Reynolds number limits for jet propulsion: A numerical study of simplified jellyfish
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 two-dimensional 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.
