On the best design for undulatory swimming

Most aquatic vertebrates swim by passing a bending wave down their bodies, a swimming mode known as undulatory propulsion. Except for very elongated swimmers like eels and lampreys, these animals have generally evolved to a similar shape: an anterior streamlined region of large volume separated from a caudal fin by a caudal peduncle of reduced cross-section. However, the link between this particular shape and the hydrodynamical constraints remains to be explored. Here, this question is addressed by seeking the optimal design for undulatory swimmers with an evolutionary algorithm. Animals of varying elliptic cross-section are considered whose motions are prescribed by arbitrary periodic curvature laws. In the elongated-body limit, reactive and resistive forces can be formulated at any cross-section, allowing the recoil motion and the mean swimming speed of a given animal to be calculated. A bi-objective optimization problem then consists of finding body shapes and corresponding motions associated with the lowest energetic costs, the highest stride lengths (which is a dimensionless measure of swimming speed) or any trade-offs between the two. For biologically relevant parameters, this optimization calculation yields two distinct 'species': one specialized in economical swimming and the other in large stride lengths. By comparing the attributes and performance of these numerically obtained swimmers with data on undulatory-swimming animals, it is argued that evolution is consistent with the selection of species with low energetic costs.