A Simplified Mathematical Model of Pumped Hydrofoils

Abstract

This paper presents a simplified mathematical model of a pumped hydrofoil (PH)—a surfboard elevated above water surface and connected to a tandem of hydrofoils by a strut. The PH is operated by a rider who stands on the surfboard and produces swinging up-and-down motions resulting in forward propulsion of the device. In the present paper, the description of the vertical motion of the PH is reduced to a linear oscillator excited by an oscillating mass coupled with the requirement that the weight is supported by dynamic lift of the foil(s). The inertial and damping influence of the hydrofoil(s) is accounted for by expanding unsteady lift force on the foil(s) in terms of kinematic parameters. The restoring term of the oscillator is associated with the phenomenon of automatic stabilization of shallowly submerged hydrofoils. The latter effect manifests itself in that when a hydrofoil approaches free surface, its lift decreases, and when it moves away from free surface, its lift increases. The analytical solution of the pumping foil mass-spring type forced oscillations equation allows one to calculate the flapping motion of the foil(s) and, thereafter, the period-averaged thrust generated by the PH. The resulting speed has been estimated on an assumption that the device enters its cruising mode when the thrust becomes equal to the drag, the latter comprising viscous, wave, and induced drag components. The model under discussion allows one to relate the main parameters of the system to its performance and, hopefully, provides further insight into the pumped hydrofoil phenomenon, its design methodology, and operation strategy. The review part of the paper focuses on two aspects of the problem: hydrodynamic behavior of the hydrofoil(s) in proximity to free water surface and their propulsion due to oscillations.