Optimal Vortex Formation as a Unifying Principle in Biological Propulsion

Transcription

1 Optimal Vortex Formation as a Unifying Principle in Biological Propulsion M. GHARIB California Institute of Technology BIOENGINEERING

2 Vortex Rings A Secondary Eruption of Mt. St. Helens, June 1980 [Photo by Robert P. VanNatta]

3 Zebrafish (Danio rerio) Embryo 1.5 mm long (100 micron) Koster, Forouhar and Gharib (2002)

4 Previous Work on Vortex Rings Early work: Helmholtz (1858), Kelvin (1869) Existence: Hill (1894), Fraenkel (1972), Norbury (1973), and others Formation: Saffman (1975, 1978), Pullin (1979), Didden (1979), Glezer (1987) Evolution and Turbulent rings: Maxworthy (1972, 1974, 1977), Glezer (1987), and others Fully-Pulsed Jets: Bremhorst et al. (e.g., 1979, 1990, and 2000) Weihs (1977)

5 Classical View Vortex ring formation is a problem of vortex sheet dynamics,, the steady state is a problem of existence,, their duration is a problem of stability,, and if there are several we have a problem of vortex interactions. -- P.G. Saffman (1981), emphasis added

6 New Motivations Understanding complicated biological flows: Aquatic Propulsion Cardiac Flows Practical Applications: Hydropropulsion /Aeropropulsion Micro jet thrusters Multi-scale Stirring and Mixing

7 Canonical Vortex Ring Generator Vortex rings can be easily generated using a piston-cylinder mechanism to produce a starting jet. D L Parameters: a) Time history of piston velocity b) L/D c) Reynolds Number d) Orifice/nozzle Geometry Can be viewed as the roll up of a half- infinite cylindrical vortex sheet.

8 Vortex Ring Formation P.S. Krueger, J.O. Dabiri and M. Gharib (2003)

9 Vortex Ring Formation

10 Vortex Ring Experiments Glezer (1981) Didden (1979) Kwon and Bernal (1989) Auerbach (1980) Schatzle (1987) Weigand and Gharib (1994) Maxworthy (1977) Sallet (1974) L/D<1 L/D<2 L/D<4 L/D<1 L/D<1 L/D<1 L/D<3 L/D<1

11 Can we generate arbitrarily large vortex rings? L D t = U D 0 ( τ ) dτ = 2 M. Gharib, E. Rambod & K. Shariff, Journal of Fluid Mechanics (1998)

12 Can we generate arbitrarily large vortex rings? L D t = U D 0 ( τ ) dτ = 2 L D = 3.8 M. Gharib, E. Rambod & K. Shariff, Journal of Fluid Mechanics (1998)

13 Can we generate arbitrarily large vortex rings? L D t = U D 0 ( τ ) dτ = 2 L D = 3.8 L D =14.5 M. Gharib, E. Rambod & K. Shariff, Journal of Fluid Mechanics (1998)

14 Can we generate arbitrarily large vortex rings? L D t = U D 0 ( τ ) dτ = 2 L D = 3.8 L D =14.5 L D 4 Vortex ring formation time Vortex pinch - off time M. Gharib, E. Rambod & K. Shariff, Journal of Fluid Mechanics (1998)

15 L/D = 2 vs. 4 u J /U L/D = 2 u J /U L/D > 4 P.S. Krueger (2001)

16 The Formation Number (F ) Formation Time U p t D = L D Total circulation Vortex circulation Circulation (cm 2 /s) M. Gharib, E. Rambod & K. Shariff, Journal of Fluid Mechanics (1998)

17 Models for Vortex Ring Pinch-off Gharib et al.. (1998): invokes Kelvin-Benjamin Variational principle Vortex ring pinch off...occurs when the source energy falls below that of a steadily translating vortex ring JET Vortex Uj < Uv M. Shusser and M. Gharib, Physics of Fluids (2000)

18 Vortex ring growth is limited by energy effects For vortex ring growth: vortex generator energy > vortex ring energy (W.T. Kelvin, 1875; T.B. Benjamin 1976) dimensionless energy α E ρ ( I ρ) Γ 3 L/D M. Gharib, E. Rambod & K. Shariff, Journal of Fluid Mechanics (1998)

19 Vortex ring growth is limited by energy effects For vortex ring growth: vortex generator energy > vortex ring energy (W.T. Kelvin, 1875; T.B. Benjamin 1976) dimensionless energy steady vortex ring energy L/D M. Gharib, E. Rambod & K. Shariff, Journal of Fluid Mechanics (1998)

