Numerical exploration of the pitching plate parameter space with application to thrust scaling

The thrust performance of a two-dimensional plate pitching harmonically in a uniform flow is assessed numerically using the OpenFOAM toolbox [1]. The mesh displacement vector associated with the rigid body motion is computed as the solution of a Laplace equation with variable diffusivity, using the appropriate mesh manipulation class of the toolbox. For a Reynolds number of 2000, the accuracy of the pressure and viscous stress distributions is assessed by comparison with reference data available for an equivalent fluid configuration. The efficiency and flexibility of the solver allows exploring large ranges of the pitching parameter space, that is the pitching frequency, amplitude and pivot-point location of the pitching plate. The forces induced by the pitching motion are computed for pitching amplitudes up to 15 circle, for Strouhal numbers varying between 0.2 and 0.5 and for different pitch pivot points. Performing a thrust scaling analysis, a classical theoretical model for the swimming of a waving plate is reliably fitted to the numerical pressure force data. The dependence of the time averaged thrust with the pitching axis is shown to be predicted accurately by a classical potential flow formula (known as Garrick's theory) for pivot points within the front quarter of the plate. The viscous drag is computed as well for the Reynolds number 2000. The time-averaged values are shown to depend on the pitching amplitude and frequency and for instance a Blasius-type scaling, sometimes used to model the viscous drag correction for oscillating two-dimensional foils in this Reynolds number range, is not reliable.