We integrate high-resolution experimental observations of a freely hopping copepod with three-dimensional numerical simulations to investigate the role of the copepod antennae in production of hydrodynamic force during hopping. The experimental 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 kinematics selected in nature in order to maximize net thrust. To examine the validity of this hypothesis we carried out numerical experiments using an anatomically realistic, tethered, virtual copepod, by prescribing two sets of antenna kinematics. In the first set, each antenna moves as a rigid, oar-like structure in a reversible manner, whereas in the second set, the antenna is made to move asymmetrically as a deformable structure as revealed by the experiments. The computed results show that for both cases the antennae are major contributors to the net thrust force during hopping, and the results also clearly demonstrate the significant hydrodynamic benefit in terms of thrust enhancement and drag reduction derived from the biologically realistic, asymmetric antenna motion. This finding is not surprising given the low local Reynolds number environment within which the antenna operates, and points to striking similarities between the copepod antenna motion and ciliary propulsion. Finally, the simulations provide the first glimpse into the complex, highly 3-D structure of copepod wakes.

Copepods are equipped with a wide range of appendages, which are deployed selectively for various modes of locomotion and feeding (Fig. 1A). Copepods use mainly their cephalic (head) appendages to generate a feeding current toward their mandibular palps during steady swimming or cruising (Koehl and Strickler, 1981; Strickler, 1975; van Duren et al., 2003) but use mainly strokes of their antennae and 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 a mate (van Duren et al., 1998). A typical hop starts with the power (effective) stroke followed by the return (recovery) stroke of the appendages to the initial position. The power stroke starts with the beating of the antenna, followed by multiple metachronal beatings of the legs while other mouth appendages stay in the retracted position (Strickler, 1975). In the return stroke all the appendages move to initial positions synchronously (Alcaraz and Strickler, 1988; Strickler, 1975; van Duren and Videler, 2003).

The hydrodynamics of a feeding copepod is significantly different from that of a hopping copepod due to drastic differences in the Reynolds number – the ratio of the inertial to viscous forces defined as:
formula
(1)
where U and L are characteristic velocity and length scales, respectively, and ν is the kinematic viscosity of the fluid (Panton, 1996) – of the flow in each case. The flow field created by the feeding copepod is characterized by relatively large viscous forces, and thus involves a low Reynolds number, with typical values in the range Re=1–10, based on the copepod length and swimming speed (Koehl and Strickler, 1981; Yen, 2000). The Reynolds number 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 the flow dynamics, which is in the so-called transitional regime (van Duren and Videler, 2003; Yen, 2000). It is well known that in the limit of Stokes flow (Re→0), entirely symmetric (reversible) kinematics – i.e. the appendage motion during the power stroke is the mirror image of that during the return stroke – produces no net force. Since time can be reversed in Stokes flow, the thrust produced during the power stroke is cancelled exactly by the drag produced during the return stroke (Panton, 1996; Purcell, 1977). Consequently, propulsion in a low Re environment is only possible if the animal employs asymmetric strokes to break the symmetry of the Stokes regime and generate net thrust during each swimming cycle. Blake (Blake, 2001) discusses three possible symmetry-breaking effects in the context of ciliary propulsion that can be deployed by an animal to propel itself in a viscosity-dominated environment: (1) the speed effect, which refers to appendage motion with higher velocity during the power stroke than during the return stroke; (2) the orientation effect, which refers to an appendage that changes its orientation during the swimming cycle such that its largest frontal area is oriented perpendicular to the direction of motion during the power stroke (to maximize thrust) and tangential to the direction of motion during the recovery stroke to minimize drag; and (3) the wall effect, which refers to possible symmetry-breaking effects due to the presence of a solid boundary in the vicinity of the animal appendages (Blake, 1974; Winet, 1973). For a copepod, all three effects could potentially become important. In our study, however, we only consider a copepod hopping in an ambient flow extending to infinity in all directions and as such, there is no contribution from exterior wall boundaries. Nevertheless, the copepods can benefit from the wall effect by moving the appendages near the body and hiding them in the body's wall boundary layer.
Fig. 1.

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

Fig. 1.

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

It has already been established in previous work that copepod appendages exhibit the speed effect during hopping or escape, i.e. the power stroke is faster than the return stroke (Alcaraz and Strickler, 1988; Lenz et al., 2004; van Duren and Videler, 2003). It has also been documented that the cephalic appendages exhibit asymmetry in the shape (orientation effect) of the power and return strokes during feeding (Gauld, 1966; Jiang and Osborn, 2004; Strickler, 1984). The motion of the copepod antennae during hopping or escape, however, has received considerably less attention in previous studies. For example, the only quantitative data available in the literature on the antennae kinematics to date are in terms of the angle and the frequency of the antennae beat, while no information is available about possible changes in shape during power and return strokes (Lenz et al., 2004; Strickler, 1975; van Duren and Videler, 2003; Yen and Strickler, 1996). Furthermore, there is a controversy in the literature regarding the role of the antennae, namely whether they are passive or active contributors in the propulsion process. Many researchers, such as Storch (Storch, 1929), have suggested that the copepod antennae motion is passive and they just bend out of the way. However, Strickler (Strickler, 1975) argues that the motion of the antennae is active since the copepod accelerates as the antennae moves before the legs start their power strokes.

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

Forces on individual appendages can be obtained via numerical simulations, provided, of course, that the virtual copepod that generates these forces has sufficient realism for the results to be biologically relevant. However, simulating the flow induced by anatomically realistic copepods poses a major challenge to even the most advanced numerical methods available today. This challenge is a result of the complex copepod body shape and the presence of multiple thin moving appendages whose detailed kinematics are complex and need to be prescribed from experimental observations. Consequently, only a handful of studies have been reported so far in the literature that have attempted to simulate copepod flows. The first attempt to simulate copepod swimming numerically was reported in a series of papers by Jiang et al. (Jiang et al., 1999; Jiang et al., 2002a; Jiang et al., 2002b; Jiang et al., 2002c). In their simulations, however, the multiple swimming appendages were neglected and their effect was collectively accounted for by distributing body forces in the equations governing the fluid motion (the Navier–Stokes equations) at grid nodes adjacent and ventral to the body. This work yielded important novel insights into the hydrodynamics of copepod swimming, but because of the simplified approach adopted to model the copepod appendages, it could neither quantify the hydrodynamic forces contributed by individual appendages nor illuminate the copepod wake structure. The first simulation of an anatomically realistic copepod with all of its major swimming appendages was reported by Gilmanov and Sotiropoulos (Gilmanov and Sotiropoulos, 2005) who employed a sharp-interface Cartesian method to model swimming of a tethered copepod. However, they carried out these simulations in order to illustrate the capabilities of their numerical method rather than study the hydrodynamics of copepod locomotion.

