An overview of a Lagrangian method for analysis of animal wake dynamics

Vortices are thought to play an important role in the mechanisms of animal swimming and flying due to their prominence in the fluid surrounding the animal. For example, vortex formation has been identified as the key mechanism that enables many insects to generate sufficient lift in flight(Maxworthy, 1979; Ellington et al., 1996; Willmott et al., 1997; Sane, 2003). These animals rely on the stably attached leading-edge vortex created by the insect wing during flapping motions; the presence of this vortex greatly enhances the forces used to hover and maneuver. Although this leading-edge vortex mechanism is not as commonly observed in swimming animals, the vortices generated during aquatic locomotion also appear to affect thrust, maneuvering and propulsive efficiency, i.e. the ratio of useful work for locomotion to the total mechanical energy input (since the motion is typically unsteady, propulsive efficiency is non-zero). Many examples can be found in the swimming of medusae, amphibians, fishes, marine mammals, etc. (e.g. Drucker and Lauder, 1999; Wilga and Lauder, 2004; Bartol, 2005; Dabiri et al., 2005; Stamhuis and Nauwelaerts,2005).

Most studies investigate momentum transfer from the animal to the fluid in the form of vortices, with the ultimate goal of quantifying locomotive forces and understanding the role of vortices in swimming and flying mechanisms. Indeed, the presence of vorticity, or the resulting circulation to be more precise, is necessary in steady locomotion (cf. Kutta–Joukowski theorem). However, vortices are not solely responsible for animal locomotion(Schultz and Webb, 2002). For example, Kanso et al. (Kanso et al.,2005) demonstrated that, in theory, vorticity/circulation need not be present at all in order for an articulated body to achieve unsteady locomotion [see Saffman (Saffman,1967) for an original proof-of-concept]. If vorticity is not necessary for unsteady locomotion, then perhaps there exist circumstances in which it is also not sufficient to achieve that locomotion. And, if the presence of vorticity is not sufficient to achieve certain modes of locomotion, then it is not likely that a study of vorticity alone can deduce the locomotive forces in those cases. These fundamental issues (including the question of whether unsteady locomotion in air and water may have actually arisen in spite of the presence of vorticity) have received relatively little attention thus far.

When an animal moves through fluid, Newton's second and third laws together dictate that the locomotive force exerted by the fluid on the animal has a magnitude equal to the rate at which the animal imparts momentum to the fluid. To be sure, viscous dissipation and vorticity cancellation will reduce the efficiency of the momentum transfer process from 100 per cent, resulting in an`information loss' in the record of locomotive dynamics contained in the wake. However, a simple viscous scaling argument shows that these effects are usually negligible on the time scale of individual stroke cycles. In particular, the distance δ over which viscosity will act during a single stroke of duration T_S isδ≈(νT_s)^1/2, where ν is the kinematic viscosity of the fluid(Rosenhead, 1963). Regions of opposite-signed vorticity (e.g. shed from the anterior and posterior edges of a fin or wing) must be within this distance(νT_s)^1/2 from each other in order to undergo vorticity cancellation and the associated information loss in the wake. For repeated swimming or flying motions at frequency f_S, the scaling is equivalentlyδ≈(ν/f_s)^1/2. Hence, information loss in the wake becomes important if the ratioδ/L≈(ν/f_s)^1/2/Lis of the order of one or larger, where L is the characteristic length scale of the appendage [the reader should recognize this as effectively the inverse square root of the Reynolds number (cf. White, 1991)]. A 1 Hz swimming motion in water (ν≈10^–2 cm²s^–1) corresponds to a characteristic viscous length scaleδ of ∼1 mm, which is substantially smaller than the length scales of most fish appendages (although not necessarily small for swimming micro-organisms). In air (ν≈10^–1 cm²s^–1) at 1 Hz, δ≈3 mm, which is also smaller than the length scales of most bird appendages. Insects may have appendage length scales of this order but will also operate at much higher frequencies, thereby reducing the length scale δ. Therefore, for the near-wake(vis-à-vis far downstream) studies of concern here in which the ratioδ/L is small, we will assume no information loss between the dynamics of the animal and the wake it generates.

