Numerical Investigation of the Water-Drop Impact on Low-Drag Airfoil Using the Euler–Euler Approach and Eulerian Wall Film Model

Abstract

The Eulerian Wall Film (EWF) model is a mathematical model employed to analyze the behavior of fluid films on a surface. The model has been widely adopted in various engineering applications due to its accuracy and efficiency. However, it is rarely applied in the aerospace field. The solution of the water-drop impact constitutes an indispensable prerequisite for the computation of ice accretion on the exterior of aircraft wings. In this study, we propose a novel approach for the estimation of water-drop impact on wing surfaces by integrating the Euler–Euler approach and EWF model. This approach is capable of furnishing a point of reference and a theoretical foundation for prospective water-drop impact experiments. Through comparison with pertinent experimental findings, the precision of the numerical simulation approach utilized in this paper is substantiated. Specifically, the research object is the NACA653-218 airfoil of the C-919 transport aircraft, for which the aerodynamic properties, water-drop collision, and liquid film flow characteristics during steady flight were simulated.

1. Introduction

When an aircraft is soaring at high altitudes, it is susceptible to freezing upon impact with supercooled water drops in the ambient environment [1]. Along with the burgeoning of China’s aerospace industry, there will be a gradual escalation in the frequency of aircraft operations and a corresponding rise in the incidence of aircraft icing events [2,3]. The phenomenon of aircraft icing manifests diverse effects on flight performance, contingent upon the specific positions of the aircraft. In particular, the leading edge of the airfoil is the region where icing events frequently transpire. The deposition of frozen water on the wing surface has the potential to adversely affect the aerodynamic properties of the aircraft, resulting in increased drag and diminished lift. This phenomenon represents a significant hazard to the safe operation of the aircraft [4]. Based on pertinent examinations, it has been demonstrated that the formation of ice on the leading edge of a wing with a chord length of 1 m can yield a diminution of the maximum lift coefficient by 20% to 50%; this phenomenon can likewise engender a decline in aerodynamic efficacy to 80%, which may severely impair the seamless operation of the aircraft [5].

At present, the software for studying wing icing mainly includes FENSAP-ICE of Canada [6], LEWICE of the United States [7], and TRAJI-CE2 of the United Kingdom [8]. There are two basic approaches for calculating the impact characteristics of water drops; one is the Lagrange approach adopted by LEWICE and the other is the Eulerian approach adopted by FENSAP-ICE [9,10]. In China, Yang et al. [11] studied the diffusion behavior of water drops impinging on the subcooled wall based on a neural network model and used machine learning algorithms to train the propagation factor and ice formation image of water-drop impact. The established model can predict the influencing factors of water-drop diffusion and surface subcooling. Chen et al. [12] have devised a tridimensional framework that can be utilized to resolve non-structured meshes; the water-drop collection coefficient and heat transfer coefficient ascertained via this framework approximate those determined by the LEWICE icing software. The surface ice configurations computed for multiple sets of NACA0012 airfoils and cylinders closely resemble the experimental findings. Yin et al. [13] have formulated a theoretical framework capable of forecasting the expansion of rime ice by analyzing the propagation properties of surface acoustic waves; empirical findings from their investigation demonstrate that the formation of ice can be accurately detected through transient modifications in sonic velocity and decay. Guo et al. [14] constructed a tridimensional icing simulation, conducted an analysis of ridge and glaze ice formations on wing blades during low velocity conditions, and computed and juxtaposed the lift–drag ratio of a sleek airfoil against that of an iced airfoil.

