xj

+∂uj

∂xi

⎛

⎝⎜⎞

⎠⎟ . (8)

antennae model of Sotiropoulos and Gilmanov (Sotiropoulos andGilmanov, 2005), in which the antennae do not fully wrap aroundthe body but remain aligned at an angle with the body (see Fig.1Cand Fig.3).

As the return stroke begins, the copepod is observed to startmoving the tips of the antennae parallel to the body forward (towardthe anterior of the body). Each antenna deforms into a U-shape withthe tip always remaining in contact with the body, as shown inFig.4B. When the tips of the U-shaped antenna reach the anteriorof the body, they begin to unfold and ultimately the contact of thetips with the body comes to an end (Fig.4C). Beyond this point theantenna continues to unfold and swing forward (Fig.4D) until itreaches its fully open, horizontal initial position. The overall motionof the antenna during the return stroke, as emerges from theexperimental observations, is reminiscent of the hand movement ofa human performing a breaststroke as well as the kinematics ofciliary propulsion (Blake, 2001).

We hypothesize, based on these experimental observations, thatthe copepod takes advantage of the flexibility of its antennae toreduce drag during the recovery stroke using the orientation andwall effects, which could also imply that the copepod antenna isnot a passive appendage but rather contributes to the production ofnet propulsive force. To test this hypothesis, we used theexperimental images to construct a new sequence of returnkinematics that closely mimics what is shown in Fig.4. Morespecifically, we model the return stroke as consisting of thefollowing four phases: (1) folding of the antennae and motionforward in contact with the body; (2) swing forward the foldedantennae around the vertical (X2) axis (for definition of axes seeFig.1C) through the joint between each antenna and the body; (3)continuing to swing forward and start unfolding; (4) unfolding tothe initial shape.

The above phases can be modeled mathematically by applyinga sequence of deformation and rotation (around the vertical X2 axis)operators to the initial, unfolded shape of the antenna. The detailedmathematical equations describing the kinematics that we use inthe model are given in Appendix B. Fig.5 shows the modeledantennae motion during the four phases of the return stroke of thevirtual copepod. By simulating the flow associated with such aflexible antenna and comparing it to that of the rigid antenna, theeffect of orientation and/or wall (shape asymmetry and close to bodymotion) on the resulting forces can be systematically examined andcontrasted with the speed effect.

Hydrodynamic forcesWe begin the presentation of computational results by firstdiscussing the forces produced by the rigid-antennae model relativeto those produced by the deformable-antennae model. The calculatedtime history of the total axial force coefficient c3 for the entirecopepod for both rigid- and deformable-antennae models, obtainedby integrating the viscous and pressure forces over the copepod bodyand each individual appendage, during the first cycle is shown in


Fig.4. A sequence of images showing theantennae during the return stroke.



Fig.6. This result should be examined closely along with Fig.3,which shows the timeline of the appendages movements. It is evidentfrom these figures that as the urosome and the antennae start tomove, a large negative (thrust) peak is produced. As these twoappendages continue to move, the magnitude of the thrust-type forcedecreases gradually until the antennae reach their maximum angulardisplacement (max in Fig.1) while the direction of motion of thetail alters sharply, starting to move downward at time t/T0.25. Afterthe power stroke of the antennae, the metachronal beating of thefour pairs of legs starts generating four successive jumps of negativeforce and prolonging the duration of the thrust regime until nearlythe end of the power stroke. The start of the return stroke is markedby the returning motion of the antennae and all of the legssynchronously, which is accompanied by a large positive (drag) peakin the total force. For the rigid antennae model the force asymptotestoward an almost constant value whereas in the deformable modelit varies with values lower than the rigid model for most of the time.The average force during the power stroke is largely negative, i.e.of thrust type, whereas it is positive, i.e. of drag type, during thereturn stroke. The net average force over the entire swimming cycleis negative, i.e. of thrust type, and equal to c3–0.046 for the rigidantennae and c3–0.342 for the deformable antennae. In other words,the magnitude of the average thrust force for the deformable modelis nearly seven times higher than that for the rigid model.

To quantify the contribution of individual appendages to the netforce acting on the entire copepod, we plotted (Fig.7) theinstantaneous force coefficient produced by the copepod body andeach individual appendage. The sudden and sharp jumps in the forcerecords were produced by the sudden stop–start of the appendagesmovement and will be discussed further in the discussion section.As expected because of the tethering of the copepod, the stationaryappendages (maxillia, maxillipeds) and the body of the tetheredcopepod do not create any appreciable thrust or drag. Note that ifthe copepod was swimming, the stationary appendages would haveproduced drag. A new finding that follows from Fig.7 is that thelarge total thrust force produced during the early stages of the powerstroke, when both the antennae and the tail move, is almost entirelydue to the power stroke of the antennae and not the tail. The beating

of the four pairs of legs is also seen to contribute to thrust duringthe power stroke. The highest thrust force is produced by therearmost pair of legs (marked as pair 78 in Fig.7), which is the firstof the legs to start moving according to the kinematics shown inFig.3 It is also evident from Fig.7 that the returning antennae alsocontribute most of the drag force during the return stroke for bothrigid and deformable antennae models.

