Wang, Zhihua
(2025)
Icing simulation and ice pattern formation based on a phase-field method.
PhD thesis, University of Nottingham.
Abstract
A phase-field model for three-phase flows with cylindrical/spherical interfaces is established by combining the Navier-Stokes (NS), the continuity, and the energy equations, with an explicit form of curvature-dependent modified Allen-Cahn (AC) and Cahn-Hilliard (CH) equations. These modified equations are proposed to solve the inconsistency of the phase-field method between flat and curved interfaces, which can result in “phase-vanishing” problems and the break of mass conservation during the phase-changing process. It is proved that the proposed model satisfies the energy dissipation law (energy stability).
Next, the icing process with three phases, i.e., air, water, and ice, is simulated on the surface of a cylinder and a sphere, respectively. It is demonstrated that the modification of the AC and CH equations remedies the inconsistency between flat and curved interfaces and the corresponding “phase-vanishing” problem. The evolution of the curved water-air and water-ice interfaces is simultaneously captured, while the volume expansion during solidification, due to the density difference between water and ice, is also monitored.
A two-dimensional icing case with bubbles is further simulated. The deformation of bubbles, as well as the evolution of the interfaces, effectively illustrate the complex interactions between different phases in the icing process with phase changes. Particularly, in the simulation of icing, the pointed tip of the icy droplet is obtained and analyzed. The influence of the temperature of the supercooled substrate and the ambient air on the droplet freezing process is studied. The results indicate that the formation of the droplet pointed tip is primarily due to the expansion in the vertical direction during the freezing process. Lower substrate temperatures can accelerate this process. Changes in air temperature have a relatively minor impact on the freezing process, mainly affecting its early stages. Moreover, the results demonstrate that the ice front transitions from an approximately horizontal shape to a concave one.
A linear stability model based on the phase-field method is further established to study the formation of ripples on the ice surface. The pattern on horizontal ice surfaces, e.g., glaciers and frozen lakes, is found to be originating from a gravity-driven instability by studying ice-water-air flows with a range of water and ice thicknesses. The results demonstrate that contrary to gravity, surface tension and viscosity act to suppress the instability. A larger value of either water thickness or ice thickness corresponds to a longer dominant wavelength of the pattern, and a favourable wavelength of 90 mm and a ratio between the maximum and minimum wavelengths of 2.86 are predicted, in agreement with observations from nature. Furthermore, the profiles of the most unstable perturbations are found to be with two peaks at the ice-water and the water-air interfaces whose ratio decreases exponentially with the water thickness and wavenumber. The linear stability theory is also applied to interpret the ripple on the icicles. The results demonstrate that an increased water flow rate corresponds to ripples with larger wavelengths, while a larger icicle radius tends to induce shorter wavelengths.
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