Projection algorithms for 2D vortex flows in complicated domains

Plane-parallel flows of an incompressible fluid in a bounded domain with minimum mean square vorticity are considered. The flow function is biharmonic function. Such flows include, for example, the stationary solution of 2D Stocks problem with a potential righthand side. If the velocity on the boundary is specified, then definition of the flow is reduced to the solution of the boundary value problem of the biharmonic equation. The projection algorithm for solving boundary value problems for the biharmonic equation in complicated domains is presented. There are used systems of functions, full on the domain boundary, creating the basis of non-grid method (method of basis potentials) of the hydrodynamic boundary value problems solution. The concept of own domain vortex – attached vortex flow of Roben - is considered. It is also considered an extended formulation of the building a plane-parallel flows problem – definition of flows by the boundary values of the flow function only when it is not necessary to set speed limits (that are, generally speaking, not known, as, for example, for Venturi funnel). The desired density of vortices belongs to the subspace of harmonic functions, obtained complete system of potentials in this subspace allows to construct a convergent projection algorithms; the density of vortices must be orthogonal to its own domain vortex. Numerical flows for the funnel domain with the condition of adhesion on the boundary and in the extended formulation are represented.
Key words: plane-parallel flows, flow function, biharmonic problem, Stocks problem, Roben potential, basis potentials method.