The EWF model can calculate the flow of the film when water drops splash and adhere to the wall surface by simplifying the fluid film [15]. When coupled with the Eulerian Multiphase Flow model, it can better reflect the collisions of gas on the wall and the flow of the film [15,16]. Ding et al. [15] studied the gas–liquid separation of supersonic separators using the Eulerian–Lagrange method coupled with the EWF model, considering the phase transition between the phases and the liquid film; they analyzed the sensitivity of the separation efficiency to the inlet pressure, non-uniform water-drop diameter, and inlet water-drop concentration, and explored the optimal inlet water-drop diameter and concentration to improve the separation efficiency of the separator. Liu et al. [17] viewed cells as water drops and captured the liquid phase by using the EWF model to form a liquid film; in this way, they simulated cell attachment. Zhou et al. [18] simulated the re-entrainment of water drops in the defogger by coupling a discrete model with the EWF model; the efficiency of defogging and the flow characteristics of the fluid film on the baffle were analyzed in combination with experiments. During the numerical simulation of a cylindrical cyclone separator, Yue et al. [19] applied the EWF model to simulate the flow behavior of the upper vortex liquid film. The EWF model was utilized to investigate falling film evaporation and convective heat transfer of a non-energetic safety cooling system, with the system’s performance evaluated based on the model [20].

Therefore, by conducting a comprehensive analysis of prior academic work, as shown in

Table 1. Academic research on EWF models in recent years.

Reference	Research Object	Research Approach
[17]	cell culture	EWF model
[19]	Gas-liquid Cylindrical Cyclones
[20]	Containment cooling system
[21]	Containment condenser
[22]	Atmospheric pressure tower
[15]	Supersonic Separator	Eulerian–Lagrangian and EWF model
[23]	Ultrasound separator
[18]	Baffle mist eliminator	Discrete Phase model and EWF model
[24]	Football field wind-driven rain
[25]	Surface of nuclear power plant containment vessel	Species Transport Model and EWF model
[26]	Natural gas station pipeline	Mixture model and EWF model

, it is evident that the implementation of the EWF model has not been frequently utilized in resolving the issue of water-drop impact on wing surfaces with consideration of the far-field effect. Simultaneously, there is little research being conducted about the liquid film on the surface of airfoils presently. Hence, the primary aim of this work is to integrate the Euler–Euler approach and the EWF model, thereby simulating the water-drop impact on the airfoil surface and its liquid film flow. Ultimately, in comparison to previous research endeavors, the novelty of the numerical simulation presented in this paper can be attributed to the following key point: a three-dimensional model of the NACA653-218 airfoil blade, possessing low resistance and utilized in the transport aircraft C919, was constructed. Through simulation, the flow characteristics of liquid film during smooth flight were analyzed.

2. Numerical Calculation Method

The turbulence selection k-ω SST model, which considers the transfer of turbulent shear stress in the turbulent viscosity, is suitable for the analog of the flow around the wing wall [27]. The particular transportation equation, the process of its solution, and the model parameters of this turbulence model can be sourced from reference [28]. In the calculation method, the SIMPLE approach is selected in pressure–velocity coupling. The setting of boundary conditions are as follows: the entrance employs velocity inlet, the outlet employs pressure outlet, and the wing surface employs wall non-slip. The present article depicts the physical model and boundary conditions, as illustrated in Figure 1. The symbol c denotes the chord length of the airfoil, while the gray region denotes the width in the spanwise direction of the model.

2.1. Euler Gas–Liquid Two-Phase Model

The core of the Lagrange method calculation is the tracking of particle motion trajectory. The solving equation is established according to the principle of Newtonian mechanics, which is suitable for two-dimensional calculation. The Eulerian method regards the liquid phase as a successive phase. By introducing the water-drop volume fraction, the continuity equation and momentum equation of the liquid phase are established to solve. It does not need to track the particle trajectory and is more fit for solving the three-dimensional wing [29]. The Eulerian model is the most sophisticated of the Euler–Euler multiphase flow models [30]. It processes a momentum equation and a mass equation for every phase and realizes linkage by pressure and exchange coefficients among each phase.

