非等温变密度糊状Stefan问题的自相似结构与改进的低马赫数焓方法
Self-similar structure of non-isothermal variable-density mushy Stefan problems and an improved low-Mach enthalpy method
浏览论文内容
中文总结 AI 辅助
本文针对非等温变密度糊状Stefan问题，构建有限$\triangle T$问题并证明其自相似结构，改进低马赫数焓方法，提升了高密比相变模拟的稳定性与精度。
中文摘要 AI 辅助
焓方法于20世纪70年代末被提出，用于在固定网格上模拟相变问题，无需显式追踪移动的相变前沿，至今仍是学术与商业软件中模拟工业熔化和凝固过程最广泛使用的方法之一。对于在单一温度下熔化或凝固的纯相变材料（PCM），焓方法引入了由固相线温度$T^{\rm sol}$和液相线温度$T^{\rm liq}$界定的人工糊状区。当数值参数$\triangle T=T^{\rm liq}-T^{\rm sol}$趋近于零时，焓方法得到的解通常被认为会收敛到经典Stefan问题的解，其中相变前沿厚度无限小。这一假设主要基于相等固液密度简化假设下的基准研究。本工作系统研究了焓方法在低密度比和高密度比相变问题中的精度及时空收敛特性。由于糊状区厚度不断减小，数值上难以实现$\triangle T\rightarrow0$的极限行为，因此我们构建并分析了焓方法求解的有限$\triangle T$糊状Stefan问题，证明该问题具有自相似结构，可将控制方程简化为包含两个未知参数的边值问题。理论分析还改进了我们此前开发的低马赫数焓方法，使其在两相密度比从$\boldsymbol{O}(1)$提升至$\boldsymbol{O}(3)$时，稳定性和精度得到增强。
英文摘要
The enthalpy method was introduced in the late 1970s to simulate phase-change problems on fixed grids without explicitly tracking the moving phase-change front. It remains one of the most widely used approaches in academic and commercial software for the simulation of industrial melting and solidification processes. For pure phase-change materials (PCMs) that melt or solidify at a single temperature, the enthalpy method introduces an artificial mushy region bounded by the solidus temperature, $T^{\rm sol}$, and the liquidus temperature, $T^{\rm liq}$. As the numerical parameter $ΔT=T^{\rm liq}- T^{\rm sol}$ approaches zero, the solution obtained with the enthalpy method is generally assumed to converge to that of the classical Stefan problem, in which the phase-change front is infinitesimally thin. This assumption is largely based on benchmark studies performed under the simplifying assumption of equal solid and liquid densities. In this work, we systematically investigate the accuracy and spatio-temporal convergence properties of the enthalpy method for both low- and high-density-ratio phase-change problems. Because the limiting behavior $ΔT\rightarrow0$ is difficult to realize numerically, owing to the diminishing thickness of the mushy region, we formulate and analyze the finite-$ΔT$ mushy Stefan problem solved by the enthalpy method. We show that this problem possesses a self-similar structure that reduces the governing equations to a boundary-value problem involving two unknown parameters. The theoretical analysis also enables improvements to our previously developed low-Mach enthalpy method, enhancing its stability and accuracy as the density ratio between the two phases increases from $\mathcal{O}(1)$ to $\mathcal{O}(3)$.
