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First published online October 17, 2008
Journal of Experimental Biology 211, 3490-3503 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.019224
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The hydrodynamics of ribbon-fin propulsion during impulsive motion

Anup A. Shirgaonkar1 and Oscar M. Curet1

Neelesh A. Patankar1,*

Malcolm A. MacIver1,2,*

1 Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA
2 Department of Biomedical Engineering, R. R. McCormick School of Engineering and Applied Science and Department of Neurobiology and Physiology, Northwestern University, Evanston, IL 60208, USA


Figure 1
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  		Fig. 1. Apteronotus albifrons, the black ghost knifefish of South America. (A) Photograph courtesy of Neil Hepworth, Practical Fishkeeping magazine. Inset shows side view of the fish to illustrate the angle of the fin with respect to the body. (B) Frame of video of the fish swimming in a water tunnel, from ventral side. (C) Model of the ribbon fin, shown together with symbols for fin length (Lfin), fin height (hfin) and two of the kinematic state variables, angular deflection ({theta}) and the vector from the rotational axis to a point on the fin (rm). The Eulerian (fluid) reference frame (x, y, z; positive direction indicated by inset) is shown together with names for velocities in the body frame fixed to the rigid fin rotation axis indicated by the red line (surge, heave, sway, roll, pitch, yaw; positive direction indicated by arrows). In this work, simulations were performed with the body-fixed frame oriented to the Eulerian frame as illustrated, and forces are with respect to a traveling wave moving in the head-to-tail direction indicated.

 

Figure 2
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  		Fig. 2. Close-up of a top view of the fluid-fin domain used for the numerical simulations. Shown here are the material points of the fin (black) and the Eulerian fluid grid (gray), with lengthwise grid spacing dx=0.375 mm and widthwise spacing of dy=0.4 mm. The inset shows the extent of the close-up with respect to the entire fluid-fin domain. Note that the material points do not conform to the fluid grid points.

 

Figure 3
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  		Fig. 3. Illustration of the digital particle image velocimetry (DPIV) setup used to visualize the flow field in the coronal (horizontal) plane of the robotic ribbon fin. The laser sheet was approximately 5 mm below the fin. The DPIV setup includes a charge couple device (CCD) camera to record the particle motion, a Nd:YAG laser light source, water tank and the robotic ribbon fin.

 

Figure 4
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  		Fig. 4. Vortex length as a function of time for the test case of the impulsively started flat plate. s is the vortex length, d is the plate length, U is the plate velocity, t is the time and T is the non-dimensional time (Taneda and Honji, 1971Go).

 

Figure 5
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  		Fig. 5. Forces on the fin from the fluid with coarse (blue), nominal (green) and fine (red) grid resolution. Colored solid lines show forces filtered to remove numerical noise at the grid-scale, and light gray lines are the raw data.

 

Figure 6
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  		Fig. 6. Vortex ring structure and x-velocity around the ribbon fin. (A) Top view, and (B) front end view of isosurface of vorticity magnitude 30 s–1, colored by the x-velocity normalized by the wave speed V (in this case V=10 cm s–1). The primary and secondary vortex rings are indicated in A (see Fig. 14A for a schematic of the primary vortex rings). (C) Bottom view, spatial correlation between an isosurface of the same vorticity magnitude as in A (yellow) and an isosurface of x-velocity at 2 cm s–1 (blue). Primary vortex rings are seen to connect across the fin surface. These results are for t=1.0 s and the baseline case: {theta}max=30 deg., f=2 Hz, {lambda}/Lfin=0.5, hfin/Lfin=0.1. Movies 1 and 2 (supplementary material) show the evolution of the streamwise jet from perspective and bottom views, respectively, for the baseline case.

 

Figure 7
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  		Fig. 7. Velocity field on a horizontal slice slightly below the lower edge of the fin (top view). The fin is shown in translucent gray color. Fin wave travels to the right. The arrows represent the velocity field and the color map indicates the x-velocity. Snapshot of flow at same time and wave parameters as for Fig. 6.

