Discrete Compatibility in Finite Diierence Methods for Viscous Incompressible Fluid Flow

Thom's vorticity condition for solving the incompressible Navier-Stokes equations is generally known as a rst-order method since the local truncation error for the value of boundary vorticity is rst order accurate. In the present paper, it is shown that convergence in the boundary vorticity is actually second order for steady problems and for time-dependent problems when t > 0. The result is proved by looking carefully at error expansions for the discretization which have been previously used to show second order convergence of interior vorticity. Numerical convergence studies connrm the results. At t = 0 the computed boundary vorticity is rst order accurate as predicted by the local truncation error. Using simple model problems for insight we predict that the size of the second order error term in the boundary condition blows up like C= p t as t ! 0. This is connrmed by careful numerical experiments. A similar phenomenon is observed for boundary vorticity computed using a primitive method based on the staggered Marker-and-Cell grid.