Fluid transport and coherent structures of translating and flapping wings

Transcription

1 Manuscript submitted to Chaos for special issue on Lagrangian Coherent Structures Fluid transport and coherent structures of translating and flapping wings Jeff D. Eldredge and Kwitae Chong Mechanical & Aerospace Engineering Department, University of California, Los Angeles Los Angeles, CA Abstract The Lagrangian coherent structures (LCS) of simple wing cross-sections in various low Reynolds number motions are extracted from high-fidelity numerical simulation data and examined in detail. The entrainment process in the wake of a translating ellipse is revealed by studying the relationship between attracting structures in the wake and upstream repelling structures, with the help of blocks of tracer particles. It is shown that a series of slender lobes in the repelling LCS project upstream from the front of the ellipse and pull fluid into the wake. Each lobe is paired with a corresponding wake vortex, into which the constituent fluid particles are folded. Flexible and rigid foils in flapping motion are studied, and the resulting differences in coherent structures are used to elucidate their differences in force generation. The clarity with which these flow structures are revealed, compared to the vorticity or velocity fields, provides new insight in the vortex shedding mechanisms that play an important role in unsteady aerodynamics. 1

2 Biological mechanisms of flying and swimming have been the subject of much recent interest, particularly with the objective of exploiting their remarkable capabilities in small-scale vehicles. One of the prevailing challenges in adapting these mechanisms for artificial use is the important role of the unsteady nonlinear fluid dynamics in determining the forces on flexible control surfaces wings, fins, etc. Reduced-order modeling of these fluid body interactions for use in control strategies is therefore an important objective. Since the force generation in these bio-inspired mechanics is strongly coupled to vortical structures both self-generated and ambient it is hoped that techniques which distill the complicated flows into the dynamics of a small number of coherent structures will provide a path toward a reduced-order description. Here, we proceed in this spirit by identifying the so-called Lagrangian coherent structures of a few canonical problems that are representative of biological locomotion small-scale rigid or flexible foils translating at large angle of attack or flapping with oscillatory kinematics using techniques for post-processing of computational flow data. It is observed here, as it has been in previous studies, that these coherent structures are relatively simple, in contrast to the complexity of the underlying flow. We use these structures to elucidate the flow transport and force production. Flow separation is often anathema for fixed aerodynamic and hydrodynamic surfaces designed for maximizing lift and minimizing drag. However, it is now accepted that many creatures airborne [8, 4, 22] and aquatic [9] exploit the transient events of separation to generate forces that greatly exceed those possible in fixed orientation to the incident flow. Most insects, in particular, generate the necessary lift by making use of the extra circulation of the leading-edge vortex formed during the early stages of a wing translating at large angle of attack [8]. It has been postulated that the flapping cycle is carefully timed to avoid the rapid decline in lift that accompanies the shedding of this vortex [21]. Separated flows, particularly those past bluff bodies, have been studied extensively with a wide range of tools [18, 23]. However, there has been comparatively little attempt to use Lagrangian 2

3 analysis to explore the transport properties of such flows. Put simply, such an analysis focuses on the relative motions of tracer particles as are common to many experimental flow visualization techniques for revealing the dominant structures in the flow. But recent extensions from dynamical systems theory have led to the development of tools for extracting such structures directly from the velocity field [10, 16]. One examines the properties of the local deformation gradient when the flow is allowed to proceed over a finite amount of time. Building from the work of Haller [10], Shadden et al. [16] provided a precise definition of the Lagrangian coherent structure (LCS) as a ridge of the finite-time Lyapunov exponent (FTLE) field. This field quantifies the change of distance between two nearby particles over the time interval; ridges, therefore, indicate curves along which two nearby particles on either side are most prone to deviate from one another. These repelling structures are analogous to the stable manifolds of a time-invariant vector field. If, in contrast, the flow is integrated backward in time, then the ridges represent the attracting Lagrangian coherent structures, which are associated with the unstable manifolds of a steady flow. Shadden et al. [16] showed that Lagrangian coherent structures are approximately material surfaces. As such, they represent dynamic transport barriers in the flow, and therefore serve as useful indicators of regions of different flow behavior. Lagrangian analysis with this tool has recently been used to reveal new features of several aerodynamically and biologically-relevant flows, obtained experimentally or computationally. Shadden et al. [15] studied the entrainment and detrainment of an empirical vortex ring as well as in the vicinity of a live jellyfish. The LCS obtained from a computational model of jellyfish swimming has been recently studied by [12] and in a low Reynolds number two-dimensional analog by Wilson et al. [24]. The entrainment regions of a sheet swimming in an inviscid fluid have been investigated by Peng and Dabiri [14] by examining the repelling LCS. The coherent structures of unsteady aerodynamic problems have also been explored with these analysis tools. Separation from an airfoil at low angle of attack and Reynolds number on the order of 10 4 has been studied by Lipinski et al. [11, 2], who used the LCS and associated particle tracking to explore the vortex formation and reattachment topology. More recently, Brunton and Rowley [1] have used LCS to visualize the wake of a flat-plate cross-section either fixed or undergoing oscillatory pitching and plunging kinematics in a free stream with Reynolds number of

