| Summary: | 碩士 === 國立臺灣大學 === 化學工程學研究所 === 88 === The development of an adaptive mesh refinement (AMR) framework and its applications to solidification problems is the main purpose of this thesis. General two-phase problems can be divided into two parts according to how we deal with the interface. One way is to transform the grid to the interface shape, and the other one uses a phase-variable to represent the situation in every control volume on a fixed-grid. We choose the later one because it’s much easier for complicated interface problems. However, we’ll get a mushy-zone instead a sharp interface while using a phase-variable-like method. Since mushy-zone will only appear in multi-component system in the real world, it usually gets poor accuracy in using a phase-variable-like method for a pure system. One way believed to get better results is to reduce the thickness of the mushy-zone. This topic also interests us and we’ll make some effort on this in the thesis.
We introduce some basic programming concepts of the AMR framework at first. Then we apply the Finite-Volume method onto this framework. Since error estimation is also an important part of AMR, we briefly discuss this topic and make some simple tests about the error estimator on a benchmark problem. In the theory part, besides reviewing the governing equations of a two-phase problem, we suggest a new formulation using a phase variable directly from a temperature function.
The examples are all about the pure material systems. On the basis of the AMR framework, we can efficiently reduce the thickness of the mushy-zone and the numerical error. Both steady-state and transient-state models are implemented and some typical problems with a known or an unknown interface are examined. The results are compared with Fluent™, a previous program and analytical solutions, as good agreements are found. The present method which uses an AMR framework for reducing the thickness of the mushy-zone and the numerical error is believed to be accurate and has great potential for complicated solidification problems. However, the convergence of the scheme is seriously affected by the heat of fusion. Although we have not found the complete solution up to now, some suggestions that may be useful are given. Applications to other two-phase systems and also the extension to three-dimensional problems are possible in the future.
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