ABSTRACT
The Helmholtz decomposition for the analysis of unsteady, viscous and inviscid, incompressible flows is reviewed, with particular emphasis on the mechanism of the vorticity generation. Computational methods based on this decomposition are presented, for both inviscid and viscous flows. It is shown that the solutions for viscous attached high-Reynolds-number flows and for inviscid flows are close to each other, provided that the Kutta-Joukowski trailing-edge condition is satisfied for inviscid flows. The incompressible-flow formulation is then extended to compressible flows. It is shown that the Helmholtz decomposition is not convenient for the boundary element analysis of compressible flows, because the rotational source terms are different from zero in the irrotational region. A new decomposition, called the Poincare decomposition, is introduced, for which the rotational source terms are equal to zero in most of the irrotational region. This makes the decomposition appealing for the boundary element solution of compressible viscous flows. Simple numerical results, for incompressible two-dimensional flows, obtained using a numerical formulation based on the Poincare decomposition are presented in order to demonstrate the feasibility of the method.
