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


Lahey, Patrick M. (1999) A fixed-grid numerical method for dendritic solidification with natural convection. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6yzn-g986.


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.

Item Type:Thesis (Dissertation (Ph.D.))
Degree Grantor:California Institute of Technology
Major Option:Applied And Computational Mathematics
Thesis Availability:Public (worldwide access)
Thesis Committee:
  • Meiron, Daniel I. (chair)
Defense Date:18 December 1998
Record Number:CaltechETD:etd-11092007-083533
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4477
Deposited By: Imported from ETD-db
Deposited On:09 Nov 2007
Last Modified:16 Apr 2021 23:05

Thesis Files

PDF (Lahey_pm_1999.pdf) - Final Version
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