4 In this paper, the Lagrangian coherent structures of the two-dimensional flow generated by a wing with simple cross-sectional shape under various motions are studied. The velocity fields of all cases are obtained from numerical simulation of the two-dimensional Navier Stokes equations, using the viscous vortex particle method [5, 6]. In the first case, an ellipse undergoes steady translation at a high angle of attack. The LCS study by Brunton and Rowley [1], who explored steadily translating as well as pitching or plunging wings, lies in the same Reynolds number regime. Here, the entrainment and detrainment of the flow are investigated by considering the joint behavior of the attracting and repelling structures, analogous to the swimming sheet study by Peng and Dabiri [14] and the airfoil study of Lipinski et al. [11, 2]. Then, the structures produced by the flapping of a flexible foil composed of two rigid ellipses connected by a torsion spring are explored. The aerodynamic performance of this foil relative to its rigid counterpart has recently been reported, and two notable cases were identified in which the flexible foil performed notably better and worse [7]. In this paper, the Lagrangian analysis of these two cases is considered in conjunction with instantaneous pressure distributions to clarify the differences. In Section 1, the basic numerical methodology for the underlying flow simulations is briefly described, and the procedure for obtaining the FTLE field is summarized from previous work. The results from the two sets of problems are described in Section 2. 1 Methodology The numerical simulations are carried out with the viscous vortex particle method (VVPM) [5], which solves the Navier Stokes equations with advecting particles that carry smooth distributions of vorticity; particles exchange strength to account for the viscous Laplacian operator. The velocity field is efficiently recovered from the vortex particle distribution by the adaptive fast multipole method [3], and the resulting temporal equations are integrated with a 4th-order Runge Kutta method. Finite-time Lyapunov exponent (FTLE) fields are typically obtained from stored velocity field data 4

5 [10, 16]. However, in the VVPM, the vortex particle strengths and positions and not the velocities are stored. In previous work [24], the procedure for obtaining the FTLE field from the vortex particle strength and position data has been described. The basic elements are merely outlined here. As in other FTLE computation procedures [16], an initial set of tracer particles is released from the vertices of a Cartesian grid at time t 0 and advected over an interval T LE, to obtain an approximation of the flow map, φ t 0+T LE t 0, t0 φ t +T LE 0+T LE t 0 (x 0 ) = x 0 + v [ φ τ t 0 (x 0 ), τ ] dτ. (1) t 0 where x 0 is the initial location, and v is the velocity field. The Jacobian of this flow map is not obtained from the usual finite differencing of the flow map on the initial grid, but by simultaneously integrating the velocity gradient along the particle trajectories, dφ t 0+T LE t0 +T LE t 0 (x 0 ) = I + dφτ t0 dx t 0 dx (x 0) v [ φ τ t 0 (x 0 ), τ ] dτ, (2) where I is the 2 2 identity tensor. The stored particle data are used to obtain the velocity and velocity gradient at any tracer particle location. As in the VVPM itself, the fast multipole method is used to accelerate this computation, though close vortex-tracer interactions are computed directly, with second-order smoothing as described in [24]. The FTLE field is defined as σ TLE = 1 T LE log λ max ( ), (3) where λ max ( ) is the maximum eigenvalue of the Cauchy Green deformation tensor,, = ( ) dφ t 0 +T LE t 0 dφ t 0 +T LE t 0, (4) dx dx and () denotes the transpose of the tensor. When T LE > 0, the FTLE field reveals repelling Lagrangian coherent structures (LCS), while when T LE < 0, attracting LCS are uncovered. 5

