A theoretical model for multiply connected wings

Abstract

Steady incompressible inviscid flow past a three-dimensional multiconnected (toroidal) aerofoil with a sharp trailing edge TE is considered, adopting for simplicity a linearized analysis of the vortex sheets that collect the released vorticity and form the trailing wake. The main purpose of the paper is to discuss the uniqueness of the bounded flow solution and the role of the eigenfunction. A generic admissible flow velocity u has an unbounded singularity at TE; and the physical flow solution requires the removal of the divergent part of u (the Kutta condition). This process yields a linear functional equation along the trailing edge involving both the normal vorticity ω released into the wake, and the multiplicative factor of the eigenfunction, a₁. Uniqueness is then shown to depend upon the topology of the trailing edge. If δTE=[empty ], as, for example, in an annular-aerofoil configuration, both ω and a₁ are uniquely determined by the Kutta condition, and the bounded flow u is unique. If δTE≠[empty ], as, for example, in a connected-wing configuration, there is an infinity of bounded flows, parametrized by a₁. Numerical results of relevance for these typical configurations are presented to show the different role of the eigenfunction in the two cases.