Vorticity conditioning in the computation of two-dimensional viscous flows

The problem of evaluating the boundary values of the vorticity in the calculation of two-dimensional viscous flows is considered. It is shown that the splitting of the fourth-order equation for the stream function into two second-order problems implies specific integral conditions which fix the abstract projection of the vorticity field with respect, to the linear manifold of the harmonic functions. These conditions are a direct consequence of the boundary conditions on the velocity, and ensure satisfaction of physically essential conservation laws for the vorticity. The discrete analogue of, the projection conditions produces as many algebraic equations as the number of boundary points and requires the solution of an equal number of Dirichlet problems. In the particular case of stationary linearized equations (Stokes equations) a direct, i.e., noniterative method of solution is obtained. Steady and unsteady computational schemes relying on the projection conditions on the vorticity are introduced and extensive numerical results of finite difference calculations of the driven-cavity model problem are reported and discussed.