Translation of Airfoil Designer’s Geometric Parameters Into Quintic Polynomials for Optimizing Thickness and Camber, with Historical Notes

Particularly in the first half of the 1900s, airfoil designers described the essence of a profile using intuitive, geometric parameters such as the leading edge radius, leading edge angle, location and magnitude of maximum thickness and camber, trailing edge thickness, trailing edge angle, and closing (boattail) angle. In this paper we provide some historical context and translate between 11 and 17 classical parameters describing the camber distribution of a profile into several abutting, quintic polynomials. Likewise, we use between 13 and 19 classical parameters to describe the thickness distribution. The method allows the designer to adjust the 24-36 classical parameters either manually or using an optimizer. Quintic polynomials are used to assure magnitude (C0), slope (C1), and curvature (C2) continuity. The quintic polynomial method is applied to ten airfoils from a variety of profile types to demonstrate its overall applicability: low-speed profiles, reflexed airfoils (S-Schlag), laminar profiles, supercritical profiles, and beaky profiles. The goal of the paper is to arrive at a method for parameterizing the thickness and camber distributions that is intuitive and works for all types of profiles, as well as lending itself for use by an optimizer without producing bumps. We end with some plans for multi-point, adjoint-based airfoil optimization.