6 2 Results In this section, the results of two different flows are described. The first flow consists of a 5:1 ellipse starting from rest and traveling to the left at constant velocity and Reynolds number 500 with an angle of attack of 60 degrees, as depicted in Figure 1. At this large angle of attack, the ellipse alternately sheds vortices from the leading and trailing edges. The second flow involves the oscillatory motion of an foil in an otherwise quiescent medium. The foil consists of two elliptical sections connected by a linear torsion spring, also shown in Figure 1. The kinematics of the upper section are prescribed with nearly sinusoidal heaving and pitching kinematics, while the lower section responds passively through deflections of the torsion spring. These kinematics are representative of hovering in many airborne insects [22]. Here, the Reynolds number is 220, based on the peak translational velocity. As mentioned in the introduction, this problem was recently studied by Eldredge and Toomey [20, 7], who performed an extensive parametric study and identified two representative cases: one in which the flexible foil performed poorly (generated weak mean lift) relative to its rigid counterpart; and another in which the flexible foil performed notably better. The LCS of these cases are explored to elucidate the essential differences between the flexible and rigid foils. 2.1 Elliptical foil in steady translation at large angle of attack The VVPM simulation was carried out with an inter-particle spacing (analogous to grid resolution) of 0.01c and time-step size 0.01c/U, where c is the chord length and U is the translational speed. Figure 2 depicts the lift force generated by the ellipse over several shedding cycles. Even at this large angle, the lift remains positive at all times. After the initial development interval, the period of oscillation settles to a value of approximately 4.6. The vortex shedding representative of one such period is illustrated by the sequence of attracting (backward-time) LCS snapshots in Figure 3, in which the ellipse is traveling to the left. An integration time of T LE U/c = 3 has been used to generate the underlying FTLE field. A ridge emanating from the rear face of the ellipse divides the leading and trailing edge vortices. The root of this dividing ridge moves downward toward the 6

7 trailing edge while the leading-edge vortex is being formed, and the ridge gets folded into the shed trailing-edge vortex. Then, the root moves upward toward the leading edge as the trailing-edge vortex is created, and gets folded into the newly-shed leading-edge vortex. Pairs of shed vortices in the wake are connected by the dividing ridge. The attracting LCS shown in Figure 3 clearly identifies the well-known flow structures of the unsteady wake of a separated flow. Similar figures have been shown recently by Brunton and Rowley [1]. It shares an intuitive connection with the streakline visualizations commonly used in experiments [18]. Less intuitive is the repelling LCS, which is shown together with the attracting LCS in Figure 4. In the forward-time FTLE field, which is used to identify the repelling structures, the FTLE values vary more significantly along ridges than in the backward-time field. Thus, it is not possible to simply use a few level sets of the FTLE to visualize the structures. It is possible to use the definition of the LCS as given by [16] by examining the zero level set of the inner product of the FTLE gradient (which is aligned with ridges) and the eigenvector corresponding to the minimum eigenvalue of the FTLE Hessian (which is normal to the ridges) to precisely extract ridges. However, the ridges are adequately clear in the gray-scale contour plot of the FTLE field shown in Figure 4, generated with T LE U/c = 5. The vortices are labeled, with TV denoting a trailing-edge vortex and LV a vortex originally formed at the leading edge. A hyperbolic point H1, along with the behavior of particles in its vicinity, has also been denoted. A large repelling LCS envelops the pair of shed vortices in the immediate wake of the ellipse. The tandem of the attracting and repelling LCS in this region resembles the boundary of a vortex ring [17, 15]. Several repelling structures some less clearly defined also lie upstream of the ellipse, divided by a strong, nearly horizontal ridge extending directly forward of the ellipse. The roles of the attracting and repelling structures can be elucidated by examining the trajectories of blocks of tracer particles, similar to the approach used by Lipinski et al. [11, 2] and Peng and Dabiri [14]. In Figure 5, five blocks of particle are released at tu/c = 5 in strategic regions of the flow and allowed to evolve until tu/c = 8. Note that the colors do not denote any meaningful difference between the blocks; rather, they are used to present a clearer view of the block deformations. The magenta particles in the wake initially straddle an intersection of the attracting and repelling LCS. 7