In this work we couple high resolution numerical simulations and experiments to systematically investigate the hydrodynamic performance of the copepod antennae during hopping. More specifically, we seek to: (1) examine whether the antenna is capable of producing net thrust; and (2) elucidate the role of the speed and orientation and/or wall effects in thrust production during the antennae motion. The approach we adopted was to construct and compare the hydrodynamic performance during hopping of two anatomically realistic, tethered virtual copepods: one with rigid antennae whose shape remains the same during power and return strokes (moving back and forth like an oar), i.e. only the speed effect is present; and the other with deformable antennae whose shape and speed change from the power to the recovery stroke, i.e. both speed and orientation and/or wall effects are present. The body shape and appendage kinematics of the former copepod model are identical to those used by Gilmanov and Sotiropoulos (Gilmanov and Sotiropoulos, 2005). The model of Gilmanov and Sotiropoulos (Gilmanov and Sotiropoulos, 2005) is anatomically realistic and was constructed using experimental data available at the time. It includes all major copepod appendages with kinematics prescribed to closely match what was reported in the literature (for details, see Gilmanov and Sotiropoulos, 2005). For the latter case, the Gilmanov and Sotiropoulos (Gilmanov and Sotiropoulos, 2005) body shape and leg shape and kinematics were retained, but the antennae kinematics and shape deformation were obtained from experimental observations of freely hopping copepods. The experiments were carried out using a high resolution cinematic dual digital holography (CDDH) technique. By comparing the forces produced by the rigid and flexible antennae models, we could discern the importance of speed effect versus orientation and/or wall effect of a hopping copepod. By integrating the pressure and viscous forces directly on the various body parts of the copepod, we calculated the magnitude and direction of force on each appendage, and their contributions to swimming. These results conclusively show that the antennae are significant contributors to hydrodynamic force during a hopping cycle. We also analyze the flow field produced by such a tethered hopping copepod, and compare it with available experimental data. It is important to note that our numerical simulations can be considered the virtual equivalent of experiments with live tethered copepods, in which the copepods have been observed to respond with rapid strokes of antennae and legs (similar to hopping) when a hydrodynamic disturbance is applied (Fields and Yen, 1997; Kiørboe et al., 1999). In our case, of course, no stimulus is necessary since the appendage kinematics is prescribed.

Fig. 2.

Optical setup of the dual-view, high-speed, in-line digital holography system.

Fig. 2.

Optical setup of the dual-view, high-speed, in-line digital holography system.

This paper is organized as follows. First, in the materials and methods section we describe briefly the experimental and numerical methods and the copepod model parameters. Then, we discuss the results of the simulations and the new kinematics for the antennae obtained by analyzing the experimental data. In the discussion section, we analyze the major findings from the simulations, and relate them to previous experimental observations. At the end, we discuss the limitations of the current work, and lay out the experimental data needed to overcome these limitations.

Experimental techniques

The swimming behavior of the copepods (Eurytemora affinis Poppe 1880) was recorded using a high-speed, dual-view, in-line, digital holography system, illustrated in Fig. 2. The principles of this technology, including a theoretical background and analysis procedures are described by Malkiel et al. (Malkiel et al., 2003) and Sheng et al. (Sheng et al., 2003; Sheng et al., 2006). Two perpendicular digital holograms were recorded simultaneously in order to maintain the same spatial resolution in all directions. The holograms were acquired at 2000 frames per second using a pair of 1 K×1 K pixels, CMOS cameras at a resolution, after magnification, of 6.8 μm pixel–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 used a red light, 660 nm, since the copepods were less sensitive in this wavelength range, unlike green light that immediately triggered a response.

The region of interest was the central 7 mm×7 mm×7 mm of glass cells whose total dimensions were 25 mm×25 mm×25 mm and 10 mm×10 mm×10 mm. The calanoid copepods, Eurytemora affinis, were collected at Chesapeake Bay, USA, a couple of hours before the experiments. They were brought to the laboratory, and kept in the same bay water during the experiments. We did not use any means to trigger their motion, did not add any food, and for the sample shown in this paper, did not add seed particles.

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

Numerical techniques

The numerical method is identical to that used in our previous work on fish-like swimming (Borazjani and Sotiropoulos, 2008; Borazjani and Sotiropoulos, 2009a), in which the complex flexible moving bodies are handled with the sharp-interface, immersed boundary method; the readers are referred to the above-mentioned papers 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 domain that contains the copepod with all the moving appendages using the hybrid Cartesian/immersed boundary (HCIB) methodology developed by our group (Gilmanov and Sotiropoulos, 2005). The HCIB method employs an unstructured, triangular mesh to track the position of a complex, moving immersed solid surface. The immersed surfaces are treated as sharp interfaces by reconstructing boundary conditions for the velocity field at grid nodes only in the immediate vicinity of the moving boundary, the so-called immersed boundary (IB) nodes, by interpolating along the local normal to the boundary. No explicit boundary conditions are required for the pressure field at the IB nodes because of the hybrid staggered–non-staggered mesh formulation (Gilmanov and Sotiropoulos, 2005). The reconstruction method has been shown to be second-order accurate on Cartesian grids with moving immersed boundaries (Borazjani and Sotiropoulos, 2008; Gilmanov and Sotiropoulos, 2005). The IB nodes at each time step are identified using an efficient ray-tracing algorithm described by Borazjani et al. (Borazjani et al., 2008).

The Navier–Stokes equations are solved on Cartesian grids using an efficient, fractional step method (Ge and Sotiropoulos, 2007). The Poisson equation is solved with the flexible generalized minimal residual method (FGMRES) method (Saad, 2003) and a multigrid as 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 by Gilmanov and Sotiropoulos (Gilmanov and Sotiropoulos, 2005) and Borazjani and Sotiropoulos (Borazjani and Sotiropoulos, 2008) for flows with moving boundaries, and successfully applied to study vortex-induced vibrations (Borazjani and Sotiropoulos, 2009b), hydrodynamics of fish-like swimming (Borazjani and Sotiropoulos, 2008; Borazjani and Sotiropoulos, 2009a), and the flow field around a tethered copepod with moving appendages (Gilmanov and Sotiropoulos, 2005).

Copepod model and the kinematics of the appendages

The copepod body was modeled with all the appendages and meshed with triangular elements needed for the sharp-interface immersed boundary method (see Fig. 1A). Two sets of simulations have been performed to study the importance of speed vesus orientation and/or wall effect: one set with rigid and another set with deformable antennae.

For the rigid-antennae model, all appendages are considered as rigid bodies that can bend at their respective hinge. The appendage movements used in this case are similar to those used in Gilmanov and Sotiropoulos (Gilmanov and Sotiropoulos, 2005), which as explained in that paper were based on experimental observations by Jeannette Yen. The power stroke starts with the stroke of the antennae posteriorly and the flap of the tail (urosome) anteriorly, in opposite directions. This is followed by the posterior-directed return stroke of the urosome to its initial position and metachronal strokes of the legs in the same direction. Next, in the return stroke the antennae and the legs move anteriorly to return to their initial position. Note that all the legs return synchronously in the return stroke. The position of each appendage at each time instant during one 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 in the 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 all appendages, except the antennae, in exactly the same manner as in the rigid-antennae model but incorporating the flexibility of the antennae to model its asymmetric motion, as observed in the experiments. The approximate shape and dynamic deformation of the two antennae during the power and return strokes were obtained by analyzing the experimental holographic movies, frame-by-frame, to develop the biologically-inspired model of antennae motion described in detail in the Results section.

It is important to point out that in the experiments, only the motion of hopping copepods was recorded and analyzed, and no attempt was made to obtain simultaneous measurements of the flow field generated by the hopping animals. In other words, the role of the experiments in this work was to reveal the motion of the copepod antennae at a resolution sufficiently high to extract biologically realistic kinematics. As such, and since no quantitative comparisons between experiments and simulations are possible, we made no attempt to develop a virtual copepod model that replicated the specific copepod species used in the experiments. Instead, we employed the generic copepod anatomy described by Gilmanov and Sotiropoulos (Gilmanov and Sotiropoulos, 2005).

Non-dimensional parameters for the tethered copepod model

We considered a virtual copepod of length L that is tethered in an initially stagnant fluid of kinematic viscosity ν. At t=0, the virtual copepod starts moving its appendages according to the prescribed kinematics, and the time it takes to complete both the power and return strokes once (i.e. the period of the appendage motion) is T. We calculate the total force on the tether and the force produced by each appendage for the first period, which represents a single hop.

