Mathematical models of ice-crystal icing

: (Alternative Format Thesis)

Abstract

Ice crystal icing (ICI) in aircraft engines is a major threat to flight safety. The complex thermodynamic and phase-change conditions involved in ICI has limited rigorous modelling of the accretion process. The models developed in this thesis include both free-boundary Stefan problems consisting of layers with distinct phases, and also enthalpy methods for mixed-phase problems. When extending the 1D models to 2D, additional complexity is introduced by spatial dynamics, and thus we consider two key strands of research: (a) numerical approaches for phase change and; (b) thin film flows on curved surfaces.

For our first strand, we consider numerical approaches formulated on enthalpy, which offer numerous advantages to ‘front-tracking’ methods where the moving boundary between phases is explicitly tracked. The piecewise definition of enthalpy, present in such formulations, effectively inserts additional nonlinearity into the governing equations, thus requiring an implicit time-evolution scheme. We develop and present a new ‘flag-update’ enthalpy method that crucially results in a linear set of equations at each time step. The equations can then be formulated as a sparse linear system, and subsequently solved using a more efficient inversion process, yielding significant improvements to computational time in 1D and 2D.

Using the new ’flag-update’ enthalpy method, we study the 1D icing problem for our developed enthalpy-based mathematical model, based on the mixed-phase accretion layers physically observed in experiments. In the 1D model, we extract scaling laws and develop asymptotic solutions, which we compare with numerical solutions.

In our second strand, we examine existing results for thin film flow on curved surfaces, and bridge the gap in literature between curvature driven flow to shear dominated flow. Using time-dependent simulations and a boundary value problem formulation, we study the existence of steady states and identify a bistability. We apply linear stability analysis to categorise this instability as a function of two key parameters: the shear magnitude, and ellipticity constant. We then consider the 2D icing problem for both our pure-phase and enthalpy-based model, tying together the different threads. The enthalpy-based model is shown to perform better than our pure-phase model and preliminary results are obtained for icing on curved surfaces, highlighting the importance of shear and treatment of the substrate temperature.

Date of Award	12 Nov 2025
Original language	English
Awarding Institution	University of Bath
Supervisor	Phil Trinh (Supervisor), Hui Tang (Supervisor) & Josh Shelton (Supervisor)