Limit of the two-dimensional linear potential theories on the propulsion of a flapping airfoil in forward flight in terms of the Reynolds and Strouhal number

A sixth-order compact finite differences scheme of the two-dimensional transportPoisson equations in terms of vorticity-stream function is formulated to solve the motion of a flapping airfoil in a noninertial reference frame, fitting a NACA-0012 profile with the Karman-Trefftz transformation. These numerical simulations have been used to estimate the validity range of the linear potential theories of Fernandez-Feria and the classical Garrick model on the propulsion of a pitching airfoil pivoting to a quarter chord length of the leading edge and a heaving airfoil in forward flight, in terms of the Reynolds, Re, and Strouhal number, St, compared with available numerical and experimental data. The Fernandez-Feria's propulsion theory shows better agreement with the numerical and experimental results than the Garrick model for the time-averaged thrust coefficient where it is found that the Fernandez-Feria's theory is limited to Re less than or similar to 4000 and St less than or similar to 0.25 for pure pitching motions and Re greater than or similar to 1000 and St less than or similar to 0.2 for pure heaving motions.