# A New Ice Accretion Model for Aircraft Icing Based on Phase-Field Method
###### Abstract
[PERSON] [21] assumed that the water film is controlled by a Couette flow driven by a shear force and the thickness of the water film is determined by the kinematic conditions at the interface between air and water, which together constitute the SWIM equations. Meanwhile, [PERSON] [22] extended the Messinger model for aircraft icing, involving solving heat equations in the ice and water layers. This method is very simple to introduce into the icing code. [PERSON] [23] extended the heat transfer model of the Myers Model by introducing the concept of critical ice thickness as the criterion to judge whether there is overflow in each ice control volume. Thus, a simple and reasonable mathematical model was established, which can directly simulate the ice accretion on three-dimensional objects. [PERSON] [24] improved the Myers model based on the local, exact solution of the unsteady Stefan problem with heat transfer in the ice layer, and introduced local air temperature to replace the constant freestream value in the Myers model. Aircraft icing is a very complex physical process, involving a phase change of the three-phase mixture of air, water and ice. Previous icing models focused on liquid water on the surface, or regarded ice and water as independent. In this study, we researched the flow and phase transition of a two-phase mixture of ice and water. The phase-field method is a powerful method to study this kind of multi-phase flow. The phase-field method treats multi-phase fluids as one fluid with variable material properties [25]. Specifically, the phase-field method utilizes order parameters such as the concentration or volume fraction of the fluids to represent different fluid phases. The order parameter is typically evolved by the Cahn-Hilliard (C-H) equation, the Allen-Cahn equation or other types of dynamic equations, so as to capture motion of the fluid interface. In order to simulate the complex flows of a fluid mixture, especially binary incompressible fluid, a phase-field model was successfully established by [PERSON] and [PERSON] [26], in which the total density and the density of each phase are constant in the model. In the same year, [PERSON], [PERSON] and [PERSON] extended the model to the framework of classical continuum mechanics and proved that it conformed to the second law of thermodynamics [27]. [PERSON] proposed a temperature-dependent free energy, and introduced the Cahn-Hilliard gradient term related to the phase-field model into the entropy function [28]. Recently, [PERSON] proposed a numerical method for a quasi-incompressible Navier-Stokes-Cahn-Hilliard (NSCH) system that satisfied the laws of thermodynamics [29]. [PERSON] et al. [30] developed a numerical simulation method for the prediction of microstructural features of aircraft icing based on the developed phase-field method. In this paper, we propose a new ice accretion model for aircraft icing based on the phase-field model. Different from previous icing models, in this study, we selected the mixture of ice and water as the research object, not just the continuous water film on the surface. Firstly, the icing problem of a two-dimensional wing is simplified. Then, based on the phase-field model, the water and ice are treated as one fluid, and the phase equation is established with the mass fraction of the mixture as the phase parameter. At the same time, according to the mass conservation and energy balance of the mixture, the mass conservation equation and energy conservation equation of the mixture are established and improved on the basis of the Messinger model and Myers model. With the establishment of the governing equations, the thickness and temperature of water film and ice layer can be determined so as to achieve the purpose of this study, to predict the aircraft ice shape under crime or glaze icing conditions. The rest of the paper is organized as follows. In Section 2, we systematically establish and complete the governing equations of the new ice accretion model based on the phase-field method.