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First published online January 31, 2007
Journal of Experimental Biology 210, 685-698 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.02692
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Non-invasive measurement of instantaneous forces during aquatic locomotion: a case study of the bluegill sunfish pectoral fin

Jifeng Peng1, John O. Dabiri1,2,*, Peter G. Madden3 and George V. Lauder3

1 Bioengineering, California Institute of Technology, Pasadena, CA 91125, USA
2 Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
3 Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA


Figure 1
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  		Fig. 1. The side and top views of a bluegill sunfish and the laser plane (shown in blue) for the DPIV experiments that generated the data used for the analysis in this paper. Note that the camera viewed flow from behind the fish at 500 frames s–1, and pectoral fin wakes thus move toward the field of view allowing a complete view of the wake at high temporal resolution.

 

Figure 2
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  		Fig. 2. A schematic drawing of a vortex boundary in a flow. Circles with inscribed arrows indicate vortex cores and their rotational sense. A pair of adjacent fluid particles close to but on different sides of the vortex boundary separate from each other faster than other arbitrary pairs of fluid particles, giving a larger value of the FTLE at the boundary. Trajectories can be followed in backward-time (A) to reveal attracting LCS, and in forward-time (B) to reveal repelling LCS boundaries.

 

Figure 3
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  		Fig. 3. Illustration of the approximation of the vortex volume. The calculated vortex boundary on the transverse plane was modeled by an ellipse, which represents the cross-section of the vortex in the transverse plane. The calculated vortex boundary and the model ellipse have the same long axis length and also the same width. The long axes of the calculated boundary and the approximated shape are parallel. The volume of the vortex is approximated by the product of the center-width w of the vortex in the laser plane view and the projected area A of the vortex ring onto a plane perpendicular to the laser sheet.

 

Figure 4
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  		Fig. 4. Velocity and vorticity fields of the pectoral fin wake in the transverse plane. (A) Early downstroke; (B) late downstroke and stroke reverse; (C) early upstroke; (D) late upstroke. Red colors represent negative or clockwise fluid rotation, while blue colors indicate positive vorticity or counterclockwise fluid rotation. The camera recorded a posterior view of the left pectoral fin and the fish body. Every other vector is shown.

 

Figure 5
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  		Fig. 5. A snapshot of color contour plots of the FTLE fields computed from DPIV. (A) Backward FTLE; (B) forward FTLE. Position coordinates are specified in mm.

 

Figure 6
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  		Fig. 6. The boundary of the vortex derived from LCS. The left solid line shows the attracting LCS from backward FTLE calculation while the right-hand-side solid line shows the repelling LCS from forward FTLE calculation. Broken lines are spline lines connecting the LCS. The fin (the curve with high brightness inside the lines) can be seen embedded inside the vortex. The attracting and repelling LCS do not intersect to give the entire vortex boundary because of the limitation in integration time T. Every other vector is shown.

 

Figure 7
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  		Fig. 7. Time evolution of the vortex boundary. Vortex boundaries at 11 different time instances are plotted from red to blue with a time interval of 30 ms.

 

Figure 8
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  		Fig. 8. Trajectory of the projection of the vortex centroid on the transverse plane. Squares: calculated data at each time instance. Solid line: spline fitting of the data using a centered moving average method with a span of five data points. Error bars indicate measurement uncertainty. The designations are the same for Figs 9, 10, 11, 12.

 

Figure 9
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  		Fig. 9. The evolution of (A) x and (B) y positions of the projection of the vortex centroid on the transverse plane. Error bars indicate measurement uncertainty. Note that due to limitations on FTLE integration time, these plots are for the first 400 ms of a 600 ms fin beat cycle; see also Figs 10, 11, 12. Squares, calculated position of the vortex centroid; solid line, spline fitting of the data.

 

Figure 10
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  		Fig. 10. The velocity of the projection of the vortex centroid on the transverse plane. (A) Horizontal component Ux; (B) vertical component Uy.

 

Figure 11
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  		Fig. 11. (A) Volume of the vortex; (B) width of the vortex; (C) cross-sectional area of the vortex; and (D) added-mass coefficient of the vortex.

 

Figure 12
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  		Fig. 12. The locomotive force (mN) in (A) the horizontal and (B) the vertical directions. Squares, calculated locomotive forces; broken line, time-averaged forces calculated using the vorticity method (Drucker and Lauder, 1999Go).

 

Figure 13
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  		Fig. 13. Analysis of the contributing factors to the locomotive force. (A) Horizontal direction; (B) vertical direction. The changes in the logarithm of each parameter to the vortex momentum are plotted as well as the total change in the vortex momentum, in order to determine the most dominant contributing parameters to the change in momentum.

 




© The Company of Biologists Ltd 2007