Relaxing the Kutta condition: A dynamic model for unsteady circulation development

The Kutta condition is a closure in two-dimensional aerodynamic modelling that ensures a unique solution to the Laplace equation governing incompressible, irrotational flow. Originally formulated for steady flows, it requires that flow exits tangentially at a sharp trailing edge, placing the rear stagnation point at the trailing edge and preventing reverse flow from the lower to the upper surface. While successful in steady regimes, the classical Kutta condition is overly restrictive for unsteady viscous cases, wherein the flow may initially round the corner before a small trailing-edge vortex forms to restore a stagnation point at the edge. In this study, the Kutta condition is reinterpreted as a dynamic process: the bound circulation is no longer constrained to equal the instantaneous Kutta circulation but is instead driven toward it over time according to a linear relaxation law. The resulting "Kutta process" introduces a relaxation parameter that controls the evolution rate of the real circulation toward its Kutta condition. To explore this new approach, starting airfoils at a constant angle of attack will be considered, considering both impulsively started airfoils (i.e., the classical Wagner problem) and airfoils undergoing finite acceleration as they ramp up to some steady velocity.