Functional morphology and virtual models: physical constraints on the design of oscillating wings, fins, legs, and feet at intermediate reynolds numbers.

Why do some animals swim by rowing appendages back and forth while others fly by flapping them up and down? One hypothesis suggests the answer lies in the sharply divergent physical environments encountered by small, slow animals, and large, fast animals. Flapping appendages allow large animals to move through a fluid environment quickly and efficiently. As size and speed decrease, however, viscous drag increasingly dominates the force balance, with negative consequences for both rowing and flapping appendages. Nevertheless, comparative data suggest that flapping does not occur in animals at Reynolds numbers (Re) less than about 15. I used a computer simulation experiment to address the question, "Below what Re is rowing more effective than flapping?" The simulation, which employed a simple quasi-steady, blade-element model of virtual oscillating appendages, has several important results. First, the mechanical efficiency of both rowing and flapping decrease dramatically with scale. Second, the performance of rowing can increase substantially by taking advantage of several dynamic shape modifications, including area and span reduction during the recovery stroke. Finally, the relative performance of rowing and flapping is dependent on the advance ratio, which is a function of the travel speed relative to the oscillation frequency. The model predicts that rowing is more efficient than flapping at Re < 20 for animals moving throughout the range of typically observed advance ratios.