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 Systems Biology
Coherent locomotion as an attracting state for a free flapping body

Communicated by Marsha J. Berger, New York University, New York, NY, June 17, 2005 (received for review December 22, 2004)
Abstract
A recent experiment [Vandenberghe, N., Zhang, J. & Childress, S. (2004) J. Fluid Mech. 506, 147–155] has shown that an axlemounted blade can spontaneously rotate when oscillated (or “flapped”) above a critical frequency in a fluid. To understand the nature of flapping locomotion we study numerically the dynamics of a simple body, flapped up and down within a viscous fluid and free to move horizontally. We show here that, at sufficiently large “frequency Reynolds number,” unidirectional locomotion emerges as an attracting state for an initially nonlocomoting body. Locomotion is generated in two stages: first, the fluid field loses symmetry by an instability similar to the classical von Kármán instability; and second, precipitous interactions with previously shed vortical structures “push” the body into locomotion. Body mass and slenderness play central and unexpected roles in each stage. Conceptually, this work demonstrates how locomotion can be transduced from the simple oscillations of a body through an interaction with its fluid environment.
A common strategy for locomotion through a fluid uses appendages, such as wings or fins, flapping perpendicularly to the direction of travel (1). This strategy is in marked difference to strategies, using propellers or screws, ciliary waves, or rowing with limbs or oars, which explicitly move fluid in the direction opposite to travel. Flapping locomotion is also never observed for microorganisms moving at low Reynolds number. We consider a rigid body of simple shape, here an ellipse of major (minor) axis length L (W), with uniform mass density ρ _{b} . The body translates with velocity u _{b} = (u_{b}, v_{b} ) (without rotation) in the infinite xy plane through a 2D viscous fluid of density ρ and viscosity μ. We specify its vertical motion as y_{b} (t) =–Acos(2π ft) (in technical parlance this is a heaving body). Hence, L is a characteristic length and Af is a characteristic velocity, which defines the frequency Reynolds number Re_{fr} = ρ(Af)L/μ. The surrounding fluid is governed by the 2D incompressible Navier–Stokes equations: with u = (u, v) the fluid velocity, p the pressure, and D/Dt the material time derivative. Eq. 1 is in nondimensional form with Re_{fr} its sole control parameter.
An important element of this study is that the horizontal motion of the body is not specified, as in many other studies (2–5), but rather is determined by the fluid force acting at its boundary through Newton's second law: where M = (ρ _{b} /ρ) · (A _{0}/L ^{2}), with A _{0} the body area, and n the outward normal. The fluid stress tensor, –p I + 2E with E the symmetric rateofstrain tensor, encodes pressure (first term) and viscous stresses (second term). Eqs. 1 and 2 together with u_{body} = u _{b} , close this coupled bodyfluid system, for which the sum of horizontal body and fluid momentum is an invariant. Newton's law has been coupled to the Navier–Stokes equations previously to study fundamental flowbody interactions such as bodies rising and falling in a fluid (6, 7) and cylinders mounted in a steady flow (8). Besides Re_{fr} , important control parameters are the body aspect ratio L/W, the dimensionless stroke amplitude A/L [set to 1/2 to simplify the problem; this value has been observed in experiments to lead to efficient flapping flight (5)], and the density ratio ρ _{b} /ρ. This last parameter is interesting as, if large, it can be considered as partially accounting for the inertia of an attached, nonoscillating body.
We have simulated this coupled bodyfluid system by using a numerical method based on the “vorticitystreamfunction” formulation of Eq. 1 and a mixed Fourier/finitedifference discretization. The vorticity, ω = v_{x} – u_{y} , measures the local rateofrotation of the fluid. Vorticity is transported and diffused according to Re_{fr} Dω/Dt = Δω, (the curl of Eq. 1). Our method is fourthorder accurate in space and secondorder in time, with the vorticity diffusion term treated implicitly. Related formulations have been developed by E and Liu (9) and Wang (2). The numerical grid conforms to and moves with the elliptical body and has an outer boundary that is typically many body diameters away from the body itself. Boundary conditions on the outer boundary simulate the fluid dynamics in the far field, by setting the values of the stream function to correspond to purely translational velocity (minus the body velocity in the physical frame) and the values of vorticity to zero (see ref. 10 for further details). In these simulations, the vertical motion of the body is specified, and the horizontal motion is studied after the imposition of perturbations in horizontal velocity (u_{b} ≈ 10^{–12} to 10^{–2}). The initial fluid velocity is zero everywhere, and the sum of horizontal body and fluid momentum is conserved to within 0.03% of the eventual steadystate magnitudes of each of the summands. Hence, any horizontal motion of the body significantly above the level of the perturbation is achieved only through a momentum exchange with the fluid.