To compare the force produced by the two antennae modelsmore closely, we plot in Fig.8 the time history of the forcecoefficient produced by the deformable antennae model with thecorresponding one from the rigid antennae model. The verticaldashed lines in this figure are used to identify the duration of thefour different phases used to model the antennae return stroke. Asexpected, during the power stroke both simulations produce nearlyidentical forces since the shape and the movement of the antennaeand all other appendages are identical in both cases. It is evidentfrom Fig.8 that this change in the shape of the folded, stationaryantennae yields a significant hydrodynamic benefit by reducingthe drag-type force in the flexible-antennae model during phase0.This change prevents the sudden stop of the antennae at max asit makes the deformable antennae to move and decelerate duringphase 0. The return stroke of the deformable model starts withphase1 when the antennae folds. Folding (phase1) and swingingforward the folded antennae (phase2) produce significantly lessdrag than swinging the rigid antennae in a reversible, oar-likemanner. The only period during which the deformable model

A B C D E

Fig.5. A sequence of images showing the deformation of the antennaeduring the return stroke, as modeled in the simulations. (A)Phase 0, (B)phase 1, (C) phase 2, (D) phase 3, (E) phase 4.

t/T

Tota

l for

ce c

oeffi

cien

t

0 0.2 0.4 0.6 0.8 1–3

–2

–1

0

1

2

Rigid antennaDeformable antenna

Fig.6. The total force coefficient for rigid and deformable antennae models.The negative values are thrust type and positive values are drag type(Re300).

0 0.2 0.4 0.6 0.8 1–3

–2

–1

0

1

2A Sudden stop

Sudden start

0 0.2 0.4 0.6 0.8 1–3

–2

–1

0

1

2

BodyTailAntennaLeg 12Leg 34Leg 56Leg 78MaxilliaMaxillipeds

BodyTailAntennaLeg 12Leg 34Leg 56Leg 78MaxilliaMaxillipeds

B

t/T

For

ce c

oeffi

cien

t

Fig.7. Force coefficient for each appendage of (A) the rigid and (B) thedeformable antennae model. The negative values are thrust type andpositive values are drag type (Re300).


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experiences higher drag force than the rigid model is during phase3when the antennae swing forward and unfold. In phase4, however,when the antennae just unfold and extend to their initial position,the antennae actually produce a significant thrust-type force as aresult of its motion in the posterior direction while deflectingtoward the straight, horizontal position.

Velocity field of the tethered copepodThe movement of the copepod appendages changes the fluidvelocity in the vicinity of the copepod appendages, creating flowstructures (eddies) which are advected downstream of the copepodwhere they will ultimately be dissipated by viscosity. Fig.9 showsthe instantaneous velocity vectors and contours of velocitymagnitude in the side view X2X3 midplane (top panel) and thedorsoventral view X1X3 midplane (bottom panel) of the copepod att/T1/3. This time instant was chosen as both the antennae and the4th (most posterior) pair of legs have finished their power strokes.The highest velocities are observed near the moving appendages.However, the flow velocity is higher in the dorsoventral view, whichis mainly created by the motion of the antennae. Furthermore, itcan be observed that the flow is symmetrical about the X2X3

symmetry midplane.

Strain rate field of the tethered copepodFig.10 shows an iso-surface of the Euclidean norm of the strainrate tensor (Eqn 8) for three instances during one hop for both therigid and deformable antennae models. Overall for both models, itcan be observed that large strain rate values occur near theappendages and become smaller downstream of the copepod. Thehigh values mainly occur in three different spatial locations: (1) thedorsoventral plane, in which the antennae move; (2) the verticalplane, in which the tail moves; and (3) the vertical plane and ventralpart of the copepod body, where the legs move. It can be observedin Fig.10A, that whereas the pockets of strain rate generated by theantennae stay almost in the same dorsoventral plane those generatedby the legs move at an angle relative to the body. Furthermore, thestrain field is similar for both antenna models in Fig.10A when thelegs are performing their power stroke and the antennae power strokehas been completed. The main difference between the rigid andflexible antennae is observed during the return stroke (Fig.10B).The symmetrical return of the rigid antennae obliterates the strainfield (hydrodynamic signal) in the dorsoventral plane in thedownstream whereas the deformable antennae cause the strain fieldto persist for longer times in this region. At the end of the hop(Fig.10C) there are more pockets of high strain in the downstream

of the copepod with deformable antennae relative to the rigid one.Therefore, the deformable antennae increase the strain field(hydrodynamic signal) longevity. This is consistent with the factthat the main purpose of a hop is to generate a hydrodynamic signalto become conspicuous without moving the animal too far from thelocation of the signal, i.e. a hop should generate a conspicuous wakewith the animal that generated the wake remaining in the middleof or very close to this wake.

