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First published online December 28, 2007
Journal of Experimental Biology 211, 280-287 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.007641
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An overview of a Lagrangian method for analysis of animal wake dynamics

Jifeng Peng1 and John O. Dabiri1,2,*

1 Bioengineering, California Institute of Technology, Pasadena, CA 91125, USA
2 Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA


Figure 1
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  		Fig. 1. Schematic diagram of a vortex boundary in a flow. (A) 3-D sketch of a vortex ring; (B) the vortex ring on its median-symmetry plane. Circles with inscribed arrows indicate vortex cores and their rotational sense. Two 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 finite-time Lyapunov exponent field (FTLE) at the boundary. Adapted from Peng et al. (Peng et al., 2007Go).

 

Figure 2
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  		Fig. 2. Contour plots of FTLE fields calculated for a moving vortex ring (propagating from right to left across the page). Left: backward-time FTLE; right: forward-time FTLE. Adapted from Shadden et al. (Shadden et al., 2006Go).

 

Figure 3
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  		Fig. 3. The boundary of the vortex derived from Lagrangian coherent structures (LCS). The left solid line shows the attracting LCS from backward FTLE calculation while the right solid line shows the repelling LCS from forward FTLE calculation. Broken lines are spline lines connecting the LCS. The fin (curved 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 due to the limitation in integration time T. Adapted from Peng et al. (Peng et al., 2007Go).

 

Figure 4
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  		Fig. 4. Time evolution of the vortex boundary. Vortex boundaries at 11 different time instances are plotted from red (t=0 ms) to blue (t=300 ms) with a time interval of 30 ms. Adapted from Peng et al. (Peng et al., 2007Go).

 

Figure 5
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  		Fig. 5. The locomotive force in (A) the horizontal and (B) the vertical directions. Squares: calculated locomotive forces. Error bars indicate uncertainty from measurement and evaluation. Solid line: spline fitting of the data. Note that due to limitations on integration time, these plots are based on the first 400 ms of a 600 ms fin beat cycle. Adapted from Peng et al. (Peng et al., 2007Go).

 




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