Fig. 3.

The position of different appendages during one cycle in terms of angles defined in Fig. 1. The power stroke starts with the stroke of the antennae posteriorly and the flap of the tail (urosome) anteriorly in opposite directions. This is followed by the posteriorly-directed return stroke of the urosome to its initial position and metachronal strokes of the legs in the same direction. Next, in the return stroke the antennae and the legs move anteriorly to return to their initiating position. Note that all the legs return synchronously in the return stroke.

Fig. 3.

The position of different appendages during one cycle in terms of angles defined in Fig. 1. The power stroke starts with the stroke of the antennae posteriorly and the flap of the tail (urosome) anteriorly in opposite directions. This is followed by the posteriorly-directed return stroke of the urosome to its initial position and metachronal strokes of the legs in the same direction. Next, in the return stroke the antennae and the legs move anteriorly to return to their initiating position. Note that all the legs return synchronously in the return stroke.

The flow field that results from the beating of the appendages of such a virtual copepod swimmer is governed by the Reynolds number of the flow equation (Eqn 1). For the present tethered copepod model, the selection of an appropriate velocity scale U for defining Re is not readily apparent. This is because there is no imposed far field velocity, as the initial flow is stagnant, and the flow field is generated by the beating of the copepod appendages. One obvious, albeit arbitrary, choice of a characteristic velocity scale for the flow is an estimate of the average appendage velocity. Note that since this velocity is transmitted to the fluid in the immediate vicinity of the copepod, a characteristic appendage velocity does provide an appropriate scale for the flow. Assuming that during hopping a representative appendage covers a distance of about one-third of the body length, we can define a characteristic appendage velocity as U=L/(3T). Using this velocity scale in the definition for Re in Eqn 1, we obtain:
formula
(2)

In this work we carried out our simulations for Re=300. To appreciate what this value of the Reynolds number means in terms of dimensional quantities, we provide the following discussion. The variation of copepod size spans one order of magnitude, with L varying with species and age in the range of L=0.5–5 mm (Boxshall and Halsey, 2004). The appendage beating frequency f=1/T of copepods also varies greatly with age, e.g. for an adult female of E. rimana, f=30 Hz, whereas for a juvenile E. rimana f=100 Hz (Yen and Strickler, 1996). Moreover, the period of appendage movement of a single copepod can change significantly from cycle to cycle, e.g. the jump duration for an adult male copepod Acartia tonsa was found to vary between 9 and 44 ms (Buskey et al., 2002). We selected values for L and T to be representative of corresponding values found in nature, by setting L=3 mm and T=10 ms (or f=100 Hz). For water with ν=10–6 m2 s–1, one obtains the aforementioned value of Re=300. Given the definition of the Reynolds number in Eqn 2, and assuming that the copepod size remains fixed, and that the kinematic viscosity of the fluid is constant, increasing the Reynolds number is equivalent to decreasing the appendage period, or increasing f, and vice versa. Alternatively, one can keep T constant and increase Re by increasing the copepod size or changing the kinematic viscosity (fluid).

Computational grid and other details

The copepod is embedded in a background Cartesian grid, a rectangular box with dimensions 6 L×4 L×6L. A fine uniform and isotropic mesh is used within a box with dimensions of 0.2 L×0.3 L×L that covers the copepod body at all time. Outside this box, which is used to enhance numerical resolution in the vicinity of the body, the mesh is stretched along all three directions toward the outer boundaries of the Cartesian domain via a hyperbolic tangent stretching function. The far field boundary conditions are applied at all outer boundaries. To examine the sensitivity of the computed solutions to mesh refinement, we carried our simulations on a coarse grid with 137×137×257 (4.7 million) nodes and a fine grid with 153×201×293 (9 million) nodes. The grid spacing in the interior, uniform mesh domain that surrounds the copepod is h=0.01 L for the coarse grid and h=0.005 L for the fine grid. The swimming period T is divided into 300 and 500 time steps for the coarse and fine meshes simulations, respectively. The sensitivity of the results to mesh refinement is discussed in the  Appendix A, where it is shown that the coarse mesh is adequate for resolving all essential feature of the flow. The deformable antennae simulation has been performed on the fine mesh for Re=300 as well.

Calculating the fluid forces on the body and appendages

The forces on the copepod body and each appendage can be calculated by integrating the pressure and the viscous forces over the respective immersed boundary surface:
formula
(3)
where p is pressure, τij is the viscous stress tensor and nj is the normal direction to the immersed boundary. The numerical details for computing the integral in Eqn 3 in the context of complex immersed boundaries can be found in Borazjani (Borazjani, 2008).
A non-dimensional force coefficient along the ith direction can be defined as follows:
formula
(4)
The average value of the force coefficient is obtained by time averaging the instantaneous values over one swimming cycle (t0 is the instant when the cycle starts):
formula
(5)

In the present simulations the copepod main axis connecting head to tail is oriented in the X3 direction (i=3; see Fig. 1). Therefore, if the fluid force on the copepod is negative it would tend to propel it forward, i.e. the force would be of thrust type. In the opposite case, the force would be of drag type.

The accuracy of our numerical procedure for calculating the pressure and viscous forces for moving boundary problems has been demonstrated in Borazjani and Sotiropoulos (Borazjani and Sotiropoulos, 2008). They simulated the flow induced by an axially vibrating cylinder and compared the results of their simulations with benchmark computational data (Dütsch et al., 1998). Excellent agreement was reported both for the total force and its two components.

Flow-field analysis and visualization tools

Our numerical simulations provide a wealth of information, the complete three-dimensional flow field generated by the tethered hopping copepod in terms of instantaneous velocity components and pressure, which needs to be analyzed and interpreted. To facilitate such analysis, we employed fluid mechanics quantities that are typically used to quantify the kinematics of a fluid element, which in general can be expressed as the superimposition of a rigid body translation, a rigid body rotation, and a deformation. These three motion components are quantified by the fluid velocity, vorticity and deformation (strain rate) fields, respectively (Kiørboe and Visser, 1999; Panton, 1996).

The vorticity vector ωi is defined as the curl of the velocity, which is related to the antisymmetric part of the velocity gradient tensor Ωij as follows:
formula
(6)
formula
(7)
Fluid element deformation is quantified in terms of the strain rate tensor Sij, which is defined as the symmetric part of the velocity gradient tensor:
formula
(8)
Velocity gradients can be perceived by the copepods through their mechanosensory ability (Kiørboe and Visser, 1999). Both strain rate and vorticity depend, of course, on the velocity gradients, but only the strain rate has been shown to correlate well with the copepod response (Kiørboe et al., 1999). Vorticity has been regularly used to visualize eddies in a specific plane of the flow field in experimental studies, and will also be used for visualization purposes in this work, but there is no direct evidence that copepods respond to vorticity (Catton et al., 2007; Kiørboe et al., 1999). We use the second invariant of the strain rate tensor ∥S∥ as measure of the strain rate at a given spatial location, where ∥ ∥ is the Euclidean matrix norm defined as follows (repeated indices imply summation):
formula
(9)

As discussed above, a major advantage of numerical simulations is that they can provide the complete description of the entire 3-D flow field. We are thus able to demonstrate the complexity of copepod flows by visualizing, for the first time, the 3-D structure of the various vortices generated by the beating appendages – in previous experiments vortical structures have been visualized so far only in a two-dimensional slice through the flow field (Catton et al., 2007; van Duren et al., 1998; van Duren and Videler, 2003; Yen and Strickler, 1996). To accomplish this we visualize the 3-D wake structure using an iso-surface of the variable q (Hunt et al., 1988). It is defined as, q=½(∥Ω∥2–∥S2), where S and Ω denote the symmetric and antisymmetic parts of the velocity gradient tensor, respectively, and ∥ ∥ is the Euclidean matrix norm. According to Hunt et al. (Hunt et al., 1988), regions where q>0, i.e. regions where the rotation rate dominates the strain rate, are occupied by vortical structures.

