Scaling law of fish undulatory propulsion

A scaling formulation is constructed for the hydrodynamic thrust by an undulatory propulsor based on the vortex dynamics. It states that the propulsion thrust is in scale with the square of Strouhal number and the relative speed of undulation and fish forward swimming, which implies an energy flux from the fish body to fluid. The higher the energy flux, the higher is the propulsion fish obtains. The results indicate that the scaling relation holds widely for different undulatory locomotion modes, where the slope factor of “thrust-energy flux” increases monotonously with the wavelength and Reynolds number. The full scaling law of the propulsion depends not only on the Strouhal number and Reynolds number, but also on the wavelength and the undulatory phase speed.