A numerical method for the vorticity-velocity Navier-Stokes equations in two and three dimensions

This paper provides a numerical method for solving the steady-state vorticity-velocity Navier-Stokes equations in two and three dimensions. The vorticity transport equation is considered together with a Poisson equation for the velocity vector, the latter equation being parabolized in time according to the false transient approach. The two vector equations are discretized in time using the implicit Euler time stepping and the delta form of Beam and Warming. A staggered-grid spatial discretization is employed in conjunction with a deferred correction procedure. Second-order-accurate central differences are used to approximate the steady-state residuals, written in conservative form for accuracy reasons, whereas upwind differences are used for the advection terms in the implicit operator, to obtain diagonally-dominant tridiagonal systems. The discrete equations are solved sequentially by means of a robust alternating direction line-Gauss-Seidel iteration procedure combined with a simple multigrid strategy. For the model driven-cavity-flow problem in two and three dimensions, the method is found to be efficient and very accurate. For the first time, the three-dimensional discrete vorticity and velocity fields, computed using a Poisson equation for the velocity vector, are both solenoidal and satisfy their mutual relationship, exactly.