Airfoil Parameterization using an Improved Class-Shape Transformation and Chebyshev Polynomials

A method for the parameterization of an arbitrary airfoil using a transformation and Chebyshev polynomial interpolation is investigated. The airfoil was transformed into a continuous function using the Class Shape Transformation. A square root spacing was used to smooth out the slope discontinuity found at the origin. This mapping reduces oscillations in the polynomial interpolation caused by the slope discontinuity at the origin. Interpolating a range of NACA 4-digit series airfoils showed that these airfoils could be accurately represented with as little as 10 polynomial terms. However, problems arise with the Class Shape Transformation when trying to parameterize non-analytically defined airfoils. The transformation expects the behavior of the leading edge to be perfectly elliptic, and any deviation from this requirement leads to the divergence of the Class Shape Transformation. As a result, parameterizing with polynomials becomes infeasible for some airfoils. To address this, a conformal mapping-based unwrapping method is suggested.