A theoretical model for multiply connected wings

Steady incompressible inviscid flow past a three-dimensional multiconnected (toroidal) aerofoil with a sharp trailing edge T E 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 pap er is to discuss the uniqueness of the bounded flow solution and the role o f the eigenfunction. A generic admissible flow velocity u has an unbounded singularity at T E, 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 vort icity omega released into the wake, and the multiplicative factor of the ei genfunction, al. Uniqueness is then shown to depend upon the topology of th e trailing edge. If partial derivative T E = 0, as, for example, in an annu lar-aerofoil configuration, both omega and al are uniquely determined by th e Kutta condition, and the bounded flow u is unique. If partial derivative TE not equal 0, as, for example, in a connected-wing configuration, there i s an infinity of bounded flows, parametrized by a(1). Numerical results of relevance for these typical configurations are presented to show the differ ent role of the eigenfunction in the two cases.