On the Optimum Efficiency of a Flapping Foil and Its Application to Valveless Pumps

Several different working principles exist to drive a valveless pump. An efficient pump may be designed by placing a flapping foil into a channel filled with fluid. In the case of inviscid flow, the pressure difference along the channel is equivalent to the thrust force acting on the propulsor. The focus in this talk is to find foil movements of highest propulsive efficiency for a given thrust. For simplicity, the effect of the channel walls on the forces is neglected. In order to treat the optimization problem analytically, the case of a slender foil placed in a free stream, flapping in time-harmonic motion with small deflections, is investigated. Novel results for the heavingand pitching amplitude and for the pivot point of the pitching motion are presented as a function of the reduced frequency σ. A surprising outcome is that in a certain range of required thrust, the reduced frequency would need to be infinitely large to obtain maximum efficiency. Furthermore, the optimum movements are accompanied by some drawbacks. An important issue is to reduce the lateral forces acting on the structure. Therefore, foil movements with zero lateral force are investigated and the deficit in efficiency is compared to the optimum foil motion. Finally, instead of the foil, an undulating wave motion is considered. The analytic result for different wave numbers k is presented and the efficiency is compared to the optimum solution of the foil.