Solidification of a binary alloy: finite-element, single-domain simulation and new benchmark solutions. (English) Zbl 1173.80310

Summary: A finite-element simulation of binary alloy solidification based on a single-domain formulation is presented and tested. Resolution of phase change is first checked by comparison with the analytical results of M. G. Worster [J. Fluid Mech. 167, 481–501 (1986; Zbl 0599.76125)] for purely diffusive solidification. Fluid dynamical processes without phase change are then tested by comparison with previous numerical studies of thermal convection in a pure fluid [G. de Vahl Davis, Int. J. Numer. Meth. Fluids 3, 249–264 (1983; Zbl 0538.76075); D.A. Mayne, A.S. Usmani, M. Crapper, Int. J. Numer. Meth. Heat Fluid Flow 10, 598–615 (2000; Zbl 0979.76049); D.C. Wan, B.S.V. Patnaik, G.W. Wei, A new benchmark quality solution for the buoyancy driven cavity by discrete singular convolution, Numer. Heat Transfer, Part B 40, No. 3, 199–228 (2001)], in a porous medium with a constant porosity [G. Lauriat, V. Prasad, Non-Darcian effects on natural convection in a vertical porous enclosure, Int. J. Heat Mass Transfer 32, 2135–2148 (1989); P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Natural convective heat transfer in an enclosure filled with fluid saturated variable porosity medium, Int. J. Heat Mass Transfer 40, 3955–3967 (1997)] and in a mixed liquid-porous medium with a spatially variable porosity [P. Nithiarasu et al. (loc. cit.); N. Zabaras, D. Samanta, Int. J. Numer. Meth. Eng. 60, 1103–1138 (2004; Zbl 1060.76581)].
Finally, new benchmark solutions for simultaneous flow through both fluid and porous domains and for convective solidification processes are presented, based on the similarity solutions in corner-flow geometries recently obtained by M. Le Bars and M.G. Worster [J. Fluid Mech. 550, 149–173 (2006; Zbl 1097.76066)]. Good agreement is found for all tests, hence validating our physical and numerical methods. More generally, the computations presented here could now be considered as standard and reliable analytical benchmarks for numerical simulations, specifically and independently testing the different processes underlying binary alloy solidification.


80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
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