morphokinematics for undulatory swimmers at intermediate Reynolds numbers

We noted in the main text the high Strouhal number corresponding to both our and Kern & Koumoutsakos (2006) optimal e cient swimmer's, and attributed this to the lower Reynolds number used in these computational works compared with the analyses in Triantafyllou et al. (1993); Taylor et al. (2003), consistent with Gazzola et al. (2014). However, both in Kern & Koumoutsakos (2006) and in the current work, the swimming frequency during the optimization is xed to T = 1. To investigate whether our high Strouhal number is an artefact of xing T , we simulated our optimal e cient swimmer with swimming periods T = 0.9 and T = 1.1. The results reported in gure 4 show that the velocity of the swimmers increase only slightly stronger than linearly with their frequencies, so that the Strouhal numbers between these three cases only change by about 2% each. The relative changes in steady-state e ciency are of the same magnitude, so that at least for this regime the e ciency is not sensitive to the swimming frequency. This is consistent with Gazzola et al. (2014), where a scaling law was derived based on uid dynamics arguments that for these simulations would predict U ∝ (T ), which agrees excellently with our results.