Convection Effects in Three-Dimensional Dendritic Growth
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Convection Effects in Three-Dimensional Dendritic Growth Yili Lu1, C. Beckermann1, and A. Karma2 1 Department of Mechanical and Industrial Engineering, University of Iowa, Iowa City, IA 52242-1527, U.S.A. 2 Department of Physics and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, MA 02115, U.S.A. ABSTRACT A phase-field model is developed to simulate free dendritic growth coupled with fluid flow for a pure material in three dimensions. The preliminary results presented here illustrate the strong influence of convection on the three-dimensional (3D) dendrite growth morphology. The knowledge of the flow and temperature fields in the melt from the simulations allows for a detailed understanding of the convection effects on dendritic growth. INTRODUCTION Dendrites are the most common microstructure found in engineering materials. The shape, size and orientation of the dendrites determine to a large extent the physical and chemical properties of cast and welded metals. While numerous experimental, numerical and analytical studies have been performed to understand dendritic growth from the melt in diffusion-controlled situations [1-10], the pattern selection and microstructure evolution are not well understood for convection-controlled growth. Convection in the melt during solidification can be caused by buoyancy forces, dendrite movement, shrinkage, or a variety of imposed flows. In the past ten years, the phase-field method [11-14] has become a popular computational tool to simulate microstructure formation in solidification. The main advantage of the phase-field method is that it avoids direct tracking of the sharp solid-liquid interfaces [15-17]. Based on an analysis of the thin interface limit, Karma and Rappel [18] proposed a computationally efficient phase-field method that allows for quantitative modeling of dendritic crystal growth. Beckermann et al. [19] employed the phase-field method to study convective effects on dendritic growth in two dimensions (2D). The numerical results show that convection can significantly alter the operating state of a dendrite tip and dendritic sidebranching. Very recently, Jeong et al. [20] investigated the effect of fluid flow on 3D dendritic growth using an adaptive-grid finite element method. They found that the flow and dendrite growth shapes in three dimensions are very different from those in two dimensions. In this paper, our previous simulations of 2D dendritic growth with convection [19] are extended to three dimensions. Preliminary results are presented that illustrate the effects of convection on dendritic growth. GOVERNING EQUATIONS The governing equations for flow and heat transfer are the same as those derived in Ref. [19], while the phase-field equation is taken from the work of Karma and Rappel [18]. The effects of flow in the phase-field equation are neglected. All equations are valid in the single-phase solid and liquid regions as well as in the diffuse interface region, where the phase-field variable, ψ , T2.2.1
varies f
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