Dynamic Scaling of Metachronal Rowing

Metachronal rowing, in which adjacent appendages stroke sequentially, is a common locomotion technique among organisms with multiple swimming appendages. Organisms using metachronal rowing range in size from tiny copepod nauplii and paramecia to large mantis shrimp and ctenophores and swim at a wide range of Reynolds numbers. However, the fluid mechanics of how this locomotion technique is used across such a wide range of scales is not well understood. Drawing data from the literature for a wide variety of organisms, we present a scaling analysis of metachronal rowing. We examine body and appendage length, advance ratio, body and appendage Reynolds numbers, and Strouhal number. The ratio of body to leg length varies widely in the range of 2-60 and does not correlate with body size. We find a linear relationship between the body and appendage Reynolds numbers across several orders of magnitude. Further, the ratio of body to appendage Reynolds number, which may be recast as the product of the advance ratio and the ratio of the body length to appendage length, governs flow interactions among appendages. At relatively high Reynolds numbers, these interactions may include vortex interception, which may increase swimming efficiency.