Quantification of the wake of rainbow trout (Oncorhynchus mykiss) using three-dimensional stereoscopic digital particle image velocimetry

Studies of locomotor hydrodynamics in fish have entered a new phase within the last decade with the application of the technique of digital particle image velocimetry (DPIV). DPIV allows quantitative visualization of flow fields created by freely moving fishes, which is an advantage over more qualitative approaches such as tracking streams of dye introduced into the flow and inference of flow patterns from kinematic data. DPIV involves visualizing flow by adding reflective, inert, near-neutrally buoyant particles to fluid and illuminating the particles in a planar section of flow using a laser light sheet. Flow-field characteristics including velocity and vorticity are calculated from successive images of particle movement captured at high frequency (Raffel et al.,1998). Wake momentum and force can be calculated from DPIV velocity vector fields (e.g. Drucker and Lauder, 1999, 2001a). DPIV has become a popular method for quantifying wakes produced by the caudal fin of fishes(Stamhuis and Videler, 1995; Müller et al., 1997; Hanke et al., 2000; Lauder, 2000; Liao and Lauder, 2000; Müller et al., 2000; Drucker and Lauder, 2001b; Nauen and Lauder, 2002).

To date all studies of aquatic locomotion using DPIV have used two-dimensional DPIV methods, in which flow is visualized separately in one plane, typically the parasagittal (xy) or frontal (xz)plane. A few studies have combined two-dimensional measurements of the three flow velocity variables, downstream (U), vertical (V) and lateral (W), calculated from separate kinematic events measured with different planar light sheet orientations to estimate the three-dimensional structure of wake flow patterns (e.g. Drucker and Lauder, 1999, 2000; Nauen and Lauder, 2002). Such three-dimensional reconstructions require study of a repetitive behavior that shows little kinematic variation among repetitions, such as pectoral fin locomotion by bluegill sunfish (Lepomis macrochirus) (Drucker and Lauder, 1999, 2000) and caudal fin locomotion by chub mackerel (Scomber japonicus)(Nauen and Lauder, 2002).

These limitations of two-dimensional analysis can be avoided by using three-dimensional stereoscopic DPIV, in which the use of two cameras with two different fields of view allows simultaneous visualization of all three velocity components using only a single light sheet orientation. While stereo-DPIV is well established in the engineering literature(Gaydon et al., 1997; Lawson and Wu, 1997; Willert, 1997; Westerweel and van Oord, 2000), this is the first application of this technique to the study of biologically generated flow patterns. Hence, the first goal of this study was to test the utility of stereo-DPIV to study wakes created by freely swimming fishes.

Our second goal was to examine the vortex wake of a salmoniform fish. This group represents a major phylogenetic grouping within ray-finned fishes for which few experimental hydrodynamic data are available, although this clade has been the subject of extensive previous research on locomotory performance. For example, the steady swimming kinematics(Stevens, 1979; Webb et al., 1984; Webb, 1988, 1993) and energetics (Webb, 1971a,b),myotomal slow muscle locomotory function(Hammond et al., 1998) and muscle power output (Coughlin,2000) of Oncorhynchus mykiss have been examined previously, although the hydrodynamics of locomotion by O. mykissremain unexplored except for the preliminary report of Blickhan et al.(1992).

Wake structure reveals important information about the timing, magnitude and direction of thrust production as well as potentially interesting data on the mechanisms of thrust production. On the basis of previous studies of trout steady swimming kinematics (Stevens,1979; Webb et al.,1984; Webb, 1988, 1993) and wake structure(Blickhan et al., 1992), it is likely that trout wakes are composed of linked vortex rings with oscillating central jet flows that are high in lateral velocity (and are thus similar to the wakes of other body—caudal fin swimmers). The expectation that trout wakes would be composed of linked vortex-ring wakes such as those previously visualized using various two-dimensional techniques(Anderson, 1996; Müller et al., 1997; Videler et al., 1999; Hanke et al., 2000; Drucker and Lauder, 2001b; Nauen and Lauder, 2002)offered an advantage in using trout rather than a completely uncharacterized system as a test of the application of stereo-DPIV to aquatic locomotory systems.