2.1.1. Solve Assumptions

Alireza et al. [31] employed both single-phase and two-phase methodologies to simulate nanofluids within microtubules. The findings of the investigation indicated that the two-phase model exhibited lower errors in comparison to the experimental data. Consequently, in numerical simulations of granular flows, the selection of a multiphase model tailored to the research subject is more precise than a single-phase alternative. Under icing weather conditions, the liquid water content (LWC) of water drops is generally only 10⁻⁶ orders of magnitude [32], so the influence of the liquid phase on the air phase can be ignored when using Eulerian multiphase flow model to compute the flow field. Therefore, the airflow field can be solved first before the water-drop impact is calculated [29]. Wing icing involves phase change flow and heat transfer [33], the sizes of water drops in the air are randomly distributed, and its shape is also affected by the surrounding aerodynamic forces. Water drops may also interact with each other. Therefore, when using the Eulerian multiphase flow model to calculate the impact characteristics of water drops, the following assumptions are made [34]:

• The liquid water drop is spherical without transfiguration or fracture;
• No water particle collision, agglomeration, or splash;
• The particle phase momentum equation does not consider the viscosity term and pressure term;
• There is no heat or mass transmission between water drop and gas;
• The effect of turbulence on water drop can be left out;
• The force exerted on the water drop is resistance and gravity, and all unstable forces can be ignored;
• The resistance exerted on the water drop is stable.

2.1.2. Control Equation

(1)

Volume fraction equation

The definition of LWC is led in by treating fluid as a continuum α_q (LWC of phase q), it is the share of space by each phase, and the volume V of phase q is obtained from the following formulae [28]:

$$V={\displaystyle \underset{V}{\int }{\alpha }_{q}dV}.$$

(1)

and

$${\displaystyle \sum _{q=1}^n{\alpha }_q=1}.$$

(2)

The actual density of phase q is:

$$\rho ={\alpha }_q{\rho }_q.$$

(3)

where ρ_q is the density of phase q.

(2)

Conservation equation

When the Euler–Euler approach is adopted to solve the water-drop motion, the continuity equation and momentum conservation equation of the water drop are as follows [35]:

$$\nabla ·({\rho }_l\alpha \mathit{u})=0.$$

(4)

$$\nabla ({\rho }_l\alpha \mathit{u}\mathit{u})={\rho }_l\alpha F+{\rho }_l\alpha ({\mathit{u}}_a-\mathit{u})·K.$$

(5)

where ρ_l is the water-drop density, α is the volume fraction of the water drop, u is the velocity vector of the water-drop, u_a is the velocity vector of the air, F is the force exerted on the water drop except resistance, and K is the transmission coefficient obtained from the following formula [35]:

$$K=\frac{18{\mu }_{a}f}{{\rho }_l{d}_p{}^2}.$$

(6)

where μ_a is the air viscosity, d_p is diameter, and f is the resistance function. According to the Schiller–Neumann model, f can be obtained from the following formula [35]:

$$f=C_{d}Re/24.$$

(7)

where C_d is the resistance index and Re is the relative Reynolds number of the water-drop diameter, which is obtained from the following formula [35]:

$$\begin{array}{l}{C}_d=\{\begin{array}{ll}24(1+0.15R{e}^{0.687})/Re& Re\le 1000\\ 0.44& Re>1000\end{array}.\\ \end{array}$$

(8)

$$Re=\frac{{\rho }_a|{\mathit{u}}_a-\mathit{u}|d_p}{{\mu }_a}.$$

(9)

where ρ_a is the air density.

When the Eulerian multiphase flow model is adopted, the influence of turbulent dispersion force on the momentum transfer of two-phase turbulence can be considered. In this paper, the diffusion in the VOF model is used. At this time, the water-drop mass conservation equation can be rewritten as [28]:

$$\nabla ·({\rho }_l\alpha \mathit{u})=\nabla \left[{\gamma }_q\nabla {\alpha }_q\right].$$

(10)

where γ_q is the diffusion coefficient of phase q. ▽[γ_q▽α_q] is the turbulent diffusion and the following conditions shall be met [28]:

$${\displaystyle \sum _{q=1}^n\nabla \left[{\gamma }_q\nabla {\alpha }_q\right]}=0.$$

(11)

The diffusion coefficient of the second phase can be determined by the turbulent viscosity μ_t,q estimate [28]:

$${\gamma }_q=\frac{{\mu }_{t,q}}{{\sigma }_q},q=2\cdot \cdot \cdot n.$$

(12)

where σ_q = 0.75.

$${\mu }_{t,q}=\frac{{\rho }_{a}k}{\omega }\frac{1}{\max \left[\frac{1}{{\alpha }^{*},}\frac{S{F}_2}{{\alpha }_1\omega }\right]}.$$

(13)

The transport equation governing the turbulent kinetic energy, denoted as k, and the specific dissipation rate, denoted as ω, as well as the control parameters and solution strategies can be obtained from the authoritative source referenced as [28].