The development of visualization and measurement techniques, especially digital particle image velocimetry (DPIV), has given researchers the ability to quantify kinematics and dynamics of the animal wake (e.g. Muller et al., 1997; Drucker and Lauder, 1999; Nauwelaerts et al., 2005; Spedding et al., 2003; Warrick et al., 2005). To interpret the wake measurements, several models have been proposed to estimate momentum transfer and evaluate locomotive forces. For example, locomotive forces experienced by the animal are calculated as the reaction to the momentum of vortex loops shed into the wake (e.g. Drucker and Lauder, 1999; Drucker and Lauder, 2001; Johansson and Lauder, 2004; Stamhuis and Nauwelaerts,2005). In these cases, the momentum of the vortex is usually measured at the time instant when the vortex ring has just detached from the animal fin/wing. The time-averaged locomotive force over the stroke cycle is then determined by dividing the momentum of the shed vortex by the time duration of the stroke cycle. In other studies, the locomotive forces have been evaluated by examining the wake far downstream, which is equivalent to taking the time average of dynamics occurring at the site of force generation(e.g. Spedding et al., 2003; Walker, 2004).

Determination of wake vortex added mass in an unsteady flow depends critically on identification of the physical boundary of the vortex. Using a concept from dynamical systems called Lagrangian coherent structures (LCS),the boundary of the vortex in a wake can be determined by tracking fluid particles in the wake and searching for material lines that are separatrices,effectively partitioning flow regions with different dynamics(Haller, 2001; Shadden et al., 2006).

In this paper, we review a recently developed analytical framework to empirically deduce unsteady swimming and flying forces based on the measurement of velocity and vortex added mass in the animal wake. Given velocity field measurements in the wake, the vortex boundary can be determined and the momentum of the wake vortex and its added mass can be calculated,leading to a quantitative evaluation of instantaneous locomotive forces. The organization of this paper is as follows: (1) the LCS approach used to identify the boundary of the wake vortex is described; (2) the momentum of the wake vortex and locomotive forces are determined based on the morphology and kinematics of the vortex; (3) implementation of the method using 2-D velocity field data from DPIV measurements is explored.

As in most fluid dynamics problems, the potential approaches for vortex boundary identification fall into one of two basic categories: Eulerian (i.e. fixed in space) or Lagrangian (i.e. moving with individual fluid particles). In most studies of animal vortex wakes, the vortex structure is determined from Eulerian data, using instantaneous vorticity or streamlines. For example,wake vortices have been previously identified by locating regions with vorticity above a given threshold (e.g. Drucker and Lauder, 1999; Drucker and Lauder, 2001; Stamhuis and Nauwelaerts,2005). Few studies use streamlines to identify vortex structures in the animal wake; streamlines are able to give a clearly defined vortex boundary in a purely steady flow (e.g. Hill's spherical vortex) but are of less use in the highly unsteady flows characteristic of swimming and flying. A method of coordinate transformation has been previously developed to expand the utility of streamlines in time-dependent flow(Dabiri and Gharib, 2004), but its use is limited to unsteady cases where a single characteristic velocity can be identified in the wake, e.g. in the isolated vortex ring-dominated wakes generated by some jellyfish, squids and salps.

An alternative is to study the wake from a Lagrangian perspective. Instead of studying the instantaneous velocity/vorticity field, fluid particle trajectories are used as the fundamental variable. By following fluid particle trajectories, vortices tend to emerge from the wake as coherent structures since, at the Reynolds numbers of relevance to animal locomotion, fluid particles remain inside a vortex over long convective time scales relative to fluid particles outside the vortex(Provenzale, 1999). An exact criterion for defining vortex boundaries in unsteady flows was introduced in a series of papers by Haller (Haller,2001; Haller,2002; Haller,2005); the boundaries are referred as Lagrangian coherent structures (LCS). Recently, an approach using the finite-time Lyapunov exponent field (FTLE) to locate LCS was developed(Shadden et al., 2005; Shadden et al., 2006). This FTLE approach is preferable because of its relative simplicity and wide compatibility with other methods used to locate LCS in time-dependent flows[e.g. hyperbolic time approach (Haller,2001)].

The mathematical derivation is briefly summarized in Appendix A; however,the reader is directed to the aforementioned papers for greater technical detail.

