We develop an enthalpy-based modeling and computational framework to quantify uncertainty in Stefan problems with an injection boundary. Inspired by airfoil icing studies, we consider a system featuring an injection boundary inducing domain changes and a free boundary separating phases, resulting in two types of moving boundaries. Our proposed enthalpy-based formulation seamlessly integrates thermal diffusion across the domain with energy fluxes at the boundaries, addressing a modified injection condition for boundary movement. Uncertainty then stems from random variations in the injection boundary. The primary focus of our Uncertainty Quantification (UQ) centers on investigating the effects of uncertainty on free boundary propagation. Through mapping to a reference domain, we derive an enthalpy-based numerical scheme tailored to the transformed coordinate system, facilitating a simple and efficient simulation. Numerical and UQ studies in one and two dimensions validate the proposed model and the extended enthalpy method. They offer intriguing insights into ice accretion and other multiphysics processes involving phase transitions.

翻译：我们开发了一个基于焓的建模与计算框架，用于量化带有注入边界的斯特藩问题中的不确定性。受机翼结冰研究的启发，我们考虑一个具有注入边界的系统，该系统引入区域变化以及一个分离相态的自由边界，从而形成两类移动边界。我们提出的基于焓的公式将整个域中的热扩散与边界处的能量通量无缝整合，并处理了用于边界移动的修正注入条件。不确定性源于注入边界中的随机变化。我们不确定性量化的主要关注点是研究不确定性对自由边界传播的影响。通过映射到参考域，我们推导出适用于变换坐标系的基于焓的数值方案，从而实现简单高效的模拟。一维和二维的数值及不确定性量化研究验证了所提出的模型和扩展的焓方法。这些研究为冰积过程及其他涉及相变的多物理场过程提供了有趣的见解。