Scaling laws of aquatic locomotion

Abstract

In recent years studies of aquatic locomotion have provided some remarkable insights into the many features of fish swimming performances. This paper derives a scaling relation of aquatic locomotion CD(Re) 2 = (Sw) 2 and its corresponding log law and power law. For power scaling law, (Sw) 2 = βnRe2−1/n , which is valid within the full spectrum of the Reynolds number Re = UL/ν from low up to high, can simply be expressed as the power law of the Reynolds number Re and the swimming number Sw = ωAL/ν as Re ∝ (Sw) σ , with σ = 2 for creeping flows, σ = 4/3 for laminar flows, σ = 10/9 and σ = 14/13 for turbulent flows. For log law this paper has derived the scaling law as Sw ∝ Re/(ln Re+1.287), which is even valid for a much wider range of the Reynolds number Re. Both power and log scaling relationships link the locomotory input variables that describe the swimmer’s gait A, ω via the swimming number Sw to the locomotory output velocity U via the longitudinal Reynolds number Re, and reveal the secret input-output relationship of aquatic locomotion at different scales of the Reynolds number.