As a proof of the concept, an example of the FTLE analysis is shown in Fig. 2(Shadden et al., 2006). A vortex ring is generated in a water tank by a piston that accelerates fluid from right to left through the open end of a cylinder. Velocity field data on the median symmetry plane of the vortex is taken by DPIV. The FTLE field is calculated for the moving vortex ring on this median symmetry plane. The analysis is carried out in both backward time and forward time. The ridges of high values of FTLE indicate the geometry of the LCS. Whereas the vortex geometry cannot be determined from inspection of the velocity field or the corresponding vorticity field, the entire vortex boundary is revealed by combining the forward- and backward-time LCS.

The momentum of the vortex wake consists of two components: the linear momentum of the fluid inside the vortex and the linear momentum of the fluid surrounding the vortex that is brought into motion as the vortex accelerates.

The latter component of momentum arises from the added mass of the wake vortex and is identical to the added mass traditionally associated with fluid surrounding solid bodies in potential flow. The added mass is dependent on the shape of the body and can be determined using the Kirchhoff potential (see Appendix B for details). The boundary of a vortex ring (or, in 2-D, a vortex dipole) can be approximated by an ellipsoid (or, in 2-D, an infinitely long cylinder with an elliptical cross-section) whose added mass is given by an analytical expression (Lamb,1932).

In this section, we discuss how to carry out the analyses described in the previous section by using a data set typically available to investigators studying animal swimming and flying: a time series of 2-D DPIV velocity fields[or equivalent computational fluid dynamics (CFD) data]. These velocity measurements are usually presented in a Eulerian frame, for which the velocity is defined at fixed locations in space and at a series of discrete instants in time. The locations in space at which velocity is measured usually form a structured grid with rectangular elements.

To determine the vortex boundary, the FTLE is calculated on a Cartesian grid defined in the region of the flow where the vortex exists. The flow map

A color contour plot of the FTLE field and the application of a threshold are usually sufficient for the purpose of identifying the LCS. A more precise,mathematical approach to extract the LCS from the FTLE field is provided by Shadden (Shadden, 2006). There, the Hessian and the gradient of the FTLE field are calculated. Since the eigenvector of the Hessian corresponding to its minimum eigenvalue is tangent to the LCS and the gradient is normal to the LCS, a scalar field can be formed by taking the inner product of the two vector fields. LCS are extracted as zero-valued level sets.

We have developed an in-house MATLAB code to analyze experimental DPIV data or CFD data and to compute the corresponding FTLE field. The software, LCS MATLAB Kit version 1.0, can be downloaded at http://dabiri.caltech.edu/software.html. A more robust C-language software for this type of calculation is MANGEN,developed by F. Lekien and C. Coulliette. This package is also available online.

Some results from an analysis of the wake generated by a bluegill sunfish pectoral fin (Peng et al.,2007) are shown here for illustration. The repelling and attracting LCS (see Appendix A) in the wake are shown in Fig. 3, superimposed on the velocity and vorticity fields. The evolution of the vortex boundary is shown in Fig. 4. These boundaries are assumed to lie on a symmetry plane. Another symmetry plane is assumed to exist normal to this first plane and equidistant from the vortex cores. The 3-D vortex structure is approximated based on this assumption of two planes of spatial symmetry. The vortex volume V_V and the velocity U_V together with the added-mass coefficient matrix Ccan then be determined and the locomotive forces evaluated. The results for the locomotive forces are shown in Fig. 5. Since two components of velocity U_V are known, forces in two directions (i.e. lateral and vertical) can be determined.

In this paper we have reviewed a framework for combining traditional DPIV measurements with a new class of Lagrangian analysis tools to analyze animal vortex wakes. The Lagrangian analysis provides clearly defined vortex boundaries in unsteady flows, a capability not offered by Eulerian analysis based on instantaneous vorticity or velocity fields. The information regarding the animal wake vortex boundary enables the determination of vortex added mass, which is a key component of locomotive forces in unsteady wakes. Using this framework, instantaneous forces, rather than time-averaged forces over a stroke cycle, can be determined. These instantaneous forces dictate important dynamics of locomotion such as the trajectory, speed and efficiency of swimming and flying.

As revealed in the example of sunfish pectoral fin wake, the fin is embedded within this wake vortex structure. Since it is known that the vortex is attached to the fin, this result suggests that the dynamical effect of the attached wake vortex on locomotion is to replace the real animal fin with an`effective appendage', whose kinematics are determined by the external forces acting on it, e.g. the force the fish exerts to move the fin through the water in the example above. Furthermore, in the limit of irrotational, inviscid flow the `effective appendage' reduces to the fin itself, and the only dynamical contribution for locomotion comes from the added mass of the fin. Thus, the`effective appendage' concept provides a bridge between theoretical studies of locomotion in inviscid flows (e.g. Kanso et al., 2005) and the dynamics of real animals.

