A review of vorticity conditions in the numerical solution of the ζ–ψ equations

In this review the conditions to be imposed on the vorticity in the calculation of two-dimensional incompressible viscous flows are discussed. Existing boundary vorticity formulas, commonly regarded as a surrogate Dirichlet boundary condition for the vorticity, are more properly interpreted as the discrete counterpart of the Neumann boundary condition for the stream function. This viewpoint helps to elucidate the algebraic equivalence of coupled numerical methods with uncoupled methods based on conditions of integral type for the vorticity. A unified understanding of several available treatments for determining correct vorticity boundary values is achieved by including in the present analysis spatial discretizations by finite differences and finite elements, coupled and uncoupled formulations of the problem as well as steady and unsteady equations. Results of some test calculations are presented to illustrate the numerical consequences of the analysis.