Resolving the Controversy Over Tin and Gallium Melting in a Rectangular Cavity Heated From the Side

Abstract

''Is the flow monocellular or multicellular?'' This question was first raised by Dantzig 13 years ago. It was about the nature of the fluid flow for the gallium melting problem in a rectangular cavity heated from the side. Following Dantzig's work, several publications have appeared on the same problem, with fewer than a handfull reporting a multicellular structure. To date, all experimental results support the monocellular structure, yet some researchers claim that the monocellular solution is incorrect. A similar controversy was reported for the problem of tin melting, suggested in 1998 as a benchmark problem. Several arguments have been suggested to explain the discrepancies, but there does not seem to be a commonly accepted answer to the problem in the scientific community. In this work, we summarize earlier works and present a grid-refinement study for several discretization schemes with emphasis on tin melting and some results for gallium melting. Simulations are carried out with the enthalpy method. The flow cell structure is analyzed in detail, while some results are provided for the heat transfer and the melting rate. Our results show that the multicellular structure is the correct numerical solution and that the flow structure has a strong influence on other features of the solution. We also provide a detailed discussion of earlier results in order to clarify important issues and bring a final answer to the controversy.

Numerical Solution of Natural Convection Problems by a Meshless Method

“…The enthalpy jump problem is partly resolved by introducing numerical smoothing of the phase change, at least enough to get stable results, but at the price of physical model accuracy. A detailed discussion on the parameter range with appearance of the flow physics based oscillations can be found in the work (Hannoun, et al, 2003). However, the results are in good agreement with the already known solutions (Gobin and Le Quéré, 2000).…”

On the importance of solid deformations in convection-dominated liquid/solid phase change of pure materials

“…Although in [26], even in the title of their paper, Hannoun et al claim to have resolved any controversy among the scientists on this experiment, in fact they do not give any explicit suggestion for improving the quality of the numerical simulations. Whereas, with the results presented in Section 5, we show that, just by including in the mathematical model the description of the mechanics of the solid phase, we reach a much more successful match with the Gau & Viskanta observation.…”

“…The analysis of the past (till very recent) literature on the two-dimensional numerical simulation of this Gau & Viskanta experiment leads to the following significant considerations: i) the multicellular flow structure of the melt convection meets expectations from fluid dynamics (Lee & Korpela [38], Derebail & Koster [17]), and from ad hoc stability analysis of melting from a side (Le Quere & Gobin [39]), ii) the multicellular flow structure of the melt convection appears combined with unphysical wavy shaped phase fronts (Dantzig [14], Stella & Giangi [48], Hannoun et al [27]), iii) certain low order numerical schemes yield one main cell melt flow and produce smoother phase fronts, surprisingly closer to experimental observation (Brent et al [6], Viswanath & Jaluria [55]), iv) some specialists, counting on highly accurate numerical discretizations, have started to explain the inconsistencies by conjecturing that the mathematical models in use for phase change with convection are not adequate for this experiment (Hannoun et al [26], Tenchev et al [51]). …”

A reference solution for phase change with convection

“…For a long time the solution of Brent et al was used for comparisons due to its rather good agreement with experimental observation and the lack of a reference solution for the problem. Recently, Hannoun et al [16] have demonstrated what several researchers have argued before [7,13,15], that the so-often mentioned solution of Brent et al [2] is actually an incorrect numerical solution due to its inappropriate level of grid convergence.…”

“…The issue of correct number of cells in the melt was raised for tin melting, as it was for gallium. In Reference [16], the issue was resolved. More recently, Mencinger [18] presented additional results for tin melting showing large discrepancies among various contributors.…”