Experimentally observed antennae kinematics

By analyzing the high-resolution reconstructed holographic movies, frame-by-frame, it was observed that during the return stroke the copepod antenna undergoes a distinctive deformation sequence, which is shown in Fig. 4. First, at the end of the power stroke the copepod is seen to fold its antennae backward, toward the posterior of the body, such that they approach and ultimately come in contact and align with the body, as shown in Fig. 4A. We can reasonably hypothesize that this final (see Fig. 4A), folded position of the antennae is hydrodynamically optimal as it reduces the drag through the orientation and wall (close to body) effects, and provides a more streamlined body shape for as long as the antennae remains folded. Obviously, this is the first major difference between the real life antennae kinematics and the kinematics assumed in the rigid-antennae model of Sotiropoulos and Gilmanov (Sotiropoulos and Gilmanov, 2005), in which the antennae do not fully wrap around the body but remain aligned at an angle with the body (see Fig. 1C and Fig. 3).

Fig. 4.

A sequence of images showing the antennae during the return stroke.

Fig. 4.

A sequence of images showing the antennae during the return stroke.

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

We hypothesize, based on these experimental observations, that the copepod takes advantage of the flexibility of its antennae to reduce drag during the recovery stroke using the orientation and wall effects, which could also imply that the copepod antenna is not a passive appendage but rather contributes to the production of net propulsive force. To test this hypothesis, we used the experimental images to construct a new sequence of return kinematics that closely mimics what is shown in Fig. 4. More specifically, we model the return stroke as consisting of the following four phases: (1) folding of the antennae and motion forward in contact with the body; (2) swing forward the folded antennae around the vertical (X2) axis (for definition of axes see Fig. 1C) through the joint between each antenna and the body; (3) continuing to swing forward and start unfolding; (4) unfolding to the initial shape.

The above phases can be modeled mathematically by applying a sequence of deformation and rotation (around the vertical X2 axis) operators to the initial, unfolded shape of the antenna. The detailed mathematical equations describing the kinematics that we use in the model are given in  Appendix B. Fig. 5 shows the modeled antennae motion during the four phases of the return stroke of the virtual copepod. By simulating the flow associated with such a flexible antenna and comparing it to that of the rigid antenna, the effect of orientation and/or wall (shape asymmetry and close to body motion) on the resulting forces can be systematically examined and contrasted with the speed effect.

Hydrodynamic forces

We begin the presentation of computational results by first discussing the forces produced by the rigid-antennae model relative to those produced by the deformable-antennae model. The calculated time history of the total axial force coefficient for the entire copepod for both rigid- and deformable-antennae models, obtained by integrating the viscous and pressure forces over the copepod body and each individual appendage, during the first cycle is shown in Fig. 6. This result should be examined closely along with Fig. 3, which shows the timeline of the appendages movements. It is evident from these figures that as the urosome and the antennae start to move, a large negative (thrust) peak is produced. As these two appendages continue to move, the magnitude of the thrust-type force decreases gradually until the antennae reach their maximum angular displacement (θmax in Fig. 1) while the direction of motion of the tail alters sharply, starting to move downward at time t/T=0.25. After the power stroke of the antennae, the metachronal beating of the four pairs of legs starts generating four successive jumps of negative force and prolonging the duration of the thrust regime until nearly the end of the power stroke. The start of the return stroke is marked by the returning motion of the antennae and all of the legs synchronously, which is accompanied by a large positive (drag) peak in the total force. For the rigid antennae model the force asymptotes toward an almost constant value whereas in the deformable model it 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 the return stroke. The net average force over the entire swimming cycle is negative, i.e. of thrust type, and equal to for the rigid antennae and for the deformable antennae. In other words, the magnitude of the average thrust force for the deformable model is nearly seven times higher than that for the rigid model.

Fig. 5.

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

Fig. 5.

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

To quantify the contribution of individual appendages to the net force acting on the entire copepod, we plotted (Fig. 7) the instantaneous force coefficient produced by the copepod body and each individual appendage. The sudden and sharp jumps in the force records were produced by the sudden stop–start of the appendages movement and will be discussed further in the discussion section. As expected because of the tethering of the copepod, the stationary appendages (maxillia, maxillipeds) and the body of the tethered copepod do not create any appreciable thrust or drag. Note that if the copepod was swimming, the stationary appendages would have produced drag. A new finding that follows from Fig. 7 is that the large total thrust force produced during the early stages of the power stroke, when both the antennae and the tail move, is almost entirely due 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 during the power stroke. The highest thrust force is produced by the rearmost pair of legs (marked as pair 78 in Fig. 7), which is the first of the legs to start moving according to the kinematics shown in Fig. 3 It is also evident from Fig. 7 that the returning antennae also contribute most of the drag force during the return stroke for both rigid and deformable antennae models.

Fig. 6.

The total force coefficient for rigid and deformable antennae models. The negative values are thrust type and positive values are drag type (Re=300).

Fig. 6.

The total force coefficient for rigid and deformable antennae models. The negative values are thrust type and positive values are drag type (Re=300).

To compare the force produced by the two antennae models more closely, we plot in Fig. 8 the time history of the force coefficient produced by the deformable antennae model with the corresponding one from the rigid antennae model. The vertical dashed lines in this figure are used to identify the duration of the four different phases used to model the antennae return stroke. As expected, during the power stroke both simulations produce nearly identical forces since the shape and the movement of the antennae and all other appendages are identical in both cases. It is evident from Fig. 8 that this change in the shape of the folded, stationary antennae yields a significant hydrodynamic benefit by reducing the drag-type force in the flexible-antennae model during phase 0. This change prevents the sudden stop of the antennae at θmax as it makes the deformable antennae to move and decelerate during phase 0. The return stroke of the deformable model starts with phase 1 when the antennae folds. Folding (phase 1) and swinging forward the folded antennae (phase 2) produce significantly less drag than swinging the rigid antennae in a reversible, oar-like manner. The only period during which the deformable model experiences higher drag force than the rigid model is during phase 3 when the antennae swing forward and unfold. In phase 4, however, when the antennae just unfold and extend to their initial position, the antennae actually produce a significant thrust-type force as a result of its motion in the posterior direction while deflecting toward the straight, horizontal position.

Fig. 7.

Force coefficient for each appendage of (A) the rigid and (B) the deformable antennae model. The negative values are thrust type and positive values are drag type (Re=300).

Fig. 7.

Force coefficient for each appendage of (A) the rigid and (B) the deformable antennae model. The negative values are thrust type and positive values are drag type (Re=300).

Fig. 8.

The comparison between force coefficients produced by a rigid and a deformable antennae (Re=300). The vertical dashed lines show the beginning and end of each phase for the deformable antenna.

Fig. 8.

The comparison between force coefficients produced by a rigid and a deformable antennae (Re=300). The vertical dashed lines show the beginning and end of each phase for the deformable antenna.

Velocity field of the tethered copepod

The movement of the copepod appendages changes the fluid velocity in the vicinity of the copepod appendages, creating flow structures (eddies) which are advected downstream of the copepod where they will ultimately be dissipated by viscosity. Fig. 9 shows the instantaneous velocity vectors and contours of velocity magnitude in the side view X2X3 midplane (top panel) and the dorsoventral view X1X3 midplane (bottom panel) of the copepod at t/T=1/3. This time instant was chosen as both the antennae and the 4th (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, which is mainly created by the motion of the antennae. Furthermore, it can be observed that the flow is symmetrical about the X2X3 symmetry midplane.