The diffusion phase D_q (q = 1) of the main phase meets the following formula [28]:

$$D_q=-{\displaystyle \sum _{p=2}^n{D}_p}.$$

(14)

2.1.3. Boundary Conditions

Boundary conditions of the flow around an airfoil are divided into the distant fields and the near wall surface field. At the distant field boundary, it is considered that the water-drop and air velocities are equal, that is, u_∞ = u_a. The volume fraction α_∞ of water drop can be obtained from the atmospheric LWC:

$${\alpha }_{\infty }=\frac{LWC}{{\rho }_l}.$$

(15)

At the near wall surface field, the airfoil surface can be divided into an impact zone and a non-impact zone according to the positive and negative of the dot product u·n of the water-drop velocity vector u and the normal vector n of the airfoil surface. As displayed in Figure 2, when u·n < 0, the water-drop velocity direction points to the airfoil wall, and the water-drop velocity and the volume fraction can be obtained by discrete control equation. When u·n > 0, the velocity of the water drop stays off from the airfoil wall, and the volume fraction is set to 0 and u·n = 0 [29].

2.2. EWF Model

When the fluid film flows on the wing surface, it will be affected by shear force, surface tension, self-viscosity force, and air pressure. Under certain circumstances, surface tension can separate the fluid film flow and form a fine flow. The EWF model can compute the flow of the fluid film for different factors, and it is assumed that the water drop will continue to flow after hitting the wall, without considering its formation of fine flow [36].

As illustrated in Figure 3, when the water drop hits the wing surface to form a thin film, the impact forms may appear as follows: 1. A small number of nearly spherical water drops directly hit the wall; 2. the water drop bounces off the wall relatively completely after hitting the wall, but the velocity changes; 3. the water drop collides with the wall and diffuses into the fluid film; 4. some water drops diffuse into the fluid film, and the other part leaves the wall with smaller water drops. The boundary conditions of the EWF model are adopted on the wing surface, taking into account the impact and attachment of water drops on the wall, which can well reflect the real situation of water drops impacting the wing wall.

2.2.1. Control Equation

The EWF model assumes that the height of the fluid film is much smaller than the curvity of the wing surface, the physical attributes of the fluid film do not change along the height direction, and the fluid film flows parallel to the wing surface.

Continuity equation of fluid film [37]:

$${\nabla }_s·\left({\rho }_l{h}_l{V}_l\right)={\stackrel{·}{m}}_s.$$

(16)

where ρ_l is water-drop density, h_l is fluid film height, ▽_s is surface gradient operator, V_l is fluid film velocity, and $\dot{m}$_s is the mass source term of water-drop collection, separation, and phase change per unit area.

Momentum conservation equation of fluid film [37]:

$${\nabla }_s·\left(h_l·V_l^2\right)=\frac{h_l·{\nabla }_s·P_L}{{\rho }_l}+g_{\tau }{h}_l+\frac{3}{2{\rho }_l}\tau f_s-\frac{3v_l}{h_l}{V}_l+\frac{\stackrel{·}{q}}{{\rho }_l}.$$

(17)

$$P_L=P_{gas}+P_h+P_{\sigma }.$$

(18)

where P_gas is the pressure of the air around the surface, P_h is part of gravity normal to the wall, P_σ is the surface tension, g_τ is the part of gravity parallel to the surface, τ is the viscous shear force at the liquid–gas boundary, f_s is the surface resistance, v_l is the kinematic viscosity of the film, and $\dot{q}$ is the momentum source term for the collection or component of water drop.

