Discrete vortex computation of separated airfoil flow

Abstract

LOW-SPEED unsteady separated flow may be computed from Eulerian finite-difference or -element representations of the Navier-Stokes equations or by Lagrangian vortex tracking methods. The latter have been used to model both two-dimensional separated airfoil flows at high Reynolds numbers1'2 as well as numerous bluff-body flows.3'5 A major difficulty with the basic vortex method is that it is inviscid; hence, separation, except at sharp edges, must be predicted by some other means. Where empirical prediction is difficult, boundary-layer calculation (viscous-inviscid matching) has been used, but it is possible to predict laminar separation too early2 unless the discrete vortex representation is started upstream of the separation point. Viscous diffusion is usually modeled by vortex blob and random walk techniques.6'7 The results presented in this synoptic were computed using the inviscid cloud-in-cell vortex method.8 This is a mixed EulerianLagrangian method in which descrete vortices are tracked through a grid on which the velocity field is computed by a finite-difference method. It has the advantage of faster computation when large numbers of vortices are present but a poorer resolution of the velocity field. Contents Cloud-in-Cell Method In this method, the circulation of the vortices shed from the body is distributed by a weighting function (bilinear in the present calculations) onto a grid. The stream function \[/ is then computed from the Poisson equation V 2 ^=— w, linking \j/ to the resulting vorticity distribution co on the grid. This equation was expressed in central difference form and solved by a combination of fast Fourier transforms and Gaussian elimination. In order to avoid interpolated boundary conditions for \I/, the airfoil was transformed into a circle of radius R using the Joukowski transformation. The computation was carried out in the circle plane on a regular n x m polar mesh (rjtBk ) , where rj=R(\+2ir/m)J and Discrete vortices were shed into the flow from the two points on the circle corresponding to the trailing edge of the airfoil and a separation point just behind the leading edge. In the present study, the flow is over an 11%-thick Joukowski airfoil at 30-deg incidence, and the leading-edge separation point was fixed empirically at 1% chord, c, from observations of the separation point on a similar airfoil