Strain rate field of the tethered copepod

Fig. 10 shows an iso-surface of the Euclidean norm of the strain rate tensor (Eqn 8) for three instances during one hop for both the rigid and deformable antennae models. Overall for both models, it can be observed that large strain rate values occur near the appendages and become smaller downstream of the copepod. The high values mainly occur in three different spatial locations: (1) the dorsoventral plane, in which the antennae move; (2) the vertical plane, in which the tail moves; and (3) the vertical plane and ventral part of the copepod body, where the legs move. It can be observed in Fig. 10A, that whereas the pockets of strain rate generated by the antennae stay almost in the same dorsoventral plane those generated by the legs move at an angle relative to the body. Furthermore, the strain field is similar for both antenna models in Fig. 10A when the legs are performing their power stroke and the antennae power stroke has been completed. The main difference between the rigid and flexible antennae is observed during the return stroke (Fig. 10B). The symmetrical return of the rigid antennae obliterates the strain field (hydrodynamic signal) in the dorsoventral plane in the downstream whereas the deformable antennae cause the strain field to 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 fact that the main purpose of a hop is to generate a hydrodynamic signal to become conspicuous without moving the animal too far from the location of the signal, i.e. a hop should generate a conspicuous wake with the animal that generated the wake remaining in the middle of or very close to this wake.

Fig. 9.

Fluid velocity vectors and velocity magnitude contours on the X2X3 midplane (top) and X1X3 of the copepod with deformable antennae at t/T=1/3 (Re=300). For clarity only every third vector has been plotted. The event depicted is when the antennae and the fourth pair of legs have completed their power strokes.

Fig. 9.

Fluid velocity vectors and velocity magnitude contours on the X2X3 midplane (top) and X1X3 of the copepod with deformable antennae at t/T=1/3 (Re=300). For clarity only every third vector has been plotted. The event depicted is when the antennae and the fourth pair of legs have completed their power strokes.

Vorticity field of the tethered copepod

Fig. 11 shows instantaneous, out-of-plane vorticity contours and 2-D streamlines plotted on the vertical plane of symmetry of the copepod at several instants in time during one cycle. These results are for the first simulated cycle, when the copepod first starts deploying its appendages in a stagnant fluid, in order to illustrate the early stage of distinct vortices created by each leg. It can be observed that during the power stroke (Fig. 11A–C) the legs create a vortex dipole with positive vorticity above the negative vorticity, whereas during the return stroke (Fig. 11D–F) they generate another dipole but with negative vorticity above the positive vorticity that interacts with the vortices from the power stroke to form a triple layer of vorticity (Fig. 11E). The vortices shed in the power stroke continue to get advected downstream even during the return stroke (Fig. 11D–F), which indicates that the net momentum transfer to the fluid is in the downstream direction. Therefore, the net force acting on the body is in the opposite direction, i.e. the net force is of thrust type. The tail generates vorticity that is also advected downstream. The vorticity generation cycle described in Fig. 11, if repeated consecutively can interact and merge with vorticity generated during previous cycles. As a result, after several cycles, a well-developed thrust-wake is formed as seen in Fig. 12.

Fig. 10.

Iso-surface of non-dimensional strain rate Euclidian norm ∥S∥=25 for (A) t/T=0.33, (B) t/T=0.66, (C) t/T=1, for the rigid antennae (left) and deformable antennae (right) (Re=300).

Fig. 10.

Iso-surface of non-dimensional strain rate Euclidian norm ∥S∥=25 for (A) t/T=0.33, (B) t/T=0.66, (C) t/T=1, for the rigid antennae (left) and deformable antennae (right) (Re=300).

In addition to the legs and the tail, the antennae also create complex structures that can best be viewed by plotting instantaneous, out-of-plane vorticity contours at the dorsoventral (horizontal) plane that contains the antenna. Such contours are shown in Fig. 13, which depicts the vorticity field for both the rigid (left) and deformable (right) antennae models. Fig. 13Ashows the antennae at the end of their power stroke for both cases. As seen in this figure, the backward-sweeping motion of the tip of each antenna creates an intense shear layer and sheds two slender layers of negative and positive vorticity, respectively, that essentially outline the path of motion of each tip. The magnitude of vorticity generated in the horizontal plane by the antennae is much higher than the vorticity generated in the vertical plane by the legs. This can be explained by the fact that each antenna has higher tip velocity than the legs due to its longer length. The resulting vorticity fields at the end of the antennae power stroke are similar for both the rigid and deformable models considering the fact that in the former case the antenna does not fold around the body in a streamlined manner, as it does in the deformable model. As one would expect, more significant differences occur during the return stroke, when the two sets of kinematics are drastically different. In the rigid model the antennae return in exactly the opposite manner to that moved during the power stroke and nearly wipe out the power-stroke vorticity field. For the deformable model, however, small scale vortical structures persist in the wake. In particular, a strong vortex dipole can be seen in Fig. 13B, just downstream of the body, the sense of rotation of which indicates thrust generation. Another distinct feature of the deformable model vorticity field is the complex structure of the shear layers that is induced by the deforming antenna as its tip advances forward towards the front of the body, as well as the vorticity structure near the fully extended antennae tips at the end of the return stroke.

Fig. 11.

Out-of-plane non-dimensional vorticity (ω1) contours with streamlines on the side view mid plane of the copepod (Re=300, deformable antennae) at different time instance: (A) t/T=0.26, the first stoke of the legs begins by the last pair; (B) t/T=0.36, the middle of the power strokes of the legs; (C) t/T=0.46, almost the end of the power strokes of the legs; (D) t/T=0.56, the return stroke of the legs has started; (E) t/T=0.66, towards the end of the return stroke of the legs; (F) t/T=0.86, after the end of the return stroke of the legs.

Fig. 11.

Out-of-plane non-dimensional vorticity (ω1) contours with streamlines on the side view mid plane of the copepod (Re=300, deformable antennae) at different time instance: (A) t/T=0.26, the first stoke of the legs begins by the last pair; (B) t/T=0.36, the middle of the power strokes of the legs; (C) t/T=0.46, almost the end of the power strokes of the legs; (D) t/T=0.56, the return stroke of the legs has started; (E) t/T=0.66, towards the end of the return stroke of the legs; (F) t/T=0.86, after the end of the return stroke of the legs.

Fig. 12.

Out-of-plane non-dimensional vorticity (ω1) contours and streamlines for the side view mid plane of the copepod after eight consecutive hops (t/T=8) showing the toroidal vortices (Re=300, rigid antennae).

Fig. 12.

Out-of-plane non-dimensional vorticity (ω1) contours and streamlines for the side view mid plane of the copepod after eight consecutive hops (t/T=8) showing the toroidal vortices (Re=300, rigid antennae).

Three-dimensional wake structure of the tethered copepod

The 2-D plots of the vorticity field cannot alone convey the richness of the three-dimensional flow generated by hopping. To elucidate the 3-D structure of the flow we visualize the instantaneous flow field using the q-criterion (Hunt et al., 1988). Fig. 14 shows the resulting visualization of the flow field for the flexible antennae model at the end of the return stroke from different view points. The movie of the wake structure can be found online as a supplementary material (Movie 1 in supplementary material). Perhaps the most striking new finding is the enormous complexity and highly 3-D structure of the hopping flow field. This complexity is due to the multiple moving appendages of copepods as opposed to a wake created say by smooth undulations of a fish (Borazjani and Sotiropoulos, 2008; Borazjani and Sotiropoulos, 2009a). The flow consists of multiple vortical structures, such as the distinct streamwise vortices emanating from the antennae tips, the lateral rib-like structures oriented perpendicularly to the body, and the multiple vortex loops and U-shaped vortices in the posterior of the body.

