The Annihilation of a Weak Discontinuity in Solidification Modelling

摘要

In this paper, a new formulation for the solution of the discontinuous isothermal solidification problem is presented. The formulation has similarities with the now classical capacitance and source methods traditionally used in commercial software. However, the new approach focuses on the solution of the governing enthalpy-transport equation rather than the governing parabolic partial differential heat equation. The advantage is that discontinuous physics can be accounted for without approximation and the arbitrariness common to classic approaches is avoided. Also introduced, is the concept of non-physical enthalpy, which unlike physical enthalpy has numerical values that are not moving-frame invariant. Understanding the behaviour of the non-physical enthalpy is central to the successful treatment of discontinuities. A particular drawback is that non-physical enthalpy is non-intuitive and new mathematical constructs are required to describe its behaviour. This involves the introduction of transport equations, which provide the new concept of relative moving-frame invariance for the non-physical enthalpy. The principal advantage however is that a unified methodology is established for the treatment of discontinuities. This is shown to establish real rigour and in many respects the formulation highlights the erroneous choices made with established classical approaches and casts in a totally new light a somewhat traditional problem. The new methodology is applied to a simple problem to provide an in-depth treatment and for ease of understanding but also to best describe the behaviour of the non-intuitive non-physical enthalpy. Demonstrated in the paper is the methods remarkable accuracy and stability.