Vorticity field of the tethered copepodFig.11 shows instantaneous, out-of-plane vorticity contours and 2-D streamlines plotted on the vertical plane of symmetry of thecopepod at several instants in time during one cycle. These resultsare for the first simulated cycle, when the copepod first startsdeploying its appendages in a stagnant fluid, in order to illustrate


0 0.2 0.4 0.6 0.8 1

–2

0

2Phase 0 1 2 3 4

t/T

For

ce c

oeffi

cien

t

Rigid antennaDeformable antenna

Fig.8. The comparison between force coefficients produced by a rigid anda deformable antennae (Re300). The vertical dashed lines show thebeginning and end of each phase for the deformable antenna.

Fig.9. Fluid velocity vectors and velocity magnitude contours on the X2X3

midplane (top) and X1X3 of the copepod with deformable antennae att/T1/3 (Re300). For clarity only every third vector has been plotted. Theevent depicted is when the antennae and the fourth pair of legs havecompleted their power strokes.



the early stage of distinct vortices created by each leg. It can beobserved that during the power stroke (Fig.11A–C) the legs createa vortex dipole with positive vorticity above the negative vorticity,whereas during the return stroke (Fig.11D–F) they generate anotherdipole but with negative vorticity above the positive vorticity thatinteracts with the vortices from the power stroke to form a triplelayer of vorticity (Fig.11E). The vortices shed in the power strokecontinue to get advected downstream even during the return stroke(Fig.11D–F), which indicates that the net momentum transfer tothe fluid is in the downstream direction. Therefore, the net forceacting on the body is in the opposite direction, i.e. the net force is

of thrust type. The tail generates vorticity that is also advecteddownstream. The vorticity generation cycle described in Fig.11, ifrepeated consecutively can interact and merge with vorticitygenerated during previous cycles. As a result, after several cycles,a well-developed thrust-wake is formed as seen in Fig.12.

In addition to the legs and the tail, the antennae also createcomplex structures that can best be viewed by plottinginstantaneous, out-of-plane vorticity contours at the dorsoventral(horizontal) plane that contains the antenna. Such contours areshown in Fig.13, which depicts the vorticity field for both therigid (left) and deformable (right) antennae models. Fig.13A

Fig.10. Iso-surface of non-dimensional strain rate Euclidian norm ||S||25 for (A) t/T0.33, (B) t/T0.66, (C) t/T1, for the rigid antennae (left) anddeformable antennae (right) (Re300).


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shows the antennae at the end of their power stroke for both cases.As seen in this figure, the backward-sweeping motion of the tipof each antenna creates an intense shear layer and sheds twoslender layers of negative and positive vorticity, respectively, thatessentially outline the path of motion of each tip. The magnitudeof vorticity generated in the horizontal plane by the antennae ismuch higher than the vorticity generated in the vertical plane bythe legs. This can be explained by the fact that each antenna hashigher tip velocity than the legs due to its longer length. Theresulting vorticity fields at the end of the antennae power strokeare similar for both the rigid and deformable models considering

the fact that in the former case the antenna does not fold aroundthe body in a streamlined manner, as it does in the deformablemodel. As one would expect, more significant differences occurduring the return stroke, when the two sets of kinematics aredrastically different. In the rigid model the antennae return inexactly the opposite manner to that moved during the power strokeand nearly wipe out the power-stroke vorticity field. For thedeformable model, however, small scale vortical structures persistin the wake. In particular, a strong vortex dipole can be seen inFig.13B, just downstream of the body, the sense of rotation ofwhich indicates thrust generation. Another distinct feature of the


Fig.11. Out-of-plane non-dimensional vorticity (1)contours with streamlines on theside view mid plane of thecopepod (Re300, deformableantennae) at different timeinstance: (A) t/T0.26, the firststoke of the legs begins by thelast pair; (B) t/T0.36, the middleof the power strokes of the legs;(C) t/T0.46, almost the end ofthe power strokes of the legs;(D) t/T0.56, the return stroke ofthe legs has started;(E) t/T0.66, towards the end ofthe return stroke of the legs;(F) t/T0.86, after the end of thereturn stroke of the legs.



deformable model vorticity field is the complex structure of theshear layers that is induced by the deforming antenna as its tipadvances forward towards the front of the body, as well as thevorticity structure near the fully extended antennae tips at the endof the return stroke.

Three-dimensional wake structure of the tethered copepodThe 2-D plots of the vorticity field cannot alone convey the richnessof the three-dimensional flow generated by hopping. To elucidatethe 3-D structure of the flow we visualize the instantaneous flowfield using the q-criterion (Hunt et al., 1988). Fig.14 shows theresulting visualization of the flow field for the flexible antennaemodel at the end of the return stroke from different view points.The movie of the wake structure can be found online as asupplementary material (Movie 1 in supplementary material).Perhaps the most striking new finding is the enormous complexityand highly 3-D structure of the hopping flow field. This complexityis due to the multiple moving appendages of copepods as opposedto a wake created say by smooth undulations of a fish (Borazjaniand Sotiropoulos, 2008; Borazjani and Sotiropoulos, 2009a). Theflow consists of multiple vortical structures, such as the distinctstreamwise vortices emanating from the antennae tips, the lateralrib-like structures oriented perpendicularly to the body, and themultiple vortex loops and U-shaped vortices in the posterior of thebody.

