1. Physical Mechanisms of Efficient Biological Propulsion
Wake Resonance

2. Vortex Shedding

3. Why Fluids is Theoretically Difficult
Navier-Stokes Equation. $1M bucks if you can solve it, no joke!
The other “solution”: DO EXPERIMENTS!

4. Larger scale vortex street (for fun)

5. Vortex Shedding Dictated by Strouhal Number
Image credit: A.H. Techet, MIT

6. Strouhal Number 𝑆𝑡= 𝑓𝐴 𝑈 = 𝐴 𝑇 /𝑈 f = flapping frequency = 1/T
𝑆𝑡= 𝑓𝐴 𝑈 = 𝐴 𝑇 /𝑈
f = flapping frequency = 1/T
A = displacement at mid- chord
U = forward speed.
St = Strouhal number
Image credit: (

7. Vortex Wake Dynamics Increasing flapping frequency
Typical vortex structures of swimming animals
(Smits et al, 2012)
(Lentink et al, 2008)

8. Oscillation vs. Undulation
Intermediate
oscillatory

9. Vortex Wake Thrust a) Ray-like robotic fin b) vorticity in 2S pattern
c) time-averaged velocity field
(Moored, 2012)

10. Wake transitions in batoid-inspired oscillating fin
Biological swimmers
2S wake
2P wake
(Dewey et al., 2012)

11. Efficient swimming: Wake Resonance
Froude Propulsive Efficiency:
Mechanical Power output =
avg. thrust x avg swimming speed
Mechanical Power input
(Moored, 2012)

12. Wake Resonance: Why it works
Initial velocity profile from vortex shedding
Final velocity profile after “roll-up”
2P wake
2S wake
“When driven at wake resonant frequency, there is increase in entrainment of momentum into the time-averaged velocity jet…” (Moored, 2014)

13. Efficient Swimming: Be a bit Flexible
Most flexible
Most rigid
(Smits et al., 2012)

14. References
1. Moored, K. W., Dewey, P. A., Boschitsch, B. M., Smits, A. J., & Haj-Hariri, H. (2014). Linear instability mechanisms leading to optimally efficient locomotion with flexible propulsors. Physics of Fluids, 26(4),
2. Dewey, P. A., Boschitsch, B. M., Moored, K. W., Stone, H. A., & Smits, A. J. (2013). Scaling laws for the thrust production of flexible pitching panels. Journal of Fluid Mechanics, 732,
3. Dewey, P. A., Carriou, A., & Smits, A. J. (2012). On the relationship between efficiency and wake structure of a batoid-inspired oscillating fin. Journal of fluid mechanics, 691,
4. Moored, K. W., Dewey, P. A., Smits, A. J., & Haj-Hariri, H. (2012). Hydrodynamic wake resonance as an underlying principle of efficient unsteady propulsion. Journal of Fluid Mechanics, 708,
5. Smits, A. J., Moored, K. W., & Dewey, P. A., (2012) The Swimming of Manta Rays, Fluid-Structure-Sound-Interactions and Control, Lecture Notes in Mechanical Engineering
6. Lentink, D., Muijres, F. T., Donker-Duyvis, F. J., & van Leeuwen, J. L. (2008). Vortex-wake interactions of a flapping foil that models animal swimming and flight. Journal of Experimental Biology, 211(2),
7. Clark, R. P., & Smits, A. J. (2006). Thrust production and wake structure of a batoid-inspired oscillating fin. Journal of fluid mechanics, 562,
8. Triantafyllou, G. S., Triantafyllou, M. S., & Grosenbaugh, M. A. (1993). Optimal thrust development in oscillating foils with application to fish propulsion. Journal of Fluids and Structures, 7(2),