Note that the 2-D out-of-plane vorticity contours in Figs 11, 12, 13 are actually the footprints of the 3-D structures seen in Fig. 14. To link 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 black lines shown in this figure mark the intersection of the q iso-surface with the symmetry plane. This figure clearly shows that the blobs of positive and negative vorticity on the 2-D plane are actually footprints of 3-D vortex tubes on the symmetry plane. These are either the heads of the hairpin vortices, which intersect the plane once leaving either a positive or a negative vorticity footprint, or vortex rings that intersect the plane twice, making both a positive and negative mark of out-of-plane vorticity, respectively. A more detailed description of the evolution of the 3-D structure can be found in Borazjani (Borazjani, 2008).

To the best of our knowledge, the results presented in this paper constitute the first systematic high-resolution numerical investigation of a tethered hopping copepod. The forces produced by each appendage have been quantified and the significance of the antennae deformation (orientation and wall effects) during the return stroke has been clearly demonstrated. Using experimental data obtained using high-resolution digital holography, the deformation of the antennae during the return stroke was modeled and compared with the rigid appendage model. We showed that the deformable antennae increase the average thrust force produced in one cycle several folds, compared with the rigid model, by drastically reducing the average drag-type force during the return stroke.

Speed versus orientation/wall effect

The experimentally observed copepod antennae motion is strikingly similar to ciliary motion (Blake, 2001), including features such as asymmetric strokes, fast beat during power stoke and slower beat in the return stroke, metachronal strokes, etc. The similarity in kinematics suggests similar mechanisms of thrust production, which could at first appear surprising given the large difference in Reynolds numbers between copepod hopping and ciliary propulsion. Note, however, that a meaningful comparison of the flow regimes of beating cilia with that of a copepod antenna requires calculating for the latter case a local Reynolds number based on the antenna diameter instead of the entire copepod length. For example, the Reynolds number based on the antenna diameter is about 6 when the Re based on the copepod length is 300. Although such low local Reynolds number might still differ from those characterizing cilia motion by orders of magnitude, the value is sufficiently low to place the problem well in the viscosity dominated regime. It is well known that the cilia regularly use wall, orientation and speed effects to produce thrust in the viscosity-dominated flow environment in which they operate (Blake, 2001) and we have now shown that the copepods do benefit from these effects.

Our simulations clearly demonstrated the importance of the orientation 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 speed effects. We showed that combining the two effects by taking advantage of the flexibility of its antennae, enables the animal to increase the average net thrust-type force it produces per cycle relative to the rigid antennae by nearly sevenfold. We did not try to separate the orientation from the wall effects of the deformable antennae in this work. However, it is possible to identify the wall effect by simulating two deformable antennae moving with identical kinematics: one that moves next to the copepod body (has wall effect) and the other that moves in the middle of the fluid (no wall effect). Such an undertaking, however, is beyond the scope of this work.

The relation between kinematics and hydrodynamic force

It is well known that copepods move in burst-like motion during hopping, i.e. they reach a high velocity (several hundred body length per second) during the power stroke, but their swimming speed decreases rapidly during the return stroke, to almost zero at the end of the return stroke (Strickler, 1975; van Duren and Videler, 2003; Yen and Strickler, 1996). Although most aquatic animals produce thrust in a pulsed rather than steady manner (Daniel, 1984), the typical variation of swimming speed within a swimming cycle encountered in nature is often significantly smaller than in copepods. For example, during steady swimming, eels shown approximately a 10% fluctuation in velocity about the mean velocity U (Müller et al., 2001; Tytell et al., 2010), whereas mullets have been found to exhibit more than 20% fluctuation (Müller et al., 1997; Nauen and Lauder, 2002). Previously we showed (Borazjani and Sotiropoulos, 2009a) that an eel-like tethered swimmer produces a hydrodynamic force time series that is much smoother than a mackerel-like tethered swimmer at the same Reynolds number, i.e. higher fluctuations of force correlates with higher fluctuations in swimming speed. Furthermore we showed (Borazjani and Sotiropoulos, 2010) that as the Reynolds number decreases the velocity fluctuations increases. This trend is consistent with the fact that at lower Re viscous forces become more effective in damping inertial forces, which are ultimately responsible for smoother swimming through fluid inertial effects. From the above discussion and given the calculated force record of the tethered copepod, it is reasonable to postulate that there are two main reasons for the larger velocity fluctuations during copepod hopping as compared with other aquatic animals: (1) copepods operate at lower Re than larger aquatic animals, such as fish, and swimming speed fluctuations are enhanced in an environment where viscous forces become important; and (2) the force record of the tethered copepod shows much larger fluctuations than other aquatic animals.

Fig. 13.

Out-of-plane non-dimensional vorticity (ω2) contours for the top view mid plane of the copepod during the return stroke at (A) t/T=0.33, (B) t/T=0.66, (C) t/T=1, for the rigid antennae (left) and deformable antennae (right) (Re=300).

Fig. 13.

Out-of-plane non-dimensional vorticity (ω2) contours for the top view mid plane of the copepod during the return stroke at (A) t/T=0.33, (B) t/T=0.66, (C) t/T=1, for the rigid antennae (left) and deformable antennae (right) (Re=300).

Our simulations showed that the return stroke of the antennae, even when they deform, creates a drag-type force. This explains the typical application of multiple strokes of legs while the antennae stay in the folded position, as has been reported in several experimental observations (Strickler, 1975; van Duren and Videler, 2003), and was also observed in the holographic movies obtained in this work. The antennae apparently remains folded while legs execute multiple strokes in order to delay incurring the large hydrodynamic penalty in terms of the drag force associated with the return of the antennae.

Two-dimensional versus three-dimensional flow fields

Our calculated flow field agrees well qualitatively with previous experimental wake visualizations. For example, Catton et al. (Catton et al., 2007) reported that the high values of velocity, vorticity and strain rate occur near the body of the copepod, which is in agreement with our findings that these quantities are indeed maximum near the moving appendages. The computed flow field is completely symmetrical about the symmetry X2X3 plane. However, Catton et al. (Catton et al., 2007) reported asymmetrical flow field for the tethered copepod but symmetrical for the free swimming copepod. They mention that the asymmetry in the tethered copepod flow field is 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 of the 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 simulations after several consecutive hops (Fig. 12) are in good agreement with the experimental observations (van Duren et al., 1998; van Duren and Videler, 2003). We have observed that the vortices produced after each power and return stroke merge and interact with the previously shed vortices, which is in agreement with the observations of van Duren and Videler (van Duren and Videler, 2003) that the next jump or the next metachronal deployment of legs started before the effect of the previous one was fully dissipated. Finally, the shape and the sense of rotation of the large vortical structure in the wake (Fig. 12) is very much consistent with that of the toroidal vortex observed in the visualization experiments of Yen and Strickler (Yen and Strickler, 1996).

Obtaining the 3-D flow field around a copepod is quite challenging (Malkiel et al., 2003). That is why most of the recent experiments 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 and Videler, 2003). Our results have shown, however, that the flow field of a hopping copepod is very complex and highly three-dimensional (e.g. see Fig. 10). Furthermore, it has already been established that the velocity field by itself is not sufficient to estimate hydrodynamic forces generated by swimming or flying and a pressure-like quantity is also required (Dabiri, 2005). Therefore, an important finding of our work is that accurate quantification of the underlying hydrodynamic forces, the energetic costs of locomotion, and the flow disturbances created by animal propulsion requires close synergy between 3-D computational modeling and laboratory experiments with PIV.