Note that the 2-D out-of-plane vorticity contours in Figs11–13are actually the footprints of the 3-D structures seen in Fig.14. Tolink the 3-D vortices seen in Fig.14 with the 2-D plot of out-of-plane vorticity, we superimposed the q iso-surface on the out-of-plane vorticity contours at the symmetry plane, in Fig.15. The blacklines shown in this figure mark the intersection of the q iso-surfacewith the symmetry plane. This figure clearly shows that the blobsof positive and negative vorticity on the 2-D plane are actuallyfootprints of 3-D vortex tubes on the symmetry plane. These areeither the heads of the hairpin vortices, which intersect the planeonce leaving either a positive or a negative vorticity footprint, orvortex rings that intersect the plane twice, making both a positiveand negative mark of out-of-plane vorticity, respectively. A moredetailed description of the evolution of the 3-D structure can befound in Borazjani (Borazjani, 2008).

DISCUSSIONTo the best of our knowledge, the results presented in this paperconstitute the first systematic high-resolution numerical investigationof a tethered hopping copepod. The forces produced by eachappendage have been quantified and the significance of the antennaedeformation (orientation and wall effects) during the return strokehas been clearly demonstrated. Using experimental data obtainedusing high-resolution digital holography, the deformation of theantennae during the return stroke was modeled and compared withthe rigid appendage model. We showed that the deformable antennaeincrease the average thrust force produced in one cycle several folds,compared with the rigid model, by drastically reducing the averagedrag-type force during the return stroke.

Speed versus orientation/wall effectThe experimentally observed copepod antennae motion is strikinglysimilar to ciliary motion (Blake, 2001), including features such asasymmetric strokes, fast beat during power stoke and slower beatin the return stroke, metachronal strokes, etc. The similarity inkinematics suggests similar mechanisms of thrust production, whichcould at first appear surprising given the large difference inReynolds numbers between copepod hopping and ciliary propulsion.Note, however, that a meaningful comparison of the flow regimesof beating cilia with that of a copepod antenna requires calculatingfor the latter case a local Reynolds number based on the antennadiameter instead of the entire copepod length. For example, theReynolds number based on the antenna diameter is about 6 whenthe Re based on the copepod length is 300. Although such low localReynolds number might still differ from those characterizing ciliamotion by orders of magnitude, the value is sufficiently low to placethe problem well in the viscosity dominated regime. It is well knownthat the cilia regularly use wall, orientation and speed effects toproduce thrust in the viscosity-dominated flow environment in whichthey operate (Blake, 2001) and we have now shown that thecopepods do benefit from these effects.

Our simulations clearly demonstrated the importance of theorientation and/or wall effects versus speed effect for the antennae,since the rigid antennae produce thrust only by the speed effect,whereas the flexible antennae use all of orientation, wall and speedeffects. We showed that combining the two effects by takingadvantage of the flexibility of its antennae, enables the animal toincrease the average net thrust-type force it produces per cyclerelative to the rigid antennae by nearly sevenfold. We did not tryto separate the orientation from the wall effects of the deformableantennae in this work. However, it is possible to identify the walleffect by simulating two deformable antennae moving with identicalkinematics: one that moves next to the copepod body (has walleffect) and the other that moves in the middle of the fluid (no walleffect). Such an undertaking, however, is beyond the scope of thiswork.

The relation between kinematics and hydrodynamic forceIt is well known that copepods move in burst-like motion duringhopping, i.e. they reach a high velocity (several hundred body lengthper second) during the power stroke, but their swimming speeddecreases rapidly during the return stroke, to almost zero at the endof the return stroke (Strickler, 1975; van Duren and Videler, 2003;Yen and Strickler, 1996). Although most aquatic animals producethrust in a pulsed rather than steady manner (Daniel, 1984), thetypical variation of swimming speed within a swimming cycleencountered in nature is often significantly smaller than in copepods.For example, during steady swimming, eels shown approximately

Fig.12. Out-of-plane non-dimensional vorticity (1) contours and streamlinesfor the side view mid plane of the copepod after eight consecutive hops(t/T8) showing the toroidal vortices (Re300, rigid antennae).


