A Rapid Prediction Model of Three-Dimensional Ice Accretion on Rotorcraft in Hover Flight

Abstract

Helicopters often operate at altitudes where cloud activity is prevalent, making them susceptible to icing hazards. Accurate and rapid prediction of ice accretion on rotors is crucial for expanding helicopter flight capabilities and enhancing flight safety. In this paper, we first introduce an improved 3-D ice accretion simulation method that accurately models runback water characteristics by considering factors such as control volume size, runback water speed, and direction. This method precisely calculates the ice accretion mass and runback water distribution. Building upon this foundation, we then present a rapid ice accretion prediction model, designed to overcome the time-consuming nature of traditional CFD frameworks. In the experimental section, our simulation methodology is applied to a hovering UH-1H rotor. A comparative analysis with experimental results reveals that the maximum absolute ice thickness error remains below 3 mm, demonstrating satisfactory computational accuracy of the proposed approach. Moreover, we demonstrate the model’s rapid prediction capabilities (achieving within a computational time of 2.66 s and a maximum ice thickness error of 7.2 mm) and implement multi-parameter predictions.

1. Introduction

Helicopters frequently operate at altitudes within cloud layers, which inevitably exposes them to complex icing conditions and the associated challenge of rotor icing. The icing certification of helicopters is strictly regulated by standards such as those from the American Federal Aviation Administration (FAA). The rotor serves as the primary component of the helicopter’s power system, and ice accretion on rotor blades directly alters their aerodynamic profile, leading to a significant reduction in aerodynamic performance [1,2,3,4,5,6]. More severely, ice shedding can damage engines, fuel tanks, and other critical systems, potentially causing catastrophic flight accidents.

Airworthiness of helicopter icing requires strict regulatory processes, including flight tests [7], spray tower tests [8,9], ice wind tunnel tests [10,11], and numerical simulations. Among them, numerical simulation has become an important means of evaluating rotor icing due to its in-depth expression of details, complete simulation state, and efficient development process.

In the early period, the numerical simulation of rotor icing was based on the Messinger model [12] which was widely applied in ice accretion calculation for aircrafts [13,14]. The main approach was to segment the rotor blades into multiple parts or extract radial sections, and then calculate the water droplets collection and ice shapes separately according to the methods for fixed wing aircrafts, and finally, the 3-D ice accretion was reconstructed based on the cross-sectional ice shape. Later on, hybrid methods of 2-D and 3-D were proposed [15,16,17,18]. These methods calculate the airflow field or the water droplet field of three dimensions, but the ice accretion is still simulated as 2-D. These 2-D or hybrid methods oversimplify the problem by neglecting 3-D effects, leading to potential inaccuracies and inefficiencies. Consequently, they are unable to accurately capture the true 3-D ice accretion.

In order to overcome this problem, Refs. [19,20] proposed 3-D simulation methods based on the Messinger model. They considered the centrifugal force effect and used the combination of centrifugal and shear forces as the distribution basis when solving the water film mass on the surface. Ref. [21] proposed an improved heat and mass transfer model that also added centrifugal force to the liquid film flow, which solved the momentum equation of the water film and obtained the flow velocity of the water film. However, this model did not consider the energy transfer of the water runback-in and runback-out, which affected the accuracy of ice accretion simulation [22].

Although CFD-based numerical simulation offers advantages over physical experiments in terms of cost and flexibility, it is often accompanied by disadvantages such as computational complexity and high time costs. To address these challenges, recent years have witnessed the development of a range of rapid prediction models utilizing machine learning and neural network techniques. These models primarily target 2-D ice accretion, leveraging neural networks to predict ice shape characteristics based on icing conditions as input. The characteristics predicted include 1-D data derived from 2-D ice shape curves [23,24] and ice thickness in the normal direction [25]. Additionally, Ref. [26] introduced a prediction model for 2-D ice shapes on any airfoil, considering both airfoil shapes and icing conditions as input. Ref. [27] proposed a novel approach based on machine learning and the Internet of Things to predict the thermal performance characteristics of a partial span wing anti-icing system. These models have demonstrated impressive prediction accuracy and opened up the possibility of rapid evaluations for airfoil anti-icing design. However, the current focus on 2-D models may limit their broader applicability.

