Locomotor forces on a swimming fish: three-dimensional vortex wake dynamics quantified using digital particle image velocimetry

Measuring the forces exerted by organisms on the fluids through which they move is a difficult proposition. On land, devices such as force plates allow direct measurement of the force applied by locomoting animals to the ground, but in the aquatic and aerial media, such devices are not applicable. As a result, a number of alternative approaches have been used in an effort to understand the mechanisms by which organisms propel themselves in fluids. For example, considerable effort has been expended in developing models of the process by which swimming and flying organisms generate locomotor force. Early theoretical models involved steady-or quasi-steady-state analyses that employed time-invariant lift and drag coefficients (e.g. Weis-Fogh, 1973; Blake, 1983). For swimming animals, such locomotor mechanisms have been termed ‘lift-based’ or ‘drag-based’ (for reviews, see Webb and Blake, 1985; Vogel, 1994). More recent models have elaborated a vortex theory of propulsion based on time-dependent circulatory forces rather than estimated force coefficients (Rayner, 1979a,b; Ellington, 1984b, c). Unsteady mechanisms, such as the ‘clap and fling’ (Lighthill, 1973; Weis-Fogh, 1973), take into account rotation as well as translation of the propulsor and acknowledge the complex time history of force development (see also Dickinson, 1994; Dickinson and Götz, 1996). Recently, the use of supercomputers to solve the unsteady Navier–Stokes equations for numerous grid points around a model animal has provided insights into the dynamics of locomotor force production (Carling et al., 1994, 1998; Liu et al., 1997, 1998). By deforming the model animal, a time-dependent estimate of the forces exerted on the fluid can be calculated. Despite these advances in computational approaches, however, the difficulty in empirically measuring forces exerted by freely moving organisms on their fluid surroundings remains a major technical challenge that has limited our ability to evaluate vortex models and to understand how animals in fluids generate thrust.

The hallmark of force production in fluids is the shedding of vorticity, which reflects a transfer of momentum into the wake. Previous attempts to study biological wake vorticity have included the tracking of dye fronts, smoke and buoyant particles introduced into the fluid through which an animal moves (for reviews, see Stamhuis and Videler, 1995; Müller et al., 1997), shadowgraphic projection (McCutchen, 1977) and Schlieren imaging (Strickler, 1975). Flow visualizations of this sort provide qualitative insight into patterns of fluid movement, but are difficult to quantify especially when flow is three-dimensional. An alternative approach is to study flow in orthogonal thin (approximately two-dimensional) sections of the three-dimensional wake. Based on the original technique of Vogel and Feder (1966), researchers have developed means of illuminating particles suspended in the flow with sheets of light to investigate vorticity in the wake of flying birds (Kokshaysky, 1979), flying insects (Grodnitsky and Morozov, 1992; Brodsky, 1994; Dickinson and Götz, 1996) and swimming fish (Müller et al., 1997). Three-dimensional clouds of particles imaged with stereophotography have also been mathematically sectioned to allow analysis of the wake of birds and bats in flight (Spedding et al., 1984; Rayner et al., 1986; Spedding, 1986). These studies have provided valuable information about wake morphology and energetics, yet involved time-intensive calculations that limited the number of propulsor strokes subjected to full quantitative analysis. Additionally, unavoidable gaps in the cloud of particles within the fluid resulted in non-uniform distributions of velocity vectors in the reported flow fields.

In the present study, we employ digital particle image velocimetry (DPIV) to examine the vortex wake shed by freely swimming fish. As opposed to the manual tracking of individual particles, this sheet illumination technique involves rapid computational processing of time-variant patterns of particle images (nearly uniformly distributed), defined by pixel intensity in digitized video fields (Raffel et al., 1998). The specific objective of this study was to use the DPIV approach to quantify the forces experienced by bluegill sunfish during labriform locomotion through an analysis of the pectoral fin wake. As a first step, we described the geometry of vortex flow downstream of the pectoral fin and examined the dynamics of vortex formation over the course of the fin-stroke cycle. Unlike previous analyses of fish swimming using particle image velocimetry or particle tracking in a single plane (Stamhuis and Videler, 1995; Müller et al., 1997), we quantified flow patterns in three perpendicular planar transections of the wake using an experimental arrangement that permitted determination of the location of the fins within each flow field. Our goal was to maximize accuracy in reconstructing wake geometry and estimating locomotor forces. By collecting DPIV data in three dimensions, we were able to quantify not only locomotor lift and thrust but also laterally directed forces exerted on the fluid to obtain a full three-dimensional resolution of the reaction forces acting on a freely swimming fish.

Bluegill sunfish (Lepomis macrochirus Rafinesque) were collected by seine from ponds in Newport Beach, CA, USA. Animals were maintained at an average temperature of 20 °C in 40 l aquaria and fed earthworms twice weekly. Eight individuals of similar size (total body length, L, 20.3±0.9 cm, mean ± S.D.) were selected for study.

Sunfish swam individually at 20 °C in the working area (28 cm×28 cm×80 cm) of the recirculating freshwater flow tank described previously by Jayne and Lauder (1995). Animals were induced to hold station at three flow speeds: 0.5 L s⁻¹, at which propulsion is achieved by oscillation of the pectoral fins alone; 1.0 L s⁻¹, the speed of transition from purely labriform swimming to combined pectoral and caudal fin oscillation (U_p-c) (see Gibb et al., 1994); and 1.5 L s⁻¹, a speed at which burst-and-coast caudal fin propulsion is supplemented by intermittent pectoral fin beating. Only steady rectilinear swimming, during which the fish maintained a speed within 5 % of the flow tank’s current speed, was considered for analysis. In addition, the fish was required to swim near the center of the volume of the working area; the minimum allowable distance between any wall and the pectoral fin tip was 5 cm. At lesser distances, hydrodynamic ‘wall effects’ (i.e. interactions of the wake with flume surfaces; Webb, 1993) were apparent from direct observation of the downstream flow. We thus ensured that only unimpeded flow structures resulting directly from movements of the fish’s pectoral fins were chosen for analysis.

