Airfoil Design in Subcritical and Supercritical Flows

The “inverse” or “design” problem in aerodynamics, which solves for the airfoil shape that induces a prescribed chordwise surface pressure subject to additional requirements on trailing edge closure, is considered in the transonic small-disturbance limit. A new formulation for the stream function ψ is suggested which uses well-set Neumann conditions on the chordwise slit, with the degree of closure dictated by a specified jump in ψ across the downstream slit emanating from the trailing edge. The boundary-value problem is solved by a type-dependent relaxation method that automatically generates closed airfoils on convergence. Computed airfoil shapes using subcritical and supercritical pressure distributions obtained from existing finite-difference analysis codes, in the latter case, with and without shockwaves, give results in reasonable agreement with the original specified shapes, and validate the basic ideas.