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.