Vorticity Boundary Conditions for the Navier-Stokes Equation in Velocity-Vorticity Formulation

Abstract

In this paper, vortex flows are computed by means of a finite difference resolution of the 2D unsteady incompressible Navier-Stokes equations cast in the vorticity function ω- velocity vector v formulation. The boundary conditions on the function ω (i.e the definition of ω as the curl of the velocity) are enforced by an influence matrix technique at each time step, a fact that enables us to enforce everywhere in the domain the definition of the function ω as the curl of the velocity and consequently the zero divergence of the velocity field. The efficiency of the proposed method is supported by the resolution of the driven cavity problem and by the resolution of the flow around a circular cylinder. In both cases, the definition of the vorticity and the continuity equation are shown to be satisfied within machine accuracy and round-off errors of library routines. In the later case the computational domain is not simply connected, a fact that implies that an additionnal condition has to be imposed.