2.2.2. Calculation Process

For the collection of water drops on solid walls, the source term can be obtained from the following equation:

$${\stackrel{·}{m}}_s=\alpha {\rho }_l{V}_{d}A$$

(19)

where V_d is the velocity of the water drop perpendicular to the surface and A is the surface area.

The momentum source term $\dot{q}$ can be calculated from the following equation:

$$\stackrel{·}{q}={\stackrel{·}{m}}_s\mathit{u}$$

(20)

Local water-drop collection coefficient β [34] is the proportion of the practical water-drop collection rate on the surface of the micro-element to the maximal possible water-drop collection rate. The Eulerian method is obtained from the following formula [35]:

$$\beta =\frac{{\alpha }_n(\mathit{u}·\mathit{n})}{V_{\infty }}, {\alpha }_n=\frac{\alpha }{{\alpha }_{\infty }}.$$

(21)

where α_n is the normalized volume fraction of the water drop, n is the unit surface vertical vector, V_∞ is the free inflow velocity, and α_∞ is the LWC of the water drop in the distant field.

The continuity equation and momentum conservation equation of the fluid film are the basis of the EWF model. The EWF model is coupled with the Eulerian multiphase flow model through the source term of the control equation. In the EWF model, the second phase will be trapped on the wall to shape a fluid film, and the mass and momentum of the film separated from the wall will be added to the second phase of multiphase flow due to the high relative velocity between the air and the film. The material set in the EWF model is water in the calculation process; the water drop collected by the wing wall will be transferred from the multiphase flow to the EWF model as the source term of the water film control equation. In this paper, the mass and momentum flux of the fixed water film is specified in the wing wall boundary conditions, which are added to the control equation as source terms to compute the water-drop collection rate and film thickness on the wall [38]. Figure 4 shows the calculation process of the approach in this paper, where C is the convergence absolute criterion for all equations.

In this work, the discretization of the first order explicit time and the first order upwind control equation is adopted in the solution algorithm of the EWF model. In the solution of pressure, a second-order accuracy approach is implemented, whereas for the solution of momentum, volume fraction, turbulent kinetic energy, specific dissipation rate, and energy, a first-order upwind scheme is employed.

The present article does not account for the effect of gravitational forces, whilst employing the following physical parameter values in the proposed solution: ρ_l = 1000 kg/m³, ρ_a = 1.225 kg/m³, μ_a = 1.79 × 10⁻⁵ kg/(m·s), P_σ = 0.07194 N/m.

2.3. 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 5a. 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 5b: 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 β of cylindrical wall and spherical surface in this paper [34]. The calculation formula is:

$$\beta ={\displaystyle \sum _{i=1}^N{w}_i{\beta }_i}.$$

(22)

In the formula, N is the number of water-drop diameters, w_i is the proportion of the diameter of phase i, and β_i is the collection coefficient of phase i.

The average volume diameters (D_mv) of a single water drop on a cylinder and a sphere are D_mv = 16 μm and D_mv = 18.6 μm, respectively, and their Langmuir-D distribution values are written in

Table 2. Langmuir-D distribution.

Ratio/%	Water-Drop Diameter/μm
	Dmv = 16	Dmv = 18.6
5	5	5.7
10	8.3	9.7
20	11.4	13.2
30	16	18.6
20	21.9	25.5
10	27.8	32.4
5	35.5	41.3

.

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 β 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 FENSAP-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°and 12°. The above three cases verify the precision of the approach for calculating the three-dimensional water-drop collection coefficient.

3. Calculation Results and Analysis

Taking the airfoil NACA653-218 of the C-212 transport aircraft as the work object, this study simulates the effects of different inflow conditions on the water-drop collection coefficient of its wing surface and different LWC on the fluid film flow when the transport aircraft is flying at an altitude of 3000 m. The research conditions are as follows: the wing chord length is 1 m, the operating pressure is 70,110 Pa, and the operating temperature is 258.65 K.

