Flat-cavity analysis of leading edge vortex separation from slender wings.

Abstract

A first-order model for the analysis of flow separation at the leading edges of slender wings is presented. The method provides the basis for a tool to be used in the analysis and design of slender marine lifting surfaces, such as highly-skewed propellers. The model represents field vorticity as free vortex sheets. Simplifying assumptions dictate that the model is most valid at small angles of attack. The separation problem is simplified using the assumptions that the separation cavities can be characterized as flat and that the transverse flow within them is of higher order, except at the cavity ends, where the flow is singular. This allows solution of the resulting nonlinear problem by iteration on only two parameters: the separation cavity length and a parameter related to the size of the opening at the inboard end of the cavity. The iteration is performed in conjunction with an integral formulation for the axial flow within the cavity, which allows the solution to be marched downstream in the sense of slender wing theory. This procedure insures that mass continuity is maintained and that there is no pressure differential across the free vortex sheet bounding the cavity as the solution is stepped between transverse planes. The system of integral equations representing each in-plane boundary-value problem is solved using a panel method. To improve convergence of the results with the number of panels, special elements are used to represent the source and vortex distributions at points where those distributions exhibit known singular behavior. The probity of the method is illustrated by application to flat delta and gothic wings. Comparison with experimental results indicates that the method gives good approximations of cavity length, pressure distributions on the wing surfaces, and sectional and global lift forces. Incorporation of the method with traditional design procedures for more general wings is illustrated by application to a delta wing of specified leading-edge radius, to a rectangular wing at shockless entry, and to a semi-elliptical wing at an angle of attack.