An empirical extension to free-streamline methods: Modeling leading-edge separation

Potential-flow methods underpin the classical theory of aerodynamics, yet this under-determined, kinematic framework relies on additional constraints in order to provide physically-correct flowfields. Most famously, the Kutta-Joukowsky condition is applied for attached flows at moderate angles of attack, yet there does not exist a classical method which may predict leading-edge separation. If a separation point is chosen a priori, methods for modeling leading-edge separation in potential flows via free streamlines trace back to the works of Helmholtz, Kirchhoff, Riabouchinsky and others. We reconsider this classical problem, without ad hoc consideration of the trailing-edge and leading-edge geometry, by augmenting the potential-flow solutions with a variational framework providing dynamics. We employ an empirically-derived upper-bound to streamlines of the flow, aided by the maximum modulus principle, in order to establish where the immersed body cannot be a streamline, and likewise use this condition to modify the conformal mapping to one which represents a physically-realizable outer solution. The NACA 0012 and the canonical flat-plate airfoil are analyzed in this framework, and contrasted with classical, empirical, and computational solutions.