Vaughan R. Voller

A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems

Abstract An enthalpy formulation based fixed grid methodology is developed for the numerical solution of convection-diffusion controlled mushy region phase-change problems. The basic feature of the proposed method lies in the representation of the latent heat of evolution, and of the flow in the solid-liquid mushy zone, by suitably chosen sources. There is complete freedom within the methodology for the definition of such sources so that a variety of phase-change situations can be modelled. A test problem of freezing in a thermal cavity under natural convection is used to demonstrate an application of the method.

ENTHALPY-POROSITY TECHNIQUE FOR MODELING CONVECTION-DIFFUSION PHASE CHANGE: APPLICATION TO THE MELTING OF A PURE METAL

The melting of pure gallium in a rectangular cavity has been numerically investigated using the enthalpy-porosity approach for modeling combined convection-diffusion phase change. The major advantage of this technique is that it allows a fixed-grid solution of the coupled momentum and energy equations to be undertaken without resorting to variable transformations. In this work, a two-dimensional dynamic model is used and the influence of laminar natural-convection flow on the melting process is considered. Excellent agreement exists between the numerical predictions and experimental results available in the literature. The enthalpy-porosity approach has been found to converge rapidly, and is capable of producing accurate results for both the position and morphology of the melt front at different times with relatively modest computational requirements. These results may be taken to be a sound validation of this technique for modeling isothermal phase changes in metallurgical systems.

Accurate solutions of moving boundary problems using the enthalpy method

Abstract After highlighting the problems associated with the conventional numerical implementations of Stefan problems using the enthalpy formulation, a simple development is described which leads to very accurate solutions. The extension of this technique to two dimensional problems is then demonstrated using a straightforward explicit method. An implicit scheme for one dimensional problems, based upon the above development, is then described which can cope with any size phase change temperature range and the influence of internal heating, simultaneously. Finally, the utility of this scheme is demonstrated by its application to a welding problem.

ERAL SOURCE-BASED METHOD FOR SOLIDIFICATION PHASE CHANGE

After a brief review of current source-based methods for modeling solidification phase change systems, a new source-based method for the treatment of latent heat evolution is presented. The essential feature of the proposed method is linearization of the discretized source term. This results in a robust and accurate computational method that can deal efficiently with a wide range of latent heat evolution mechanisms (i.e., liquid fraction temperature relationships). The proposed method is illustrated on application to a test problem in which various liquid fraction temperature relationships are employed.

The modelling of heat, mass and solute transport in solidification systems

Abstract The aim of this paper is to explore the range of possible one-phase models of binary alloy solidification. Starting from a general two-phase description, based on the two-fluid model, three limiting cases are identified which result in one-phase models of binary systems. Each of these models can be readily implemented in standard single phase flow numerical codes. Differences between predictions from these models are examined. In particular, the effects of the models on the predicted macro-segregation patterns are evaluated.

FAST IMPLICIT FINITE-DIFFERENCE METHOD FOR THE ANALYSIS OF PHASE CHANGE PROBLEMS

This paper develops a rapid implicit solution technique for the enthalpy formulation of conduction controlled phase change problems. Initially, three existing implicit enthalpy schemes are introduced. A new enthalpy solution scheme, requiring no under- or over-relaxation, is then developed. The previous three schemes and the new scheme are tested on a range of problems in one and two dimensions. A comparison of CPU times shows that the new scheme is between 1.5 and 10 times faster than the previous schemes.

A generalized Exner equation for sediment mass balance

[1] The advance of morphodynamics research into new areas has led to a proliferation of forms of sediment mass balance equation. Without a general equation it is often difficult to know what these problem-specific versions of sediment mass balance leave out. To address this, we derive a general form of the standard Exner equation for sediment mass balance that includes effects of tectonic uplift and subsidence, soil formation and creep, compaction, and chemical precipitation and dissolution. The complete equation, (17), allows for independent evolution of two critical interfaces: that between bedrock and sediment or soil and that between sediment and flow. By eliminating terms from the general equation it is straightforward to derive mass balance equations applicable to a wide range of problems such as short-term bed evolution, basin evolution, bedrock uplift and soil formation, and carbonate precipitation and transport. Dropping terms makes explicit what is not being considered in a given problem and can be done by inspection or by a formal scaling analysis of the terms. Scaling analysis leads directly to dimensionless numbers that measure the relative importance of terms in the equation, for example, the relative influence of spatial versus temporal changes in sediment load on bed evolution. Combining scaling analysis with time averaging shows how the relative importance of terms in the equation can change with timescale; for example, the term representing bed evolution due to temporal change in sediment load tends to zero as timescale increases.

Fluvio-deltaic sedimentation: A generalized Stefan problem

We present a model of sedimentation in a subsiding fluvio-deltaic basin with steady sediment supply and unsteady base level. We demonstrate that mass transfer in a fluvio-deltaic basin is analogous to heat transfer in a generalized Stefan problem, where the basin’s shoreline represents the phase front. We obtain a numerical solution to the governing equations for sediment transport and deposition in this system via an extension of a deforminggrid technique from the phase-change literature. Through modication of the heat-balance integral method, we also develop a semi-analytical solution, which agrees well with the numerical solution. We construct a space of dimensionless groups for the basin and perform a systematic exploration of this space to illustrate the influence of each group on the shoreline trajectory. Our model results suggest that all subsiding fluvio-deltaic basins exhibit a standard autoretreat shoreline trajectory in which a brief period of shoreline advance is followed by an extended period of shoreline retreat. Base-level cycling produces a shoreline response that varies relative to the autoretreat signal. Contrary to previous studies, we fail to observe either a strong phase shift between shoreline and base level or a pronounced attenuation of the amplitude of shoreline response as the frequency of base-level cycling decreases. However, the amplitude of shoreline response to base-level cycling is a function of the basin’s age.

A general enthalpy method for modeling solidification processes

In the present work, a general implicit source-based enthalpy method is presented for the analysis of solidification systems. The proposed approach is both robust and efficient. The performance of the method is illustrated by application to a number of problems taken from recent metallurgical literature.

Towards a general numerical scheme for solidification systems

A central problem in the numerical treatment of the solidification of alloys is the coupling between the temperature and concentration fields. In this paper, governing equations and relationships that describe the temperature-solute coupling in a multicomponent alloy are presented. An overview of previous numerical coupling schemes is outlined. Following the presentation of a mixed explicit/implicit discretization of the governing equations a new numerical temperature-solute coupling scheme is developed. This new scheme can model the solidification of multicomponent alloys for a wide range of local scale behaviors. The performance of the scheme is tested on comparing numerical predictions for a ternary eutectic alloy with results obtained from a similarity solution.