The third goal of this study was to examine two specific features of the salmoniform vortex wake in the light of current data on the function of the homocercal caudal fin in teleost fishes (for a review, see Lauder, 2000). Previous studies of more derived clades (Gibb et al., 1999; Lauder,2000; Nauen and Lauder,2002) have suggested that the homocercal caudal fin, which is externally symmetrical in shape, can generate lift forces due to asymmetrical movements that direct flow dorsally or ventrally. We quantified wake flow patterns to determine whether the homocercal caudal fin of rainbow trout was directing flow dorsally or ventrally, and thus producing lift, during steady swimming. In addition, previous studies of more derived clades using two-dimensional DPIV have demonstrated that lateral flow velocities are typically higher than the downstream flow velocities, resulting in surprisingly low caudal fin mechanical performance ratios of approximately 0.35 (out of a possible maximum value of 1) for taxa as varied as bluegill sunfish Lepomis macrochirus(Drucker and Lauder, 2001b) and chub mackerel Scomber japonicus(Nauen and Lauder, 2002). We used the simultaneously measured magnitudes of U, V and Wproduced by O. mykiss during steady swimming to determine whether the ratio of downstream to total wake flow magnitude is similar to the previously reported caudal fin mechanical performance measurements for more derived species. Values of Froude propulsive efficiency, an index of overall locomotory efficiency, were also calculated from the velocity measurements for comparison both with values from other studies and with values of caudal fin mechanical performance.

Rainbow trout, Oncorhynchus mykiss (Walbaum), were purchased from a local fish hatchery. The trout were fed commercial trout food and housed in 1200 l tanks at a water temperature of 15°C with a 12 h: 12 h light:dark photoperiod. Four fish with a total body length (BL) range of 16.5-21.5 cm and a mean body length of 18.9±1.0 cm (mean ±S.E.M., N=4) were used.

Experiments were conducted using a 600 l recirculating flow tank. The flow tank has a working area of 82 cm×28 cm×28 cm (length × width× height). Water temperature was 15±1°C. Flow speed in the tank was controlled using a variable-speed motor. Details of the flow tank and flow speed calibration have been presented previously(Jayne et al., 1996; Gibb et al., 1999). A flow tank was used to control the speed of the fish and to provide a relatively small field of view for the images (approximately 15 cm×7 cm, length× height) for a close view of the wake (see Figs 2, 3).

Fish swam at 1.2 BLs^-1, which matches the speed used for several previous two-dimensional DPIV studies of caudal fin locomotion(Liao and Lauder, 2000; Drucker and Lauder, 2001b; Nauen and Lauder, 2002) and is close to speeds examined in other two-dimensional DPIV studies of caudal fin locomotion (Müller et al., 1997, 2001). The relatively low flow velocities also facilitated the calculation of out-of-plane particle movement(see below). Only video sequences during which the trout swam steadily and did not drift either vertically or laterally were analyzed. Steady swimming behavior was determined by visually evaluating the xy position of the dorsal and ventral tips of the caudal fin as it moved through the light sheet during each stroke. Error is estimated to be of the order of 2 mm, which is substantially less than 5% of the field of view of the cameras (15 cm×7 cm).

As in previous two-dimensional DPIV studies conducted in this laboratory(e.g. Drucker and Lauder, 2000; Lauder, 2000; Wilga and Lauder, 2000; Nauen and Lauder, 2001, 2002), flow was visualized by seeding the water with near-neutrally buoyant 12 μm diameter silver-coated glass beads (density 1.3 g cm^-3, Potters Industries, USA) that reflected light from an 8 W argon-ion laser(Fig. 1). The laser light sheet was approximately 2 mm×7 cm×15 cm (depth × height ×width). It was positioned in the center of the working section of the flow tank, and oriented in the parasagittal (xy) plane. Data were collected when the fish was positioned so that the caudal fin beat through the light sheet in mid-stroke.

A key characteristic of the stereo-DPIV arrangement is the use of two cameras (Fig. 1) to view the light sheet. For two-dimensional measurements (see studies cited above), a single camera is positioned perpendicular to the light sheet. Stereo-DPIV measurements require two synchronized cameras both positioned at a wide angle to the light sheet (Fig. 1), so that each camera records a different, but overlapping, perspective of the flow. In the present study, two Redlake Motionscope model PCI 500 digital video cameras were used. Images of 480×420 pixels resolution were recorded at a frequency of 250 Hz.