The grid generation method and grid independence inspection method of the NACA653-218 airfoil are the same as those of the NACA0012 airfoil in Section 2; the turbulent y+ on the wall is less than 1. The C-type structured grid is adopted in this study. The total number of grids is 197,000 with 220,242 nodes. The grid is displayed in Figure 7.

3.1. Effect of Different Inflow Conditions on Water-Drop Collection on Airfoil Wall

Under the same conditions, this paper compares and analyzes the calculated results of the water-drop collection coefficient with those calculated by FENSAP-ICE. In the solution of the airflow field, the turbulence model used by FENSAP-ICE and the k-ω SST model adopted in this paper are different. The FENSAP plate of FENSAP-ICE adopts the Spalart–Allmaras model; this model is devised for aeronautical employment concerned with wall boundary fluidity. The principle of the water-drop impact on the wing calculated by the WATER-DROP3D plate of FENSAP-ICE is the same as that in this paper. They are all the Euler–Euler approach: water drops are regarded as a successive phase concerning the LWC of the water drop with the continuity equation and momentum equation of the water drop solved.

In this study, the different influencing factors employed to compute the water-drop collection coefficient on the airfoil wall and the basis for their selection are shown in

Table 3. The basis for the numerical selection of different inflow conditions in this paper.

Inflow Condition	Value Selection	Select by
AOA of the wing (°)	0, 3, 6, 12	the AOA is greater than 0° to generate lift
Incoming velocity (m/s)	70, 84, 100	Cruise speed of transport aircraft
Water-drop diameter (μm)	16, 25, 40	Large supercooled water-drops are not considered
LWC (g/m3)	0.2, 0.4, 0.6, 1	Iced climatic conditions

.

(a)

Effect of AOA on water-drop collection coefficient

When the AOA of the wing is greater than 0°, lift will be generated, and the lift coefficient of the aircraft will also rise with the increase of the AOA within a certain range. The AOAs selected in this paper are 0°, 3°, 6°, and 12°.

When the inflow speed is 84 m/s with a water-drop diameter of 16 μm, the water-drop collection coefficients calculated in this paper at different AOAs are displayed in Figure 8a, and the calculation results of FENSAP-ICE under the same conditions are written in Figure 8b.

From the illustration, it is evident that the results obtained via both methodologies exhibit conformity. As the AOA escalates from 0° to 12°, the overall trend of the collection coefficient of water drops is downward. Notably, the water-drop impact threshold on the lower surface of the airfoil increases gradually, while that on the upper surface decreases correspondingly. The upper surface water-drop impact limit moves downwards from position 0.03 m to position 0 m (displacement of 16.7% of the total airfoil height), and the lower surface moves downwards from position 0 m to position −0.05 m (27.8% of the airfoil height). This implies that the range of water-drop impact shifts towards the lower surface and the extent of downward movement is greater.

(b)

Effect of incoming flow velocity on water-drop collection coefficient

The maximal cruising speed of the C-212 transport aircraft is 100 m/s, and the economic cruising speed is 84 m/s. Therefore, the incoming flow speeds selected in this study are 70 m/s, 84 m/s, and 100 m/s.

When other conditions are fixed, the value of y+ on the wing wall is proportional to the incoming flow velocity. In this paper, the grid width of the first boundary layer on the wing wall is determined based on the flow velocity of 100 m/s. Therefore, the divided grid also conforms to the calculation conditions at 84 m/s and 70 m/s.

Figure 9 illustrates the water-drop collection coefficients obtained from the present study and FENSAP-ICE software for various inlet velocities at an AOA of 6° and a water-drop diameter of 16 μm. The results indicate that the maximum local water-drop collection coefficient and the water-drop impact threshold on the wing exhibit a gradual increase with the increase in incoming flow velocity. However, the magnitude of the increase is not significant. As the inflow velocity increases from 70 m/s to 100 m/s, the peak of water-drop collection coefficient only increases from 0.45 to 0.5 and the impact limit of the upper and lower surfaces expands very little.