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a 10% fluctuation in velocity about the mean velocity U (Müller etal., 2001; Tytell et al., 2010), whereas mullets have been found toexhibit more than 20% fluctuation (Müller et al., 1997; Nauen andLauder, 2002). Previously we showed (Borazjani and Sotiropoulos,2009a) that an eel-like tethered swimmer produces a hydrodynamicforce time series that is much smoother than a mackerel-like tetheredswimmer at the same Reynolds number, i.e. higher fluctuations offorce correlates with higher fluctuations in swimming speed.Furthermore we showed (Borazjani and Sotiropoulos, 2010) that asthe Reynolds number decreases the velocity fluctuations increases.This trend is consistent with the fact that at lower Re viscous forcesbecome more effective in damping inertial forces, which areultimately responsible for smoother swimming through fluid inertialeffects. From the above discussion and given the calculated forcerecord of the tethered copepod, it is reasonable to postulate thatthere are two main reasons for the larger velocity fluctuations duringcopepod hopping as compared with other aquatic animals: (1)copepods operate at lower Re than larger aquatic animals, such asfish, and swimming speed fluctuations are enhanced in anenvironment where viscous forces become important; and (2) the

force record of the tethered copepod shows much larger fluctuationsthan other aquatic animals.

Our simulations showed that the return stroke of the antennae,even when they deform, creates a drag-type force. This explains thetypical application of multiple strokes of legs while the antennaestay in the folded position, as has been reported in severalexperimental observations (Strickler, 1975; van Duren and Videler,2003), and was also observed in the holographic movies obtainedin this work. The antennae apparently remains folded while legsexecute multiple strokes in order to delay incurring the largehydrodynamic penalty in terms of the drag force associated withthe return of the antennae.

Two-dimensional versus three-dimensional flow fieldsOur calculated flow field agrees well qualitatively with previousexperimental wake visualizations. For example, Catton et al. (Cattonet al., 2007) reported that the high values of velocity, vorticity andstrain rate occur near the body of the copepod, which is in agreementwith our findings that these quantities are indeed maximum nearthe moving appendages. The computed flow field is completely


Fig.13. Out-of-plane non-dimensional vorticity(2) contours for the top view mid plane of thecopepod during the return stroke at (A)t/T0.33, (B) t/T0.66, (C) t/T1, for the rigidantennae (left) and deformable antennae (right)(Re300).



symmetrical about the symmetry X2X3 plane. However, Catton etal. (Catton et al., 2007) reported asymmetrical flow field for thetethered copepod but symmetrical for the free swimming copepod.They mention that the asymmetry in the tethered copepod flow fieldis probably due to the effect of the tether on the cephalic appendages.We have also observed individual vortices produced by each leg,which were not observed in the experiments probably because ofthe lower resolution of their experiments relative to our simulations(Catton et al., 2007; van Duren et al., 1998; van Duren and Videler,2003). However, the large vortices that emerged in the simulationsafter several consecutive hops (Fig.12) are in good agreement withthe experimental observations (van Duren et al., 1998; van Durenand Videler, 2003). We have observed that the vortices producedafter each power and return stroke merge and interact with thepreviously shed vortices, which is in agreement with the observationsof van Duren and Videler (van Duren and Videler, 2003) that thenext jump or the next metachronal deployment of legs started beforethe effect of the previous one was fully dissipated. Finally, the shapeand the sense of rotation of the large vortical structure in the wake(Fig.12) is very much consistent with that of the toroidal vortexobserved in the visualization experiments of Yen and Strickler (Yenand Strickler, 1996).

Obtaining the 3-D flow field around a copepod is quitechallenging (Malkiel et al., 2003). That is why most of the recentexperiments used 2-D particle image velocimetry (PIV)measurements to quantify the flow field (Catton et al., 2007;Stamhuis and Videler, 1995; van Duren et al., 1998; van Duren andVideler, 2003). Our results have shown, however, that the flow fieldof a hopping copepod is very complex and highly three-dimensional(e.g. see Fig.10). Furthermore, it has already been established thatthe velocity field by itself is not sufficient to estimate hydrodynamicforces generated by swimming or flying and a pressure-like quantityis also required (Dabiri, 2005). Therefore, an important finding ofour work is that accurate quantification of the underlyinghydrodynamic forces, the energetic costs of locomotion, and theflow disturbances created by animal propulsion requires closesynergy between 3-D computational modeling and laboratoryexperiments with PIV.

Limitations of current simulations and future workIn this work there is a virtual tether that holds the copepod in placeand instead of moving through the fluid the fluid moves over itsbody. Tethered copepods have been used in the experiments todetermine the hydrodynamic forces by measuring the force on thetether (Alcaraz and Strickler, 1988; Lenz et al., 2004). Tethering isbelieved to not change the appendages kinematics of copepods(Hwang et al., 1993; van Duren and Videler, 2003) except that thefrequency of motion has been observed to decrease for tetheredcopepods relative to the free swimming ones (Lenz et al., 2004;Svetlichnyy, 1987). As discussed in van Duren and Videler (vanDuren and Videler, 2003), the tethering does not affect the flowfield that much if the copepod is foraging or hopping. In foraging,copepods create a feeding current and use their body drag andnegative buoyancy as a ‘natural tether’ (Strickler, 1982). Thehopping movement is designed to shift a bulk of water withoutdisplacing the animal very much (van Duren and Videler, 2003).However, in escape responses, during which the animal moves asfast as possible away from the danger, the tether effects on the flowstructure cannot be ignored. In fact, as shown by the recent workby Catton et al. (Catton et al., 2007), the force on the tether modifiesthe flow field irreversibly, i.e. the tether does have an effect on the3-D wake in this work. The tether also affects the hydrodynamic

forces by absorbing the forces produced by the individualappendages whereas these forces in a free swimming copepod tendto accelerate or decelerate the copepod. In the tethered model, thenon-moving appendages do not produce a significant force, similarto a stationary cylinder in a stagnant fluid does not produce anydrag. However, for a freely swimming copepod, the non-movingappendages will produce drag. Nevertheless, our conclusion

