## 1 Introduction  
Aircraft icing is one of the important issues that civil aviation manufacturers must focus on. The freezing process of supercooled droplets is usually considered in icing research [1; 2; 3]. Since the early 1990s, the failure events of commercial aircraft engine caused by high-altitude ice crystals have attracted the attention of researchers [4]. In 2011, NASA devised a research framework for engine icing under the conditions of high ice-water content and conducted relevant research [5]. In 2012, the European Union launched a research project on high-altitude ice crystals [6]. In 2014, supercooled large droplets and ice crystals were included in the airworthiness certification in FAR 25-140 and Amendment 33-34 of the United States Federal Aviation Regulations. It is obvious that the study of ice crystal icing is extremely important. Ice crystals occur in clouds above 6700 m. The maximum water content of ice crystals can be up to 9 g/m\({}^{3}\), and their maximum diameter can reach 200 \(\upmu\)m [7], which is quite different from conventional supercooled droplets. When only ice crystals hit the windward surface of the aircraft, the ice crystals will break and rebound, and will not form an ice accretion on the surface. When ice crystals melt in the process of movement or form a mixed phase with supercooled droplets, ice crystals will form ice through heat and mass transfer with the liquid water film on the surface [8; 9; 10]. Ice crystal icing destroys the original aerodynamic shape of the airfoil, resulting in a reduction in lift and drag and a deterioration in aircraft flight performance. As the meteorological conditions of ice crystals are worse than those of supercooled droplets, the icing hazard of ice crystals is more serious [11]. There is a large difference between the freezing of ice crystals and supercooleddroplets, and the issue of ice crystals icing is a key issue in aircraft/engine design and airworthiness verification. Many scholars have carried out research on ice crystal icing and made research achievements. [PERSON] et al. [12; 13] developed a mixed phase icing model considering air, supercooled droplets, and ice crystals, to simulate ice growth. In the absence of experimental results, an NACA0012 airfoil and cylinder were used to verify the validity of the model. [PERSON] et al. [14] proposed a semi-empirical model considering the influence of ice crystals on the conservation of ice mass and momentum. By adjusting the model parameters, the predicted results were made consistent with the experimental data. [PERSON] et al. [15] used the Eulerian method to calculate ice crystal motion trajectories and simulated ice shapes after ice crystals impacted. [PERSON] et al. [16] studied mixed phase icing based on [PERSON]'s secondary development. [PERSON] et al. [17; 18] suggested that ice crystals affect the airflow and that the two-way coupling method should be adopted in theory. However, only the one-way coupling method was used to study the motion characteristics of nonspherical ice crystals. [PERSON] et al. [19] calculated and analyzed the difference between one-way coupling and two-way coupling. It was found that the decrease in temperature was more pronounced when two-way coupling was used. It was pointed out that two-way coupling was required for multiphase flow calculation with ice crystals. It can be seen that there have been many research works on ice crystal icing, and the prediction process of ice crystal icing includes the calculation of the air flow field, ice crystal motion trajectory, and ice crystal icing model. The ice crystal motion trajectory is an important step in the icing calculation. The collection characteristics of ice crystals, including the impact location and amount of impacts, are the input conditions for icing calculation and directly affect the final icing range and amount. Therefore, how to accurately predict ice crystal trajectory is an important factor for high-precision numerical simulation of icing.