First, we find that if the body flaps at sufficiently small Re_{fr} the state of no horizontal body motion is stable; horizontal motions imposed transiently on the flapping body rapidly dissipate and the fluid resumes a leftright symmetric dynamics (in qualitative appearance, like Fig. 1A ). At higher Re_{fr} the body dynamics is dramatically different. Fig. 1 (and Movie 1, which is published as supporting information on the PNAS web site) shows a simulation for an elliptical body (of aspect ratio 5:1) that illustrates the stages leading to “takeoff.” At early times (0 < t < 0.5), during an imposed horizontal perturbation (u_{b} ≈ 10^{–2}), the flow appears very nearly symmetric (Fig. 1 A ). However, when the horizontal motion evolves freely under fluid forcing (t > 0.5), this initial perturbation grows in time and the fluid motions become increasingly asymmetric as the body both sheds vorticity into the fluid and interacts with vorticity already there (Fig. 1B ).
Fig. 1F shows the body and surrounding vorticity field after 16 flapping periods, by which time the body is moving steadily to the left and leaving behind a staggered array of positive and negative vortices. As is typical of flapping flight (4), a vortex is shed near the trailing edge of the body into the fluid during an upstroke (positive vorticity) or downstroke (negative vorticity). During a halfstroke the developing leadingedge vortex remains bound to the body and typically merges into and strengthens the trailingedge vortex of the same sign that is produced during the following halfstroke. The vortex wake resembles a von Kármán vortex street behind a bluff body in a flow, but with vortices of reversed sign (11); this “inverse von Kármán street” has been observed in numerous experimental and numerical studies of flapping foils and in the carangiform mode of fish swimming (12).
In numerous simulations we have observed characteristic events that move the system into locomotion; an example is shown in Fig. 1C . Into its sixth cycle of flapping the body is moving upward and shedding oppositely signed vortices from its two edges. However, the right edge is colliding with a positively signed vortex that was shed from the opposite edge as the body was flapped downward on the previous halfcycle. This “free” vortex, at one point a full body length away, now forms part of a vortex dipole that subsequently carries fluid momentum away from the body. The horizontal fluid stresses around the body (Fig. 1D ) show the effect of this interaction. There is a large suction force at the left edge, associated with vorticity creation. In symmetric flapping, an opposite suction force would act at the right edge, but this force has been reduced by the formation of the vortex dipole. This imbalance in suction force seems to initiate the locomotion of the body, and after a nearly identical dipole ejection on the next upward stroke, the body settles into steady locomotion. In other simulations locomotion is instead initiated by the collision of an incipient leading edge vortex with a vortex of the same sign. This merging strengthens the resultant shed vortex, leading to a larger, unbalanced suction force at the leading edge.
Horizontal fluid forces show clearly the importance of these collisions in the transition to locomotion. The dynamics unfold in three stages (Fig. 1E ). First, by studying carefully the growth of horizontal velocity from very small initial amplitudes it can be shown that the initial growth of horizontal velocity (0 ≲ t ≲ 3) is exponential in time; this is the signature of a linear instability. Second is a transition period (3 ≲ t ≲ 8) typified by complex, largeamplitude temporal and spatial dynamics, during which the viscous and (dominant) pressure forces act generally in the same direction. The transition period ends with one or more vortex collisions, the timing of which depends on how the asymmetric flow field develops, and can be sensitive to initial conditions. Finally, as locomotion is achieved (t ≳ 8), the forces move rapidly toward a 1/2periodic dynamics with cycleaveraged force on the body essentially zero. Pressure forces are predominantly in the direction of motion, and viscous forces are in the opposite direction. We also find that the cycleaveraged “input power” (Fig. 1E , green line) required to move the body vertically increases substantially with the loss of flow symmetry, but is followed by a drop of nearly 50% with the onset of locomotion.
The initial loss of flow symmetry is similar to the classical von Kármán instability of the symmetric wake behind a bluff body, as both lead to alternating shedding of vortices above a critical Reynolds number (13). We have systematically studied the response to small initial horizontal velocities for vertically oscillated, elliptical bodies and find clear evidence for either exponential growth from (or decay to) the zero horizontal state as a function of Re_{fr} . Fig. 2 shows the calculated growth rate with an apparently sharp transition to instability occurring at moderate Re_{fr} . This transition Re_{fr} is apparently independent of mass ratio ρ _{b} /ρ, suggesting that the instability is that of the fluid flow alone and not of the coupled bodyfluid system.
Body shape and mass are central to whether and how a coherently locomoting state is achieved. Fig. 3 A and B (and Movie 2, which is published as supporting information on the PNAS web site) shows the trajectory and integrated horizontal forces on a more slender body (10:1). We find generally that thinner bodies move more directly and smoothly into steady locomotion, as shown by the predominantly singlesigned fluid forces in the transition period (4 ≲ t ≲ 9). Also, viscous forces play a larger role. Fig. 3C (and Movie 3, which is published as supporting information on the PNAS web site) illustrates the effect of body mass on the attainment of steady locomotion. Fig. 3C shows the apparently chaotic dynamics of a body that is lighter (“neutrally buoyant”) than that in Fig. 1, but otherwise identical in shape and Re_{fr} . The motion of this light body is highly sensitive to the instantaneous fluid forces on it, and thus it cannot sustain locomotion against a period of drag, even with the additional inertia caused by the added mass effect [the increased force per unit body acceleration required to accelerate a body in fluid versus a massless medium (14)]. However, light, yet more slender, bodies seem less susceptible to fluid force fluctuations.
