Optimal vorticity conditions for the node-centred finite-difference discretization of the second-order vorticity-velocity equations

The present paper considers the 2D vorticity–velocity Navier–Stokes equations written as a second-order system, when a node-centred finite-difference discretization and a uniform Cartesian grid are employed. For such a formulation common vorticity boundary conditions yield much inferior solutions to those obtained in the node-centred vorticity–stream function or staggered-grid vorticity–velocity formulations. However, we demonstrate that these three formulations are formally equivalent in the sense that they all identically satisfy: (i) the node-centred finite-difference form of the continuity equation; (ii) the node-centred finite-difference form of the vorticity definition with respect to the mid-cell or staggered velocity components; and (iii) the cell-centred finite-volume (integral) form of the vorticity definition with respect to the nodal values of the velocity components. This last property naturally provides the “optimal” boundary conditions for the wall vorticity in the node-centred vorticity–velocity formulation. Numerical solutions to the driven cavity flow problem are provided which confirm the equivalence of the three formulations.