Numerical validation of simple non-stationary models for self-propelled pitching foils

Abstract

High-resolution numerical simulations of the self propelled locomotion of two-dimensional pitching foils are used to assess simplified models based on linear potential theory for the fluid-foil interaction. These models are very useful because they provide simple analytical estimations of the swimming velocity, among other relevant features of the aquatic locomotion of fishes and underwater robotic devices propelled by flapping foils. In particular, we consider a pitching foil self-propelled from two different models of the unsteady thrust force based on linear potential theory, both complemented with a new simple model for the unsteady viscous friction obtained from the present full-numerical simulations, valid in a wide range of Reynolds numbers (103≲Re≲104) of interest for many natural and robotic swimmers. The resulting ordinary differential equation for the swimming velocity is easily integrated numerically, comparing favorably with the full-numerical simulations for small pitch amplitudes (Strouhal numbers St≲0.25) and the above range of Reynolds numbers. Further, when the swimming velocity is small, simple approximate solutions of the dynamic model equation are obtained, whose pitch amplitude validity range is more limited than the numerical solution of the model as the Reynolds number and the foil mass ratio increase, becoming negligibly small when the frequency-based Reynolds number is well over 104. Although both thrust models yield similar quantitative results, they predict qualitatively different dependencies of the swimming velocity on the different non-dimensional parameters.