20 Vortex ring growth is limited by energy effects For vortex ring growth: vortex generator energy > vortex ring energy (W.T. Kelvin, 1875; T.B. Benjamin 1976) vortex generator energy dimensionless energy steady vortex ring energy L/D M. Gharib, E. Rambod & K. Shariff, Journal of Fluid Mechanics (1998)

21 Vortex ring growth is limited by energy effects For vortex ring growth: vortex generator energy > vortex ring energy (W.T. Kelvin, 1875; T.B. Benjamin 1976) vortex generator energy dimensionless energy L/D 4 steady vortex ring energy vortex ring growth trailing jet formation L/D M. Gharib, E. Rambod & K. Shariff, Journal of Fluid Mechanics (1998)

22 Models for Vortex Ring Pinch-off α E ρ ( I ρ) Γ 3 steady vortex ring energy 0.33 M. Gharib, E. Rambod & K. Shariff, Journal of Fluid Mechanics (1998)

23 Models for Vortex Ring Pinch-off Mohseni (1998): combines Norbury vortex model and slug model approximation for ring translational velocity U vortex = 0.5U piston Linden and Turner (2001): combines Norbury vortex model and volume conservation approximation Ω jet = Ω vortex

24 Models for Vortex Ring Pinch-off Mohseni (1998): combines Norbury vortex model and slug model approximation for ring translational velocity U vortex = 0.5U piston Linden and Turner (2001): combines Norbury vortex model and volume conservation approximation Ω jet = Ω vortex However, entrained fluid by the leading vortex can reach over 50 percent of its total volume (Ω vortex ) J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (2004)

25 Physical Implications of Pinch-Off Pinch-Off Maximum vortex ring strength (energy) Maximum fluid entrainment per vortex ring and maximum vortex ring velocity Maximum thrust per pulse?

26 Total Impulse of Starting Jets I(N*s) Measured I Reference Line Slope of I U I U = I U ( ) t t p 2 p uj ( r, τ ) dadτ 0 L/D A P.S. Krueger and M. Gharib, Physics of Fluids (2003)

27 Average Force of Starting Jets 14 Average Thrust During a Pulse (grams) L/D P.S. Krueger and M. Gharib, Physics of Fluids (2003)

28 Added and Entrained Mass W Entrained Fluid Added Mass Vortex Bubble Ejected Fluid I ( U + I p = mejected + mentrained + madded )W Experiments m ejectedw < I U [ L/D = 2.0, NS Ramp] P.S. Krueger and M. Gharib, Physics of Fluids (2003)

29 Limiting physical processes dictate optimal parameters for vortex ring formation Experiments demonstrate correlation between vortex ring pinch-off and maximum mass and momentum transfer

30 Do biological systems exploit vortex ring formation for optimal fluid transport? LA LV

31 Do biological systems exploit vortex ring formation for optimal fluid transport? RA LV B. Lin and M. Gharib (2003) J.O. Dabiri et al. (2004)

32 Previous attempts to verify optimal vortex formation in biological systems have been inconclusive L D L D where D = 1 t t 0 ( ) D τ dτ I.K. Bartol et al., Journal of Experimental Biology (2001)

33 Previous attempts to verify optimal vortex formation in biological systems have been inconclusive Jet exit diameter, D Leading Vortex Trailing Jet L/D I.K. Bartol et al., Journal of Experimental Biology (2001)

34 Previous attempts to verify optimal vortex formation in biological systems have been inconclusive Measurements in the literature Squid: L/D = 4-7, 10-40, 34, 87 Salps: L/D = 3-4, 6.7, Jellyfish: L/D = Are we missing something?

35 Biological Factors Affecting Pinch-off Co-flow Swimming and flying animals generate vortices in a free-stream flow

36 Biological Factors Affecting Pinch-off Co-flow Swimming and flying animals generate vortices in a free-stream flow Temporal exit diameter variation e.g. squid (Bartol et al., 2001) Parallels time-dependent flap kinematics in other locomotion modes

37 A proper analysis must include fluid-structure interactions External interactions: Flow past the vortex generator Flow past the propulsor can affect vortex ring formation P.S. Krueger, J.O. Dabiri and M. Gharib, Physics of Fluids (2003)

38 A proper analysis must include fluid-structure interactions External interactions: Flow past the vortex generator P.S. Krueger, J.O. Dabiri and M. Gharib, Physics of Fluids (2003)

39 Effects of Co-flow Up = 11.4 cm/s Up = 5.5 cm/s 2.5 F F = (U external +U jet )T/D R v = U external /U jet R v P.S. Krueger, J.O. Dabiri and M. Gharib, Physics of Fluids (2003)

40 Effects of Co-flow Up = 11.4 cm/s Up = 5.5 cm/s 2.5 F Bartol et al., Dabiri et al., R v

41 Effects of Co-flow Animals can exploit external flow in other flow-related functions (i.e. feeding and maneuvering) feeding via vortex ring fluid entrainment See Gallery of Fluid Motion Video #14 (Dabiri et al.)

