E1 Propulsion by Whirlwind Generation

Topic

Vortex-based propulsion is the fundamental hydrodynamic mechanism of undulatory swimming in fish. As the caudal fin beats from side to side within the fluid, it generates counter-rotating vortices that shed alternately from each end of the fin's trailing edge, forming a vortex street known as a reverse von Kármán street. This reverse vortex street differs from the passive von Kármán street that forms behind a rigid body and generates drag; the reverse street, by contrast, injects momentum backward into the fluid, creating a propulsive jet that—in accordance with Newton's third law—drives the fish forward.

The hydraulic impulse of each of these vortices is proportional to the vortex circulation—a measure of the fluid's rotation around the vortex—and the fluid density. The frequency and amplitude of the caudal fin beat determine the circulation of each shed vortex and, consequently, the average thrust the fish derives from vortex-based propulsion.

To compare caudal fin kinematics across fish of varying sizes and speeds, researchers use a dimensionless quantity known as the Strouhal number; this combines beat frequency, peak-to-peak tail amplitude, and swimming speed into a single unitless value. Empirical observations show that fish swim with a Strouhal number between 0.25 and 0.35—precisely the range where, according to Triantafyllou's theorem, thrust generation via the vortex street achieves maximum efficiency. Hydrodynamic propulsive efficiency—also known as Froude efficiency—compares the useful power a fish expends generating thrust with the total power delivered to the fluid, including the kinetic energy left in the wake that does not contribute to thrust. Fast-swimming, fusiform fish like tuna achieve efficiencies of between 80% and 90% by operating near the optimal Strouhal number range described earlier. In vortex-based propulsion, the expected behavior is that thrust increases with the square of the fin-beating speed, and that for every cruising speed there is an optimal beat frequency that maximizes efficiency; this establishes the link between the beat frequency and amplitude—introduced at the start of this model—and the ultimate performance of vortex-based propulsion.

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