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A parabolic trailing-edge flap is defined as a parabolic deflection of the airfoil geometry aft of a hinge point. Whereas a traditional flap deflection causes a discontinuous camber-line slope at the hinge point, a parabolic deflection produces a camber line that is first-order continuous at the hinge point. The geometry manipulation of a parabolic flap is mathematically defined such that it can be applied to any airfoil with a known camber line and thickness distribution. Small-angle and small-camber approximations are used to find analytical predictions for the ideal section flap effectiveness as well as the section pitching moment in comparison to a traditional flap. Results of the parabolic flap are compared to those of a traditional flap producing equivalent lift using thin airfoil theory and the vortex-panel method. It is shown that the ideal section flap effectiveness for a parabolic flap can be 33-50% higher than that of a traditional flap, depending on the flap-chord fraction. Additionally, a parabolic flap will produce a change in pitching moment 5-50% greater than that of a traditional flap for a given change in lift. Results may be applied in the design of modern morphing wings, for which complex flap deflections can be produced. Nomenclature A,B,C: Constant Coefficients used in Eq. (17); c: Airfoil Chord Length; cf: Flap Chord Length; L C ~ : Airfoil Lift Coefficient; α , ~ L C : Airfoil Lift Slope;

Geometric Definition and Ideal Aerodynamic Performance of Parabolic Trailing-Edge Flaps