42 A proper analysis must include fluid-structure interactions Internal interactions: Dynamical effects of a variable exit D(t) J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

43 A proper analysis must include fluid-structure interactions Internal interactions: Dynamical effects of a variable exit D(t) Possible effects 1) As a source of additional vorticity J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

44 A proper analysis must include fluid-structure interactions Internal interactions: Dynamical effects of a variable exit D(t) Possible effects 1) As a source of additional vorticity 2) As a manipulator of existing vorticity J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

45 A proper analysis must include fluid-structure interactions Internal interactions: Dynamical effects of a variable exit D(t) Possible effects 1) As a source of additional vorticity 2) As a manipulator of existing vorticity These effects are obscured when the time-dependent D(t) is replaced by D J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

46 A new technique replicates fluid-structure interactions in variable-diameter jet flows rotating drum variable-diameter elastic nozzle J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

47 A new technique replicates fluid-structure interactions in variable-diameter jet flows rotating drum variable-diameter elastic nozzle pulley array J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

48 A new technique replicates fluid-structure interactions in variable-diameter jet flows J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

49 A new index for vortex formation properly accounts for time-dependent boundary conditions DYNAMIC FORMATION TIME Increase the vortex formation time incrementally: ( L D) * ( U( τ ) D( τ )) τ J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

50 A new index for vortex formation properly accounts for time-dependent boundary conditions DYNAMIC FORMATION TIME Increase the vortex formation time incrementally: ( L D) * ( U( τ ) D( τ )) τ Integrating from flow initiation at τ = 0 to flow termination at τ = t: t ( L D) = ( ( τ ) ( τ ) * U D d 0 τ J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

51 Nondimensional Analysis Normalized Circulation Γ* = Normalized Time Γ ( U D ) e U 2 e e T* = t 0 U D e e ( τ ) ( τ ) dτ = U D e e t J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

52 Vortex formation time is unaffected by temporal increases in jet diameter (L/D)* = 4.0 ± 0.5 for all dd(t)/dt > 0 tested J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

53 J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

54 Vortex formation time is unaffected by temporal increases in jet diameter (L/D)* = 4.0 ± 0.5 for all dd(t)/dt > 0 tested However impulse I ~ x ωdv down stream increases with dd(t)/dt J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

55 Vortex formation time is unaffected by temporal increases in jet diameter (L/D)* = 4.0 ± 0.5 for all dd(t)/dt > 0 tested However impulse I ~ x ωdv down stream increases with dd(t)/dt J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

56 Vortex formation time is unaffected by temporal increases in jet diameter (L/D)* = 4.0 ± 0.5 for all dd(t)/dt > 0 tested However impulse I ~ x ωdv down stream increases with dd(t)/dt 2 R R piston increasing dd (t )/dt 0.5 Vorticity (s -1 ) J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

57 J.O. Dabiri and M. Gharib, Journal of Fluid Mechanics (to appear)

58 These results suggest the first set of engineering strategies for optimal fluid transport During fluid ejection at (L/D)* < 4: I ~ D 2 L * < 4 ( D) (J.O. Dabiri & M. Gharib, J Fluid Mech) J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

59 These results suggest the first set of engineering strategies for optimal fluid transport During fluid ejection at (L/D)* < 4: I ~ D 2 L * < 4 ( D) (J.O. Dabiri & M. Gharib, J Fluid Mech) If total vortex formation time (L/D)* 4, fluid is transported with maximum efficiency (P.S. Krueger & M. Gharib, Phys Fluids) J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

60 These results suggest the first set of engineering strategies for optimal fluid transport During fluid ejection at (L/D)* < 4: I ~ D 2 L * < 4 ( D) (J.O. Dabiri & M. Gharib, J Fluid Mech) If total vortex formation time (L/D)* 4, fluid is transported with maximum efficiency (P.S. Krueger & M. Gharib, Phys Fluids) During fluid ejection at (L/D)* > 4: I ( L D) ~ D 4 * > 4 (J.O. Dabiri & M. Gharib, J Fluid Mech) J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