Fig. 2 shows that thinner bodies have a smaller critical Re_{fr} for the symmetrybreaking instability. Furthermore, for the 10:1 and 20:1 bodies the growth rate approaches zero more smoothly at the critical Re_{fr} . Fig. 2 Inset shows the average body velocity U (in the locomoting state) for a 10:1 body, expressed as the translational Reynolds number, Re_{U} = ρUL/μ, over a broad range of Re_{fr} . There are three points worth noting. First, there is an apparently linear onset of translational motion at a positive . Second, the putative bifurcation point is greater than the critical Re_{fr} of symmetry breaking. Third, Re_{U} increases monotonically with Re_{fr} and increases with apparent linearity for large Re_{fr} . For large Re_{fr} the corresponding Strouhal numbers, St = 2Re_{fr} /Re_{U,} have an asymptotic value slightly <0.2, which is at the lower end of the range (0.2 < St < 0.4) observed for efficient animal locomotion (15).
The examples of Fig. 3 give a sense of the physical constraints that delineate the regime of coherent locomotion accessible from rest. We have quantified this further by scanning the (Re_{fr} , ρ _{b} /ρ) parameter space for a 10:1 body aspect ratio (Fig. 4). As expected, below a transition Re_{fr} (apparently independent of ρ _{b} /ρ) we find that the state of no horizontal motion is apparently stable (Movie 4, which is published as supporting information on the PNAS web site). Above the transition Re_{fr} there is an intermediate region in which the body does not coherently locomote, but engages in a variety of periodic, quasiperiodic, or apparently chaotic dynamics. The periodic dynamics (Movie 5, which is published as supporting information on the PNAS web site) consist of regular horizontal oscillations, typically locked to a multiple of the flapping period. For small mass ratios, an increase in Re_{fr} brings a progressive loss of periodicity, leading to the irregular dynamics exemplified by Fig. 3C . For larger mass ratios, an increase of Re_{fr} leads to a locomoting state that shows decreased St with increased Re_{fr} (consistent with Fig. 2 Inset). Increasing the aspect ratio of the body decreases the ranges, in both Re_{fr} and ρ _{b} /ρ, of the various nonlocomoting states, and eases the transition to locomotion.
These results demonstrate how locomotion can arise “naturally” as an attracting dynamical state. We find that the interaction of the body with previously shed vortices can both enhance thrust during locomotion and provide the initial thrust that drives the body into locomotion. It has been conjectured that hovering insects enjoy enhanced lift because of the collision of their wings with previously shed vortices (16). Elaborations of our study include questions about the consequence of an attached and towed body (whose effect is approximated in the mass of the flapping wing), optimality in wing shapes and motions to obtain locomotion (including explicitly broken foreaft symmetry), and the dynamics of bodies with purely internal forcing. The general 3D case introduces additional degrees of freedom that make the problem far more complex than we have considered here. However, simulations of 2D flapping airfoils undergoing prescribed motions can show good quantitative agreement with experimental studies of 3D finitespan wings, for the case when the vortex shedding is induced primarily by the reversal of the stroke direction (17), as it is in the present study. Complementary to the present study are recent numerical studies (4, 5) for flapping bodies whose translational motion is specified, and which identify a moderate Reynolds number below which thrust forces rapidly decrease. Our results show that spontaneous rotation of a flapped, axlemounted wing seen in experiment (18) is observed in the more physically relevant regime of translation and demonstrate the critical effects of body mass and shape. Our study suggests, too, that the higher critical flapping frequency and the subcritical, hysteretic transition to rotation observed in the experiment may have resulted from axlebearing friction. Taken together, those studies (4, 5, 18) and ours may also provide a fluiddynamical explanation as to why some organisms change their locomotory strategy with increases in swimming speed [e.g., the mollusk Clione antarctica moves from using ciliary bands to flapping wings (19)] and suggest one possible origin of animal locomotion through the interaction of an oscillating body or appendage with its fluid environment.
Acknowledgments
We thank S. Childress, N. Vandenberghe, and J. Zhang for fruitful discussions. This work was supported by Department of Energy Grant DEFG0200ER25053 and National Science Foundation Grant DMS9980069.
Footnotes

↵ † To whom correspondence should be addressed. Email: alben{at}deas.harvard.edu.

Author contributions: S.A. and M.S. designed research; S.A. performed research; S.A. and M.S. analyzed data; and S.A. and M.S. wrote the paper.
 Copyright © 2005, The National Academy of Sciences
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