Fig.14. Different views of the vortical structures visualized by the iso-surfaces of q-criterion around the tethered copepod with deformableantennae at t/T1 (Re300). See supplementary material available online(Movie 1 in supplementary material) for a movie of the wake structure.


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regarding the importance of speed and orientation effects is notchanged by the tether, since both the hydrodynamic force producedby the rigid and deformable antennae were compared under the sameconditions (i.e. tethered and similar Re).

Another limitation of our current work is that all four pairs oflegs have been assumed to move back and forth in a reversiblemanner (symmetric stroke), retaining the same surface area duringthe return and power strokes. Such a treatment can underestimatethe forces produced by the legs since these appendages are hairy,and copepods are known to increase their surface area during thepower stroke and decrease it during the return stroke (Lenz et al.,2004). Moreover, the asymmetric motion of the legs during thepower and return strokes is evident in the movies of live copepodswe obtained for this work (Fig.16). As seen in Fig.16, when the

copepod is viewed from above, the movie frames clearly show thelegs to be visible during the power stroke, extending laterally fromthe body. Conversely, they are not visible during the return stroke,i.e. the legs are apparently being returned underneath the body ratherthan extending outwards as in the power stroke. We made no attemptto improve the leg model in this work, but this is obviously an areawhere our virtual copepod model can be greatly improved in thefuture. Note that the asymmetric stroke can have much larger impacton the forces produced by the legs than by the antennae since thelegs are smaller, i.e. they operate in lower local Re (based on theirsize) environment, and the lower the Re the higher the effect of theasymmetric stroke. This might be the reason why we found that,regardless of whether it is rigid or flexible, the antenna was thegreatest contributor to both thrust- and drag-type forces during thepower and return stroke, respectively. Nevertheless, the legs in ourmodel produced distinguishable peaks in the force record duringthe power stroke. The fourth pair of legs was the largest contributorto thrust among the all the pairs as it moved faster than the otherpairs of legs (see Fig.3) and the second largest after the antennae.

In spite of the limitation of our model in terms of leg modeling,there is evidence that the antennae are capable of producingsignificant amount of thrust. First, Strickler (Strickler, 1975), whoobserved free-swimming copepods during escape, reported theacceleration of his copepod to a velocity of 1.5cms–1 during themovement of antennae but before the leg strokes. He emphasizedthat the antennae stroke is definitely part of the power stroke andthe antennae are not just bending out of the way as suggestedpreviously by Storch (Storch, 1929). Second, in the work of Lenzet al. (Lenz et al., 2004), in which the total force on a tetheredcopepod was measured, during the early stages of the escaperesponse, when only the antennae and the tail were moving, aconsiderable amount of thrust force was recorded (more than halfof the force record peak). However, the largest peak in their forcerecord appeared when two pairs of legs plus antennae, tail and othermouthpart appendages were moving, which still does not show thatthe legs create more power than the antennae. Nevertheless, suchevidence suggests that the movement of the antennae is at least partlyactive, i.e. the antennae have musculature that can produce thrustduring the jumps. In fact, the work of Boxshall (Boxshall, 1985)shows that the copepod antennae indeed have muscle. However,since in this work the deformation was prescribed and not calculatedbased on a fluid–structure interaction, the extent to whichdeformation is produced passively by the flow or actively by themuscles cannot be determined. Gill and Crisp (Gill and Crisp, 1985)have shown that the removal of copepods antennae does not preventthe escape response. However, they did not comment on theeffectiveness of the escape without antennae. Nevertheless, theyreport that the removal of both antennae increased the number oftimes the tail alone was flicked, which could suggest that the copepodneeds to compensate for the loss of its antennae by repeateddeployment of other appendages.

Another limitation of our current work is the non-smooth motionof the appendages. As evident in Fig.7 the sudden stop of allappendages during the end of the power stroke in the current modelproduces a significant amount of drag. Similarly, the sudden startof the appendages at the beginning of the return stroke produces asignificant amount of drag. By contrast, the sudden start of theappendages at beginning of power stroke and their sudden stop atthe end of the return stroke creates a large thrust-type force.Therefore, it can be hypothesized that to achieve better performancethe copepods’ appendages should gradually decelerate and accelerateduring the end of the power stroke and the beginning of the return


Fig.16. Cinematic dual digital holography of a copepod during hopping: thelegs are visible from the top view during the power stroke of legs (A) butnot during the return stroke (B).

