Hydrodynamics of a fish-like body undulation mechanism: Scaling laws and regimes for vortex wake modes

A comprehensive two-dimensional numerical investigation has been undertaken to calculate the energetic cost of propulsion and the various flow transitions of a fish-like body undulation mechanism based on a National Advisory Committee for Aeronautics 0012 hydrofoil. This covers a wide range of Strouhal $(0\le St\le 1.4)$ and Reynolds $(500\le Re\le 5000)$ numbers from simulations based on a level-set function immersed-interface method. It is found that the time-averaged thrust coefficient displays a quadratic relationship with increasing St, and increases significantly with Re. Additionally, the time-averaged input power coefficient exhibits a cubic dependence with increasing St but is independent of Re. Both St dependences agree with those previously observed experimentally and numerically for an oscillating foil; however, for similar ranges of governing parameters, comparisons suggest that the body undulation mechanism possesses a higher propulsive efficiency. The $St\propto R{e}^{-0.19}$ scaling for the drag-to-thrust transition is consistent with that found for a wide variety of fish and birds in nature. Interestingly, for cases with an undulation wave-speed below the free-stream speed, the time-averaged drag coefficient is found to be higher than that of a stationary hydrofoil at the same Re. Furthermore, the time-averaged input power coefficient is negative, indicating the potential for the undulation mechanism to extract energy from the free-stream. Eight different wake patterns/transitions are documented for the parameter space; these have been assembled into a wake-regime parameter-space map. The present findings should aid in predicting and understanding different hydrodynamic forces and wake patterns for undulating kinematics.