## 2 Numerical Calculation Method
### Verification of Calculation Results of Water-Drop Collection Coefficient  
Wing icing is caused by supercooled water drops in the air hitting the wing wall to condense. The collection efficiency of water drops on the wing wall is the preliminary work of studying wing icing and is the premise of calculating mass flow and heat flow in icing. In this paper, a three-dimensional cylinder [39], sphere [40], and NACA0012 literature [39] airfoil are chosen to testify the effectiveness of the approach, and the accuracy of calculating the water-drop collection coefficient is verified by comparing it with empirical evidence obtained under identical circumstances. For the NACA0012 airfoil, the mesh in the vicinity of the wing was subjected to encryption to minimize the y+ value of the wing wall turbulence to below 1. The resultant distribution of y+ values is illustrated in Figure 4(a). At the same time, to guarantee that the computed outcomes are not affected by the grid, the independence test of the grid is carried out with the lift coefficient of the airfoil as a variable. The calculation outcomes of diverse grid numbers are displayed in Figure 4(b): when the grid number exceeds 130,000, the effect of the grid number on the calculation results can be ignored. Therefore, to decrease the calculation cost while ensuring the calculation accuracy, the final number of grids used to calculate the NACA0012 airfoil is 184,000. To enhance the precision of calculation, the Langmuir-D distribution of water-drop diameter is used to compute the comprehensive water-drop collection coefficient \(\beta\) of cylindrical wall and spherical surface in this paper [34]. The calculation formula is:  
\[\beta=\sum_{i=1}^{N}w_{i}\beta_{i}. \tag{22}\]  
In the formula, \(N\) is the number of water-drop diameters, \(w_{i}\) is the proportion of the diameter of phase \(i\), and \(\beta_{i}\) is the collection coefficient of phase \(i\). The average volume diameters (\(D_{mw}\)) of a single water drop on a cylinder and a sphere are \(D_{mw}\) = 16 \(\upmu\)m and \(D_{mw}\) = 18.6 \(\upmu\)m, respectively, and their Langmuir-D distribution values are written in Table 2.  
Figure 5: Verification of \(y^{*}\) value and grid independence of the NACA0012 airfoil. The validation comparison is displayed in Figure 6. The abscissa in the picture is the coordinate in the y-axis direction (height direction) of the three-dimensional wall, while the ordinate is the water-drop collection coefficient \(\beta\) on the wall. As shown in the figure, the three-dimensional wall water-drop collection coefficient calculated in our study is close to the test value. As displayed in Figure 6b, the distribution curve of the spherical water-drop collection coefficient calculated using a single water-drop diameter coincides with the computed result of its Langmuir-D distribution, while Figure 6a displays that the comprehensive water-drop collection coefficient on the cylinder surface is closer to the test value than that of a single water-drop diameter. It can be shown from the picture that the impact limit computed using Langmuir-D distribution on the cylinder surface is closer to the test value; therefore, the result calculated by Langmuir-D distribution is more accurate. The literature data [39] used to verify the wing water-drop collection coefficient are computed by using the icing software LEWICE. At the same time, this paper also calculates the water-drop collection coefficient through FENSP-ICE under the same conditions to compare the three. Figure 6c displays the comparison of the airfoil water-drop collection coefficient while the angle of attack (AOA) is 0\({}^{\circ}\)and 12\({}^{\circ}\). The above three cases verify the precision of the approach for calculating the three-dimensional water-drop collection coefficient.