In this paper, we propose a rapid prediction method for 3-D rotor ice accretion, aiming to overcome the limitations of both high-fidelity CFD simulations and existing two-dimensional data-driven models. Our approach utilizes the POD method to reduce the dimensionality of high-dimensional ice accretion data from CFD simulations, and employ the GA-BPNN method for training. This approach enables us to achieve rapid prediction of three-dimensional ice accretion on rotor surfaces. The ice shape dataset for this model is generated using an improved Messinger method that comprehensively takes into account the grid scale, direction, and magnitude of the airflow, as well as centrifugal forces.

The rest of this paper is arranged as follows. In Section 2, an improved 3-D ice accretion simulation method considering the time effect and centrifugation of runback water is proposed. In Section 3, a fast prediction model of rotor icing based on POD and GA-BPNN methods is presented. In Section 4, the accuracy of the ice accretion simulation method is verified with the experiment results. In Section 5, the prediction capability of the proposed model is shown. The paper is summarized in Section 6.

2. Improved 3-D Ice Accretion Simulation Method

2.1. The Messinger Model

In the Messinger model, it is necessary to solve the equations of mass of overflow water and energy conservation, so as to obtain the mass of final frozen water on the surface and then calculate the corresponding ice shape. For the structured mesh, the corresponding equations of mass conservation and energy conservation are as follows,

$${\dot{m}}_{\mathrm{im}}+{\displaystyle \sum _{nb}{\dot{m}}_{\mathrm{in},nb}}-{\dot{m}}_{\mathrm{va}}-{\displaystyle \sum _{nb}{\dot{m}}_{\mathrm{out},nb}}={\dot{m}}_{\mathrm{i}}$$

(1)

$${\dot{H}}_{\mathrm{i}}+{\dot{H}}_{\mathrm{va}}+{\displaystyle \sum _{nb}{\dot{H}}_{\mathrm{out},nb}}-{\displaystyle \sum _{nb}{\dot{H}}_{\mathrm{in},nb}}-{\dot{H}}_{\mathrm{im}}={\dot{Q}}_{\mathrm{f}}-{\dot{Q}}_{\mathrm{c}}$$

(2)

where $\dot{m}$ is the mass flow rate and $\dot{H}$ is the energy. The subscripts ‘im’, ‘va’, and ‘i’ denote the impingement, evaporation, and icing respectively, ‘in’ and ‘out’ are the runback water flowing from the upstream control volume and flowing out to the downstream control volume. Additionally, ${\dot{Q}}_{\mathrm{f}}$ is the heat due to airflow friction, ${\dot{Q}}_{\mathrm{c}}$ is the heat transfer due to the convection, the subscript $nb$ represents the four directions ($\mathrm{e},\mathrm{w},\mathrm{n},\mathrm{s}$) of the control volume. Figure 1 displays the mass and heat transfer of a control volume on the rotor surface.

2.2. Improved 3-D Ice Accretion Numerical Method Considering Runback Water Characteristics

The frozen water mass serves as the cornerstone for acquiring ice accretion, and it is intricately linked to the runback water. To enhance the precision of calculating the runback characteristics of the water film, we account for the temporal dynamics of the runback water. Specifically, we take into consideration the magnitude and direction of the water film, along with the surface grid size. Additionally, we incorporate the influence of centrifugal force on the water film. Details are presented below.

