Scaling laws of aquatic locomotion

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In recent years studies of aquatic locomotion have provided some remarkable insights into the many features of fish swimmingperformances. This paper derives a scaling relation of aquatic locomotion $C_\text{D}(\mathrm{\textit{Re}})^2=(\mathrm{\textit{Sw}})^2$ and its corresponding log law and power law. For power scaling law, $(\mathrm{\textit{Sw}})^2={\beta_\textit{n}}\mathrm{\textit{Re}}^{2-1/\textit{n}}$, which is valid within the full spectrum of the Reynolds number $\mathrm{\textit{Re}}=UL/\nu$ from low up to high, can simply be expressed as the power law of the Reynolds number $\mathrm{\textit{Re}}$ and the swimming number $\mathrm{\textit{Sw}}=\omega AL/\nu$ as $\mathrm{\textit{Re}}\propto (\mathrm{\textit{Sw}})^\sigma$, with $\sigma=2$ for creeping flows, $\sigma=4/3$ for laminar flows, $\sigma=10/9$ and $\sigma=14/13$ for turbulent flows. For log law this paper has derived the scaling law as $\mathrm{\textit{Sw}}\propto \mathrm{\textit{Re}}/(\ln \mathrm{\textit{Re}}+1.287)$, 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 $\mathrm{\textit{Sw}}$ to the locomotory output velocity $U$ via the longitudinal Reynolds number $\mathrm{\textit{Re}}$, and reveal the secret input-output relationship of aquatic locomotion at different scales of the Reynolds number.