The paper presents a Chebyshev-Fourier collocation method for solving the unsteady 3D Navier-Stokes equations in a cylindrical domain. The numerical scheme uses primitive variables and the incompressibility constraint is satisfied by applying iteratively a correction to the pressure field. The method, due to Cahodt and Chabard ( h t . j . numer methodrfluih, 8, 869-895 (1988)) and originally developed in the framework of finite elements, is checked with respect to the present highader approach. Several tests are carried out in Cartesian geometries, successively 2D and 3D. then a comparison is perfoxmed in a cylindrical domain with two different sets of radial collocation nodes: Gauss-Lobatto nodes and Gauss-Radau points. Although quite acceptable results are obtained with the latter chain, a general decrease in efficiency is noticeable in the collocation method. This is interpreted as the consequence of two factors: the collocation fornulation is not symmetric and the Fourier analysis, used BS heuristic guide by Cahouet and Chabard, loses its efficiency in a non-equidistant grid, especially in a cylindrical geometry.
We present an application to the study of thennosolutal convection induced by unidirectional solidification of a binary alloy. The latter grows h m a Pb-30?hTl liquid phase in a cylindrical crucible comsponding to the vertical Bridgman upward configuration. We study the influence of the flow patterns on the crystal composition.