A semi-inverse design technique for leading edge slats

Abstract

A design technique for analytically generating a leading edge slat which will induce the modulating field required to match a specified pressure distribution on the nose of an elliptical airfoil was developed. This planar potential flow solution can be readily generalized to the design of slats to prevent boundary-layer separation at the nose of an arbitrary airfoil. The technique is described as semi-inverse because the singularity representation for the slat is constrained so that only realistic slat shapes will be generated. The elliptical airfoil is mapped to a half-plane. In this domain, the slat is represented by a finite series of distributed singularities on an inclined chord line which is placed along a zero order nose flow streamline. These distributed singularities correspond to the singular and regular camber and thickness modes of thin airfoil theory. A suitable slat chord location in the half-plane is selected by examination of the distribution of the specified modulating velocity. For a fixed slat location, the slat-induced velocity field can be written explicitly in terms of the unknown series coefficients. A least squares matching to the specified modulating field is used to select the coefficients. The velocity distribution along the chord line is integrated to determine the slat surface streamlines which are then transformed back to the ellipse plane. The digital computer program for the semi-inverse solution can be executed rapidly. Once an appropriate slat chord location in the half-plane has been selected, an accurate matching of the specified pressure distribution on the airfoil can be achieved. The airfoil nose flow calculated by the semi-inverse solution agrees very closely with the flow computed by the Douglas-Neumann direct solution for the same slat geometry. The airfoil nose flow is very sensitive to the distribution of camber and thickness along the chord line. In some test cases, a modified semi-inverse solution was required in which the minimum acceptable slat thickness was prescribed and a restricted inversion solution was conducted to select the camber mode coefficients required to match the specified modulating field. For a thin airfoil with a severe nose suction peak, a small, thin, highly-cambered slat which is located close to the airfoil nose is desirable.