 

Figure 8
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  		Fig. 8. Isosurfaces of positive (yellow) and negative (blue) axial vorticity ({omega}x), side view. Note the counter-rotating axial vortex pairs attached to the tip of the ribbon fin (see Fig. 14B for a simplified schematic). Snapshot of the flow at the same time for the same wave parameters as for Fig. 6.

 

Figure 9
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  		Fig. 9. Horizontal slice of y-velocity (top view). Blue indicates velocity out of the plane of the paper, producing a net upward heave force. The fin is shown in translucent gray color. The fin wave travels to the right. Snapshot of the flow at the same time for the same wave parameters as for Fig. 6.

 

Figure 10
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  		Fig. 10. Digital particle image velocimetry (DPIV) results for the robotic ribbon fin at three different time instances. The ribbon fin is shown in translucent gray. The colormap indicates the x-velocity normalized by the wave speed V. In this case V=0.5 m s–1. (A) DPIV image and computed flow field at t=2.67 s (after four complete wave cycles). (B1), (B2) and (B3) flow field for the robotic ribbon fin at t=2.67, 2.80 and 3.00 s, respectively, equivalent to 0, 20 and 50% of one wave cycle. Velocity values at and above 50% of the wave speed are mapped to red to highlight the streamwise jet, which includes velocities as high as 90% of the wave speed.

 

Figure 11
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  		Fig. 11. Computed fin forces as a function of time for the baseline case, f=2 Hz, {theta}max=30 deg., {lambda}/Lfin=0.5, hfin/Lfin=0.1.

 

Figure 12
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  		Fig. 12. Trends in surge and heave forces with respect to frequency, maximum angular deflection, fin aspect ratio and specific wavelength. In each panel, the parameter shown on the x-axis is varied while keeping other parameters of the baseline case constant ({theta}max=30 deg., f=2 Hz, {lambda}/Lfin=0.5, hfin/Lfin=0.1). The gray area shows the root mean square deviation of the force for one cycle. (A) Simulation Set 1, (B) simulation Set 2, (C) simulation Set 3 and (D) simulation Set 4; surge to heave crossover is at {lambda}/Lfin=1.5. The estimated error based on grid sensitivity and domain length sensitivity is 10% of the mean value.

 

Figure 13
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  		Fig. 13. Tow drag of a 18.5 cm cast of a black ghost knifefish at 10, 12 and 15 cm s–1. Error bars indicate standard deviation. The broken line shows drag when towing the fish tail first. Values are slightly horizontally offset to show the standard deviation bars. Solid gray line indicates thrust values estimated from Blake (Blake, 1983Go) for a 15 cm Xenomystus nigri, a knifefish with a body shape similar to that of the black ghost knifefish.

 

Figure 14
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  		Fig. 14. Schematic of the primary flow features constituting the mechanisms of surge and heave force generation. (A) Surge: streamwise jet with the associated vortex rings. Position of vortex rings do not necessarily follow a regular pattern. See also Fig. 6C and Fig. 7. (B) Vortex tubes created along the edge of the ribbon fin shown in Fig. 8. Yellow and blue indicate the sign of vorticity, which alternates according to the direction of the fin tip movement.

 

Figure 15
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  		Fig. 15. Scaling of the surge force. Computational results are shown with a dot, and the curve fit with a black line. The error is defined as the root mean square error in mN for each curve fit. Estimation of surge force for a stationary fin using eqn 11 in Lighthill and Blake (Lighthill and Blake, 1990Go) are show in gray. In each panel, the factor shown on the x-axis is varied while keeping other parameters of the baseline case constant ({theta}max=30 deg., f=2 Hz, {lambda}/Lfin=0.5, hfin/Lfin=0.1). (A) linear fit of the natural log-transformed data for frequency (Hz), (B) linear fit of the natural log-transformed data for {theta}max (deg.) (C) linear fit of the log-transformed data for hfin/Lfin, (D) Curve fit of surge force as a function of specific wavelength.

 

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© The Company of Biologists Ltd 2008