Date of Award

December 2012

Degree Type


Degree Name

Doctor of Philosophy



First Advisor

Krishna M. Pillai

Committee Members

Pradeep K. Rohatgi, Michael J. Nosonovsky, Emmanuel Y.A. Wornyoh, Behzad Niroumand


Die-Casting Process, Lubrication Approximation, Numerical Analysis


Casting of thin wall parts has become a reality of the die cast industry today. Computational fluid dynamics analyses are an integral part of the production development process. Typically the three-dimensional Navier-Stokes equations coupled with the energy equation have to be solved in order to gain an understanding of the flow and solidification patterns, position of the flow front, as well as location of the solid-liquid interface as a function of time during the process of cavity filling and solidification. A typical solution of the governing equations for a thin-wall casting requires large number of computational cells, and as a result, takes impractically long time to generate a solution. Using the Hele-Shaw flow modelling approach, solution of the flow problem in a thin cavity can be simplified by neglecting the out-of-plane flow. As a further benefit, the problem is reduced from a three-dimensional problem to a two-dimensional one. But the Hele-Shaw approximation requires that viscous forces in the flow are much higher than its inertia forces, and in such a case, the Navier-Stokes equation reduces to the Reynolds's lubrication equation. However, owing to the fast injection speed of the die-cast process, the inertial forces cannot be neglected. Therefore the lubrication equation has to be modified to include the inertial effects of the flow.

In this PhD thesis, a fast numerical algorithm is developed for modeling the steady-state and transient flows of liquid metal accompanied by solidification in a thin cavity. The described problem is closely related to the cold-chamber, high-pressure die-cast process and in particular to the metal flow phenomenon observed in thin ventilation channels.

Using the fact that the rate of metal flow in the channel is much higher than the solid-liquid interface velocity, a novel numerical algorithm is developed by treating the metal flow as steady at a given time-step while treating the heat transfer along the thickness direction as transient. The flow in the thin cavity is treated as two- dimensional after integrating the momentum and continuity equations over the thickness of the channel, while the heat transfer is modelled as a one-dimensional phenomenon in the thickness direction. The staggered grid arrangement is used to discretize the flow governing equations and the resulting set of partial differential equations is solved using the SIMPLE(Semi-Implicit Method for Pressure-Linked Equations) algorithm. The thickness direction heat-transfer problem accompanied by phase change is solved using a control volume formulation. The location and shape of the solid-liquid interface are found using the Stefan condition as a part of the solution. The simulations results are found to compare well with the predictions of the commercial software FLOW3D® that solves the full three-dimensional set of flow and heat transfer equations accompanied with solidification.

The proposed numerical algorithm was also applied to solve a transient metal-filling and solidification problem in thin channels. The presence of a moving solid-liquid interface introduces a non-linearity in the resulting set of flow equations, which are now solved iteratively. Once again, a good match with the predictions of FLOW3D® was observed.

These two studies indicate that the proposed inertia-modified Reynolds's lubrication equations accompanied by through-the-thickness heat loss and solidification models can be successfully implemented to provide a quick analysis of flow and solidification of liquid metals in thin channel during the die cast process. Such simulation results, obtained with tremendous savings in CPU time, can be used to provide a quick, initial analysis during the design of the ventilation channels of a die-cast die.

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