Significant discrepancies can exist between the vortex boundary dictated by LCS and the spatial distribution of vorticity in the flow (e.g. Fig. 3). There are two primary sources for this disagreement. First, viscous diffusion occurring at these finite Reynolds numbers enables vorticity to cross the flow boundaries defined by the LCS, even when these boundaries form perfect barriers to fluid transport. A similar effect has been observed in studies of isolated vortex rings (Dabiri and Gharib, 2004; Shadden et al., 2006), and in the kinematics of boundary layer vorticity that defines `displacement thickness' in steady flows (Rosenhead,1963). The second and probably more dominant effect is that of appendage rotation, e.g. the anteroventral rotation of the sunfish pectoral fin during its downstroke. Haller (Haller,2005) has shown that in flows with global rotation, the vorticity field can be a poor indicator of vortex boundaries. Hence, in these unsteady flows it is possible to lose a correlation between the wake vortex boundary dictated by the LCS and the spatial distribution of vorticity. The dynamical effect of vorticity external to the LCS is a topic of ongoing study.

When the present analytical framework is applied to 3-D measurements, it can give much more accurate force estimates than in 2-D studies. With 3-D DPIV data, the boundary of the vortex can be determined directly instead of being approximated. The volume and added mass of the vortex can then be accurately quantified without making assumptions on the vortex structure. Identification of 3-D vortex boundaries would also enable evaluation of the effect of rotational added mass, which is neglected in the present study. Although 3-D flow visualization and measurement techniques have been developed and implemented (e.g. Pereira et al.,2000), they are not yet in use in studies of animal swimming and flying. In the meantime, the approach described here can be used to interpret 2-D DPIV measurement data already available to most researchers in the field. Estimation of instantaneous, unsteady locomotive forces can be made from these 2-D DPIV data, as shown in the example above. A potential improvement using current 2-D DPIV is simultaneous data collection in multiple perpendicular planes, which can give additional 2-D geometric information regarding the 3-D vortex structure.

At this juncture, it is fair to ask how closely the estimates that are currently deduced from 2-D measurements – both in the present work and elsewhere in this field of study – agree with the `correct' answer. Validation using numerically defined canonical flow fields is the goal of an ongoing study. Validation within the context of animal swimming would be difficult at present since it is not yet possible to simulate fully coupled fluid-structure dynamics of self-propelled animals where the body motion (e.g. flow-induced fin deformation) is solved iteratively rather than being fixed or prescribed a priori. Comparison with experimental measurements is also difficult in the absence of instantaneous fluid dynamic measurements(velocity field, forces and moments) that can be made simultaneously with recordings of instantaneous body dynamics. Until then, we can only achieve agreement amongst the various methods for force estimation. Recent results(e.g. Peng et al., 2007)suggest that the instantaneous force estimation techniques described here are compatible with time-averaged forces deduced from traditional vorticity studies and with qualitative observations of animal body dynamics.

It is important to note that though the FTLE

The integration time |T| is chosen according to the particular flow being analyzed. If a smaller integration time is used, then less of the boundary is revealed, whereas if a longer integration time is used, more of the boundary is revealed (i.e. relative differences in fluid particle behavior become more apparent when observed over longer periods of time). Generally, if the integration time |T| is sufficiently long, the repelling and the attracting LCS usually intersect to give the boundary of the vortex [in cases where a vortex is known to be present; cf. Fig. 6 in Shadden et al.(Shadden et al., 2006)]. A larger integration time |T| also gives LCS with higher spatial resolution. However, the choice of |T| is sometimes limited in practice by the availability of data.

The authors thank C. P. Ellington and J. L. van Leeuwen for organizing the conference session in which this paper was originally presented in August 2006; M. S. Gordon for comments on the manuscript; and G. L. Brown, J. E. Marsden and P. Moin for enlightening discussions. The authors are also grateful to the anonymous referee for valuable suggestions that have led to several improvements in the manuscript. This research is funded by a grant from the Ocean Sciences Division, Biological Oceanography Program at NSF (OCE 0623475) to J.O.D.