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First published online January 30, 2009
Journal of Experimental Biology 212, 576-592 (2009)
Published by The Company of Biologists 2009
doi: 10.1242/jeb.025007

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Numerical investigation of the hydrodynamics of anguilliform swimming in the transitional and inertial flow regimes

Iman Borazjani and Fotis Sotiropoulos^*

St Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55402, USA

View larger version (10K): [in this window] [in a new window]   Fig. 1. The anguilliform virtual swimmer, created from a CT scan of a lamprey and meshed with triangular elements as needed by the sharp-interface immersed boundary method, from (A) side view, (B) top view and (C) perspective.

View larger version (15K): [in this window] [in a new window]   Fig. 2. (A) The amplitude envelope for the virtual anguilliform swimmer and (B) midlines of the virtual anguilliform swimmer according to Eqn 3 for several time instants during one tail-beat cycle.

View larger version (35K): [in this window] [in a new window]   Fig. 3. Time history of the axial force coefficient (CF) normalized by the rigid body drag coefficient for different St at Re=4000 of (A) the anguilliform virtual swimmer compared with (B) the carangiform virtual swimmer (Borazjani and Sortiropoulos, 2008). Positive and negative values indicate that the force is of thrust- and drag-type, respectively.

View larger version (19K): [in this window] [in a new window]   Fig. 4. Time history of the axial force coefficient (CF, broken lines) and its pressure (Cp, dotted lines) and viscous (Cv, solid lines) components at Re=4000 and critical St*=0.62 for the anguilliform virtual swimmer (in red) compared with the carangiform virtual swimmer at St*=0.6 (in black) (Borazjani and Sortiropoulos, 2008). Positive and negative values indicate that the force is of thrust- and drag-type, respectively.

View larger version (12K): [in this window] [in a new window]   Fig. 5. Comparison of root-mean-square (rms) of axial force coefficient (CF) fluctuations normalized by the rigid body drag coefficient for the virtual anguilliform swimmer (filled symbols) and the carangiform swimmer (open symbols).

View larger version (10K): [in this window] [in a new window]   Fig. 6. Effect of Reynolds and Strouhal numbers on the mean axial force coefficient (CF) produced by the tethered lamprey. The axial force coefficient is time-averaged and normalized by the rigid body drag coefficient. The lower broken horizontal line shows the rigid body drag coefficient, and the upper broken horizontal line shows the zero mean axial force coefficient, i.e. self-propulsion limit.

View larger version (9K): [in this window] [in a new window]   Fig. 7. Effect of the amplitude envelope a(z) at different Reynolds and Strouhal numbers on the mean axial force coefficient produced by the tethered lamprey. The axial force coefficient is time-averaged and normalized by the rigid body drag coefficient. The broken horizontal line shows the zero mean axial force coefficient, i.e. self-propulsion limit.

View larger version (8K): [in this window] [in a new window]   Fig. 8. Schematic showing anguilliform propulsion as an undulatory pump. U, swimming velocity; V, traveling wave speed; Ur, is the relative velocity as observed from the shown element (shaded in blue); Un, component of relative velocity in the normal direction; Ut, component of relative velocity in the tangential direction; Fn, normal force; Ft, tangential force; Tn, part of normal force in the direction of swimming; Tt, part of tangential force in the direction of swimming; Tnet, net force in the direction of swimming.

View larger version (4K): [in this window] [in a new window]   Fig. 9. Schematic showing carangiform propulsion as a heaving and pitching foil. U, swimming velocity; Utail, the lateral velocity of the tail; Ur, relative velocity; FL, lift force; FD, drag force; TL, part of the lift force in the direction of swimming; TD, part of the drag force in the direction of swimming; Tnet, inet force in the direction of swimming.

View larger version (14K): [in this window] [in a new window]   Fig. 10. Variation of the skin (friction) drag, form and total drag with Strouhal number at two Reynolds numbers: Re=300 (A) and Re=4000 (B). The drag forces are calculated using Eqn 9. All the drag coefficients are normalized by the rigid body drag coefficient.

View larger version (19K): [in this window] [in a new window]   Fig. 11. Pressure contours and streamlines in mid-plane of the fish relative to a frame moving with the body traveling wave speed V (Re=4000). (A) Flow does not separate for the straight rigid fish. (B) Flow separates for St=0.2, U/V=1.39. (C) Flow does not separate for St=0.4, U/V=0.69.

View larger version (11K): [in this window] [in a new window]   Fig. 12. Sketch of the wake behind a steady undulatory swimmer. (A) A single row of vortices observed behind carangiform swimmers. (B) A double row of vortices observed behind anguilliform swimmers. The circles indicate the shed vortices, with the arrowheads indicating their rotational sense. The arrows indicate the jet flows.

View larger version (46K): [in this window] [in a new window]   Fig. 13. Calculated out-of-plane vorticity contours with velocity vectors for the anguilliform swimmer at Re=, St=0.45 on the horizontal (x1–x3) mid-plane. For clarity, only every third velocity vector is plotted.

View larger version (31K): [in this window] [in a new window]   Fig. 14. Instantaneous streamlines with vorticity contours for anguilliform swimmer showing: (A) a single-row regular Karman street (Re=4000, St=0.2, F=0.0027); (B) a double-row thrust-type wake (Re=, St=0.45, F=–3x10–5); (C) a double-row drag-type wake (Re=4000, St=0.7, F=–0.0009); and (D) a double-row drag-type wake (Re=4000, St=0.62, F=–6x10–6). The red arrows show the general direction of the wake flow.

View larger version (31K): [in this window] [in a new window]   Fig. 15. Three-dimensional (3-D) vortical structures visualized using the q-criterion showing 3-D wake structures simulated for the Re=300 case. (A) Double-row wake at St=1.1 (F=–0.0043); (B) single-row wake at St=0.2 (F=–0.0157).

View larger version (24K): [in this window] [in a new window]   Fig. 16. Three-dimensional (3-D) vortical structures visualized using the q-criterion showing 3-D wake structures simulated for the Re=4000 case. (A) Double-row wake at St=0.7 (F=0.0009); (B) single-row wake at St=0.2 (F=–0.0027).

View larger version (19K): [in this window] [in a new window]   Fig. 17. Three-dimensional (3-D) vortical structures visualized using the q-criterion showing 3-D wake structures simulated for the inviscid case. (A) Double-row wake at St=0.5 (F=0.0002); (B) single-row wake at St=0.2 (F=–0.0003).

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