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First published online August 8, 2008
Journal of Experimental Biology 211, 2669-2677 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.015883
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The `upstream wake' of swimming and flying animals and its correlation with propulsive efficiency

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 flexible plate model swimmer. s is the curve length and {Psi}(t,s) is the tangential angle. LE, leading edge (s=–1); TE, trailing edge (s=1).

 

Figure 2
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  		Fig. 2. Flow field from behind an eel. (A) Digital particle image velocimetry (DPIV) measurement (reproduced with permission from Tytell, 2007Go). (B) Simulation using the vortex sheet method. The key feature of the wake claimed by Tytell and Lauder (Tytell and Lauder, 2004Go), i.e. the strong lateral jet rather than a downstream jet, is consistent between the simulation and the measurement.

 

Figure 3
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  		Fig. 3. Schematic of particle separation at the boundary of a vortex ring. (A) Fluid particle pairs straddling the vortex boundary have a larger separation rate in backward time, indicating larger values of backward-time FTLE (finite-time Lyapunov exponent) at the front boundary of the vortex ring. (B) Fluid particle pairs straddling the vortex boundary have a larger separation rate in forward time, indicating larger values of forward-time FTLE at the rear boundary of the vortex ring. x and y are particle trajectories; T is integration time.

 

Figure 4
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  		Fig. 4. The effect of integration time on FTLE (finite-time Lyapunov exponent) and LCS (Lagrangian Coherent Structure) calculation for a vortex ring. (A) integration time T=0.4 s; (B) T=1.2 s; (C) T=2.0 s; (D) T=2.8 s. With longer integration time, the FTLE ridges become sharper, i.e. LCS resolve into clearly defined thin lines.

 

Figure 5
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  		Fig. 5. Locomotion of the model swimmer. The swimmer (S) begins at rest (A) and flaps its flexible body to propel itself forward. The vortex wake generated by the swimming motion (blue curve) and the resulting forward motion of the swimmer are shown after two (B), four (C) and six (D) swimming cycles. See Movie 2 in supplementary material.

 

Figure 6
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  		Fig. 6. Finite-time Lyapunov exponent (FTLE) fields of the flow created by the model swimmer (S). Results are presented in a reference frame fixed on the swimmer; the swimmer moves from right to left in a laboratory reference frame (see Fig. 5). Top panel, time t=0; second panel, t=T/5; third panel, t=(2/5)T; bottom panel, t=(4/5)T, where T is the duration of a single swimming cycle. (A) Backward-time FTLE field. The ridge of large FTLE values (solid blue curve) identifies the attracting LCS. (B) Forward-time FTLE field. The ridge of large FTLE values (solid red curve) identifies the repelling LCS. See Movie 3 in supplementary material.

 

Figure 7
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  		Fig. 7. Temporal evolution of the upstream wake. (A) Time t=0; (B) t=(7/5)T; (C) t =(14/5)T, where T is the duration of a single swimming cycle. See Movie 4 in supplementary material. Fluid particles in each of three adjacent repelling LCS (red curves) are labeled magenta or green in order to track their evolution. After interaction with the swimmer (shown here in a reference frame fixed on the swimmer; the swimmer moves from right to left in a laboratory reference frame; see Fig. 5), the fluid particles in the repelling LCS are shown to comprise the subsequent downstream wake, illustrated by the attracting LCS (blue curves). Fluid particles initially separated by the repelling LCS do not mix and are only moderately deformed. S, swimmer; w(x), the width of the upstream wake; RLCS, repelling LCS; ALCS, attracting LCS.

 

Figure 8
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  		Fig. 8. Temporal evolution of arbitrary upstream fluid parcels. Fluid particles in adjacent regions with a horizontal (A,B) or vertical (C,D) interface are labeled magenta or green in order to track their evolution. The interfaces in both cases do not coincide with the repelling LCS (red curves). After interaction with the swimmer (shown here in a reference frame fixed on the swimmer; the swimmer moves from right to left in a laboratory reference frame; see Fig. 5), the particles in the adjacent parcels exhibit substantial deformation and mixing in the vicinity of the attracting LCS (blue curves). In both cases, the parcels do not indicate the full extent of the downstream wake, in contrast to the repelling LCS parcels in Fig. 7. S, swimmer; RLCS, repelling LCS; ALCS, attracting LCS. (A,C) Time t=0; (B,D) t=(14/5)T, where T is the duration of a single swimming cycle. See Movies 5 and 6 in supplementary material.

 

Figure 9
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  		Fig. 9. Total force acting on the swimmer (A) and the velocity of the swimmer (B). The swimmer starts from rest. As it approaches a steady state, i.e. the mean velocity over a stroke cycle approaches a constant, the time average of force approaches zero.

 

Figure 10
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  		Fig. 10. Efficiency based on mass flow rate (Eqn 6) of the model swimmer versus Strouhal number St=fA/U. Red, kinematics 1; black, kinematics 2; green, kinematics 3. Error bars indicate uncertainty in measurement of upstream wake width.

 

Figure 11
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  		Fig. 11. Comparison of measures for swimming efficiency. Plots A, B and C show kinematics 1, 2 and 3, respectively. Solid lines: efficiency based on the width of the upstream fluid structure; error bars indicate uncertainty in measurement of upstream wake width; broken lines: efficiency calculated by replacing upstream wake width with stroke amplitude of the swimmer (peak-to-peak excursion at trailing edge).

 

Figure 12
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  		Fig. 12. The width of the wake. (A) Wake width of a swimmer swimming at a constant velocity (2 BL s–1) with different tail amplitudes. (B) Wake width of a swimmer swimming at different velocities with the same body kinematics (flapping amplitudes of 1.44 BL).

 

Figure 13
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  		Fig. 13. Efficiency of the swimmer versus modified Strouhal number St=fw/U. Red, kinematics 1; black, kinematics 2; green, kinematics 3.

 

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© The Company of Biologists Ltd 2008