A digital particle image velocimetry system was developed to visualize and analyze quantitatively fluid flow in the pectoral fin’s wake (Fig. 1A). Water in the flow tank was seeded at a density of 14 mg l⁻¹ with silver-coated glass spheres (mean diameter 12 μm, density 1.3 g cm⁻³). Although slightly negatively buoyant, the spheres remained suspended in the water as a cloud of reflective particles during the course of each experiment, and particle sinking as flow traversed the working area was negligible. Light from a 3 W continuous-wave argon-ion laser was focused into a sheet 1–2 mm thick and 10 cm wide to illuminate near-fin particle motion. Two synchronized high-speed video cameras (NAC HSV-500) were used to record images of the swimming fish and its wake at 250 fields s⁻¹. One camera imaged particle reflections, while a second provided a perpendicular view of the left pectoral fin’s position in relation to the laser sheet. The signals from the cameras were recorded as split-screen images, facilitating simultaneous quantification of particle motion and the location and orientation of the left fin within the flow field (Fig. 1B).

To elucidate the three-dimensional structure of the wake, a series of front-surface mirrors was used to align the light sheet, in separate experiments, in three orthogonal planes: frontal (XZ), parasagittal (XY) and transverse (YZ) (Fig. 2). Thus, data presented below for perpendicular light sheets were not collected simultaneously. For each laser orientation, flow around the fin was examined in a variety of planar sections ranging from the fin base to the distal fin tip. Data presented here are from planes positioned approximately half-way along the span of the abducted pectoral fin, the location at which wake vortices were best defined. DPIV data were collected from five fish to establish general vortex flow patterns. Fifty-five pectoral fin strokes performed by two of these individuals at 0.5 L s⁻¹ were selected for a detailed analysis of the wake produced by slow labriform swimming. In total, 143 image pairs were extracted from these strokes and analyzed quantitatively to characterize planar flow patterns near the fin over the course of the fin-beat cycle. An additional 40 fin beats and 117 image pairs were examined at 1.0 and 1.5 L s⁻¹ to allow quantification of vortex flow at higher speeds.

After review of video tapes to select scenes of steady swimming, pairs of successive video images (separated by the framing period Δt=4 ms) were digitized at eight bits per pixel (grayscale) and downloaded to hard disk. The portion of each digital field available for processing (i.e. containing particle images) was 440 horizontal × 480 vertical pixels. DPIV recordings were quantitatively evaluated by means of two-frame cross-correlation (Raffel et al., 1998). Image interrogation was performed computationally using a 20×20 grid (Insight v. 3.0 software, TSI Inc., St Paul, MN, USA). The displacement of particle images over Δt was analyzed by cross-correlation to yield two-dimensional flow fields, typically 6–8 cm on each side, each consisting of 400 uniformly distributed water velocity vectors. Analysis of these vector diagrams at selected times during the fin beat, and in different perpendicular planes, allowed reconstruction of the time course of development and three-dimensional geometry of the pectoral fin’s wake. Reconstruction of the wake from non-simultaneous orthogonal flow fields (Fig. 2) assumes low among-stroke (intra-individual) variation in the three-dimensional pectoral fin kinematics of Lepomis macrochirus, as documented by Gibb et al. (1994).

Post-processing of DPIV data began with validation of the velocity vectors. A dynamic mean value algorithm was used to reject each vector whose velocity markedly exceeded the average velocity of its eight nearest neighbors (Raffel et al., 1998). Validated vectors that clearly misrepresented the actual flow were detected by visual inspection of each vector field and were deleted manually. When they occurred, such erroneous vectors were at the edges of the data field and near the fish’s fin (i.e. when light-sheet images included the fin itself), whose velocity (both magnitude and direction) most often differed from that of the surrounding fluid. Gaps in the vector field arising from validation were filled by interpolation in a 3×3 vector neighborhood (least-squares fit estimate). The overall measurement accuracy of the DPIV system was assessed by comparing the average free-stream velocity calculated by cross-correlation with that determined by tracking individual particles in the video record. In 20 fin beats examined, the DPIV estimation of velocity was within 5 %, on average, of the true velocity.

Velocity vector maps were displayed graphically, and the average free-stream velocity U_f (i.e. the speed of the current distant from the wake) was subtracted from each vector in the frontal and parasagittal matrices to reveal vortical flow structures. Since the current passed perpendicularly through the transverse light sheet (Fig. 2C), it was not necessary to subtract U_f in this plane. At swimming speeds greater than 0.5 L s⁻¹, increased flow normal to the transverse light sheet resulted in out-of-plane loss of particle images over Δt, which greatly limited the possibility of valid correlation peak detection. Accordingly, vortex geometry at 1.0 and 1.5 L s⁻¹ was analyzed in frontal and parasagittal slices of the wake only.

To compare the morphology of vortices shed by the pectoral fin with theoretical vortical structures, velocity gradients across the wake were examined. For each orthogonal laser plane, the pair of axes defining the flow field (X–Z, X–Y or Y–Z) was rotated through an angle Ψ, measured between the horizontal axis of the vector map and the longitudinal axis of the wake structure of interest (see Fig. 5C). Thus, the original perpendicular components of each velocity vector in a two-dimensional slice through the wake were transformed to components oriented along the rotated axes (X′, Y′, Z′) (see equation 2 in Spedding, 1986). These new components of velocity were plotted against position along the wake’s rotated major axes to yield flow velocity profiles. From such profiles, meristic variables describing the linear dimensions of the wake were measured (see Results).

To evaluate the balance of horizontal and vertical forces acting on the sunfish during constant-speed pectoral fin locomotion, total body drag (D) and weight (F_g) were determined empirically and compared with propulsive forces calculated from DPIV data. Sunfish were lightly anesthetized with tricaine methanesulfonate (0.3 g l to eliminate movement of the body or fins and to allow a loop of suture (6-0 gauge silk) to be passed around the dentary symphysis and upper lip. The use of live animals allowed the natural mucus covering on the body to remain intact. Two anesthetized fish (L=21–22 cm), used previously for DPIV experiments, were towed individually from a force transducer suspended in the flow tank at a current speed of 0.5 L s⁻¹ to measure D. To determine F_g, eight sunfish, including four used for the DPIV experiments, were anesthetized and weighed under water to the nearest 0.01 g using a pan balance.

