Mechanical Intelligence in Propulsion via Flexible Caudal Fins

Fish swim with flexible fins that stand in stark contrast to the rigid propulsors of engineered vehicles, though it is unclear whether this flexibility endows animals with a performance advantage. Using numerical simulations of the mechanics of flow-structure interaction (FSI), we found that flexible fins are up to 70% more efficient than rigid fins. By comparing the dynamics of FSI, we find that the power requirements of rigid fins can be largely attributed to their propensity to generate high-magnitude lateral forces. In contrast, flexible fins achieve high efficiency by a mechanism known as local-force redirection where deformations orient fluid forces in fore-aft and dorso-ventral directions to reduce the power demand of generating lateral forces. These deformations occur at moments in the tail-beat cycle when large velocities and pressure differentials are generated due to the mechanics of the fin and surrounding flow.

💡 Research Summary

The paper investigates how the flexibility of a fish’s caudal fin influences propulsive performance, using high‑fidelity fluid‑structure interaction (FSI) simulations. Three fin models are constructed: a perfectly rigid fin (Fin‑R) and two compliant membranes with thickness‑to‑chord ratios h* = 0.04 (intermediate flexibility, Fin‑IF) and h* = 0.02 (high flexibility, Fin‑HF). All fins are driven by sinusoidal lateral displacement and pitch at the peduncle, with Strouhal numbers ranging from 0.3 to 0.5, a regime typical of efficient swimming in both fish and flying animals.

The simulations reveal that flexible fins can achieve up to a 70 % increase in propulsive efficiency compared with the rigid counterpart, even when delivering comparable thrust (C_T ≈ 0.75). Efficiency gains are traced to reductions in the power required to overcome lateral hydrodynamic forces. By decomposing the instantaneous power into a lateral‑force × lateral‑velocity term and a pitch‑moment × pitch‑rate term, the authors show that for the rigid fin roughly 97 % of the power is spent on lateral forces, whereas for the most flexible fin this fraction drops to about 77 %. Consequently, the flexible fin spends a larger share of its power on pitching work, but the overall power consumption is still lower because the lateral‑force term is dramatically reduced.

Three mechanisms are examined to explain this reduction. The first, a “pitch‑moment reduction” hypothesis, suggests that flexible deformation shifts high‑pressure regions forward, lowering the moment about the peduncle. The simulation data, however, indicate that while the leading‑edge vortex (LEV) remains attached longer on flexible fins, the pressure distribution actually increases pitching moments, making this mechanism ineffective for efficiency gains.

The second mechanism, “force‑velocity phase mismatch,” proposes that flexibility changes the timing between force generation and fin velocity. Power is the product of force and velocity; if peak lateral forces occur when the fin’s lateral velocity is low, less power is required. Time histories of surface‑averaged pressure differentials and fin velocities confirm that flexible fins indeed align force peaks with lower velocity phases, reducing the instantaneous power spikes.

The third and most decisive mechanism is “local‑force redirection.” As the flexible membrane deforms, the normals of surface elements rotate, redirecting fluid pressure from the lateral direction toward fore‑aft and dorso‑ventral directions. Visualizations of spanwise vorticity show that flexible fins maintain more uniform LEVs along dorsal and ventral edges and generate a pronounced dorsoventral vortex at the trailing edge each half‑stroke. This deformation effectively reorients pressure forces, diminishing the lateral component that dominates power consumption. The result is a substantial drop in the lateral‑force term of the power balance.

A focused “iso‑thrust” comparison (same thrust coefficient C_T ≈ 0.5) further isolates the effect: flexible fins achieve the same thrust with markedly lower power, confirming that the efficiency advantage stems from the altered force‑velocity relationship and force redirection rather than from reduced pitching work.

The authors frame these findings within the concept of “mechanical intelligence”: passive material compliance enables the fin to exploit fluid‑structure interactions without active sensing or complex control, automatically achieving a near‑optimal propulsion strategy. This insight has direct implications for bio‑inspired underwater robotics. By embedding appropriate flexibility into propulsors, designers can reduce reliance on sophisticated control algorithms while attaining higher energetic efficiency, mirroring the natural solution evolved by fish.

Future work suggested includes exploring a broader range of flexural stiffness distributions, asymmetric fin geometries, and coupling with realistic muscle actuation models, as well as experimental validation of the simulated mechanisms. Overall, the study provides a rigorous quantitative foundation for leveraging passive flexibility as a design principle in efficient aquatic propulsion.