8 Such an intersection represents a hyperbolic point (H1), and as expected, the particle block is compressed in the direction of the repelling ridge and extended along the attracting curve. Most of these particles get wrapped into the counter-clockwise-rotating vortex (TV1), but the particles that originated outside the repelling boundary remain outside, pulled downward along the external portion of the attracting LCS. The neighboring block depicted in dark green lies completely inside the boundary. Many of the particles in this block lie above and to the left of the dark repelling ridge that hooks around TV1, and these particles are folded into that vortex. Those that originate outside that hook-shaped ridge, however, get drawn into the right vortex (LV1). Thus, the behaviors of particles initially in the wake are clearly demarcated by the combined attracting and repelling LCS. Particles initially in front of the ellipse, in contrast, exhibit more complicated behavior as they are pushed into the wake by the advancing ellipse. Some representative blocks are shown in Figure 5. The particles in blue lie mostly above the strong central repelling ridge projecting in front of the ellipse. Most of these particles travel over the ellipse and are entrained into the leading-edge vortex (LV2) formed during the depicted time interval, whereas the few blue particles initially below the ridge join the green block of particles that travel under the ellipse. Save for a few strays, the green block is split into two primary groups: one group is entrained (along with the black particles starting just ahead of the ellipse) into the previously-shed trailing edge vortex TV1, while the others are pulled into the region between LV2 and the newly-created trailing-edge vortex TV2. Beyond tu/c = 8 (not shown in Figure 5), this latter group is further divided between these two new vortices. The attracting LCS are clarified by the forward marching of particle blocks, but the repelling LCS are more naturally defined by blocks tracked backward in time (since the repelling structures become attracting in this direction). Four such blocks are released at tu/c = 8 and tracked both backward (to tu/c = 5) and forward (to tu/c = 8.5) in time. In Figure 6, the progression of these sets of particles are followed in forward time. At tu/c = 5, one can clearly see the particles aligned with dominant repelling ridges, and the roles of these ridges becomes clear as the particles evolve. Cyan particles initially wrapped up in LV1 at tu/c = 5 are detrained along the repelling LCS that transects the vortex pair. The other cyan particles initially outside the vortex boundary remain 8

9 outside. At tu/c = 8, the block straddles a hyperbolic point at the intersection of the attracting and repelling LCS at the top of the vortex pair. At tu/c = 8.5, the left half of this block is starting to be entrained downward into the counterclockwise vortex. The green block, initially (at tu/c = 5) stretched along the repelling LCS on either side of the hyperbolic point on the lower boundary of this wake vortex pair, is compressed along this ridge and then extended along the attracting LCS. The blue particles are initially split, aligned with two of the repelling LCS in front of the ellipse. One of these LCS, and the associated particles, is pulled above the ellipse into the wake, while the other travels below the ellipse. Both sets of particles are drawn toward the attracting ridge that extends from the rear of the ellipse. The block at tu/c = 8 sits across a hyperbolic point, and the particles behavior is determined by their orientation with respect to this point. Most are drawn upward into the leading-edge vortex, while a few are pulled downward toward the older trailingedge vortex. The final set of particles, depicted in orange, initially reside in a slender finger-shaped lobe that extends directly in front of the ellipse. These particles are pulled towards the ellipse and under it, and are eventually entrained into the newly-created trailing edge vortex. In summary, the repelling coherent structures consist of a series of lobes some nested that project from the front of the ellipse. Each of these lobes is the extension of a repelling LCS in the wake, which itself is paired with an attracting LCS and the associated wake vortex. These projecting lobes thus represent the pockets of fluid entrained into each wake vortex. The lobes are pulled toward the ellipse and into the wake as their companion vortices grow and shed. It is noted that these upstream structures bear some qualitative resemblance to those identified recently by Peng and Dabiri [14] in an inviscid model of a swimming sheet. This process has also been elegantly visualized at low angle of attack and higher Reynolds number by Lipinski et al. [11] and Cardwell & Mohseni [2]. 2.2 Flexible and rigid flapping foils In this section, the LCS of flapping foils both flexible and rigid are examined for two different sets of kinematics. The two cases are simulated with inter-particle spacing of 0.013c (where c is 9

10 the length of each constituent body) and time-step size 0.02c/U max, where U max is the maximum translational velocity. With help of the general knowledge of LCS and lobe structures obtained from the steady translating ellipse in Section 2.1, we can examine the more complex flow structures of the flapping foil by considering LCS and trajectories of tracer particles. A similar investigation to Section 2.1 has been carried out for the first case, whose amplitude of oscillatory translation is A 0 = 11.2c and whose maximum rate of pitching occurs at the instant of translation reversal (i.e. zero pitching phase lead). The pitching lead body varies between ±45 degrees from the vertical axis. The attracting and repelling LCS are shown in Figure 7, at which instant the foil is translating to the right. This snapshot bears some qualitative resemblance to the LCS plots for the ellipse in steady translation: the attracting LCS in the wake exhibit the leading and trailing edge vortices, and the repelling LCS consist of ridges in the wake that transect the attracting structures, along with a strong ridge projecting in front of the foil. Two important hyperbolic points are labeled, along with arrows indicating the direction of particle transport in their vicinity. The ridges are less clearly defined because the FTLE integration interval (±6c/U max ) encompasses a substantial fraction of the flapping period (33.8c/U max ), including acceleration and deceleration of the foil. As revealed by the results of the steady translating foil, the tracer particles are pushed toward hyperbolic points along the repelling LCS and then stretched along the attracting LCS. This is clarified in Figure 8, in which blue particles located above the strong repelling LCS that projects upstream from the front face of the upper body travel over the body and get pulled into the region immediately behind the foil. Light green particles that sit below the blue particles travel below the body and get stretched along the attracting LCS connecting to TV1 at t/t = Some of the dark green particles initially located above the repelling LCS which separates the two attracting LCS in the wake get pulled into the shed leading edge vortex LV1 and others below the repelling LCS get stretched along the trailing edge vortex TV1. The magenta particles straddle hyperbolic points H1 and H2. Some of these particles get pulled toward vortex LV1, and others get drawn toward the attracting LCS that stretches between H1 and H2 as these two points move apart from each other. It is interesting to note that the lower right portion of magenta particles and lower 10

