Scaling laws of aquatic locomotion

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 C-D(Re)(2) = (Sw)(2) and its corresponding log law and power law. For power scaling law, (Sw)(2) = beta Re-n(2-1/n), which is valid within the full spectrum of the Reynolds number Re = UL/nu from low up to high, can simply be expressed as the power law of the Reynolds number Re and the swimming number Sw = omega AL/nu as Re proportional to (Sw)(sigma), with sigma = 2 for creeping flows, sigma = 4/3 for laminar flows, sigma = 10/9 and s = 14/13 for turbulent flows. For log law this paper has derived the scaling law as Sw proportional to 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 swimmers gait A, omega 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.