for the vorticity. Furthermore, although a second-orderaccurate compact discretization [5,6]
of such a vorticity
The present paper considers the 2D vorticity-velocity Navier-Stokes equations written as a second-order system, when a nodedefinition provides satisfactory solutions for a wide range centred finite-difference discretization and a uniform Cartesian grid of the Reynolds number [6], the discrete counterpart of are employed. For such a formulation common vorticity boundary div V ϭ 0 obtained from such solutions goes to zero less conditions yield much inferior solutions to those obtained in the than linearly with the mesh size h for h as small as 0.005.
node-centred vorticity-stream function or staggered-grid vorticity-Nevertheless, the greater simplicity of a node-centred velocity formulations. However, we demonstrate that these three formulations are formally equivalent in the sense that they all identischeme, particularly in view of the 3D equations and when cally satisfy: (i) the node-centred finite-difference form of the conticombined with a multigrid (see Refs. [6,7] for details) nuity equation; (ii) the node-centred finite-difference form of the justifies a continuing effort to circumvent the aforemenvorticity definition with respect to the mid-cell or staggered velocity tioned difficulties.
components; and (iii) the cell-centred finite-volume (integral) form of the vorticity definition with respect to the nodal values of the
To this end, the following numerical experiment was velocity components. This last property naturally provides the ''optiperformed. The 2D node-centred vorticity-velocity equamal'' boundary conditions for the wall vorticity in the node-centred tions (Ͷ, u, v) were solved using the vorticity boundary vorticity-velocity formulation. Numerical solutions to the driven values obtained from the solution of the Ͷ, equations cavity flow problem are provided which confirm the equivalence on the same grid. The entire vorticity field was identical of the three formulations.