11 portion of blue particles at t/t = 0.97 are initially located inside the same repelling LCS lobe which envelops the rear of the foil and passes through H2. Those particles are eventually pulled along the attracting LCS of the new leading-edge vortex LV2 located just behind the foil at t/t = 1.09 in Figure 8. In the sequence depicted in Figure 8 for the flexible foil, one notes that a small portion of green particles are pulled into the gap between the two bodies. In [19] it was found that the gap has little effect on the aerodynamic performance of the wing. This conclusion extends to the major flow structures, as can be verified by comparing the attracting LCS (now shown in gray scale rather than red) generated by a rigid flapping foil with and without a gap. In Figure 9 the attracting LCS of a rigid flapping foil with a gap shows almost the same behavior as a foil without a gap. In particular, at the moment when the fluid gets pulled into the gap at t/t = 1.03, the leading edge vortex of the foil with a gap has exactly the same configuration and strength of the case without a gap, despite the presence of a small LCS just aft of the gap. Eldredge and Toomey [7] found that the flexible foil generates significantly less lift with these kinematics compared to its rigid counterpart. This is evident in the lift history depicted in Figure 10, in which time has been scaled by the stroke period T = f 1. The mean lift of the last strokes of the flexible foil and the rigid foil is 5.89 and 11.92, respectively, when scaled by ρ f f 2 A 2 0 c (where ρ f is fluid density). It is interesting to inquire whether these differences in aerodynamic performance can be qualitatively connected with the behavior of the flow structures. The attracting LCS of the two foils, depicted in Figure 11 during the latter stages of translation in one half-stroke, reveal significant differences in the wake structures of these foils. After shedding the starting trailing edge vortex, the rigid foil generates a leading-edge vortex and a second trailing-edge vortex, distinctly separated by a dominant ridge that joins the foil with the previous shed vortices. The flexible foil, in contrast, deflects in response to the flapping kinematics, which prevents further trailing-edge vortices beyond the initial one, and stunts the formation of the leading-edge vortex. In fact, the stunted leading-edge vortex is prematurely shed by the flapping foil, as shown in Figure 12 at t/t = 1.53, in which the foil is now in the following half-stroke and traveling to the left. The leading-edge vortex of the flexible foil has been shed into the wake, while the same vortex 11

12 is retained by the rigid foil. At this instant, Figure 10 shows that the lift on the rigid foil is nearly at its peak, while the flexible foil actually has negative lift. These lift values are reflected in the surface pressure distribution. The rigid foil has strong suction on the upper surface due to the presence of the leading-edge vortex, as well as strong upward pressure on the lower surface from its leftward motion. The flexible foil, in stark contrast, only has mild suction on the leading part of the upper surface, a downforce on the rear portion, and a downward suction on the lower surface due to the inverted camber of the deflected foil. In both cases, the local attracting LCS provides an indication of the expected surface pressures in their vicinity. In the second set of kinematics, the translational amplitude is lower, A 0 = 5.6c, and the pitching phase lead is 3π/8. Eldredge and Toomey [7] showed that, at this lower translation amplitude, the flexible foil generates nearly invariant mean lift over a wide range of pitching phase leads, while the lift of the rigid foil is much more sensitive to this phase. At the phase lead studied here, the mean lift over the last stroke of the rigid foil (3.00) is less than one third of that on the flexible foil (9.55). The lift history shown in Figure 13 shows that the lift of the rigid foil changes character after around two strokes, exhibiting large negative incursions, whereas the lift signature of the flexible foil remains largely unchanged. This can be qualitatively explained by examining the attracting LCS at several instants, depicted in Figure 14 over one half-stroke (during which the lift generated by the foils is nearly the same) and 15 for one stroke later (in which the rigid-foil lift has dropped significantly). In the first sequence, the flexible foil generates a series of trailing-edge vortices: the first one due to the pitching and the second one a starting vortex of opposite sign when the foil reverses direction. These vortices pair and propel themselves horizontally and to the right of the foil. The rigid foil, in contrast, generates two trailing-edge vortices of the same sign during the foil pitching. It is the second of these two that pairs with the starting vortex, and the resulting trajectory of the pair is notably more vertically upward. The consequence of the different trajectories taken by these pairs is evident in the following sequence, shown in Figure 15. Previously-shed vortices have not escaped from the vicinity of the rigid foil, and these vortices have left a notable residue in the backward-time FTLE field in the immediate vicinity of the foil. The flexible foil, in contrast, interacts notably less with its previously-generated 12

