# A New Ice Accretion Model for Aircraft Icing Based on Phase-Field Method
## 2 Materials and Methods
### Two-Dimensional Form of Governing Equation
\(s\) is the surface distance. \(Prt\) is a Prandtl number, which is approximated as a constant 0.9. \(\text{Re}_{k}\) and \(\text{Re}_{c}\) are, respectively, the Reynolds number at the rough surface and the critical Reynolds number. \(C_{f}\) is the friction coefficient between the air flow and the wall surface. \(St_{k}\) is a rough Stanton number, which can be obtained by  
\[St_{k}=0.8\cdot[\frac{\cdot Ve\cdot k_{s}C_{f}}{2v_{a}}]^{-0.2}Prt^{-0.44}, \tag{28}\]  
where \(k_{s}\) is the roughness height. The solution process of the phase-field ice accretion model is as follows: the liquid water temperature \(T_{w}\) is solved by the energy balance equation, and the freezing coefficient is determined by the liquid water temperature \(T_{w}\), so as to solve the phase equation and the mass conservation equation. The final ice thickness and the mass fraction of ice are obtained by iteration. Figure 3: Diagram of water film and ice. Figure 7 shows the distribution of water film thickness under four cases, from Case 1 to Case 4, calculated by the phase-field model. It can be seen that when the ambient temperature is relatively low, the icing type is rim ice. At this time, the water film thickness is almost 0, and the droplets freeze immediately when hitting the surface. When the icing temperature is high, there will be water film distribution on the surface. The water film thickness at the leading edge is larger, and the more it flows downstream, the thinner the water film is. This is due to the large amount of water droplets collected at the leading edge and the small surface convective heat transfer coefficient. With the flow of the water film, the liquid water gradually freezes into ice, so the water film gradually decreases. At the same time, it can be seen from the figure that with the increase in freezing temperature, the water film thickness increases gradually. This is because the higher the temperature is, the lower the freezing coefficient of water drops is, and more liquid water forms on the surface, resulting in the increase in water film thickness. Figure 8a shows the comparison of icing shapes in the case of rim ice. It can be seen that the ice shapes can be well simulated by both LEWICE and the phase-field model in this study. The freezing temperature is relatively low at this time, and the droplet freezes immediately after impacting on the surface. The liquid water on the surface does not flow, so the simulation accuracy is relatively high. Meanwhile, the stagnation thickness from the phase-field model is closer to that of the experiment than that from LEWICE. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Verification** & **Pressure** & **MVD** & **Chord** & **AOA** & **Mach** & **LWC** & **Temperature** & **Time** \\
**Case** & **Pa** & \(\mu\)**m** & **m** & **deg** & **Number** & **g/m\({}^{3}\)** & **K** & **s** \\ \hline Case 1 & 101,325 & 20 & 0.5334 & 4 & 0.197 & 1.0 & 244.8 & 360 \\ Case 2 & 101,325 & 20 & 0.5334 & 4 & 0.197 & 0.5 & 260 & 360 \\ Case 3 & 101,325 & 20 & 0.5334 & 4 & 0.197 & 1.0 & 262.05 & 360 \\ Case 4 & 101,325 & 20 & 0.5334 & 4 & 0.197 & 1.0 & 267.55 & 360 \\ Case 5 & 101,325 & 19 & 0.5351 & 0 & 0.292 & 0.75 & 263.25 & 600 \\ Case 6 & 101,325 & 30 & 0.5334 & 4 & 0.313 & 1.8 & 263.07 & 360 \\ Case 7 & 101,325 & 20 & 0.5334 & 4 & 0.207 & 1.0 & 259.85 & 360 \\ Case 8 & 101,325 & 20 & 0.5334 & 0 & 0.178 & 2.1 & 263.45 & 300 \\ \hline \end{tabular}
\end{table} 
Table 1: Icing conditions for verification cases. Figure 6: Grid independence test. Figure 8b,c show the comparison of ice shapes when the incoming temperature is between \(-11\)\({}^{\circ}\)C and \(-15\)\({}^{\circ}\)C, which can be considered as mixed ice. It can be seen from the figures that the ice shapes simulated by LEWICE still have a large deviation. The simulation results of the phase-field model are closer to the experimental values, especially the prediction of the top ice horn. Therefore, the accuracy of the phase-field model is higher than that of LEWICE under this freezing weather condition.