E8 Hydrodynamics of Shoals

Topic

Schooling hydrodynamics describes how the interaction between the flow fields generated by individual fish—when swimming in close proximity—can reduce each fish's energy expenditure and create collective vortex structures that benefit those swimming downstream of the others. This phenomenon is the aquatic equivalent of the drafting effect utilized by cyclists in a peloton: a fish swimming in another's wake can exploit the vortices left behind to extract energy from the fluid, rather than expending its own energy by imparting it to the fluid, as occurs during normal swimming.

When a fish swims within the wake of counter-rotating vortices left by another fish—and positions itself correctly within that wake—it can adopt a swimming mode known as "Kármán-gaiting." In this mode, the fish synchronizes the movement of its caudal fin with the frequency of the wake's vortices, extracting energy from them much like a surfer draws energy from a wave instead of having to generate their own propulsion.

This hydromechanical effect of extracting energy from the lead fish's wake requires two simultaneous conditions: the distance between the two fish must remain within an optimal range—specifically, between three and five-tenths of the fish's own body length—and the tail-beat frequency of the following fish must be synchronized (in phase) with the vortices in the lead fish's wake. The reduction in transport cost achieved by a follower fish through this schooling formation can be modeled using potential theory applied to vortex interactions: the velocity induced by the lead fish's wake vortices upon the follower fish increases the latter's effective forward speed without requiring additional energy, thereby reducing its transport cost by 10% to 40% compared to swimming alone.

When this same effect is repeated across all fish in the school, the resulting collective vorticity is no longer limited to the scale of a single fish; instead, it forms coherent structures larger than the length of any individual fish. These structures influence the pressure field of the entire group, rather than just that of isolated leader-follower pairs.

Within this schooling hydrodynamic framework, the most advantageous position for a fish is not directly behind the leader, but rather one that is laterally offset and slightly behind—positioned along the flank of the wake vortices, where the velocity induced on the follower fish reaches its maximum value.

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