For each orthogonal velocity vector field, the angle φ between the horizontal axis and the mean orientation of the central water jet of the vortex ring was measured. The force arising in reaction to the wake of the left pectoral fin was then resolved into its perpendicular components running parallel to the X, Y and Z axes. The upstream component of force acting on the fin F_x (thrust) was calculated independently in the frontal and parasagittal planes as ⁠, and the upward component F_y (lift) was calculated from the parasagittal and transverse planes as ⁠. These values were doubled to account for the action of both fins and to allow an estimate of the total thrust and lift experienced by the whole animal over the course of the fin stroke. The medially directed component of the reaction force on each fin F_z was calculated from the frontal and transverse planes as and ⁠, respectively (Fig. 2; see equations 9, 10 in Spedding, 1987).

Labriform locomotion in Lepomis macrochirus involves an oscillatory cycle of (1) anteroventral fin movement (downstroke), (2) rotation of the fin around its long axis at the end of downstroke (stroke reversal, or fin ‘flip’), (3) posterodorsal fin movement (upstroke), and (4) a kinematic pause period during which the fin is held flush against the body. DPIV flow fields reveal that, at 0.5L s⁻¹, the downstroke and stroke reversal are the primary generators of wake vorticity. During the downstroke, a free starting vortex is seen immediately downstream of the dorsal margin of the pectoral fin in the parasagittal and transverse planes (Fig. 3A, upper vortices; Fig. 4, P1, T1). At the end of the downstroke, and as the stroke reversal begins, a stopping vortex becomes visible near the ventral edge of the fin (Fig. 3A, lower vortices; Fig. 4, P2, T2). Within the frontal plane, only a single vortex is observable during the downstroke, but a second vortex appears as the fin rotates and begins to return towards the body (Figs 3A,B, 4, F1, F2). In all three planes, completion of the stroke cycle results in a discrete pair of counterrotating vortices introduced into the wake. Between each vortex pair, and strengthened by the rotational fluid motion, is a jet flow oriented posteriorly (downstream), ventrally and laterally (Fig. 3B). During the downstroke, the developing jet flow has a distinct spanwise component: velocity vectors near the fin are oriented parallel to the fin rays (Fig. 3A, frontal). Water encountering the anterior fin margin early in the stroke is drawn posterolaterally along the fin to contribute to vortex F1.

Plots of vorticity components calculated from planar velocity fields illustrate the opposite-sign rotation of the vortex pair in each perpendicular plane (Fig. 4). The observed vortices are consistent with slices through a toroidal structure and indicate that each fin sheds one complete vortex ring per stroke cycle at 0.5 L s⁻¹ (see Fig. 8E). A hierarchical analysis of variance (ANOVA) with paired vortices nested within laser plane orientation indicated that circulation magnitude does not differ significantly between starting and stopping vortices (d.f.=3 between vortices within planes; P=0.65). The average of clockwise and counterclockwise vortex strength (Table 1, г) was therefore used in each plane for calculations involving total ring circulation (equations 3, 5). Vortex ring momentum angle (Table 1, Ψ), the angle of inclination of the line connecting the two vortex centers, became acute within 200–300 ms of the start of the fin beat (Figs 3, 4). Mean momentum angles measured at the end of the stroke in each flow plane differed significantly from one another (Bonferroni multiple-comparison test at α=0.0167), indicating an oblique orientation of the vortex ring and its central water jet in space.

Both theory (Milne-Thomson, 1966) and experimental studies of man-made vortex rings in fluid (Maxworthy, 1977; Raffel et al., 1998) provide details of wake morphology with which the sunfish’s wake may be compared. Fig. 5A shows three flow velocity distributions across planar sections of an ideal, axisymmetric vortex ring. For the purpose of illustration, these distributions are shown for the parasagittal (XY) plane, within which are the rotated X′ axis, the longitudinal axis of the ring traversing both vortex core centers, and the perpendicular Y′ axis. The components of velocity parallel to X′ and Y′ are defined as u′ and v′, respectively. Thus, in Fig. 5A, the top flow profile v′(X′) illustrates variation in the water velocity component perpendicular to the longitudinal axis of the ring at positions along X′, the middle profile v′(Y′) is the distribution of velocity oriented along the ring’s central axis, and the bottom profile u′(Y′) depicts the relationship between velocity parallel to X′ and position along the perpendicular axis traversing a single vortex core. In each theoretical profile, velocities attain their greatest magnitude at the edges of the vortex cores in solid-body rotation and in the region between the vortices, reflecting the ring’s central water jet. Examples of velocity gradients determined empirically from pectoral fin flow data at 0.5L s⁻¹ in the parasagittal plane are given in Fig. 5B.

Velocity gradients across the sunfish’s wake also serve to define the geometry of the interior of the vortex ring. From profiles of v′(X′) in the parasagittal plane, and corresponding profiles in the frontal and transverse planes, vortex ring radius R was taken as half the distance between the points of intersection of the plotted curve with the long axis of the ring (following Spedding, 1986). Measurements of R in the three perpendicular planes differed by no more than 16 % (2.9 mm) on average at 0.5 L s⁻¹, and the vortex ring was therefore assumed to be circular in projected area (A=πR²). Neither R nor A demonstrated a significant dependence upon the orientation of the flow-field plane (Table 1). Vortex core radius R_o, an indication of ring thickness, was measured in each plane from profiles of the type shown at the bottom of Fig. 5B as half the distance along Y′ between the maximum and minimum values of u′ (following Spedding, 1986; Müller et al., 1997). A hierarchical ANOVA with paired vortices nested within laser plane orientation indicated that R_o does not differ significantly between starting and stopping vortices (d.f.=3 between vortices within planes; P=0.50). Averages of clockwise and counterclockwise vortex core radii in each plane are reported in Table 1. On the basis of measurements pooled across the three planes, the vortex ring at the end of the stroke period (0.5 L s⁻¹) had a mean area of 10.2 cm² with R=1.8 cm and mean R_o=1.1 cm.