Limitations of current simulations and future work

In this work there is a virtual tether that holds the copepod in place and instead of moving through the fluid the fluid moves over its body. Tethered copepods have been used in the experiments to determine the hydrodynamic forces by measuring the force on the tether (Alcaraz and Strickler, 1988; Lenz et al., 2004). Tethering is believed to not change the appendages kinematics of copepods (Hwang et al., 1993; van Duren and Videler, 2003) except that the frequency of motion has been observed to decrease for tethered copepods relative to the free swimming ones (Lenz et al., 2004; Svetlichnyy, 1987). As discussed in van Duren and Videler (van Duren and Videler, 2003), the tethering does not affect the flow field that much if the copepod is foraging or hopping. In foraging, copepods create a feeding current and use their body drag and negative buoyancy as a ‘natural tether’ (Strickler, 1982). The hopping movement is designed to shift a bulk of water without displacing the animal very much (van Duren and Videler, 2003). However, in escape responses, during which the animal moves as fast as possible away from the danger, the tether effects on the flow structure cannot be ignored. In fact, as shown by the recent work by Catton et al. (Catton et al., 2007), the force on the tether modifies the flow field irreversibly, i.e. the tether does have an effect on the 3-D wake in this work. The tether also affects the hydrodynamic forces by absorbing the forces produced by the individual appendages whereas these forces in a free swimming copepod tend to accelerate or decelerate the copepod. In the tethered model, the non-moving appendages do not produce a significant force, similar to a stationary cylinder in a stagnant fluid does not produce any drag. However, for a freely swimming copepod, the non-moving appendages will produce drag. Nevertheless, our conclusion regarding the importance of speed and orientation effects is not changed by the tether, since both the hydrodynamic force produced by the rigid and deformable antennae were compared under the same conditions (i.e. tethered and similar Re).

Fig. 14.

Different views of the vortical structures visualized by the iso-surfaces of q-criterion around the tethered copepod with deformable antennae at t/T=1 (Re=300). See supplementary material available online (Movie 1 in supplementary material) for a movie of the wake structure.

Fig. 14.

Different views of the vortical structures visualized by the iso-surfaces of q-criterion around the tethered copepod with deformable antennae at t/T=1 (Re=300). See supplementary material available online (Movie 1 in supplementary material) for a movie of the wake structure.

Fig. 15.

The vortical structures visualized by the iso-surfaces of q-criterion superimposed on the out-of-plane non-dimensional vorticity (ω1) contours on the midplane of the tethered copepod with deformable antenna at t/T=1 (Re=300).

Fig. 15.

The vortical structures visualized by the iso-surfaces of q-criterion superimposed on the out-of-plane non-dimensional vorticity (ω1) contours on the midplane of the tethered copepod with deformable antenna at t/T=1 (Re=300).

Another limitation of our current work is that all four pairs of legs have been assumed to move back and forth in a reversible manner (symmetric stroke), retaining the same surface area during the return and power strokes. Such a treatment can underestimate the forces produced by the legs since these appendages are hairy, and copepods are known to increase their surface area during the power stroke and decrease it during the return stroke (Lenz et al., 2004). Moreover, the asymmetric motion of the legs during the power and return strokes is evident in the movies of live copepods we obtained for this work (Fig. 16). As seen in Fig. 16, when the copepod is viewed from above, the movie frames clearly show the legs to be visible during the power stroke, extending laterally from the body. Conversely, they are not visible during the return stroke, i.e. the legs are apparently being returned underneath the body rather than extending outwards as in the power stroke. We made no attempt to improve the leg model in this work, but this is obviously an area where our virtual copepod model can be greatly improved in the future. Note that the asymmetric stroke can have much larger impact on the forces produced by the legs than by the antennae since the legs are smaller, i.e. they operate in lower local Re (based on their size) environment, and the lower the Re the higher the effect of the asymmetric stroke. This might be the reason why we found that, regardless of whether it is rigid or flexible, the antenna was the greatest contributor to both thrust- and drag-type forces during the power and return stroke, respectively. Nevertheless, the legs in our model produced distinguishable peaks in the force record during the power stroke. The fourth pair of legs was the largest contributor to thrust among the all the pairs as it moved faster than the other pairs of legs (see Fig. 3) and the second largest after the antennae.

Fig. 16.

Cinematic dual digital holography of a copepod during hopping: the legs are visible from the top view during the power stroke of legs (A) but not during the return stroke (B).

Fig. 16.

Cinematic dual digital holography of a copepod during hopping: the legs are visible from the top view during the power stroke of legs (A) but not during the return stroke (B).

In spite of the limitation of our model in terms of leg modeling, there is evidence that the antennae are capable of producing significant amount of thrust. First, Strickler (Strickler, 1975), who observed free-swimming copepods during escape, reported the acceleration of his copepod to a velocity of 1.5 cm s–1 during the movement of antennae but before the leg strokes. He emphasized that the antennae stroke is definitely part of the power stroke and the antennae are not just bending out of the way as suggested previously by Storch (Storch, 1929). Second, in the work of Lenz et al. (Lenz et al., 2004), in which the total force on a tethered copepod was measured, during the early stages of the escape response, when only the antennae and the tail were moving, a considerable amount of thrust force was recorded (more than half of the force record peak). However, the largest peak in their force record appeared when two pairs of legs plus antennae, tail and other mouthpart appendages were moving, which still does not show that the legs create more power than the antennae. Nevertheless, such evidence suggests that the movement of the antennae is at least partly active, i.e. the antennae have musculature that can produce thrust during 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 calculated based on a fluid–structure interaction, the extent to which deformation is produced passively by the flow or actively by the muscles cannot be determined. Gill and Crisp (Gill and Crisp, 1985) have shown that the removal of copepods antennae does not prevent the escape response. However, they did not comment on the effectiveness of the escape without antennae. Nevertheless, they report that the removal of both antennae increased the number of times the tail alone was flicked, which could suggest that the copepod needs to compensate for the loss of its antennae by repeated deployment of other appendages.

Another limitation of our current work is the non-smooth motion of the appendages. As evident in Fig. 7 the sudden stop of all appendages during the end of the power stroke in the current model produces a significant amount of drag. Similarly, the sudden start of the appendages at the beginning of the return stroke produces a significant amount of drag. By contrast, the sudden start of the appendages at beginning of power stroke and their sudden stop at the end of the return stroke creates a large thrust-type force. Therefore, it can be hypothesized that to achieve better performance the copepods' appendages should gradually decelerate and accelerate during the end of the power stroke and the beginning of the return stroke, respectively, and start and stop suddenly at the beginning of the power stroke and end of return stroke, respectively. In fact, the experiments have shown that the copepods' appendages do indeed accelerate fast and decelerate gradually at beginning and end of power stroke as seen for example in Fig. 4 of Lenz et al. (Lenz et al., 2004) and Fig. 3 of Alcaraz and Strickler (Alcaraz and Strickler, 1988). The appendages also accelerate gradually at the beginning of the return stroke (Alcaraz and Strickler, 1988; Lenz et al., 2004). The results shown in Fig. 7 provide clear evidence in support of the importance of such acceleration and deceleration in appendage kinematics. Furthermore, this suggests that the kinematics we 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 and change in the surface area during the power and return stroke. Furthermore, in the current model all the appendages including the legs start and stop suddenly. To overcome these limitations better experimental data is required. However, capturing the acceleration and deceleration of appendages is quite challenging experimentally because the power and return strokes take about 2–5 and 6–10 ms, respectively, and with a high speed camera with 1000 frames per second only a few frames per stroke can be captured, e.g. only 2–5 frames per stroke has been captured in the recent experiments (Lenz et al., 2004; van Duren and Videler, 2003). Furthermore, multiple views are required to obtain the 3-D asymmetric motion of the legs and other appendages. In addition, some information may still be lost when the appendage is located behind the tail or the body. Repeatability is also an issue in experiments, even for the same copepod. Therefore, obtaining detailed kinematics and other quantities such as porosity of the hairy appendages and their effective surface area, which is required to build more realistic models, is not possible with the current technology, so major experimental breakthroughs are required. Note that even the asymmetric motion of the antennae in this work was restricted to the 2-D horizontal plane and the antennae did not move in the vertical direction.

