Scaling and performance of simultaneously heaving and pitching foils

We consider the propulsive performance of an unsteady heaving and pitching foil, experimentally studying an extensive parameter space of motion amplitudes, frequencies, and phase offsets between the heave and pitch motions. The phase offset $\varphi$ between the heaving and pitching motions proves to be a critical parameter in determining the dynamics of the foil and its propulsive performance. To maximize thrust, the heave and pitch motions need to be nearly in phase ($\varphi ={330}^{\circ }$), but to maximize efficiency, the pitch motion needs to lag the heave motion ($\varphi ={270}^{\circ }$), corresponding to slicing motions with a minimal angle of attack. We also present scaling relations, developed from lift-based and added mass forces, which collapse our experimental data. Using the scaling relations as a guide, we find increases in performance when foil amplitudes (specifically pitch) increase while maintaining a modest angle of attack.