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Function of the heterocercal tail in sharks: quantitative wake dynamics during steady horizontal swimming and vertical maneuvering

C. D. Wilga^1,* and G. V. Lauder²

¹ Department of Biological Sciences, University of Rhode Island, 100 Flagg Road, Kingston, RI 02881, USA
² Museum of Comparative Zoology, Harvard University, Cambridge, MA 02138, USA

View larger version (28K): [in a new window]   Fig. 1. Schematic summary of two alternative models illustrating the forces acting on the body of a shark during steady horizontal swimming. (A) Modified version of the classical model (with data on body angle and pectoral fin function incorporated from Wilga and Lauder, 2000, 2001) in which the beating of the tail is proposed to generate an upward lift force (Ftail) that generates a torque around the center of mass (shaded circle). Force on the water is directed posteroventrally (Fwater), and an equal and opposite reaction force is directed anterodorsally, dorsal to the center of mass (Freaction). Torques generated by the tail are countered by equal and opposite torques resulting from lift forces produced by the body (Fbody), which has a positive angle of attack during horizontal locomotion. The net upward lift forces are balanced by the weight (Fweight) of the negatively buoyant shark. The pectoral fins do not generate lift during steady horizontal locomotion (Wilga and Lauder, 2000, 2001) and, hence, no forces are shown acting on these fins. (B) Modified version of the model of Thomson (1976) (to include our previously published data on shark body angle and pectoral fin function) in which the tail generates a reaction force that is directed anteriorly through the center of mass.

View larger version (131K): [in a new window]   Fig. 2. Synchronized video images illustrating lateral views of (A) Triakis semifasciata (top) and (B) Chiloscyllium punctatum (bottom) during steady horizontal swimming to show body angle and position relative to the edge of the laser sheet (left) and the vertical laser sheet with tail and particles (right). Scale bars, 2 cm. The X and Y axes are marked.

View larger version (16K): [in a new window]   Fig. 3. Schematic summary illustrating body and wake variables measured relative to the horizontal: body angle, from a line drawn along the ventral body surface; path of motion of the center of mass; tail angle between the caudal peduncle and dorsal tail lobe; ring axis angle from a line extending between the two centers of vorticity; and mean vortex jet angle. Angle measurements from the variables of interest (dotted lines) to the horizontal (dashed line) are indicated by the curved solid lines. Angles above the horizontal are considered positive and those below the horizontal negative. Ring axis angle was measured from 0 to 180 °.

View larger version (63K): [in a new window]   Fig. 4. DPIV analysis of the wake of the tail of a representative Triakis semifasciata (A) and Chiloscyllium punctatum (B) during steady horizontal locomotion. On the left is a tracing of the tail depicting its position relative to single the shed vortex ring visible in this vertical section of the wake. The color plot to the right shows fluid vorticity, with superimposed black velocity vectors representing the results of DPIV calculations based on particle displacements. A strong jet, indicated by the larger velocity vectors, passes between two counter-rotating vortices representing a slice through the vortex ring shed from the tail at the end of each beat. The white dashed line indicates the ring axis cycle. Note that a green color indicates no fluid rotation, a blue color reflects clockwise fluid rotation and a red/yellow color indicates counterclockwise fluid rotation. To assist in visualizing jet flow, a mean horizontal flow of U=19 and U=24 cm s-1 was subtracted from each vector for T. semifasciata and C. punctatum, respectively.

View larger version (35K): [in a new window]   Fig. 9. Schematic diagram of the force balance on Triakis semifasciata swimming at 1.0 Ls-1, where L is total body length, while holding position (also representative of Chiloscyllium punctatum), rising and sinking in the water column. The yellow circle represents the center of mass, and vectors indicate forces F exerted by the fish on the fluid. Lift forces are generated by the ventral body surface, both anterior and posterior to the center of mass. The jet produced by the beating of the tail maintains a constant angle relative to body angle and path angle and results in an anterodorsally directed reaction force oriented dorsal to the center of mass during all three behaviors, supporting the classical model. Data on pectoral fin function are from Wilga and Lauder (2000, 2001). Tail vortex jet angles are predicted mean values from Table 2.

View larger version (20K): [in a new window]   Fig. 5. Plots of jet angle versus body angle in (A) Triakis semifasciata and (B) Chiloscyllium punctatum for steady horizontal locomotion (holding position) only. The solid line for C. punctatum indicates a significant linear regression (y=-58.287+1.452x; P=0.008, r2=0.432).

View larger version (23K): [in a new window]   Fig. 7. Plots of body angle versus (A) tail angle, (B) jet angle and (C) ring axis angle in Triakis semifasciata. The solid lines indicate a significant linear regression, while the dotted lines represent the predicted relationships (see text for discussion). The lack of significance of the tail versus body angle regression (P=0.731, r2=0.003) indicates that the sharks are not altering their tail angle as body angle changes, but instead are maintaining a constant angular relationship regardless of locomotor behavior. Jet angle decreases with increasing body angle (y=-17-1.087x; P<0.001, r2=0.312) at the same rate as the predicted parallel relationship, indicating that the vortex jet is generated at a constant angle to the body regardless of body position. Ring axis angle increases with body angle at the same rate as the predicted perpendicular relationship (y=107+1.280x; P<0.001, r2=0.401). Circles, triangles and squares represent holding, rising and sinking, respectively.

View larger version (23K): [in a new window]   Fig. 8. Plots of jet angle versus (A) path of motion angle and (B) ring axis angle for Triakis semifasciata. The solid lines indicate a significant linear regression, while the dotted lines represent the predicted relationships (see text for discussion). Jet angle has a significant negative correlation with path angle (y=-29-1.045x; P<0.001, r2=0.257) and parallels the predicted relationship in which vortex jets are oriented in a direction opposite to the path of motion. Jet angle decreases with increasing ring axis angle at a slower rate than expected assuming a perpendicular relationship (y=34.720-0.519x; P<0.001, r2=0.291). Circles, triangles and squares represent holding, rising and sinking, respectively.

View larger version (49K): [in a new window]   Fig. 6. Representative DPIV analyses of the wake of the tail of Triakis semifasciata while (A) rising and (B) sinking in the water column. Each color plot shows fluid vorticity, with superimposed black velocity vectors representing the results of DPIV calculations as in Fig. 4. During rising behavior, the direction of the jet is similar to that while holding vertical position during horizontal locomotion, although the ring axis angle is inclined more horizontally. In contrast, during sinking behavior, the fluid jet is significantly more horizontal and the ring axis angle is significantly more vertically inclined. To assist in visualizing the flow pattern, a mean horizontal flow of U=19 and U=24 cm s-1 was subtracted from each vector for rising and sinking, respectively. The white dashed line indicates the ring axis cycle.