A second novel characteristic of the stereo-DPIV arrangement is the camera lens position and orientation. Lenses are uncoupled from the body housing the camera sensor and tilted relative to the camera to achieve the Scheimpflüg stereoscopic configuration(Fig. 1B) in which the object plane (in this case, the light sheet), the image plane and the lens principal plane intersect in a common line. Tilting the lens of each camera relative to the camera's CCD sensor (using a custom-designed mounting bracket) makes the plane of best focus the plane of the light sheet, rather than the plane perpendicular to the camera's optical axis. This results in images that are bright and in good focus across the field of view for accurate vector calculations.

The camera arrangement and Scheimpflüg lens orientation introduce perspective distortion. For example, the cameras view a rectangular calibration grid illuminated by the light sheet as a trapezoid(Fig. 2A). This image distortion is corrected by recording simultaneous views of a three-dimensional calibration grid (Fig. 2A). Calibration marker points in a square grid with a fiducial mark at grid center(Fig. 2A) define the coordinate system and match calibration image points to target locations in the fluid. Alternating rows of calibration marks are offset in depth by 1 mm for calibration in the z direction (this offset is not visible in the Fig. 2A calibration images). Images of the target from each camera are analyzed, providing a pair of locations for each marker on the target: the x, y and zlocation of the marker point on the target in the fluid and its x and y pixel location in the image. A polynomial equation mapping function is created based on calibration grid points using a least-squares method; this equation is used to determine the spatial dimensions of the flow field. Use of the calibration grid and image mapping functions had the added advantage of aligning the images during processing so that the field of view of the cameras did not have to be aligned pixel to pixel. Areas seen by each individual camera that do not overlap the view from the other camera (approximately 15%of the field of view, Fig. 2A)are discarded prior to calculation of the three-dimensional vector field. Stereo-DPIV calibration and the required calculations are discussed extensively in the engineering literature(Gaydon et al., 1997; Lawson and Wu, 1997; Willert, 1997; Westerweel and van Oord, 2000) and will not be repeated here.

The first step in calculating the three-dimensional vector field is two-dimensional analysis of the simultaneously collected left and right images(Fig. 2B). Using Insight software (v. 3.3, TSI, Inc., USA), sequential pairs of video images (4 ms apart in time) were analyzed by subdividing the images into a series of interrogation areas and comparing pixel intensity in these areas using two-frame cross-correlation analysis. The areas of flow analyzed were typically rectangular in shape. The cross-correlation analysis used a Hart algorithm (Hart, 2000), a spot size of 64 pixels and a peak:noise ratio of 1.5. DPIV calculation methods have been previously described in detail(Willert and Gharib, 1991; Anderson, 1996; Raffel et al., 1998). The two components of velocity nominally perpendicular to the camera optical axis (in our case, U and V) are calculated from the two-dimensional recorded images (Fig. 2B). Note that the two-dimensional vectors are calculated in a symmetrical, aligned grid(Fig. 2B) based on the calibration grid (Fig. 2A).

The out-of-plane velocity component (in our case, W, Fig. 2C, D) is calculated using the two-dimensional vector files (Fig. 2B) and a set of four polynomial equations that determine the W component at each location in the two-dimensional vector fields. Interpolation of the right and left vector fields provides particle image displacements; fluid displacement is solved for using a least-squares method.

To visualize wake geometry, downstream free velocity(U_o) was subtracted from each three-dimensional velocity vector (see, for example, Fig. 3). Downstream rather than upstream velocity was subtracted because the presence of the fish affects the flow field. The method for subtracting U_o was based on the fact that the present visualizations showed defined, oscillating, posteriolaterally oriented jet flows in the trout wakes such as are associated with linked vortex ring wakes produced by body—caudal fin swimmers (e.g. Müller et al., 1997; Wolfgang et al., 1999; Drucker and Lauder, 2001b; Nauen and Lauder, 2002). In addition, previous studies of trout steady swimming kinematics(Stevens, 1979; Webb et al., 1984; Webb, 1988, 1993) and wake flow(Blickhan et al., 1992)indicate that the jet flows seen here are associated with a linked vortex-ring wake. The height of linked vortex-ring wakes produced during steady swimming is typically equal to caudal fin span(Blickhan et al., 1992; Müller et al., 2000; Nauen and Lauder, 2002). Thus,flow above and below the caudal fin represented free-stream velocity, and we evaluated free-stream velocity magnitudes in regions of the stereo-DPIV images above and below the caudal fin.

