Hydrodynamic wake resonance as an underlying principle of efficient unsteady propulsion

A linear spatial stability analysis is performed on the velocity profiles measured in the wake of an actively flexible robotic elliptical fin to find the frequency of maximum spatial growth, that is, the hydrodynamic resonant frequency of the time-averaged jet. It is found that: (i) optima in propulsive efficiency occur when the driving frequency of a flapping fin matches the resonant frequency of the jet profile; (ii) there can be multiple wake resonant frequencies and modes corresponding to multiple peaks in efficiency; and (iii) some wake structures transition from one pattern to another when the wake instability mode transitions. A theoretical framework, termed wake resonance theory, is developed and utilized to explain the mechanics and energetics of unsteady self-propulsion. Experimental data are used to validate the theory. The analysis, although one-dimensional, captures the performance exhibited by a three-dimensional propulsor, showing the robustness and broad applicability of the technique.