Transit time difference and equal or non-equal transit time theory for airfoils with lift

The transit time difference of fluid particles moving along the upper and lower surfaces of a lift-producing airfoil is studied here both theoretically and numerically. We show that, under thin airfoil assumption and for potential flow, the transit time difference is equal to the circulation divided by the square of the inflow velocity and the lift coefficient is equal to half of the number of chords travelled by the airfoil during the transit time difference. An analysis of transit time difference for very thick airfoil (c.f. very large angle of attack) suggests the transit time may change sign beyond thin airfoil assumption, a conclusion supported by an example of flow with an attached vortex. Thus, fluid particles may transit the upper surface with less, equal and more time than those transiting the lower surface for lift producing airfoils, depending on the configuration of flow structure and geometry.