Fig.15. The vortical structures visualized by the iso-surfaces of q-criterionsuperimposed on the out-of-plane non-dimensional vorticity (1) contourson the midplane of the tethered copepod with deformable antenna at t/T1(Re300).



stroke, respectively, and start and stop suddenly at the beginningof the power stroke and end of return stroke, respectively. In fact,the experiments have shown that the copepods’ appendages doindeed accelerate fast and decelerate gradually at beginning and endof power stroke as seen for example in Fig.4 of Lenz et al. (Lenzet al., 2004) and Fig.3 of Alcaraz and Strickler (Alcaraz andStrickler, 1988). The appendages also accelerate gradually at thebeginning of the return stroke (Alcaraz and Strickler, 1988; Lenzet al., 2004). The results shown in Fig.7 provide clear evidence insupport of the importance of such acceleration and deceleration inappendage kinematics. Furthermore, this suggests that the kinematicswe used in our model can over-estimate the drag-type force.

In summary, the leg movements in this work were symmetric,and we have not studied the effect of asymmetric leg motion andchange in the surface area during the power and return stroke.Furthermore, in the current model all the appendages including thelegs start and stop suddenly. To overcome these limitations betterexperimental data is required. However, capturing the accelerationand deceleration of appendages is quite challenging experimentallybecause the power and return strokes take about 2–5 and 6–10ms,respectively, and with a high speed camera with 1000 frames persecond only a few frames per stroke can be captured, e.g. only 2–5frames per stroke has been captured in the recent experiments (Lenzet al., 2004; van Duren and Videler, 2003). Furthermore, multipleviews are required to obtain the 3-D asymmetric motion of the legsand other appendages. In addition, some information may still belost when the appendage is located behind the tail or the body.Repeatability is also an issue in experiments, even for the samecopepod. Therefore, obtaining detailed kinematics and otherquantities such as porosity of the hairy appendages and their effectivesurface area, which is required to build more realistic models, isnot possible with the current technology, so major experimentalbreakthroughs are required. Note that even the asymmetric motionof the antennae in this work was restricted to the 2-D horizontalplane and the antennae did not move in the vertical direction.

Supported by more detailed information from experiments, thepresent virtual copepod model can be made biologically far morerealistic, and evolve into a powerful computational tool forinvestigating the hydrodynamics of copepod swimming. As partof our future work, we intend to continue and further exploit thecoupled computational and experimental research paradigmadopted in this work in order to refine the description of theappendage kinematics in the model. With the improvedkinematics, the model will be extended to simulate self-propelledcopepod swimming to explore important issues regarding thedifferences between freely-swimming and tethered copepodhydrodynamics. Simulating self-propelled copepods requiresimplementation of a full, fluid-structure interaction algorithm inour numerical approach. Such an algorithm has already beendeveloped and successfully applied, and validated, forcardiovascular flows (Borazjani et al., 2008), and self-propelledaquatic fish-like swimming (Borazjani and Sotiropoulos, 2010).

APPENDIX AGrid sensitivity study of the numerical solutions

Fig.A1 compares the total force obtained from tethered simulationswith rigid appendages at Re300 for the coarse and the fine meshes.This figure shows that the total force obtained on the two grids arein good agreement with each other and show the same trend.Nevertheless, the total force from the fine mesh result is muchsmoother and does not exhibit the sharp oscillations and peaksobserved in the coarse mesh result. This finding is similar to our

previous work (Borazjani et al., 2008), where we showed that gridrefinement reduces spurious high frequency oscillations of the force,which are an inherent feature of the reconstruction algorithm usedin sharp-interface, immersed boundary methods. The average totalforces in the second cycle obtained from the force history shownin this figure are c3–1.1163�10–1 and –1.8213�10–1 for the fineand coarse mesh, respectively. Even though there is a significantdiscrepancy between these two predictions, it is evident from Fig.A1that this discrepancy arises from the large, spurious spikes of thecoarse-mesh force record occurring when the various appendagesbegin to move or stop suddenly. The results obtained on the twomeshes are in good qualitative and quantitative agreement duringthe rest of the cycle, which suggests that even the coarse mesh usedin this study is adequate for capturing the most important aspectsof the hop flow fields. Similar conclusions are also derived bycomparing the coarse and fine mesh instantaneous flow fields interms of vorticity contours and general flow patterns.

