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A fixed-grid numerical method for dendritic solidification with natural convection.

机译：自然对流树枝状凝固的固定网格数值方法。

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摘要

The solidification of a material into an undercooled melt occurs quite frequently in material processing applications. The interface between the solid and liquid phases in such cases is inherently unstable. This instability can lead to the formation of dendritic growth patterns which may significantly impact the microstructure of the resulting solid. Because the microstructure of materials notably influences their macroscopic properties, there is significant interest in understanding and controlling the formation and evolution of dendrites.; For many years, material scientists have sought to develop a predictive theory that could relate the observed dimensions and characteristics of dendrites to the thermal and fluid dynamics conditions that prevailed during their formation. To date, no such general theory exists. The problem is a difficult one, both from an experimental and mathematical standpoint.; In this work, we develop an accurate numerical method capable of simulating dendritic solidification both with and without natural convection effects. The scheme explicitly tracks and parametrizes the interface between the liquid and solid phases using a series of independent marker particles. Due to the release of latent heat, the derivatives of the temperature of a growing dendrite are discontinuous across the interface. As a consequence, great care is required when discretizing the derivatives at nodes adjacent to the interface. We use a generalized version of LeVeque and Li's immersed interface method to accurately compute the spatial derivatives. We also develop an accurate one-step time marching scheme for problems with derivatives that jump discontinuously across a moving interface. The method is notable because it does not require that the same time discretization scheme be applied to every term in the governing equation.
机译：在材料加工应用中，将材料固化成过冷的熔体非常频繁。在这种情况下，固相和液相之间的界面固有地不稳定。这种不稳定性会导致形成树枝状生长图案，这可能会严重影响所得固体的微观结构。因为材料的微观结构会显着影响其宏观性能，所以人们对理解和控制枝晶的形成和演化有着极大的兴趣。多年来，材料科学家一直在寻求发展一种预测理论，该理论可以将观察到的树枝状晶体的尺寸和特征与其形成过程中普遍存在的热力学和流体动力学条件联系起来。迄今为止，还没有这样的一般理论。从实验和数学的角度来看，这个问题都是一个难题。在这项工作中，我们开发了一种精确的数值方法，能够模拟具有和不具有自然对流效应的树枝状凝固。该方案使用一系列独立的标记粒子明确跟踪和参数化液相和固相之间的界面。由于释放潜热，正在生长的树枝状晶体的温度导数在界面上不连续。结果，当在与接口相邻的节点处离散导数时，需要格外小心。我们使用LeVeque和Li的沉浸式界面方法的广义版本来精确计算空间导数。我们还针对衍生品在移动界面上不连续地跳跃的问题，开发了一种精确的一步式时间行进方案。该方法之所以显着，是因为它不需要对控制方程式中的每个项应用相同的时间离散方案。

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  • 作者

    Lahey, Patrick Morris.;

  • 作者单位

    California Institute of Technology.;

  • 授予单位 California Institute of Technology.;
  • 学科 Mathematics.; Physics Condensed Matter.; Engineering Materials Science.
  • 学位 Ph.D.
  • 年度 1998
  • 页码 250 p.
  • 总页数 250
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;工程材料学;
  • 关键词

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