61 These results suggest the first set of engineering strategies for optimal fluid transport During fluid ejection at (L/D)* < 4: I ~ D 2 L * < 4 ( D) Transport fluid with dd(t)/dt > 0 (J.O. Dabiri & M. Gharib, J Fluid Mech) If total vortex formation time (L/D)* 4, fluid is transported with maximum efficiency (P.S. Krueger & M. Gharib, Phys Fluids) During fluid ejection at (L/D)* > 4: I ( L D) ~ D 4 * > 4 (J.O. Dabiri & M. Gharib, J Fluid Mech) J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

62 These results suggest the first set of engineering strategies for optimal fluid transport During fluid ejection at (L/D)* < 4: I ~ D 2 L * < 4 ( D) Transport fluid with dd(t)/dt > 0 (J.O. Dabiri & M. Gharib, J Fluid Mech) If total vortex formation time (L/D)* 4, fluid is transported with maximum efficiency (P.S. Krueger & M. Gharib, Phys Fluids) (L/D)* = 4? no During fluid ejection at (L/D)* > 4: I ( L D) ~ D 4 * > 4 (J.O. Dabiri & M. Gharib, J Fluid Mech) J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

63 These results suggest the first set of engineering strategies for optimal fluid transport During fluid ejection at (L/D)* < 4: I ~ D 2 L * < 4 ( D) Transport fluid with dd(t)/dt > 0 (J.O. Dabiri & M. Gharib, J Fluid Mech) If total vortex formation time (L/D)* 4, fluid is transported with maximum efficiency (P.S. Krueger & M. Gharib, Phys Fluids) During fluid ejection at (L/D)* > 4: I ( L D) ~ D 4 * > 4 (J.O. Dabiri & M. Gharib, J Fluid Mech) (L/D)* = 4? yes Transport complete? no J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

64 These results suggest the first set of engineering strategies for optimal fluid transport During fluid ejection at (L/D)* < 4: I ~ D 2 L * < 4 ( D) Transport fluid with dd(t)/dt > 0 (J.O. Dabiri & M. Gharib, J Fluid Mech) If total vortex formation time (L/D)* 4, fluid is transported with maximum efficiency (P.S. Krueger & M. Gharib, Phys Fluids) During fluid ejection at (L/D)* > 4: I ( L D) ~ D 4 * > 4 (J.O. Dabiri & M. Gharib, J Fluid Mech) (L/D)* = 4? yes Transport complete? yes Efficiency maximized no J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

65 These results suggest the first set of engineering strategies for optimal fluid transport During fluid ejection at (L/D)* < 4: I ~ D 2 L * < 4 ( D) (J.O. Dabiri & M. Gharib, J Fluid Mech) Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 If total vortex formation time (L/D)* 4, fluid is transported with maximum efficiency (P.S. Krueger & M. Gharib, Phys Fluids) During fluid ejection at (L/D)* > 4: I ( L D) ~ D 4 * > 4 (J.O. Dabiri & M. Gharib, J Fluid Mech) (L/D)* = 4? yes Transport complete? yes Efficiency maximized no no J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

66 These results suggest the first set of engineering strategies for optimal fluid transport During fluid ejection at (L/D)* < 4: I ~ D 2 L * < 4 ( D) Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 (J.O. Dabiri & M. Gharib, J Fluid Mech) If total vortex formation time (L/D)* 4, fluid is transported with maximum efficiency (L/D)* = 4? yes no (P.S. Krueger & M. Gharib, Phys Fluids) During fluid ejection at (L/D)* > 4: I ( L D) ~ D 4 * > 4 (J.O. Dabiri & M. Gharib, J Fluid Mech) Transport complete? yes Efficiency maximized no Transport complete? no J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

67 These results suggest the first set of engineering strategies for optimal fluid transport During fluid ejection at (L/D)* < 4: I ~ D 2 L * < 4 ( D) Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 (J.O. Dabiri & M. Gharib, J Fluid Mech) If total vortex formation time (L/D)* 4, fluid is transported with maximum efficiency (L/D)* = 4? yes no (P.S. Krueger & M. Gharib, Phys Fluids) During fluid ejection at (L/D)* > 4: I ( L D) ~ D 4 * > 4 (J.O. Dabiri & M. Gharib, J Fluid Mech) Transport complete? yes Efficiency maximized no Transport complete? yes Impulse maximized no J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

68 Are these strategies observed in biological systems? Animal swimming revisited Squid rely on jet flow for high-speed swimming and escaping predation J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

69 Are these strategies observed in biological systems? Animal swimming revisited Squid rely on jet flow for high-speed swimming and escaping predation impulse maximization J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

