A theory of the flow past a slender delta wing with leading edge separation

Abstract

The flow at incidence to a slender delta wing with a sharp leading edge usually separates along this edge, i.e. a vortex layer extends from the edge into the main flow. This layer rolls up above and inboard of the leading edge to form a region of high vorticity which strongly influences the flow pattern. This paper gives a theoretical treatment, more complete than those hitherto available, of this type of flow. A potential flow model is constructed, in which the vortex layer is replaced by a vortex sheet of spiral form, and the problem is then reduced to a two-dimensional one by the use of slender-body theory and the assumption of a conical velocity field. The boundary conditions expressing that the vortex sheet is a stream surface and sustains no pressure difference determine, in principle, its shape and strength. In practice, the inner part of the spiral and the finite, outer part of the spiral, which joins the inner part to the leading edge, are treated separately. The inner part is regarded as small and a solution is given for it which is asymptotically correct as the centre of the spiral is approached. The outer part is replaced by a sheet whose shape and strength depend on a finite number of parameters; these parameters are determined by applying the boundary conditions at isolated points. Results are given for the shape of the sheet and the pressures, loadings and forces on the wing, as functions of the ratio of the incidence to the aspect ratio.