Theoretical Model for Airfoils to Suppress Leading-Edge Separation

Abstract

A theoretical study is presented for an airfoil tted with an upper-surface ap undergoing leading-edge stall. Instead of delaying or circumventing ow separation, ow is allowed to separate tangentially from an airfoil with a portion of its suction side truncated. The separation streamline is, thus, made to reattach smoothly at the tip of a streamlined forward-facing ap, which joins the airfoil tangentially to eschew any unnecessary stagnated ow. Different from conventional free-streamline models, the pressure along the bounding streamline is not assumed constant in the present model. Rather, nite velocities (whose values are unknown a priori) are achieved at separation, reattachment, and the trailing edge of the airfoil by utilizing mathematical singularities such as a vortex located inside the hollow airfoil. Furthermore, the condition of a nite pressure gradient at reattachment as deduced from available experimental data is proposed. The resulting lift is found to be larger than that of the basic Joukowsky airfoil for a wide range of angle of attack because the bounding streamline increases the airfoil camber. The modi ed pro le may be considered as a new family of Joukowsky airfoil. Nomenclature= radius of circle in Z 1 plane R V = radial distance of vortex r = real constant U, V = uniform upstream velocities W = complex velocity x, y = coordinates Z , Z 1 , Z 1,0 , = complex numbers Z 2 , Z 3 , Z 4 Z 4, , Z 4, = complex numbers a = angle of attack a , b , d , h 0 , = angles h , h , r C 1 , C 2 = vortex strengths e , l = positive constants f = complex plane K = positive constant q = density