On the forced flow around a rigid flapping foil

Abstract

The two dimensional incompressible viscous flow past a flapping rigid foil immersed in a uniform stream is studied using a lattice-Boltzmann model. When the foil’s center of mass is fixed in space, numerical results reproduce the transition from the von Kármán (vKm) to the inverted von Kármán wake [T. Schnipper, A. Andersen, and T. Bohr, “Vortex wakes of a flapping foil,” J. Fluid Mech. 633, 411 (2009) and A. Das, R. K. Shukla, and R. N. Govardhan, “Existence of a sharp transition in the peak propulsive efficiency of a low pitching foil,” J. Fluid Mech. 800, 307 (2016)]. Beyond the inverted vKm transition, the foil was released. The numerical results show that the hydrodynamic forces on the flapper are oscillatory functions of time with amplitudes and mean values that scale with the square of the Strouhal number, defined with either the flapping amplitude or the flapper length that decays an order of magnitude when the foil is freed to swim. Upstream swimming consisted of a uniform horizontal motion and a vertical heaving. The swimming speed showed a linear dependence on the Strouhal number, defined with the amplitude of oscillation of the foil tip. As a consequence, thrust generated by the free flapper is related to the square of the swimming speed for moderate Reynolds numbers.