generalized Stokes problem
Chebyshev spectral method
vertical Bridgman problem
Wiley InterScience Backfile Collection 1832-2000
Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
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.
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