Hydrodynamic forces acting on the bluegill sunfish during pectoral fin swimming at 0.5 L s⁻¹ are summarized in Table 2. Each component of locomotor force calculated from the wake depended directly upon the orientation of the central momentum jet of the vortex ring. Mean jet orientation varied significantly with flow-field plane (Table 1, φ), a result that Bonferroni multiple-comparison tests (α=0.0167) attributed to the marked difference in φ between the transverse plane and the other planes. Since force arises from changes in fluid momentum, it was necessary to quantify, for each plane, the component of momentum carried into the wake by the vortex ring as well as the period of locomotor force production. Average ring M exhibited considerable variation both among and within flow-field planes; significant differences among planes were not detected (Table 1). The pectoral fin-beat period averaged 0.83±0.04 s (mean ± S.E.M., N=18) at 0.5 L s⁻¹.

Independent estimates of F_x from frontal-and parasagittal-plane flow data were not significantly different, averaging 5.56±1.14 mN per fin (mean ± S.E.M., N=10). Similar agreement was observed for F_y obtained from the parasagittal and transverse planes (pooled mean ± S.E.M.=1.62±0.25 mN, N=12) (Table 2). To account for the contributions of both left and right pectoral fins, mean values of F_x and F_y were doubled to yield the total reaction forces on the animal. These wake-derived forces were very closely balanced by empirically determined counter-forces. The average total thrust force (11.12 mN) and lift force (3.24 mN) calculated from the wake did not differ significantly from the respective resistive forces of body drag, as measured from towing experiments (10.58 mN) and submerged body weight (3.37 mN) (Table 2).

F_z pooled across planes was 6.96±0.88 mN per fin (mean ± S.E.M., N=12). This medially directed component of force, assumed to cancel on opposite sides of the animal, averaged 125 % of F_x per fin and exceeded mean F_y by a factor of greater than four.

As viewed in the parasagittal plane, the wake at 1.0 L s⁻¹ consisted of a downstroke vortex pair (Fig. 6B, P1, P2), analogous to that formed at 0.5 L s⁻¹ (Fig. 4B), and a new upstroke vortex pair created as the fin adducts (Fig. 6B, P3, P4). Between each pair, a region of linear jet flow was observed. The parasagittal flow-field data therefore reflect the production of two (at least partially complete) vortex rings over the course of the fin-beat cycle (see Fig. 9C). The downstroke ring, like that measured at 0.5 L s⁻¹, lay at an acute angle (Ψ=75.1±5.8 °, mean ± S.E.M., N=6); the upstroke ring was approximately vertical in orientation (96.7±9.1 °, N=6) with a downstream-directed momentum jet. At 1.5 L s⁻¹, clockwise circulation produced in the parasagittal plane at the end of the downstroke and at the beginning of the upstroke coalesced into a single intense vortex (Fig. 6D, P2+P3). The strength of this combined vortex (г=—0.0053±0.0007 m² s⁻¹, mean ± S.E.M., N=5) was very nearly equal to the sum of mean г for P2 (—0.0026 m² s⁻¹) and P3 (—0.0033 m² s⁻¹) shed individually at 1.0 L s⁻¹. For the purposes of calculation (equations 3, 5), vortex ring strength at 1.5 L s⁻¹ was taken as the average of г<εnτ>magnitude for P1 (downstroke ring) or P4 (upstroke ring) and half the magnitude of г for P2+P3. The downstroke vortex ring at the fastest swimming speed studied had a rather shallow momentum angle (26.9±5.1 °, N=5), whereas for the upstroke ring this angle was obtuse (132.2±10.5 °, N=5), resulting in a fluid jet oriented downstream and slightly above the horizontal (Fig. 6D).

Concurrent with the development of P1 and P2 at both 1.0 and 1.5 L s⁻¹ was the appearance of paired vortices in the frontal plane (Fig. 6A,C, F1, F2). However, no additional vorticity was detected in the frontal plane during the upstroke. Over the complete range of swimming speeds studied, the orientation of the frontal vortex pair changed little (cf. Figs 4, 6), but ring radius measured from this plane declined steadily (Fig. 7A). Downstroke ring R and total ring г determined in the parasagittal plane showed a less direct relationship with speed (Fig. 7B,C). The total fluid momentum injected into the wake by each pectoral fin stroke, and the associated thrust force, increased up to the pectoral–caudal gait transition speed, and above U_p-c plateaued or declined slightly (Fig. 7D,E).

Three lines of evidence obtained from DPIV experiments support the generation of a vortex ring wake by the pectoral fins of bluegill sunfish. First, the simplest geometric interpretation of paired, counterrotating vortices produced concurrently in each of three perpendicular flow fields (Fig. 3) is the transection of a three-dimensional vortex tube closed upon itself. Helmholtz’s theorem, a conservation law governing the behavior of fluids (Milne-Thomson, 1966, p. 169), in fact requires that vortex filaments either arise and terminate on solid boundaries (in this case, the fin or flank of the animal) or form a closed vortex ring. Second, Helmholtz’s theorem states that the magnitude of circulation is equal around any two circuits (i.e. cross-sectional paths) along a vortex tube (Fung, 1990, p. 91). The lack of a significant difference in г both between vortices within each plane and among different planes (Table 1) satisfies the theoretical condition for a vortex ring. Third, the resemblance between the pectoral-fin wake profiles measured for sunfish (Fig. 5B) and the velocity gradients for ideal, axisymmetric vortex rings (Fig. 5A) suggests the production of a grossly symmetrical vortex toroid. Flow profiles similar to those in Fig. 5B have been defined for thin parasagittal sections of the wake of birds (e.g. Spedding et al., 1984; Spedding, 1986) and demonstrated in frontal sections of the wake of a fish’s tail (Müller et al., 1997), supporting the existence of the vortex ring wake behind animals moving through fluids.

