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First published online December 28, 2007
Journal of Experimental Biology 211, 206-214 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.012161

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Biorobotic insights into how animals swim

Promode R. Bandyopadhyay^*, David N. Beal and Alberico Menozzi

Naval Undersea Warfare Center, Newport, RI 02841, USA

View larger version (49K): [in this window] [in a new window]   Fig. 1. Photograph showing a 30 cm-span fin attached to the roll/pitch mechanism. (Dimensions are in cm.)

View larger version (2K): [in this window] [in a new window]   Fig. 2. Schematic diagram of a fin positioned normal to uniform flow showing drag producing `cross-flow bluff-body drag' vortices. These are formed as the angle of attack of the fin increases well above 0°. The 90° situation is shown. The rolling and pitching motion of the fin helps retain the vortices over the fin, thereby delaying stall and enhancing lift due to the low pressures in their cores.

View larger version (20K): [in this window] [in a new window]   Fig. 3. Schematic diagram of the variables in a rolling and pitching fin. For definitions, see List of symbols and abbreviations.

View larger version (10K): [in this window] [in a new window]   Fig. 4. Comparison of the measurements of unsteady coefficients of lift (red) and drag (blue) with the cross-flow vortex model. The steady measurements are also included. Tow speed is 1.34 m s–1. Motion parameters are 0=30°, 0=15°, f=1.25 Hz, U=1.34 m s–1, span=20 cm, Bias=0°, St=0.26. St is Strouhal number defined as 2f0Ravg/U, where f is the frequency of oscillation.

View larger version (19K): [in this window] [in a new window]   Fig. 5. Comparison of the measurements of unsteady time signatures of forces in the forward and transverse directions and of power with the cross-flow vortex model for the case shown in Fig. 4. Tow speed is 1.34 m s–1. The fin kinematics are shown at the top.

View larger version (10K): [in this window] [in a new window]   Fig. 6. Comparison of the measurements of unsteady coefficients of lift (red) and drag (blue) with the cross-flow vortex model. Tow speed is 0.46 m s–1. The steady measurements are also included. Motion parameters are 0=30°, 0=35°, f=1.25 Hz, U=0.46 m s–1, span=20 cm, Bias=0°, St=0.78.

View larger version (18K): [in this window] [in a new window]   Fig. 7. Comparison of the measurements of unsteady time signatures of forces in the forward and transverse directions and of power with the cross-flow vortex model for the case shown in Fig. 6. Tow speed is 0.46 m s–1. The fin kinematics are shown at the top.

View larger version (7K): [in this window] [in a new window]   Fig. 8. Measurements of the roll torque and roll velocity at a tow speed of 1.34 m s–1 for the case shown in Fig. 4 (blue) and a tow speed of 0.46 m s–1 for the case shown in Fig. 6 (green). Observe the presence of hysteresis in the latter in comparison with the former. The hysteresis is attributable to the slower development of the Kutta condition at lower speed.

View larger version (16K): [in this window] [in a new window]   Fig. 9. Comparison of the averaged measurements of instantaneous lift forces with the cross-flow vortex model. The steady fin measurements are also included. Blue and green data indicate tests with a 20 or 30 cm span, respectively. The inset expands the data up to an angle of attack of 20° to clarify the validity of the model up to angles where stall occurs in the steady case.

View larger version (16K): [in this window] [in a new window]   Fig. 10. Comparison of the averaged measurements of instantaneous drag forces with the cross-flow vortex model. The steady fin measurements are also included. Blue and green data indicate tests with a 20 or 30 cm span, respectively. The inset expands the data up to an angle of attack of 20° where stall occurs in the steady case (Fig. 9) to clarify that the cross-flow vortex model does not account for viscous drag.

View larger version (9K): [in this window] [in a new window]   Fig. 11. Time sequence trace of the search for the highest efficiency during hovering. The green circles track the highest level of efficiency reached as yet during the scheme. Two flapping cycles were tested for each oscillation parameter set. Total search time is 4 min.

View larger version (11K): [in this window] [in a new window]   Fig. 12. Measurements of efficiency versus coefficient of X-force. The blue data were first collected in a systematic matrix study over a predefined range of oscillation parameters. This data gathering was spread over about 1.5 years, which is common in conventional experimental procedures where the hydrodynamic characteristics and the models of control laws are determined before a vehicle design is carried out. The symbols denote the carriage speed, where x is U=0, o is 0.46, + is 0.83, and * is 1.34 m s–1. The numbered dots denote trial numbers from the random search algorithm; green dots denote trials with a pitch bias greater than 5°, and red dots denote runs after which the bias had converged to less than 5°. Observe that from an arbitrary starting point, the algorithm rapidly reaches the point of highest efficiency for hovering as denoted by the x symbols. The search for maximum efficiency converges with any initial random selection of oscillation parameters. The rapid search method also works well when the fin is towed.

View larger version (200K): [in this window] [in a new window]   Fig. 13. PIV and laser cross-sectional end view of sunfish pectoral fin station keeping in a stream of speed 8.5 cm s–1 (from G. V. Lauder and P. G. A. Madden, personal communication, 2005); total fish length is 17 cm. The fin is in abduction phase. Observe the formation of contrarotating vortices at the fin tips. We hypothesize that they are cross-sections of two different stall vortices as shown in Fig. 14.

View larger version (86K): [in this window] [in a new window]   Fig. 14. Proposed dynamic stall vortex pairs shown in color in the sunfish pectoral fin formed during outstroke when the fin is undergoing `cupping' motion. The fin picture is from Lauder et al. (Lauder et al., 2007). The total fish length is 17 cm and the stream speed is 8.5 cm s–1. The fish is maintaining its station in a `uniform' stream, shown by the vertical arrow, while the fin is turning upstream and the spanwise edges are curling inward. The stall vortices locally augment the pressure difference across the two leading edges and could help cancel perturbations in the vertical directions. The co-flowing jets on both sides of the fish could be vectored appropriately to hold station laterally and provide some thrust. The vertical arrow shows the stream direction.

View larger version (12K): [in this window] [in a new window]   Fig.·15. Comparison of measurements of the variation in fin pitch angle with time during one cycle (red and blue diamonds) in bluegill sunfish pectoral fins (Lauder et al., 2007) with our rigid fin data. Sunfish pectoral fin: roll amplitude, 40.8°; frequency, 1.0·Hz; pitch amplitude, 44.8°.