APPENDIX BThe mathematical model of the flexible antennae

As discussed in the Materials and methods section, the copepodbody, including the antennae is meshed with triangular elementsneeded for the sharp-interface immersed boundary method (seeFig.1C). The triangular mesh nodes are tracked by their Lagrangianposition vector r during the prescribed motion of the appendages.The flexible antennae return stroke obtained from the experimentsis modeled at any instant in time by applying a sequence ofdeformation and rotation operators to the initial (fully opened with0 shown in Fig.1C) antenna shape. Deformation and rotationfunctions are applied sequentially at any instant to the initial shapeof the antennae as follows:

r RD(rinit) , (A1)

where r and rinit are the new and initial position vectors of thetriangular mesh nodes, respectively, and D and R are thedeformation and rotation operators, respectively. The deformationoperator is defined as follows:

where, sign is the sign function, and A, b and Linit are the parametersof the model. A is the amplitude of the deformation, which hasthe form of half of a sine wave, Linit is the initial total length ofone arm of the antennae, and b is half of the wavelength of the

D(rinit ) =

x1 = bx1init

x2 = x2init

x3 = x3init − sign(x1) Asin(π x1

init / Linit )

⎧

⎨⎪⎪

⎩⎪⎪

, (A2)

1 1.2 1.4 1.6 1.8 2–3

–2

–1

0

1

2

Fine mesh

Coarse mesh

t/T

Tota

l for

ce c

oeffi

cien

t

Fig. A1. Comparison of total force coefficient on the tethered copepod withrigid antennae obtained on the fine and the coarse grids (Re300).


3034

sine wave. The swinging is achieved by applying the rotationmatrix around the X2 axis, which affects only x1 and x3 values, toswing forward the deformed shape with an angle relative to theinitial position, similar to the rigid antenna motion around thevertical (X2) axis through the joint between each antenna and thebody (Fig.1C):

The model parameter values A, b and change in time to achievethe desired shape and they are defined as follows during each phase:

Phase 0: t0<t<t1

b 1

Phase 1: t1<t<t2

A Linit / 8b

max+add .

Phase 2: t2<t<t3

b 0.4

A Linit / 8b

Phase 3: t3<t<t4

A Linit / 8b

Phase 4: t4<t<T

b 1

= 0 .

In the above equations t is the time in the cycle, max70deg,add15deg, Linit0.75, t0/T0.25, td0/T1/3, t1/T0.48, t2/T2/3,

Rθ =cosθ 0 − sinθ

0 1 0

sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

. (A3)

A =Linit / 8

t − t0t0d − t0

for t ≤ t0d

Linit / 8 for t > t0d

⎧

⎨⎪

⎩⎪

(A4)

θ =θmax + θadd

t − t0t0d − t0

for t ≤ t0d

θmax + θadd for t > t0d

⎧

⎨⎪

⎩⎪

. (A5)

b =0.4 − 1

t2 − t1(t − t1) + 1 (A6)

. θ = θmax + θadd −θmax + θadd

teR − tsR(t − tsR ) (A7)

b =0.4 − 1

t3 − t4(t − t4 ) + 1 (A8)

. θ = θmax + θadd −θmax + θadd

teR − tsR(t − tsR ) (A9)

A =Linit / 8b

t4 − T(t − T ) (A10)

tsR/T2/3, t3/T0.833, teR/T0.95 and t4/T0.967. Fig.A2 plots the and parameter b values during all the phases of the return strokeaccording to the above equations.

LIST OF SYMBOLSci ith component of the time-averaged non-dimensional force

coefficient vectorci(t) ith component of the non-dimensional force coefficient vector

at time tCDDH cinematic dual digital holographyD the deformation operatorf1/T frequency of appendages motionFGMRES flexible generalized minimal residualFi(t) ith component of the force vector at time th grid spacingHCIB hybrid Cartesian/immersed boundaryIB immersed boundaryL length of the copepodnj jth component of the normal vector to the immersed

boundaryp pressurePETSc portable, extensible toolkit for scientific computationPIV particle image velocimetryq q-criteriar Lagrangian position vectorrinit initial Lagrangian position vectorRe L2/3T Reynolds numberR rotation operatorsign sign functionSij strain rate tensort timeT period of appendages motionUL/(3T) velocity scale angular displacement of an antennaadd additional angular displacement of the deformable relative to

the rigid antennamax maximum angular displacement of the rigid antenna kinematic viscosityij viscous stress tensori ith component of the vorticity vectorij antisymmetric part of the velocity gradient tensor|| || Euclidean matrix norm


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

θb

Phase 1 Phase 3Phase 2Phase 0

t1t0

td0

td0

tRe

tRs

t2 t3 t4

t /T

Fig. A2. The values of (radians) and parameter b for the deformableantenna motion. The dashed vertical lines indicate the beginning andending of each phase. The t0, t1, t2, t3 and t4 indicate the beginning ofphase 0, 1, 2, 3 and 4, respectively. tR

s and tRe indicate the start and end of

the swinging (rotation) of the antennae, respectively.



ACKNOWLEDGEMENTSThis work was supported by NSF grant 0625976 and the MinnesotaSupercomputing Institute. The experiments were funded by NSF under grant nos.OCE-0402792 and CTS 0625571. Funding for instrumentation was provided byNSF, MRI grant CTS0079674. We are also grateful to the anonymous reviewerswhose insightful comments have helped to greatly improve this manuscript.

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