Although few significant differences in ring morphology were detected across planes (Table 1), asymmetries in the wake undoubtedly exist. R measured at 0.5 L s⁻¹ from the parasagittal plane, for instance, was 0.17–0.30 cm less on average than that from the other planes, raising the possibility that the vortex ring is shed in a slightly elliptical, rather than circular, configuration. Possible deformation of the ring at higher swimming speeds is also suggested by minor discrepancies between planar estimates of R (Fig. 7A,B). Vortices visualized in planar sections of the wake appear approximately circular (Fig. 3), yet regions of concentrated vorticity do not (Fig. 4), indicating that flow velocities are not uniform throughout the wake structure. Further geometrical asymmetry was observed in the orientation of the ring’s momentum jet. In a perfectly symmetrical vortex ring, the central momentum flow extends perpendicularly from the ring plane such that the momentum angle and the jet angle are complementary. For sunfish, the sum of the magnitudes of these angles does approximate 90 ° in the parasagittal and transverse planes, but consistently falls short of 90 ° in the frontal plane (by an average of 12 °: Table 1; Fig. 3B, frontal). This divergence from the expectations of an ideal ring is probably related to the necessity of generating a strong downstream component of jet flow to experience forward thrust as a reaction.

There is no a priori reason to expect the sunfish to produce a strictly symmetrical ring, as observed in studies of non-biological vortex flows (e.g. ejection of fluid by a piston from a circular nozzle; see Maxworthy, 1972, 1977; Didden, 1979; Raffel et al., 1998). The wake behind a swimming animal is the product of rapidly accelerating and reciprocating hydrofoils which are asymmetric in shape, change over time in contour and surface area, and move in a complex three-dimensional manner. Unsteady flows of this type are highly unlikely to resemble more controlled, man-made wakes. The pectoral fin of bluegill sunfish, as in most teleost fishes, possesses a relatively stiff leading edge that is under direct muscular control during locomotion (Gosline, 1971; Gibb et al., 1994; Westneat, 1996; Drucker and Jensen, 1997). In contrast, the posterior fin margin appears to be mostly passive during the fin beat at low speeds (Gibb et al., 1994). Differing mechanical properties of the anterior and posterior fin margins, coupled with complex fin motion, may generate a somewhat elliptical vortex ring. The wake of the bluegill sunfish may be most accurately compared with a ‘vortex loop’ (Dickinson and Götz, 1996), a closed vortex tube obeying the required physical conservation laws but possessing a non-circular projected area and lacking an internal plane of symmetry.

On the basis of patterns of fin motion and the observed fluid flows quantified using DPIV, we suggest the following interpretation of vortex wake dynamics during the fin-beat cycle (Fig. 8). On the downstroke, as the fin’s leading edge peels away from the body, fluid on the lateral side of the fin rushes into the gap created between the medial fin surface and the body. As the remainder of the fin is abducted, a trailing vortex bubble is produced, extending from the fin’s margin. Sectioning this loop early in its development in the parasagittal plane reveals a free counterclockwise starting vortex (Fig. 8A, P1). In accordance with Kelvin’s theorem, clockwise bound circulation develops simultaneously, representing the future downstroke stopping vortex. Bound vorticity was not visualized in this study, but can be modeled as an attached leading-edge vortex (Fig. 8A, a; see Dickinson and Götz, 1993; Ellington et al., 1996). A free counterclockwise vortex visible in the frontal plane (Fig. 3A; Fig. 8A, F1) is consistent with a growing vortex tube. At the end of the downstroke, the initially formed medial leg of the vortex loop has been convected downstream approximately 1–2 fin-chord lengths. Two factors contribute to the early appearance of free clockwise fluid rotation in the wake (as viewed in the parasagittal plane). First, the acceleration reaction (Daniel, 1984) resists deceleration of the pectoral fin at the end of the downstroke and causes entrained fluid to flow from the dorsomedial to the ventrolateral side of the fin. Separation of water at the fin edge results in the rolling-up of negative vorticity (Fig. 8B, b). A similar phenomenon is apparent at the end of each half-stroke in the tail wake of swimming fishes (Lighthill, 1975; Triantafyllou and Triantafyllou, 1995), where deceleration of the tail causes fluid to move around the trailing edge and contribute to vortex shedding. Second, clockwise rotation of the fin at the stroke reversal is hypothesized to strengthen bound circulation of the same rotational sense, as demonstrated for insect wing models (Dickinson, 1994). During the fin flip, accumulated bound vorticity (the sum of flows a and b in Fig. 8B) is forced to the ventrolateral side of the fin and introduced into the wake (see Maxworthy, 1979; Dickinson and Götz, 1996). Thus, by the early stages of stroke reversal, a second free (stopping) vortex is visible in the parasagittal plane (Fig. 3A, lower vortex; Fig. 8C, P2).

The general pattern of vortex production on the upstroke mirrors that of the downstroke. As the fin returns towards the body, a clockwise starting vortex is shed from the trailing surface of the fin (Fig. 8D, P3) and is matched by a counterclockwise vortex attached at the leading edge (Fig. 8D, P4). However, the vortical flow actually visualized in the wake of the upstroke varies directly with swimming speed. For all speeds studied, the downstroke stopping vortex has the same rotational sense as the upstroke starting vortex. A number of kinematic and hydrodynamic factors determine whether these two vortices are shed as separate discrete structures or a single fused element. The absence of a pause period after the stroke reversal at 0.5 L s⁻¹ means that P3 begins to develop immediately after P2 is shed into the wake (Fig. 8D). In this low-speed current (approximately 10 cm s⁻¹), P2 has little time to travel downstream from P3, and these vortices therefore remain in close proximity at the beginning of the upstroke. Accordingly, they are predicted to fuse into a single ‘stopping–starting’ vortex (Fig. 8E, P2+P3) (Brodsky, 1991, 1994). The combined strength of this free negative circulation counteracts the development of positive bound circulation on the return stroke (by the Wagner effect; see Ellington, 1984a; Dickinson, 1996). Since the upstroke at 0.5 L s⁻¹ is relatively slow (mean T_u=0.30 s; Fig. 7G), P4 vorticity, which was never visualized in our experiments at low speed, is expected to be weak (see Dickinson and Götz, 1996, p. 2102) and thus greatly diminished by that of P2+P3. At the end of the upstroke, P1 and P2+P3 are the major flow structures visible in the parasagittal plane (Figs 3B, 8E) and, together with flow patterns in perpendicular planes (e.g. Fig. 3B, frontal; Fig. 8E, F1, F2), reflect the introduction of a single vortex ring into the wake.