13 structures. This lack of interaction ensures that lift generated by the flexible foil is more consistent from half-stroke to half-stroke compared to its rigid counterpart. In addition to the trailing-edge vortex system, both foils generate a leading-edge vortex which is recaptured during the following half-stroke. However, there are subtle differences in this recapture process, particularly evident at t/t = 1.42 and at t/t = The vortex is accommodated into the wake of the flexible foil by a slight rearward deflection of the lower foil section, whereas it lingers longer in front of the rigid wing. These differences are further explored in Figure 16, where the surface pressure distribution and local attracting LCS are shown at t/t = At this instant, the lift on the rigid foil has reached a negative trough, whereas the flexible foil maintains positive lift. Despite the flexible foil having only a small deflection, the pressure on the two foils has notable differences. In particular, the flexible foil has mild upward force on the trailing portion of the lower side, whereas the pressure on the rigid foil has opposite sign in this region. The attracting LCS plots in Figure 16, magnified from the corresponding frames of Figure 15, show clearly that the recaptured leading-edge vortex is somewhat smaller in front of the flexible foil than the rigid foil, and the wake structures are oriented differently. On the rigid foil, this vortex exerts a downward suction on the lower section, which provides the dominant contribution to the negative lift. On the flexible foil, the suction is weaker and is more than offset by an upward force arising in reaction to downward recoil of this passive component. 3 Conclusions In this paper, the Lagrangian coherent structures produced by foils in steady translation and oscillatory flapping have been explored. In both cases, tracer particles have elucidated the role of the repelling coherent structures, particularly the series of lobes that project from the front of the foil and intersect with attracting structures in the wake. These lobes demarcate the regions from which fluid is entrained into the wake vortices. These findings may have important consequences for the response of a wing or fin to incident disturbances (e.g. gusts, wakes of upstream wings or fins), since the nature in which the wing or fin receives such a disturbance has a notable effect on 13

14 the forces it generates. The flow structures are depicted with much better definition in the FTLE field compared to the vorticity or velocity fields, as can be confirmed by comparing with the previously reported results [7]. In flapping foils, the improved definition provided by the LCS has been used to elucidate the differences in force generation between flexible and rigid foils. In particular, it has been shown that the global behavior of the attracting LCS is qualitatively connected with the lift generated by the foil. For large amplitude flapping, a flexible foil tends to prematurely shed its leading-edge vortex revealed clearly by the LCS which leads to a rapid fall in lift. For lower amplitude flapping, a flexible foil that pitches prior to stroke reversal generates consistently high lift, whereas a rigid wing experiences a large overall drop. The LCS analysis clearly reveals differences in pairing behavior of the shed vortices in the two foils, and consequently the rigid foil shows greater tendency to interact with its own wake. Though we have attempted to use instantaneous pressure distributions to make this connection between flow structures and force generation more explicit, the relationship has only been described qualitatively in this paper. It would clearly be desirable to develop a more quantitative connection between the LCS and force generation. Attempts of this sort have been made recently, by indirectly accounting for forces through momentum changes in the wake [13]. With further progress in this regard, LCS identification can be an important element in the ongoing quest for reduced-order modeling of unsteady aerodynamic problems. Support for this work by the National Science Foundation, under award CBET , is gratefully acknowledged. References [1] S. L. Brunton and C. W. Rowley. Modeling the unsteady aerodynamic forces on small-scale wings. AIAA Paper ,

