Collocation Method for Convective Flow Induced by Directional Solidification in a Cylinder

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 Cahouët and Chabard (Int. j. numer. methods fluids,8, 869–895 (1988)) and originally developed in the framework of finite elements, is checked with respect to the present high-order approach. Several tests are carried out in Cartesian geometries, successively 2D and 3D, then a comparison is performed 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 formulation is not symmetric and the Fourier analysis, used as heuristic guide by CahouMt and Chabard, loses its efficiency in a non-equidistant grid, especially in a cylindrical geometry. We present an application to the study of thermosolutal convection induced by unidirectional solidification of a binary alloy. The latter grows from a Pb–30%Tl liquid phase in a cylindrical crucible corresponding to the vertical Bridgman upward configuration. We study the influence of the flow patterns on the crystal composition.