Spectral discretization of the axisymmetric vorticity, velocity and pressure formulation of the Navier-Stokes problem

We deal in this work with the nonlinear Navier-Stokes equations set in a three-dimensional axisymmetric bounded domain. The boundary conditions that we consider are given on the normal component of the velocity and the tangential component of the vorticity. Such conditions occur in a large number of flows and we are led to write a vorticity-velocity-pressure formulation. Under assumptions on the data of the problem, the three-dimensional problem is reduced in a two-dimensional one. For the discretization, we use the spectral methods which are well-adapted here. We prove the well-posedness of the obtained formulations and we derive optimal error estimates on the three unknowns. The results of the numerical experiments for known functions and a given data corresponding to a Poiseuille type flow are coherent with the theoretical ones. (C) 2012 Elsevier B.V. All rights reserved.