A wing that is heaving and pitching simultaneously may extract energy from an oncoming flow, thus acting as a turbine. The theoretical performance of such a concept is investigated here through unsteady two-dimensional laminar-flow simulations using the commercial finite volume computational fluid dynamics code FLUENT. Computations are performed in the heaving reference frame of the airfoil, thus leaving only the pitching motion of the airfoil to be dealt with through a rigid-body mesh rotation and a circular nonconformal sliding interface. Unsteady aerodynamics basics of the oscillating airfoil are first exposed, with a description of the operating regimes. Effects of unsteadiness are stressed and the inadequacy of a quasi-steady approach to take them into account is exposed. We present a mapping of power-extraction efficiency for a single oscillating airfoil in the frequency and pitchin-gamplitude domain:​ 0 < fc/U_∞ < 0:​25 and 0 < θ₀ < 90 deg for a NACA 0015 airfoil at a Reynolds number of Re = 1100, a heaving amplitude of one chord (H₀ = c), and a pitching axis at the third chord (x_p = c/3). Remarkably, efficiency as high as 34% is observed, as well as a large parametric region at θ₀ > 55 deg in which efficiencies are higher than 20%. Results from a parametric study are then provided and discussed. It is found that motion-related parameters such as heaving amplitude and frequency have the strongest effects on airfoil performances, whereas geometry and viscous parameters turn out to play a secondary role.