Skin friction on a flapping plate in uniform flow.

To calculate the energy costs of swimming or flying, it is crucial to evaluate the drag force originating from skin friction. This topic seems not to have received a definite answer, given the difficulty in measuring accurately the friction drag along objects in movement. The incoming flow along a flat plate in a flapping normal motion has been considered, as limit case of a yawed cylinder in uniform flow, and applying the laminar boundary layer assumption it is demonstrated that the longitudinal drag scales as the square root of the normal velocity component. This lends credit to the assumption that a swimming-like motion may induce a drag increase because of the compression of the boundary layer, which is known as the 'Bone-Lighthill boundary-layer thinning hypothesis'. The boundary-layer model however cannot predict the genuine three-dimensional flow dynamics and in particular the friction at the leeward side of the plate. A multi-domain, parallel, compact finite-differences Navier-Stokes solution procedure is considered, capable of solving the full problem. The time-dependent flow dynamics is analysed and the general trends predicted by the simplified model are confirmed, with however differences in the magnitude of the friction coefficient. A tentative skin friction formula is proposed for flow states along a plate moving at steady as well as periodic normal velocities.