Patterns of upstroke vorticity observed at higher swimming speeds can be understood in the light of more rapid and forceful fin beats and an increase in near-fin free-stream velocity. At 1.0 L s⁻¹, the downstroke period is on average 100 ms shorter than at 0.5 L s⁻¹ (Fig. 7G). As a result, the downstroke vortex ring self-convects more rapidly away from the pectoral fin. A twofold increase in the velocity of the current of the flow tank passing over the fin further speeds the passage of the wake downstream. Accordingly, clockwise vorticity shed at the stroke reversal (Fig. 9A, P2) is hypothesized to travel far enough away from the fin to allow upstroke circulation of the same sign to develop separately (Fig. 9B, P3). Mean fin upstroke duration at 1.0 L s⁻¹ is 20 % shorter than that at 0.5 L s⁻¹ (Fig. 7G) and is also expected to influence flow patterns in the wake. A faster, and presumably stronger, upstroke should yield a well-developed leading-edge vortex (Fig. 9B, P4) which, despite its antagonistic interaction with P3, is ultimately shed into the wake (Fig. 9C). The failure of P3 to fuse with P2 (cf. Fig. 8D,E), and the accompanying reduction in P3 circulation, may contribute to the survival of P4 and its detection in DPIV flow fields (Fig. 6B). Elevated current velocity at the highest swimming speed, 1.5 L s⁻¹, should, as at 1.0 L s⁻¹, promote the shedding of P2 and P3 as distinct vortices. However, the markedly reduced upstroke period (mean 95 ms shorter than at 1.0 L s⁻¹, and 79 % of T_d at 1.5 L s⁻¹; Fig. 7G) appears instead to cause the fusion of these structures into a stopping–starting vortex (Fig. 9E). The augmented magnitude of г for P2+P3 must reduce, but does not annihilate, the opposite-sign г of P4 (Fig. 9F).

The abrupt deceleration of the fin as it returns to the body at the end of the upstroke will cause the release of remaining bound circulation into the wake. In the light of the observation that the parasagittal plane contains at least three centers of vorticity (Fig. 6B,D), yet only a single vortex pair ultimately appears in the frontal plane (Fig. 6A,C), we conclude that the two vortex rings evident at higher swimming speeds must be linked, rather than discrete. A lateral linkage at 1.0 L s⁻¹ and a broad ventrolateral linkage at 1.5 L s⁻¹, as depicted in Fig. 9C and 9F, respectively, are consistent with the experimentally determined orthogonal flow patterns. The absence of a third vortex, with counterclockwise rotation, in the frontal plane implies that the medial leg of the upstroke loop is not fully closed upon itself at the end of the upstroke. In the subsequent kinematic pause period, during which the fin remains adducted against the body, the shed upstroke vortex filament may slide posteriorly along the fish’s flank and form a complete loop once downstream of the tail (a similar phenomenon has been hypothesized for Drosophila melanogaster flight by Dickinson and Götz, 1996). This hypothesis could be tested by subjecting to DPIV analysis particle images more posteriorly situated than those obtained for this study.

The saltatory kinematics of the pectoral fin stroke (i.e. fin movement interrupted by pause) precludes the formation of a continuous, coupled-chain type of wake proposed for some insect wings (Brodsky, 1991, 1994) and fish tails (Videler, 1993; Rayner, 1995). At the low swimming speed of 0.5 L s⁻¹, each fin beat results in a single torus containing all large-scale vorticity generated by the fin (Fig. 8). At higher swimming speeds, a linked-pair vortex wake is produced consisting of discrete, double-ring structures shed during each fin stroke (Fig. 9) (cf. ‘disconnected chain’ of Brodsky, 1991, and ‘V-like connected rings’ of Grodnitsky and Morozov, 1992, in the wake of insects).

Average lift and thrust forces calculated from the wake of the sunfish do not differ significantly from the corresponding resistive forces of body weight and drag (Table 2; Fig. 10). Such a force balance is indeed the expected condition during steady locomotion (Daniel and Webb, 1987; Webb, 1988), but has not been demonstrated previously for swimming animals using empirical data. Attempts to resolve theoretically derived locomotor forces on fishes have met with mixed success (Lighthill, 1971; Magnuson, 1978). The force equilibrium observed for Lepomis macrochirus at 0.5 L s⁻¹ indicates that the DPIV system employed here successfully detects the major vortical structures shed during swimming and allows accurate estimates of the momentum injected into the wake, validating the technique for the study of unsteady aquatic force production. In addition, in the case of L. macrochirus, this result allows assessment of the contribution of the pectoral fin to thrust and lift at speeds at which paired-fin forces are out of balance (i.e. at and above U_p-c). The relationship between swimming speed and measured force can be interpreted in the light of the changing morphology of the fish’s wake. Between 0.5 and 1.0 L s⁻¹, the approximately fivefold increase in forward thrust (Fig. 7E) is attributed to a second, downstream-facing vortex ring formed on the upstroke (Fig. 6B). The plateauing of thrust at U_p-c is mirrored by the observation that pectoral fin-beat frequency in bluegill sunfish peaks at or near the speed of caudal fin recruitment (Gibb et al., 1994; see also Drucker and Jensen, 1996). Downstroke vortex rings at 0.5 and 1.0 L s⁻¹ exhibit similar jet angles in the parasagittal plane (Figs 4B, 6B), and net lift therefore remains approximately constant (Fig. 7F). The doubling of lift between 1.0 and 1.5 L s⁻¹ (Fig. 7F) stems directly from the increased tilt of the downstroke ring and its central downwash (Fig. 6D). The generation of additional upward force by the pectoral fins at high speeds may be necessary to counter the growing negative lift associated with the fish’s anterior body profile or positive lift generated posteriorly by oscillation of the tail (Lauder et al., 1996).

