On numerical modelling of a three-dimensional flow past a wing with external flow suction and on the effect of flow suction on trailing vortices

Abstract

In this paper, we propose a numerical method of calculating a three-dimensionalunsteady ow past a thin wing in the presence of ow suction on the wing surface through a point hole or a narrow slot. The proposed method is a further development of the discrete vortex frame method that allows one to model a ow past bodies and the evolution of a vortex wake behind them in the framework of an ideal incompressible uid model.When developing a mathematical model that takes into account the ow suction from the wing surface we rst consider the problem of a potential steady ow past a wing (without a vortex wake). We give the physical formulation of the problem assuming that the normal component of the uid velocity on the wing surface must be zero everywhere beyond the suction point (suction line), while the uid ow through the wing surface on the side of suction is nonzero. To satisfy this physical condition we propose to pose a mathematical boundary value problem for the velocity eld potential in which the normal derivative of an unknown function on one of the sides of the surface is a delta function concentrated at the suction point (on the suction line). We use the Neumann boundary value problem theory with generalized boundary conditions, which was extended in [4]. On the basis of this theory, we prove the solvability of the posed boundary value problem, investigate the behaviour of the arising velocity eld in a neighbourhood of the suction point, propose and justify the numerical method of solving the problem. Further the developed method of solving the stationary problem is empirically coupled with the known method of the numerical solution to the unstationary problem of ow past a wing with vortex wake evolution.Using the developed mathematical model, we numerically modelled a ow past a rectangular wing in the presence of ow suction through a narrow slot or a point hole. We investigated the vortex wake structure at various positions of a ow suction device. In the numerical modelling it was found that at certain positions of ow suction devices there is a loss of the stability of trailing vortices and their destuction process begins at distances from a wing, which are several times smaller than those without suction.¤ N. E. Zhukovsky Air Force Engineering Academy,