The magnitudes and directions of U, V and W were determined for 10 vectors in the central jet flow of the wake (Figs 2, 3) created by each stroke of the caudal fin. Six to eight consecutive strokes were analyzed for each individual. The ratio of the magnitudes of U/(U+V+W), which is referred to hereafter as U/total wake flow, was calculated from the flow values. Froude propulsive efficiency (η_p) was also calculated as 2U_o(U-U_o)/U²+V²+W²-U_o²)based on Blake (1983). Single-factor analyses of variance (ANOVAs) were performed using SuperANOVA(version 1.11) to determine the effect of individual on the measured parameters, the ratio of U/total wake flow and η_p. All F-values were calculated as described by Zar(1984).

During steady swimming at 1.2 BL s^-1, the wake of Oncorhynchus mykiss contained a directed jet of relatively high-velocity flow that alternated in direction to the left and right sides of the fish with the change in direction of the caudal fin at the end of each stroke (Figs 2C,D, 3). Stereo-DPIV methods produced three-dimensional views of the jets with their relatively high-velocity lateral flow components; the overlapping field of view area of approximately 90 cm² was characterized by more than 750 vectors(Fig. 3).

A series of jets was visible in the field of view of the cameras (e.g. Fig. 3B), which allows an evaluation of jet dynamics. The time-dependence of lateral (W) flows is depicted for an individual of 16.5 cm BL swimming at 1.2 BL s^-1 (19.8 cm s^-1). A wake flow of approximately 6.6 cm in length, which was created over 0.33 s, is visible in a single image. Lateral flows are strong and consistent and showed defined directional reversals that resulted from the change in movement direction of the caudal fin with each stroke.

Average vertical flows (V) were low in magnitude (typically less than 1 cm s^-1, Fig. 4A) and variable in direction(Fig. 4A). The magnitude and direction of V were dependent on individual (ANOVA, d.f.=3,286, F=6.9, P=0.0002). One individual (no. 2) typically produced small vertical flows in the upward direction (or positive flows, Fig. 4A). The other three individuals produced small vertical flows that were positive or negative(Fig. 4A) depending on the particular tail beat.

The magnitude of downstream flow (U) was also dependent on individual (ANOVA, d.f.=3,286, F=51.0, P=0.0001). Uflows, of the order of 3-6 cm s^-1 above free-stream velocity(Fig. 4A), were relatively substantial. Lateral flows (W), were typically the largest in magnitude, with average values of 4-6 cm s^-1(Fig. 4A), and were dependent on individual (ANOVA, d.f.=3,286, F=6.7, P=0.0002).

The ratio of U/total wake flow (a flow index of mechanical performance) was also dependent on individual (ANOVA, d.f.=3,286, F=15.6, P=0.0001). Mean U/total wake flow ratios for each individual (Fig. 4B)varied from 0.32±0.012 to 0.45±0.014 (mean ± S.E.M., N=60-80). Froude efficiency (η_p) was independent of individual (ANOVA, d.f.=3,286, F=1.67, P=0.17). On average,η _p was 0.74±0.01 (mean ± S.E.M., N=290).

Stereo-DPIV is a unique method for the simultaneous calculation of U,V and W velocity components for flow fields created by free-swimming animals. In this case, the flow fields examined are wakes created by caudal fin locomotion by Oncorhynchus mykiss. Caudal fin wakes have been the subject of many studies using two-dimensional DPIV techniques (Stamhuis and Videler,1995; Müller et al.,1997; Hanke et al.,2000; Lauder,2000; Liao and Lauder,2000; Müller et al., 2000, 2001; Drucker and Lauder, 2001b; Nauen and Lauder, 2002) but have not previously been visualized in three-dimensions. Characteristic of most of these wakes is the formation of linked vortex rings with laterally oriented central jet flow. Such jet flows were created during slow, steady,forward swimming by O. mykiss and were visible in high resolution(>750 vectors) using stereo-DPIV.

