Exception to Triantafyllou's Strouhal number rule of flapping

Triantafyllou and Triantafyllou (Sci. Amer. 1995) have shown that fish caudal fins have a preferred Strouhal number of 0.25-0.35 for efficient swimming. Strouhal number is defined as \textit{fA/U}, where $f$ is flapping frequency, $A$ is the peak-to-peak flapping amplitude at the tip of the caudal fin, and $U$ is stream velocity. Although this preference was attributed to efficient swimming, they did not measure the efficiency of fish swimming. Later Biorobotic experiments by Bandyopadhyay et al. (JFE 2000) have suggested that while Strouhal number is the dominant factor, another yet unidentified factor is also involved in efficiency. Rohr and Fish (JEB 2004) have shown that in captive cetaceans the most common range of Strouhal number is 0.20-0.30---slightly lower than that given by Triantafyllou and Triantafyllou. We have carried out the measurements of efficiency and forces produced by a single penguin-like fin. We show that for a single fin, in the range of maximum efficiency of 0.55-0.60, the Strouhal number is 0.28-0.55. Here Strouhal number is defined with amplitude $A$ equal to the arc length traversed by the point on the fin which divides the swept area in two. In addition to Strouhal number, the pitch amplitude also determines the regime of high efficiency, with the peak of efficiency seen at lower Strouhal numbers for low pitch amplitudes and at higher Strouhal numbers at higher pitch amplitudes.