(c)

Effect of water-drop diameter on water-drop collection coefficient

The diameter of water drops in the cloud is generally between 2 μm and 50 μm. A water drop with an average diameter of less than 15 μm can easily bypass the airfoil wall, and the amount of icing caused by it can be ignored for the icing of the whole wing. In addition, this paper doesn’t consider the splash of large supercooled water drops with a diameter of more than 50 μm [41], so the diameters of water drops selected in this paper are 16 μm, 25 μm, and 40 μm.

As depicted in Figure 10, when subjected to specific conditions (i.e., an AOA of 6° and an inlet velocity of 84 m/s), an increase in water-drop diameter leads to an escalation in the maximal local water-drop collection coefficient, accompanied by an expansion in the range of water-drop impact. This phenomenon arises due to the heightened mass and inertia of water-drops, rendering it increasingly challenging to deviate from their motion trajectory. By comparing the results with those illustrated in Figure 9, it can be inferred that, within a certain range, the water-drop diameter exerts a more significant impact on water-drop collection on the wing wall than the inlet velocity. As the diameter of water-drops increases from 16 μm to 40 μm, the impact limit on the upper surface of the airfoil extends from position 0.01 m to position 0.025 m (with a movement distance of approximately 8.3% of the total height of the airfoil), and the impact limit on the lower surface extends from position −0.045 m to position −0.075 m (with an increase of approximately 16.7%). The impact range of water drops increased from 0.055 m to 0.1 m, increasing by 81.8%. It can be seen that the influence of supercooled water droplets with large diameter in the upper air on the icing of airfoil leading edge is far greater than the speed of the aircraft. Therefore, based on the preliminary research in this paper, the accurate measurement of supercooled water drops is a research topic that needs to be focused on in the future aircraft anti icing and deicing.

(d)

Effect of LWC on water-drop collection coefficient

Because most of the frozen LWC is less than 1 g/m³, when the average LWC exceeds 0.12 g/m³ and the diameter of water drops in stratiform clouds is 14 μm, the aircraft will produce slight icing, so the LWCs selected in this calculation are 0.2 g/m³, 0.4 g/m³, 0.6 g/m³, and 1 g/m³.

According to Figure 11, it can be observed that the local water-drop collection coefficient remains consistent despite variations in LWC, when the AOA is fixed at 6°, inflow velocity is at 84 m/s, and the diameter of the water-drop is 16 μm. Therefore, it can be inferred that wall water-drop collection is not influenced by LWC.

3.2. Analysis of Flow Field Characteristics

As demonstrated by Figure 12, it becomes evident that under the condition of an AOA of 0°, the gas–liquid motion trajectory surrounding the airfoil and the limit of water-drop impact on the wing are both symmetrically distributed. Due to the alterations in AOA, the location at which the fluid comes into contact with the airfoil surface undergoes a corresponding adjustment. As illustrated in Figure 12a,b, a discernible trend is observed in the displacement of the gas–liquid impact area along the airfoil’s leading edge as the AOA increases from 0° to 9°. Figure 12c illustrates that the region of high LWC encircling the airfoil face also increases with an elevation in AOA. Furthermore, the zone with the highest LWC of water drops progressively expands towards the trailing edge of the lower surface of the airfoil. As depicted in Figure 12d, it is evident that the threshold for water-drop impact position has undergone a notable downward shift. Based on the streamline diagram, it is evident that the air bypasses the wings along the wall, and a significant portion of water drops also bypass the surface. Only a minor fraction of the water drops made contact with a small section of the leading edge of the wing. It is clear that air and diminutive water drops are notably susceptible to flow perturbations, resulting in trajectory deviations, whereas certain macroscopic droplets exhibit direct impingement upon the wing wall, owing to their larger inertia. After passing over the wing’s leading edge, the fluid stream proceeds smoothly along the wing’s surface without making direct contact again. Based on the data presented in Figure 12d, it can be observed that the water-drop impact threshold is confined to a limited area of the wing’s leading edge. This phenomenon accounts for the predominant occurrence of wing icing in this particular region.