Previous studies have estimated three-dimensional wake geometry using two-dimensional DPIV methods (see, for example, Drucker and Lauder, 1999; Nauen and Lauder, 2002), which requires combining two-dimensional measurements made in at least two different planes (typically the parasagittal and frontal planes) measured during independent kinematic events. This combination of measurements requires study of a repetitive behavior with low kinematic variation. Such requirements are unnecessary with stereo-DPIV, however, as simultaneous calculation of U,V and W allows for three-dimensional visualization from measurements made using a single light sheet orientation. The 1-2 mm thickness of the light sheet used here limited the present W measurements to a subset of the total wake volume. Nevertheless, stereo-DPIV has high potential for three-dimensional flow visualization of non-repetitive, highly variable behaviors for which combining separate measurements from orthogonally oriented light sheets is not suitable.

The swimming speed of 1.2 BL s^-1 studied here has been used in previous studies of wake structure produced by caudal fin locomotion(Liao and Lauder, 2000; Drucker and Lauder, 2001b; Nauen and Lauder, 2002) and is close to speeds examined in other two-dimensional DPIV studies of caudal fin locomotion (Müller et al., 1997, 2001), which allows for direct comparison of the present wake structure data with wake structures described for other species. The data of Webb et al.(1984) indicate that the speed of 1.2 BL s^-1 is well within the aerobic capacity limits of O. mykiss.

Stereo-DIPV revealed a strong, oscillating jet with a large lateral flow component in the wake of Oncorhynchus mykiss swimming steadily at 1.2 BL s^-1. Such jets are typically associated with linked vortex ring wakes produced by body—caudal-fin locomotors (e.g. Müller et al., 1997; Wolfgang et al., 1999; Drucker and Lauder, 2001b; Nauen and Lauder, 2002). The magnitudes and directions of U, V and W flows in the wake correspond to the magnitude of reaction forces to the jet flow that powers locomotion. Of particular interest are the magnitude and direction of vertical flows produced by the caudal fin that may act to rotate the fish about its anteriorly positioned center of mass and the ratio of U/total wake flow, which is a flow index of caudal fin mechanical performance.

Vertical flows measured here were very small in magnitude and were variable in direction. O. mykiss, a member of the Salmoniformes, has a symmetrically shaped, or homocercal, forked caudal fin. It is traditionally assumed that caudal fins of this shape move symmetrically and generate forces in the horizontal plane only (for a review, see Lauder, 2000). The present data, and study of more derived teleosts with forked homocercal caudal fins,do not, however, support this assumption. During relatively slow, steady swimming, the caudal fin of bluegill sunfish Lepomis macrochirustilts relative to the vertical (probably because of intrinsic tail muscle activity; Lauder, 2000) and produces ventrally inclined flows (Lauder,2000). The homocercal forked caudal fin of chub mackerel Scomber japonicus also tilts relative to vertical during steady swimming (Gibb et al., 1999)and produces ventrally inclined flows and thus lift forces that tend to lower the head (Nauen and Lauder,2002).

Vertical flows produced by homocercal caudal fins tend to be small in magnitude. For example, vertical forces are less than 3% of the total force production of slightly larger mackerel (23±1 cm fork length, mean± S.E.M., N=4) swimming at 1.2 and 2.2 fork lengths s^-1 (Nauen and Lauder,2002). The low-magnitude V flows measured in the present study are thus similar to those for the more derived species(Lauder, 2000; Nauen and Lauder, 2002). The present data indicate, however, that, unlike L. macrochirus(Lauder, 2000) and S. japonicus (Nauen and Lauder,2002), O. mykiss shows variable caudal fin vertical orientations during slow steady swimming. Indeed, on average, the flows produced by three out of the four individuals studied here were directed slightly upwards. This indicates the production of a lift force that would tend to raise the head, which is the opposite of previous findings.

There is also typically a strong lateral component to jet flow produced by caudal fin locomotion (Stamhuis and Videler, 1995; Müller et al., 1997; Hanke et al.,2000; Lauder,2000; Liao and Lauder,2000; Müller et al., 2000, 2001; Drucker and Lauder, 2001b; Nauen and Lauder, 2002), which signifies relatively strong lateral force production. Studies of both bluegill sunfish (Drucker and Lauder,2001b) and chub mackerel(Nauen and Lauder, 2002)indicated that for animals swimming freely at a speed of approximately 1 BL s^-1, lateral forces are approximately twice the magnitude of thrust forces. W flows seen in the present study were typically greater in magnitude than U, although by a factor of less than two (Fig. 4).

