Effect of the pivot point location on the propulsive performance of a pitching foil

Numerical simulations of the flow around a NACA0012 pitching foil are presented to investigate the effect of the pivot axis location on its propulsive performance for a Reynolds number Re = 16 000. The pivot point position a is varied between the foil's leading edge (a = -1) and its trailing edge (a = 1). A wide range of values of the reduced frequency k is covered in the present analysis for several (small) values of the pitch amplitude. Simple scaling laws for the thrust, power, and efficiency are derived in terms of the pitching axis location, the Strouhal number St and the pitch amplitude alpha(0), with very good agreement with the numerical results for a less than or similar to 1/4 and St greater than or similar to 0.4. For a greater than or similar to 1/4 the flow is aperiodic and generates relatively less thrust with low propulsive efficiency. Although the power input presents a minimum at a = 1/4, the maximum efficiency is reached for -1 <= a <= 0, with the optimum pivot point location mainly depending on St and the offset drag C-D0, a parameter introduced in the scaling of the thrust coefficient. For C-D0/St(2) << 1, the maximum efficiency, which decays as k(-1), is reached when the foil pitches about its middle point (a = 0), being practically independent of the offset drag C-D0. As St decreases, the optimum location of the pivot point moves upstream towards the leading edge, and the scaling laws provide quite simple estimations of the maximum propulsive efficiency for any location a of the pivot axis. The highest efficiency is found for a foil pitching about its leading edge (a = -1), with an estimated value of about (1/4)root sin alpha(0)/C-D0 and generated for a Strouhal number about (2/3)root C-D0.