15 [2] B. M. Cardwell and K. Mohseni. Vortex Shedding over a Two-Dimensional Airfoil: Where the Particles Come from. AIAA J., 46: , [3] J. Carrier, L. Greengard, and V. Rokhlin. A fast adaptive multipole algorithm for particle simulations. SIAM J. Sci. Stat. Comput., 9(4): , [4] M. H. Dickinson and K. G. Götz. Unsteady aerodynamic performance of model wings at low Reynolds numbers. J. Exp. Biol., 174:45 64, [5] J. D. Eldredge. Numerical simulation of the fluid dynamics of 2D rigid body motion with the vortex particle method. J. Comput. Phys., 221: , [6] J. D. Eldredge. Dynamically coupled fluid-body interactions in vorticity-based numerical simulations. J. Comput. Phys., 227: , [7] J. D. Eldredge and J. Toomey. On the roles of chord-wise flexibility in a flapping wing with hovering kinematics. Submitted to J. Fluid Mech. [8] C. P. Ellington. The aerodynamics of hovering insect flight. IV. Aerodynamic mechanisms. Phil. Trans. R. Soc. Lond. B, 305:79 113, [9] F. E. Fish and G. V. Lauder. Passive and active flow control by swimming fishes and mammals. Annu. Rev. Fluid Mech., 38: , [10] G. Haller. Lagrangian coherent structures from approximate velocity data. Phys. Fluids, 14(6): , [11] D. Lipinski, B. Cardwell, and K. Mohseni. A Lagrangian analysis of a two-dimensional airfoil with vortex shedding. J. Phys. A, 41(34):344011, [12] D. Lipinski and K. Mohseni. Flow structures and fluid transport for the hydromedusae Sarsia tubulosa and Aequorea victoria. J. Exp. Biol., 212(15): , [13] J. Peng and J. O. Dabiri. An overview of a Lagrangian method for analysis of animal wake dynamics. J. Exp. Biol., 211: , [14] J. Peng and J. O. Dabiri. The upstream wake of swimming and flying animals and its correlation with propulsive efficiency. J. Exp. Biol., 211: ,

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17 List of Captions 1. Kinematics of steady translating ellipse (left) and oscillatory flapping foil (right). 2. Lift history on ellipse in steady translation. 3. Attracting LCS of ellipse in steady translation, from tu/c = 5 to 8.5, in 0.5 increments. Panels should be read from top to bottom, then left to right. 4. Repelling (gray) and attracting (red) LCS of an ellipse in steady translation at time tu/c = Wing in steady translation, with different sets of tracer particles. Repelling (gray) and attracting (red) LCS, and instantaneous tracer particle locations, at times tu/c = 5 (top), 5.5 (center) and 6 (bottom). 5. (cont d) tu/c = 6.5 (top), 7 (center) and 7.5 (bottom). 5. (cont d) tu/c = LCS of ellipse in steady translation. Repelling (gray) and attracting (red) LCS, and instantaneous tracer particle locations, at times tu/c = 5 (top), 5.5 (center) and 6 (bottom). 6. (cont d) tu/c = 6.5 (top), 7 (center) and 7.5 (bottom). 6. (cont d) tu/c = 8 (top), 8.5 (bottom). 7. Repelling (gray) and attracting (red) LCS of a flexible flapping foil with A 0 /c = 11.2 and zero pitching phase lead at t/t = LCS of flexible flapping foils with A 0 /c = 11.2 and zero pitching phase lead. Repelling (gray) and attracting (red) LCS, and instantaneous tracer particle locations, at times t/t = 0.97 (top), 0.99 (center) and 1.00 (bottom). 8. (cont d) t/t = 1.01 (top), 1.03 (center) and 1.04 (bottom). 8. (cont d) t/t = 1.06 (top), 1.07 (center) and 1.09 (bottom). 17

18 9. Attracting LCS of rigid flapping foils without gap (left column) and with gap (right column) with A 0 /c = 11.2 and zero pitching phase lead. Rows correspond to times t/t = 1.03 and Lift generated by a flexible ( ) and rigid ( ) foil with amplitude A 0 /c = 11.2 and zero pitching phase lead. 11. Attracting LCS of flexible (left column) and rigid (right column) flapping foils with A 0 /c = 11.2 and zero pitching phase lead. Rows correspond to times t/t = 0.97, 1.03, 1.09, 1.15, Surface pressure distribution (top) and attracting LCS (bottom) at t/t = 1.53 for flexible (left) and rigid (right) foils with amplitude A 0 /c = 11.2 and zero pitching phase lead. 13. Lift generated by a flexible ( ) and rigid ( ) foil with amplitude A 0 /c = 5.6 and pitching phase lead of 3π/ Attracting LCS of flexible (left column) and rigid (right column) flapping foils with A 0 /c = 5.6 and pitching phase lead of 3π/8. Rows correspond to times t/t = 1.06, 1.18, 1.30, 1.42, Attracting LCS of flexible (left column) and rigid (right column) flapping foils with A 0 /c = 5.6 and pitching phase lead of 3π/8. Rows correspond to times t/t = 2.13, 2.24, 2.36, 2.48, Surface pressure distribution (top) and attracting LCS (bottom) at t/t = 2.36 for flexible (left) and rigid (right) foils with amplitude A 0 /c = 5.6 and pitching phase lead of 3π/8. 18

