Uncertainty Quantification of Phase Transition Problems with an Injection Boundary

TL;DR

This work tackles uncertainty quantification for Stefan-type phase transition problems with an injection boundary, a scenario modeling ice accretion on surfaces. It develops an enthalpy-based framework that couples diffusion with a energy-consistent injection boundary condition $k abla U oldsymbol{n} + \frac{dL}{dt} H = \eta \frac{dL}{dt}$, together with a coordinate transform to map the moving domain to a fixed reference. An extended enthalpy method is then used to numerically solve the transformed problem in 1D (and extended to 2D), enabling implicit handling of moving interfaces without explicit front tracking. Uncertainty is propagated via non-intrusive generalized polynomial chaos with stochastic collocation to parameters controlling the boundary influx, particularly the injection rate and incoming energy, yielding insights into how random fluctuations affect the location and morphology of the phase boundary. Numerical experiments in 1D and 2D reveal how injection uncertainties shape icing regimes and ice shapes, including non-Gaussian output distributions, highlighting the framework's potential for robust icing predictions and multiphysics phase-transition modeling.

Abstract

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.

Figures (15)

• Figure 1: Model system for the 2D phase transition problem with an injection boundary.
• Figure 2: Enthalpy curve. The curve depicts the enthalpy change as a function of temperature. Below the critical temperature $T_m$, the curve reflects the enthalpy change of the solid phase growths gradually with temperature. At $T_m$, a sharp discontinuity marks the absorption of latent heat during the solid-to-liquid phase transition. Beyond $T_m$, the enthalpy of the liquid phase exhibits a similar ascent with respect to temperature increments.
• Figure 3: Model system for the 1D phase transition problem with an injection boundary.
• Figure 4: The transformed system compared to the original elementary system for the phase transition problem with an injection boundary in one-dimension. The reference system is depicted.
• Figure 5: The transformed system compared to the original elementary system for the phase transition problem with an injection boundary in two-dimension. The reference system is depicted.