70 Are these strategies observed in biological systems? Animal swimming revisited Squid rely on jet flow for high-speed swimming and escaping predation impulse maximization Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 (L/D)* = 4? no yes Transport complete? no Transport complete? no yes yes Efficiency maximized Impulse maximized J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

71 Are these strategies observed in biological systems? Animal swimming revisited Squid rely on jet flow for high-speed swimming and escaping predation impulse maximization Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 D (cm) (L/D)* = 4? no 0.4 yes 0.3 Transport complete? yes Efficiency maximized no Transport complete? yes Impulse maximized no Leading Vortex Trailing Jet Formation Time, (L /D ) and (L /D )* J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

72 Are these strategies observed in biological systems? The optimal ejection strategy depends on cardiac health efficiency and/or impulse maximization Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 (L/D)* = 4? no yes Transport complete? no Transport complete? no yes yes Efficiency maximized Impulse maximized J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

73 Are these strategies observed in biological systems? The optimal ejection strategy depends on cardiac health efficiency and/or impulse maximization Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 D/D 2 (L/D)* = 4? no 1.5 yes Transport complete? no Transport complete? no 1 Normal 50 BPM Normal 75 BPM yes Efficiency maximized yes Impulse maximized 0.5 Ischemia Reduced E/A Mitral Stenosis Formation Time, (L /D )* J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

74 Are these strategies observed in biological systems? The optimal ejection strategy depends on cardiac health efficiency and/or impulse maximization Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 D/D 2 (L/D)* = 4? no 1.5 yes Transport complete? no Transport complete? no 1 Normal 50 BPM Normal 75 BPM yes Efficiency maximized yes Impulse maximized 0.5 Ischemia Reduced E/A Mitral Stenosis Formation Time, (L /D )* J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

75 Are these strategies observed in biological systems? The optimal ejection strategy depends on cardiac health efficiency and/or impulse maximization Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 D/D 2 (L/D)* = 4? no 1.5 yes Transport complete? no Transport complete? no 1 Normal 50 BPM Normal 75 BPM yes Efficiency maximized yes Impulse maximized 0.5 Ischemia Reduced E/A Mitral Stenosis Formation Time, (L /D )* J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

76 Are these strategies observed in biological systems? The optimal ejection strategy depends on cardiac health efficiency and/or impulse maximization Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 D/D 2 (L/D)* = 4? no 1.5 yes Transport complete? no Transport complete? no 1 Normal 50 BPM Normal 75 BPM yes Efficiency maximized yes Impulse maximized 0.5 Ischemia Reduced E/A Mitral Stenosis Formation Time, (L /D )* J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

77 Are these strategies observed in biological systems? The optimal ejection strategy depends on cardiac health efficiency and/or impulse maximization Transport fluid with dd(t)/dt > 0 Transport fluid with dd(t)/dt < 0 D/D 2 (L/D)* = 4? no 1.5 (L /D )* total = 63 yes Transport complete? yes Efficiency maximized no Transport complete? yes Impulse maximized no Normal 50 BPM Normal 50 BPM Normal 75 BPM Normal 75 BPM Ischemia Ischemia Reduced E/A Reduced E/A Mitral Stenosis Mitral Stenosis Formation Time, (L /D )* J.O. Dabiri and M. Gharib, Proceedings of the Royal Society B (submitted)

78 L/D=UT/D Age (Years) FN(1995) DCM (1995) FN=VTI/D(g,97) Stenotic Prolaps Small Annulus Stiff Annulus High "A" MR Pregnant

79 Conclusions Do biological systems exploit vortex ring formation for optimal fluid transport?

80 Conclusions Do biological systems exploit Yes, both mobile and

81 Comments The vortex ring motif studied here is not limited to jet-based fluid transport Flapping birds (Rayner, 1988) fish (Drucker, 2000) Undulating Paddling eels (Tytell, 2004) frogs (Johansson, 2004)

82 Comments therefore, in order to better understand the physics and evolutionary incentives behind other vortex-based mechanisms, we need To include realistic boundary and flow conditions such as compliance and co-flow To investigate physics of individual events of vortex formation in the context of Dynamic Formation Time rather than Strouhal frequency

83 Acknowledgements Special Thanks to John O. Dabiri And JOD Paul S. Krueger (Southern Methodist) PSK K. Mohseni (Colorado-Boulder), I. Bartol (Old Dominion), Michael Shusser (Technion), Moshe Rosenfeld (Tel Aviv), S. Colin (Roger Williams), J. Costello (Providence) Supported by National Science Foundation and the Office of Naval Research