A consistent balance of forces computed from wake measurements has been elusive for animals locomoting in fluid. Calculations of lift based on the estimated wake geometry of birds, for example, have often fallen short of values necessary to maintain steady flight against the force of gravity (but see Spedding, 1987). In reconstructing three-dimensional vortex lines in the wake of flying pigeons (Columba livia) and jackdaws (Corvus monedula), the elegant studies of Spedding et al. (1984) and Spedding (1986) showed that shed vortex rings were paradoxically too small and too weak to provide weight support. Since ring radii measured in orthogonal planar sections of the wake were comparable, the assumption of a more-or-less symmetrical ring is justified. Non-midline and oblique transections of the vortex loop, however, are possible errors that could result in deviations in R and г to which calculations of momentum are sensitive. Such types of error are the unavoidable consequence of studying unsteady fluid flows produced by freely moving animals in which the shape, orientation and path of travel of the wake are out of the experimenter’s control. For a fish swimming by axial undulation, Müller et al. (1997) recently found that thrust power calculated from flow velocities in frontal-plane transections of the tail’s wake exceeded that expected from theory by a factor of three. In this case, circular ring geometry was selected in the absence of complete three-dimensional flow-field information. Since many variations of the vortex ring morphology are theoretically possible from oscillating biofoils (Brodsky, 1994; Rayner, 1995; Dickinson, 1996), the assumption of a particular wake structure based on single-plane views of downstream flow may lead to inaccurate estimations of propulsive force.

For some swimming animals, the locomotor force balance has been proposed to involve an unsteady ‘squeeze’ mechanism for the supplementation of thrust. Acceleration of the animal’s propulsor against the body at the end of the propulsive stroke is thought to eject a rearward jet of water whose reaction provides a forward force (Daniel and Meyhöfer, 1989). This mechanism has been cited for the pectoral fins of fish as a possible means of increasing the total thrust generated per fin beat (Geerlink, 1983). For the bluegill sunfish, however, the pectoral fins are adducted against the body at the end of the upstroke at comparatively low speed. In addition, direct observations of particle motion in the wake of the pectoral fins at the end of the upstroke do not reveal the presence of a strictly downstream-oriented jet (Fig. 3B). For these reasons, we conclude that, for pectoral fin propulsion in sunfish, the squeeze force does not substantially augment thrust produced by vortex ring shedding.

In the present study, locomotor forces were calculated as averages over the course of the fin-stroke duration. Upon completion of the upstroke, remaining bound vorticity is shed from the fin in the form of stopping vortices that migrate away from their starting vortex counterparts (e.g. Fig. 9B,C, P3 and P4). By the end of the kinematic pause period, free counterrotating vortices are no longer in close proximity. Accordingly, the mutual antagonism of regions of opposite-sign vorticity in the wake, as described by the Wagner effect, is much reduced, and circulation attains its full steady-state magnitude (Dickinson and Götz, 1996, p. 2103). Forces estimated over T are therefore the most informative for resolution of the overall force balance on the animal. However, forces acting on a swimming fish are undoubtedly out of equilibrium at any particular moment during the stride. At high swimming speeds, for example, sunfish generate positive lift on the pectoral fin downstroke and negative lift on the upstroke, as indicated by the tilted orientation of vortex rings in the wake (Fig. 6D). DPIV analysis of flow-field patterns over increasingly small time intervals throughout the fin-beat period would allow estimation of instantaneous locomotor forces (equation 4), a necessary step in determining the time course of force development for labriform swimming.

Animals locomoting in air must continuously generate lift, a constraint apparent in the form of their wake. For many vertebrates (Kokshaysky, 1979; Spedding et al., 1984; Spedding, 1986) and insects (Grodnitsky and Morozov, 1992, 1993; Dickinson and Götz, 1996), vorticity is shed predominantly on the wing downstroke. Vortex ring momentum angles are quite shallow (e.g. Spedding et al., 1984), reflecting an earthward fluid jet. The upstroke of these animals is thought to perform little or no useful aerodynamic function (Rayner, 1979a; but see Brodsky, 1994; Ellington, 1995). In contrast, buoyant animals swimming in water are faced with a fundamentally different primary requirement: the production of forward thrust throughout the propulsive cycle. Animals such as the bluegill sunfish that are neutrally buoyant (or nearly so) are freed from the necessity of reserving a large portion of the impulse of each stroke for counteracting the force of gravity. Accordingly, a diversity of mechanisms to supplement thrust may be employed (Daniel and Webb, 1987). The pectoral fin of Lepomis macrochirus operates at Reynolds numbers of 4.6×10³ to 1.4×10⁴ between 0.5 and 1.5 L s⁻¹, values within the range at which the acceleration reaction makes its largest contribution to locomotor force (Webb, 1988). Augmentation of circulation at the end of the downstroke by the rolling up of fluid entrained by the decelerating fin (Fig. 8B, flow b) may be characteristic of biological vortex flows in the aquatic medium. For animals moving in the far less dense medium of air, acceleration reaction effects on fluid flow will generally be less pronounced (Daniel and Webb, 1987). At high swimming speeds, L. macrochirus sheds vorticity on both half-strokes. The addition of a second vortex ring, with a downstream-facing wash, on the upstroke (Fig. 6B,D) is a further means of increasing force for forward propulsion.