Supported by more detailed information from experiments, the present virtual copepod model can be made biologically far more realistic, and evolve into a powerful computational tool for investigating the hydrodynamics of copepod swimming. As part of our future work, we intend to continue and further exploit the coupled computational and experimental research paradigm adopted in this work in order to refine the description of the appendage kinematics in the model. With the improved kinematics, the model will be extended to simulate self-propelled copepod swimming to explore important issues regarding the differences between freely-swimming and tethered copepod hydrodynamics. Simulating self-propelled copepods requires implementation of a full, fluid-structure interaction algorithm in our numerical approach. Such an algorithm has already been developed and successfully applied, and validated, for cardiovascular flows (Borazjani et al., 2008), and self-propelled aquatic fish-like swimming (Borazjani and Sotiropoulos, 2010).

APPENDIX A

Grid sensitivity study of the numerical solutions

Fig. A1 compares the total force obtained from tethered simulations with rigid appendages at Re=300 for the coarse and the fine meshes. This figure shows that the total force obtained on the two grids are in good agreement with each other and show the same trend. Nevertheless, the total force from the fine mesh result is much smoother and does not exhibit the sharp oscillations and peaks observed in the coarse mesh result. This finding is similar to our previous work (Borazjani et al., 2008), where we showed that grid refinement reduces spurious high frequency oscillations of the force, which are an inherent feature of the reconstruction algorithm used in sharp-interface, immersed boundary methods. The average total forces in the second cycle obtained from the force history shown in this figure are and –1.8213×10–1 for the fine and coarse mesh, respectively. Even though there is a significant discrepancy between these two predictions, it is evident from Fig. A1 that this discrepancy arises from the large, spurious spikes of the coarse-mesh force record occurring when the various appendages begin to move or stop suddenly. The results obtained on the two meshes are in good qualitative and quantitative agreement during the rest of the cycle, which suggests that even the coarse mesh used in this study is adequate for capturing the most important aspects of the hop flow fields. Similar conclusions are also derived by comparing the coarse and fine mesh instantaneous flow fields in terms of vorticity contours and general flow patterns.

Fig. A1.

Comparison of total force coefficient on the tethered copepod with rigid antennae obtained on the fine and the coarse grids (Re=300).

Fig. A1.

Comparison of total force coefficient on the tethered copepod with rigid antennae obtained on the fine and the coarse grids (Re=300).

APPENDIX B

The mathematical model of the flexible antennae

As discussed in the Materials and methods section, the copepod body, including the antennae is meshed with triangular elements needed for the sharp-interface immersed boundary method (see Fig. 1C). The triangular mesh nodes are tracked by their Lagrangian position vector r during the prescribed motion of the appendages. The flexible antennae return stroke obtained from the experiments is modeled at any instant in time by applying a sequence of deformation and rotation operators to the initial (fully opened with θ=0 shown in Fig. 1C) antenna shape. Deformation and rotation functions are applied sequentially at any instant to the initial shape of the antennae as follows:
formula
(A1)
where r and rinit are the new and initial position vectors of the triangular mesh nodes, respectively, and D and Rθ are the deformation and rotation operators, respectively. The deformation operator is defined as follows:
formula
(A2)
where, sign is the sign function, and A, b and Linit are the parameters of the model. A is the amplitude of the deformation, which has the form of half of a sine wave, Linit is the initial total length of one arm of the antennae, and b is half of the wavelength of the sine wave. The swinging is achieved by applying the rotation matrix around the X2 axis, which affects only x1 and x3 values, to swing forward the deformed shape with an angle θ relative to the initial position, similar to the rigid antenna motion around the vertical (X2) axis through the joint between each antenna and the body (Fig. 1C):
formula
(A3)

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

Phase 0: t0<t<t1
formula
(A4)
formula
(A5)
Phase 1: t1<t<t2
formula
(A6)
Phase 2: t2<t<t3
formula
(A7)
Phase 3: t3<t<t4
formula
(A8)
formula
(A9)
Phase 4: t4<t<T
formula
(A10)

In the above equations t is the time in the cycle, θmax=70 deg, θadd=15 deg, Linit=0.75, t0/T=0.25, , t1/T=0.48, t2/T=2/3, , t3/T=0.833, and t4/T=0.967. Fig. A2 plots the θ and parameter b values during all the phases of the return stroke according to the above equations.

Fig. A2.

The values of θ (radians) and parameter b for the deformable antenna motion. The dashed vertical lines indicate the beginning and ending of each phase. The t0, t1, t2, t3 and t4 indicate the beginning of phase 0, 1, 2, 3 and 4, respectively. and indicate the start and end of the swinging (rotation) of the antennae, respectively.

Fig. A2.

The values of θ (radians) and parameter b for the deformable antenna motion. The dashed vertical lines indicate the beginning and ending of each phase. The t0, t1, t2, t3 and t4 indicate the beginning of phase 0, 1, 2, 3 and 4, respectively. and indicate the start and end of the swinging (rotation) of the antennae, respectively.

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

     
  • ith component of the time-averaged non-dimensional force coefficient vector

  •  
  • ci(t)

    ith component of the non-dimensional force coefficient vector at time t

  •  
  • CDDH

    cinematic dual digital holography

  •  
  • D

    the deformation operator

  •  
  • f=1/T

    frequency of appendages motion

  •  
  • FGMRES

    flexible generalized minimal residual

  •  
  • Fi(t)

    ith component of the force vector at time t

  •  
  • h

    grid spacing

  •  
  • HCIB

    hybrid Cartesian/immersed boundary

  •  
  • IB

    immersed boundary

  •  
  • L

    length of the copepod

  •  
  • nj

    jth component of the normal vector to the immersed boundary

  •  
  • p

    pressure

  •  
  • PETSc

    portable, extensible toolkit for scientific computation

  •  
  • PIV

    particle image velocimetry

  •  
  • q

    q-criteria

  •  
  • r

    Lagrangian position vector

  •  
  • rinit

    initial Lagrangian position vector

  •  
  • Re =L2/3Tν

    Reynolds number

  •  
  • Rθ

    rotation operator

  •  
  • sign

    sign function

  •  
  • Sij

    strain rate tensor

  •  
  • t

    time

  •  
  • T

    period of appendages motion

  •  
  • U=L/(3T)

    velocity scale

  •  
  • θ

    angular displacement of an antenna

  •  
  • θadd

    additional angular displacement of the deformable relative to the rigid antenna

  •  
  • θmax

    maximum angular displacement of the rigid antenna

  •  
  • ν

    kinematic viscosity

  •  
  • τij

    viscous stress tensor

  •  
  • ωi

    ith component of the vorticity vector

  •  
  • Ωij

    antisymmetric part of the velocity gradient tensor

  •  
  • ∥ ∥

    Euclidean matrix norm

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