Research Topics in Solidification Heat Transfer


Current work is focused on experimental and numerical studies of alloy solidification in crystal growth and the processing of binary-alloy Metal Matrix Composites (MMCs). Various items of experimental equipment and a cluster of workstations and servers are available for this work.

Complete details of current research may be found from our publications.

The following topics are explained in detail below:

1. Fixed-Grid 3D Front Tracking Algorithm

Moving interfaces between dissimilar phases are encountered in diverse applications such as melting and solidification, evaporation and condensation, flame fronts in combustion, and materials processing. The solid-liquid or liquid-vapor interfaces may in general be highly distorted. The shape of the interface is not known a priori, but instead must be obtained by an accurate tracking of the interface in response to the computed thermal, compositional or flow fields, in addition to a solution of the governing equations. Numerical diffusion at the advancing interface which blurs the sharpness of the phase front has limited tracking attempts thus far to simplified cases. The alternative has been to treat the two phases with a phase-averaged, single-domain approach.

In view of the inadequacy of the interface capturing methods in the literature, we have formulated a new easily implemented methodology for accurate interface tracking in three dimensions on a fixed grid in such a way as to allow the imposition of realistic boundary conditions at the interface. Our approach was motivated by the desire to develop a simple and elegant methodology which allows the complex physics at the interface to be modeled, without involving computationally intensive codes.

Some results obtained using the new front tracking algorithm (110K total) are available for perusal.

This algorithm is currently being employed to simulate solidification problems. Interface stability and kinetics will eventually be studied using this algorithm.

2. Isolated and Interacting Bubble Dynamics.

Additionally, we have formulated a modified volume-of-fluid (VOF) method for interface tracking in three dimensions to account for surface tension effects in the modeling of gas bubble dynamics.

Bubble dynamics has been studied in the literature using front-tracking and volume-tracking approaches. Significant modifications to the volume-of-fluid (VOF) method have been proposed in recent times which have improved its implementation. However, the extension of the method to three dimensions has introduced serious complications. In joint work with Dr. Li Chen of the Australian CSIRO, Professor John Reizes of University of Technology, Sydney, and Professor E. Leonardi of University of New South Wales, we have formulated a modified VOF method which accomplishes in three dimensions the tracking of an interface between immiscible liquids. We use this method to elucidate the development of the shape of a single, isolated bubble under the influence of gravity. We have obtained results in two and three dimensions that are in excellent agreement with experimental results in the literature. We have also been able to successfully model multiple bubble interactions, interactions of the bubbles with the container walls, and the effects of evaporation at the liquid-vapor interface. Our modified VOF method is able to identify and physically treat detailed features such as bubble deformation, cusp-formation, and bubble breakup and merging. We expect to apply this method to track solid-liquid interfaces in our materials-processing work in the near future.

Results are provided for simulations of isolated and interacting bubbles.

3. Alloy Solidification

A program for the numerical simulation of the solidification of a dilute binary alloy in 2D has been created. The algorithm used the robust and efficient enthalpy method to solve the energy equation throughout the entire domain. A new formulation for the concentration equation is employed which enables the effects of solute segregation to be incorporated. The solution scheme for natural convection in the melt was originally developed by Professors E. Leonardi and G. de Vahl Davis at the University of New South Wales in Sydney, Australia. This scheme is highly stable and utilizes the vorticity -vector potential formulaiton of the constitutive equations. Convection due to both thermal and solutal gradients is considered.

Since the enthalpy method is employed, the front location is not known explicitly, but rather is recovered a posteriori from the enthalpy values. Currently the code considers the solid/liquid front to be a sharp interface, but the nature of the solution scheme is such that it lends itself towards generalization to include a mushy zone.

The algorithm has been used to simulate the directional solidification of a pure material, the directional solidification of a dilute Sn-Bi alloy and for Space crystal growth (described below). In the future, this algorithm will be used to simulate the solidification of GaGe binary alloy.

Results for the simulation of a dilute Sn-Bi alloy are available.

Results for experiments into the directional solidification of aqueous ammonium chloride solutions are also available.

4. Modeling of Crystal Growth Under Terrestrial and Microgravity Conditions

This is a cooperative Australia-USA-France research project known as MEPHISTO, an acronym for Matériel pour l'Etude des Phénomènes Intéressant la Solidification sur Terre et en Orbite. The instrument, based on the Bridgman process for crystal growth, has been designed and developed by CNES (the French National Space Agency). Up to seven flights are expected on NASA's United States Microgravity Payload (USMP) carrier with a frequency of about one mission per year or year and a half.

In collaboration with Professors G. de Vahl Davis and E. Leonardi of the University of New South Wales in Sydney, Australia, our contribution to the MEPHISTO project involves the computational modeling of crystal growth under earth and microgravity conditions. Our work at the University of Wisconsin-Milwaukee on this project was funded by the NASA Glenn Research Center. The aims of the modeling work for MEPHISTO include the study of solute-rich alloys with low partition coefficients; the development of a model for crystal growth in the presence of supercooling; the study of anisotropic crystals; the study of inhomogeneous ingots; and a study of solidification in the presence of a free surface. The work addresses the qualitative and quantitative effects of microgravity on convection, interface morphology and stability, and segregation in the solidification of alloys. We are currently implementing our new interface tracking approach to accurately model the crystal-growth interface in order to achieve these objectives.

Fully transient numerical simulations of microgravity Bridgman crystal growth have been performed. The results show the effect of thermo-solutal convection on the process. The algorithm employed was an extension of that described for alloy solidification above.

Velocity vectors, isotherms and concentration values for a simulation of Bridgman growth for Bi-1.0 at.% Sn.

Impact of including a capillary tube in the domain and solute profiles.

Experimental results for the Bridgman growth of Succinonitrile and a Succinonitrile-Acetone alloy are also available.

5. Microscale Simulation of MMC solidification.

A program to calculate the solidification of a binary alloy in the presence of various fibers has been written. The code is completely general, in so far as any number of fibers at any location throughout the simulation domain may be specified. Another key feature of the algorithm is its efficiency. At the present time, convection effects are ignored, but the algorithm lends itself to the easy inclusion of convection effects and this is being introduced in ongoing work.

Thus far, single fiber, multiple in-line fiber cases and cases involving staggered fibers and different fiber spacings have been simulated and the results published.

Results for in-line fibers.

Results for staggered fibers and different fiber spacings.

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