High-magnitude W relative to U (and low-magnitude V) resulted in U/total wake flow ratios in the range 0.32-0.45. This ratio is a flow-based estimate of true mechanical performance(which is the ratio of thrust/total force). Mechanical performance ratios of the order of 0.3-0.4 have been determined using two-dimensional DPIV for caudal fin locomotion by slightly larger (approximately 23 cm BL)bluegill sunfish (Drucker and Lauder,2001b) and chub mackerel(Nauen and Lauder, 2002)swimming at approximately 1 BL s^-1. These values are consistent with the present three-dimensional DPIV data for O. mykiss. Thus, data collected using two methods, from three distantly related euteleost taxa with a variety of body and fin shapes and a range of swimming performance capacities, indicate a high lateral force component during slow, steady caudal fin locomotion. It is possible that the relatively low mechanical performance of these taxa reflects an inherent limitation to caudal fin propulsion as a posteriorly propagated wave of lateral bending. Supporting this hypothesis is the finding of Triantafyllou et al.(1991) of optimum mechanical performance values of 0.2 for a foil oscillating transversely through uniform incoming flow (which is a physical model of caudal fin locomotion).

In contrast to these caudal fin mechanical performance values, our calculated mean Froude propulsive efficiency (η_p) of 0.74 was relatively high. The η_p value of 0.74 determined here for the single relative speed of 1.2 BL s^-1 is somewhat larger than values determined previously using kinematic data. For example, Webb(1971a) calculated aη _p value of 0.54 using larger trout (30 cm BL) than those studied here swimming at speeds of approximately 1.25 BLs^-1. Values of η_p approached 0.7 as swimming speed increased to approximately 1.7-1.9 BL s^-1 (see Table 11 in Webb, 1971a). In that study,η _p was calculated as the ratio of fish swimming speed to the speed of the body propulsive wave, on the basis that this is the ratio between the momentum added to the water by the propulsive wave and the amount of kinetic energy necessary to accelerate that water(Webb, 1971b). A later study(Webb et al., 1984) confirmed the conclusion that η_p increases with increasing speed for O. mykiss. An efficiency value of 0.7 was, however, previously determined for O. mykiss on the basis of power measurements calculated from automatic particle tracking and laser-Doppler anemometry techniques (Blickhan et al.,1992); however, few data and methodology details from that study are available for direct comparison with the present study.

The above range of η_p values suggests a relatively high propulsive efficiency for O. mykiss at slow cruising speeds, in contrast to the caudal fin mechanical performance values of 0.32-0.45. Theη _p ratio reflects whole-body propulsive efficiency, including,for example, the volume of water accelerated by the fish (see the discussion in Blake, 1983), whereas caudal fin mechanical performance simply reflects the proportion of wake flow that is thrust. The finding that η_p is much greater than caudal fin mechanical performance suggests the involvement of additional factors (such as body shape and propulsive wave amplitude and frequency), which might result in relatively high overall propulsive efficiencies even though the performance of any given propulsor is relatively low.

The use of stereo-DPIV to determine simultaneously all three velocity components allowed for calculations of both η_p and caudal fin mechanical efficiency for single kinematic events in the present study (rather than averaging over multiple events as is necessary with two-dimensional DPIV,Drucker and Lauder, 1999, 2000; Nauen and Lauder, 2002). The homocercal caudal fin shows flexibility in performance during steady forward locomotion with increasing speed, particularly in regard to vertical force production and increases in thrust/total force production as a function of speed (Nauen and Lauder,2002). Future studies of the homocercal caudal fin of teleost fishes under a wider range of hydrodynamic conditions (including unsteady and maneuvering locomotion) are necessary to ascertain the extent of functional versatility in the caudal fins of different taxa. Including calculations of both η_p and caudal fin mechanical performance will allow a better understanding of the interplay between caudal fin performance and locomotor efficiency.

The authors thank Laura Farrell for assistance with obtaining trout and conducting the experiments. We also thank Laura Farrell, Ellen Freund, Tonia Hsieh, Jimmy Liao and especially Eric Tytell for helpful comments during the course of the study. Comments from two anonymous reviewers improved the manuscript. Support was provided by NSF grant IBN-9807012 to G.V.L.