Math @ Duke

Publications [#316992] of Gregory J. Herschlag
Papers Published
 Herschlag, G; Miller, LA, Reynolds number limits for jet propulsion: A numerical study of
simplified jellyfish
(October, 2010) [1010.3357v1]
(last updated on 2018/10/16)
Abstract: The Scallop Theorem states that reciprocal methods of locomotion, such as jet
propulsion or paddling, will not work in Stokes flow (Reynolds number = 0). In
nature the effective limit of jet propulsion is still in the range where
inertial forces are significant. It appears that almost all animals that use
jet propulsion swim at Reynolds numbers (Re) of about 5 or more. Juvenile squid
and octopods hatch from the egg already swimming in this inertial regime. The
limitations of jet propulsion at intermediate Re is explored here using the
immersed boundary method to solve the twodimensional Navier Stokes equations
coupled to the motion of a simplified jellyfish. The contraction and expansion
kinematics are prescribed, but the forward and backward swimming motions of the
idealized jellyfish are emergent properties determined by the resulting fluid
dynamics. Simulations are performed for both an oblate bell shape using a
paddling mode of swimming and a prolate bell shape using jet propulsion.
Average forward velocities and work put into the system are calculated for
Reynolds numbers between 1 and 320. The results show that forward velocities
rapidly decay with decreasing Re for all bell shapes when Re < 10. Similarly,
the work required to generate the pulsing motion increases significantly for Re
< 10. When compared actual organisms, the swimming velocities and vortex
separation patterns for the model prolate agree with those observed in Nemopsis
bachei. The forward swimming velocities of the model oblate jellyfish after two
pulse cycles are comparable to those reported for Aurelia aurita, but
discrepancies are observed in the vortex dynamics between when the 2D model
oblate jellyfish and the organism.


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