Differences in body and propulsor shape among animals moving through fluids have additional important implications for mechanisms of force production. Unlike most flying birds and insects whose body depth is small compared with wingspan, the bluegill sunfish possesses a deep, laterally compressed body that isolates the relatively short pectoral fins on either side of the animal. This anatomical configuration suggests two hypotheses relevant to the design of fishes for maneuvering. First, a deep body, characteristic of many perciform fishes, may function as a flow barrier to separate the vortex rings generated by the left and right pectoral fins and hence limit their hydrodynamic interactions (Dickinson, 1996). Independent control of the strength and orientation of left-and right-side vortices may be a key factor in allowing rapid, precise turns associated with locomotion in near-shore vegetated habitats or with a planktivorous feeding mode in which frequent changes in body position occur to maintain orientation towards prey. Second, separation of the wake produced by opposite-side fins may confer an advantage in the maintenance of body stability during locomotion. The medially directed force F_z experienced by each fin, contributing neither to lift nor thrust, was unexpectedly large in bluegill, exceeding both mean F_y and F_x per fin (Table 2; Fig. 10). Large, medially oriented reaction forces exerted independently on either side of the animal should assist in stabilizing the body during rectilinear locomotion against rolling moments induced by flow turbulence. Such forces are critical for fishes that are hydrostatically unstable because their center of buoyancy is not located at the center of mass. Even in fishes such as bluegill whose centers of mass and buoyancy coincide (Webb and Weihs, 1994), corrective fin movements generating inward reaction forces are expected in order to offset destabilizing perturbations from local flow.

An additional consequence of the anatomy of swimming fishes is that traditional ‘clap and fling’ mechanisms cannot function as envisioned for flying animals (Weis-Fogh, 1973). The amplitude of the pectoral fin beat is low compared with the height of the body in virtually all teleost fishes, and the left and right fins therefore do not meet (i.e. clap) at the end of either half-stroke. Fishes whose paired fins do extend below the ventral margin of the body (exemplified by the gar Lepisosteus and other basal actinopterygians) or above the dorsal margin (e.g. the butterflyfish Pantodon) do not rely on these appendages for propulsion and swim instead primarily by undulation of the trunk and tail. Since hydrodynamic interactions between opposite-side fins are precluded, single vortex loops extending from one side of the fish to the other are unlikely (see Fig. 7A–E in Dickinson, 1996). We predict that the wake of most (if not all) labriform swimmers consists of a series of individual vortex rings (Fig. 8) or linked pairs (Fig. 9) generated separately by each pectoral fin (cf. the ‘double chain’ wake of Brodsky, 1991).

In applying digital particle image velocimetry to studies of moving animals, several practical issues need to be addressed. First, it is critical to have control over animal speed and position relative to the laser light sheet. This can be accomplished by using recirculating flumes or wind tunnels, so that animal speed is known, and by using a second camera to quantify the position of the propulsive appendage relative to the light sheet (Fig. 1). Without knowledge of the location of the propulsor with respect to the flow field, it is impossible to determine what region of the propulsive surface is interacting with the visualized plane of flow. The use of flumes also allows positioning of the animal away from walls and solid surfaces, which can interfere with wake patterns. The study of fish moving in shallow trays, for example, is likely to produce artifactual results due to the close proximity of the bottom of the tank and upper water surface. Second, the use of at least two orthogonal laser plane orientations permits reconstruction of the gross three-dimensional structure of the flow, a task not possible from single-plane analyses alone. The use of three orthogonal laser planes allows flow in three perpendicular directions to be examined and independent estimates of force components to be compared (Table 2). Third, the use of relatively high-speed imaging hardware (greater than 200 fields s⁻¹) allows characterization of the time-dependent nature of biological flow and visualization of the development of vortices around and behind animal appendages. Fourth, the use of automated cross-correlation algorithms for processing particle images yields a complete and evenly spaced matrix of velocity vectors that greatly facilitates calculation of vortex geometry and circulation and removes the uncertainty that arises from tracking individual particles by hand.

Once these practical concerns have been resolved, the applications of DPIV to questions of unsteady biological fluid flow are numerous and represent an exciting area of future research. Assumptions about propulsor shape and kinematics, inherent to mechanical and computational modelling of locomotion (Liu et al., 1996; Van den Berg and Ellington, 1997; Carling et al., 1998), are circumvented by direct flow visualization behind unrestrained animals. Estimation of propulsive force from wake velocity fields both provides insight into the unsteady mechanisms of locomotion and allows calculation of quantities describing locomotor performance. For example, the mechanical efficiency (η) of the pectoral propulsor may be determined as the ratio of forward thrust to total force generated per stride. For Lepomis macrochirus swimming at 0.5 L s⁻¹, η calculated from mean forces pooled across planes (Table 2) was 0.39, a value exceeding the theoretical estimate (0.25) of Blake (1979) for the pectoral fins of angelfish (Pterophyllum eimekei). Also of interest for cross-taxonomic comparisons are stride-specific work and power, which can be calculated from wake-derived forces. Over a mean stride length of 0.082 m, the average work performed by the sunfish to swim forward at 0.5 L s⁻¹ was 0.91 mJ, and the average mechanical power produced by both fins during the stride period was 1.17 mW. Apart from the study of Müller et al. (1997), energetic calculations based upon empirical measurements of the wake of other swimming fishes have yet to be reported.

The DPIV technique further provides an approach for evaluating quantitatively the functional design of propulsive surfaces in organisms moving through fluids. Differences in external shape and patterns of body movement in fishes have long been used for assessment of locomotor design and for reconstruction of the patterns and processes associated with the evolution of locomotor function (Breder, 1926; Affleck, 1950; Gosline, 1971; Thomson and Simanek, 1977). Such indirect assessments can now be coupled with direct measurements of the effects of body form and motion on fluid flow and calculations of the force exerted by animal propulsors.

We thank A. F. Bennett, C. Connon, M. Dickinson, D. Dunn-Rankin, L. Ferry-Graham, A. Gibb, G. Gillis, J. Hicks, R. Josephson, J. Liao, J. Nauen, J. Posner, M. Triantafyllou and C. Wilga for helpful discussions and technical assistance. The manuscript was improved by the comments of two anonymous reviewers. Special thanks to S. Anderson for developing circulation analysis software. Supported by NSF DBI-9750321 to E.G.D. and NSF IBN-9807012 to G.V.L.