3.3. Analysis of Fluid Film Flow Characteristics

Despite the dearth of empirical evidence on the flow attributes of wing-shaped liquid films, Wang et al. [42] have substantiated the precision of utilizing the Euler–Euler approach integrated with the EWF model in resolving overflow pressure drop and liquid film thickness through both empirical investigations and numerical simulations of a columnar gas–liquid cyclone separator. Thus, this paper posits that the pertinent flow characteristics of the wing-shaped liquid film derived from this approach possess a certain degree of referential significance.

Figure 13b exhibits a range of fluid film thicknesses (h_l) within distinct LWC conditions in the vicinity of the airfoil, whereas Figure 13a presents the distribution of these values on the y-axis. The presented diagrams depict a positive correlation between the LWC and the thickness of the film on the wing. It is evident that as the LWC increases, so does the film thickness. Furthermore, the data indicate that the increase in film thickness is more prominent near the leading edge of the wing. As shown in the Figure 13a, the wing height is approximately 0.18 m, and the magnitude of the increase in film thickness from position −0.04 m on the lower surface and position 0.03 m on the upper surface to the leading edge position (accounting for 38.9% of the total wing height) is significantly increased. When the LWC increases from 0.2 g/m³ to 1 g/m³, the film thickness increases from 0.1 μm to 0.5 μm.

According to the data presented in Figure 13c, an increase in LWC results in an augmentation of the flowing velocity of the fluid film on the wall, which can be attributed to the effect of the flow field. To facilitate the comparative analysis, this paper expands the wing model by two times in the y-axis direction, resulting in Figure 13d. From the illustration, it is evident that the augmentation of the LWC results in a progressive expansion of the fluid film’s flow spectrum along the airfoil wall.

This paper is computed at an AOA of 6 degrees; hence, the flow coverage of the fluid layer on the lower surface of the wing is greater compared to that on the upper surface. From the Figure 13d, it can be seen that when the LWC increases from 0.2 g/m³ to 1 g/m³, the film flowing velocity has the following trends on the airfoil: in the altitude direction, the upper surface limit of the film velocity streamline extends from position 0.03 m to position 0.07 m, and the lower surface limit extends from position −0.04 m to position −0.08 m (lowest end of the airfoil). In the chord length direction, the increasing magnitude in the streamline limit of the liquid film velocity on the lower surface is greater, increasing from the initial 0.15 m position to the 0.4 m position (chord length 1 m), while the streamline limit on the upper surface only increases from the 0.05 m position to the 0.15 m position.

As such, while the wall water-drop collection coefficient remains unaffected by the LWC, its elevation results in an increase in film thickness, flow velocity, and scope, consequently leading to an augmented water impacting the wing surface. This phenomenon bears noteworthy implications on the icing pattern distribution across the wing surface.

4. Conclusions

Based on the Euler–Euler approach and the EWF model, we studied both the dashing features of water drops and the flowing features of fluid film flow on a three-dimensional wall. The following conclusions were reached:

(1)

As the AOA increases, the impact position of the water drop moves downward. The impact of the inflow velocity on drop impact is relatively small. The diameter of water drops significantly affects their impact. When the diameter of water drops increases from 16 microns to 40 microns, the impact range of water drops increases by 81.8%. The LWC does not affect the water-drop collection coefficient.

(2)

By comparing the flow fields at 0° and 9° AOAs, it was found that incoming air bypasses the airfoil surface, and only a small portion of water drops directly impacts the leading edge of the wing. The increase in AOA causes a deviation in particle trajectory, an increase in LWC near the wall and its extension to the trailing edge of the airfoil.

(3)

When the LWC increases from 0.2 g/m³ to 1 g/m³, the film thickness increases from 0.1 μm to 0.5 μm. This phenomenon mainly occurs in the 38.9% area of the leading edge (along the airfoil height). At an AOA of 6°, the film velocity streamline flows 25% chord length on the lower surface and 10% chord length on the upper surface (horizontally).