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First published online May 2, 2008
Journal of Experimental Biology 211, 1541-1558 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.015644
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Numerical investigation of the hydrodynamics of carangiform swimming in the transitional and inertial flow regimes

Iman Borazjani and Fotis Sotiropoulos*

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


Figure 1
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  		Fig. 1. (A,B) Different views of the virtual carangiform swimmer closely modeled after a mackerel and meshed with triangular elements as needed for the sharp-interface immersed boundary numerical method.

 

Figure 2
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  		Fig. 2. Midlines of the fish for different instants during one tail beat cycle (A) and the amplitude envelope profile (B).

 

Figure 3
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  		Fig. 3. Time history of the force coefficient normalized by the rigid body drag for different St at Re=4000. Positive and negative values indicate that the net force is of thrust- and drag-type, respectively.

 

Figure 4
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  		Fig. 4. Effect of Re and St on the mean force coefficient produced by the tethered fish. The force coefficient is time-averaged and normalized by the rigid body drag coefficient. The lower broken line shows the rigid body drag coefficient and the upper broken line shows the zero mean force coefficient, i.e. self-propulsion limit.

 

Figure 5
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  		Fig. 5. Variation of the skin (friction) drag, form drag and total drag with Strouhal number at two Reynolds numbers: Re=300 (A) and Re=4000 (B). The drag forces are calculated using Eqn 17.

 

Figure 6
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  		Fig. 6. Pressure contours and streamlines on the midplane of the fish relative to a frame moving with the body wave phase speed V (Re=300). (A) Rigid body (St=0). (B) Flow separates for St=0.1, U/V=2.11. (C) Flow does not separate for St=0.3, U/V=0.7.

 

Figure 7
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  		Fig. 7. Calculated out-of-plane vorticity contours with velocity vectors for the Re={infty}, St=0.26 case (A) on the horizontal (x1–x3) mid-plane and (B) the vertical (x2–x3) plane.

 

Figure 8
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  		Fig. 8. Instantaneous streamlines with vorticity contours showing (A) a single row regular Karman street (Re=4000, St=0.2); (B) singe row reverse Karman street (Re={infty}, St=0.26); and (C) double row reverse Karman street (Re=4000, St=0.7). The red arrows show the general direction of the wake flow.

 

Figure 9
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  		Fig. 9. Instantaneous streamlines and vorticity contours at the horizontal mid-plane for St=0.3 highlighting the effect of Reynolds number on the wake structure. (A) Re=300; (B) Re=4000 (middle); (C) Re={infty}.

 

Figure 10
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  		Fig. 10. Three-dimensional (3D) vortical structures visualized using the q-criterion showing 3D wake structures simulated for the Re=300 case. (A) Double row wake at St=1.2; (B) single row wake at St=0.3.

 

Figure 11
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  		Fig. 11. Three-dimensional (3D) vortical structures visualized using the q-criterion showing 3D wake structures simulated for the Re=4000 case. (A) Double row wake at St=0.7; (B) single row wake at St=0.2.

 

Figure 12
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  		Fig. 12. Three-dimensional (3D) vortical structures visualized using the q-criterion showing 3D wake structures simulated for the inviscid case. (A) Double row wake at St=0.7; (B) single row wake at St=0.26.

 

Figure 13
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  		Fig. 13. The iso-surfaces of q cut with (A) x1–x3 and (B) x2–x3 mid-planes showing the footprints of the wake structure using the out-of-plane vorticity contours. The inset (C) gives a closer look at the structures cut with both x1–x3 and x2–x3 mid-planes (Re=4000, St=0.7).

 

Figure 14
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  		Fig. A1. Time history of the inline force coefficient (solid lines) and its pressure (dotted lines) and viscous (broken lines) components from the present computations (red lines) compared with the computations of Dutsch et al. (Dutsch et al., 1998Go) (black lines).

 

Figure 15
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  		Fig. A2. Comparison of the inline velocity component (u) profile at position x1=–0.6D for three different phase angles ({phi}=2{pi}ft) obtained from the present study (CFD; solid line) and the experimental measurements of Dutsch et al. (Dutsch et al., 1998Go) (Exp; square symbols).

 

Figure 16
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  		Fig. A3. Contours of vorticity at four different phase angles ({phi}=2{pi}ft): (A) {phi}=0°; (B) {phi}=96°; (C) {phi}=192°; (D) {phi}=288°. Broken lines indicate negative values.

 

Figure 17
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  		Fig. A4. The effect of grid refinement ({Delta}x) on the force coefficient and St* for Re=4000.

 

Figure 18
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  		Fig. A5. Variation of the error {epsilon} in the calculation of the force coefficient CF (absolute value of the difference between CF on a given grid minus the value calculated on the finest mesh with {Delta}x=0.004) with grid size in the log–log scale (Re=4000, St=0.5). The error is reduced with a slope slightly larger than 2nd order accuracy.

 

Figure 19
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  		Fig. A6. Time history of force coefficient normalized by the rigid body drag coefficient for three different grid sizes at Re=4000 and St=0.5. Positive and negative values indicate that the net force is of thrust- and drag-type, respectively.

 

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© The Company of Biologists Ltd 2008