19 Figure 1: Kinematics of steady translating ellipse (left) and oscillatory flapping foil (right). 19

20 F y /ρu 2 c tu/c Figure 2: Lift history on ellipse in steady translation. 20

21 Figure 3: Attracting LCS of ellipse in steady translation, from tu/c = 5 to 8.5, in 0.5 increments. Panels should be read from top to bottom, then left to right. 21

22 Figure 4: Repelling (gray) and attracting (red) LCS of a ellipse in steady translation at time tu/c = 5. 22

23 Figure 5: Wing in steady translation, with different sets of tracer particles. Repelling (gray) and attracting (red) LCS, and instantaneous tracer particle locations, at times tu/c = 5 (top), 5.5 (center) and 6 (bottom). 23

24 Figure 5: (cont d) tu/c = 6.5 (top), 7 (center) and 7.5 (bottom). 24

25 Figure 5: (cont d) tu/c = 8. 25

26 Figure 6: LCS of ellipse in steady translation. Repelling (gray) and attracting (red) LCS, and instantaneous tracer particle locations, at times tu/c = 5 (top), 5.5 (center) and 6 (bottom). 26

27 Figure 6: (cont d) tu/c = 6.5 (top), 7 (center) and 7.5 (bottom). 27

28 Figure 6: (cont d) tu/c = 8 (top), 8.5 (bottom). 28

29 Figure 7: Repelling (gray) and attracting (red) LCS of a flexible flapping foil with A 0 /c = 11.2 and zero pitching phase lead at t/t =

30 Figure 8: LCS of flexible flapping foils with A 0 /c = 11.2 and zero pitching phase lead. Repelling (gray) and attracting (red) LCS, and instantaneous tracer particle locations, at times t/t = 0.97 (top), 0.99 (center) and 1.00 (bottom). 30

31 Figure 8: (cont d) t/t = 1.01 (top), 1.03 (center) and 1.04 (bottom). 31

32 Figure 8: (cont d) t/t = 1.06 (top), 1.07 (center) and 1.09 (bottom). 32

33 Figure 9: Attracting LCS of rigid flapping foils without gap (left column) and with gap (right column) with A 0 /c = 11.2 and zero pitching phase lead. Rows correspond to times t/t = 1.03 and

34 F y / ρ f c f 2 A t/t Figure 10: Lift generated by a flexible ( ) and rigid ( ) foil with amplitude A 0 /c = 11.2 and zero pitching phase lead. 34

35 Figure 11: Attracting LCS of flexible (left column) and rigid (right column) flapping foils with A 0 /c = 11.2 and zero pitching phase lead. Rows correspond to times t/t = 0.97, 1.03, 1.09, 1.15,

36 Figure 12: Surface pressure distribution (top) and attracting LCS (bottom) at t/t = 1.53 for flexible (left) and rigid (right) foils with amplitude A 0 /c = 11.2 and zero pitching phase lead. 36

37 F y / ρ f c f 2 A t/t Figure 13: Lift generated by a flexible ( ) and rigid ( ) foil with amplitude A 0 /c = 5.6 and pitching phase lead of 3π/8. 37

38 Figure 14: Attracting LCS of flexible (left column) and rigid (right column) flapping foils with A 0 /c = 5.6 and pitching phase lead of 3π/8. Rows correspond to times t/t = 1.06, 1.18, 1.30, 1.42,

39 Figure 15: Attracting LCS of flexible (left column) and rigid (right column) flapping foils with A 0 /c = 5.6 and pitching phase lead of 3π/8. Rows correspond to times t/t = 2.13, 2.24, 2.36, 2.48,

40 Figure 16: Surface pressure distribution (top) and attracting LCS (bottom) at t/t = 2.36 for flexible (left) and rigid (right) foils with amplitude A 0 /c = 5.6 and pitching phase lead of 3π/8. 40