After evaporation and freezing, some water may be left in the control volume; we call it active water and denote its mass flow rate as ${\dot{m}}_{act}$. This part is in the liquid phase and will be included in the mass allocation. At the beginning, it has the form as

$${\dot{m}}_{\mathrm{act}}=\left({\dot{m}}_{\mathrm{im}}+{\displaystyle \sum _{nb}{\dot{m}}_{\mathrm{in},nb}}-{\displaystyle \sum _{nb}{\dot{m}}_{\mathrm{out},nb}}\right)-{\dot{m}}_{\mathrm{va}}-{\dot{m}}_{\mathrm{i}}$$

(3)

We can find that the mass flow rate of inflow and outflow of water runback, even evaporation and freezing, are all related with ${\dot{m}}_{act}$, so it is a key intermediate variable. ${\dot{m}}_{\mathrm{act}}$ satisfies the governing equations of the flow,

$${\displaystyle \frac{\partial {\dot{m}}_{\mathrm{act}}}{\partial t}}+\nabla ·({\dot{m}}_{\mathrm{act}}{V}_{\mathrm{d}})=0$$

(4)

where $V_{\mathrm{d}}$ and $t$ are the velocity and motion time of the water film, respectively.

By the finite volume method, we integrate Equation (4) to obtain

$${\displaystyle \iiint \frac{\partial {\dot{m}}_{\mathrm{act}}}{\partial t}\mathrm{d}x\mathrm{d}y\mathrm{d}z}+{\displaystyle ∯{\dot{m}}_{\mathrm{act}}{V}_{\mathrm{d}}·n\mathrm{d}s}=0$$

(5)

The second term on the left of Equation (5) is the sum mass of water entering and leaving the control volume, which can be discretized as follows,

$${\displaystyle \iiint \frac{\partial {\dot{m}}_{act}}{\partial t}\mathrm{d}x\mathrm{d}y\mathrm{d}z}={\displaystyle \frac{{\dot{m}}_{act}^{n+1}-{\dot{m}}_{act}^n}{\mathrm{\Delta }\mathrm{t}}}\tilde{\mathrm{\Omega }}$$

(6)

$${\displaystyle ∯{\dot{m}}_{\mathrm{act}}{V}_{\mathrm{d}}·n\mathrm{d}s}={\dot{m}}_{\mathrm{act}}^n{\displaystyle \sum _{nb}{V}_{\mathrm{d}}·n_{nb}{S}_{nb}}$$

(7)

where $n$ and $n+1$ represent current and next iterative steps, and $\mathrm{\Delta }\mathrm{t}$ is the time step. In addition, $n_{nb}$ and $S_{nb}$ denote the normal vector and area of the side face, respectively, and $\tilde{\mathrm{\Omega }}$ represents the volume of the control volume.

Combining Equations (5)–(7), we can get,

$$\begin{array}{ll}\hfill {\dot{m}}_{act}^{n+1}-{\dot{m}}_{act}^n& =-\mathrm{\Delta }\mathrm{t}\cdot {\displaystyle \frac{{\dot{m}}_{act}^n}{\tilde{\mathrm{\Omega }}}}{\displaystyle \sum _{nb}{V}_{\mathrm{d}}·n_{nb}{S}_{nb}}\hfill \\ \hfill & =-\mathrm{\Delta }\mathrm{t}\cdot {\dot{m}}_{act}^n{\displaystyle \sum _{nb}\frac{V_{\mathrm{d}}·n_{nb}{S}_{nb}}{\tilde{\mathrm{\Omega }}}}.\hfill \end{array}$$

(8)

For the frozen surface, all the grid heights of vertical planes are taken as the same, and the water escaping into the air is ignored. Then Equation (8) is simplified as

$${\dot{m}}_{act}^{n+1}-{\dot{m}}_{act}^n=-\mathrm{\Delta }\mathrm{t}\cdot {\dot{m}}_{act}^n{\displaystyle \sum _{nb}\frac{v_{nb}\cdot l_{nb}}{\tilde{S}}}$$

(9)

where $l_{nb}$ is the boundary length of adjacent grids on the frozen surface, $\tilde{S}$ is area of the surface grid, and $v_{nb}=V_{\mathrm{d}}·n_{nb}$ is the projection of velocity $V_{\mathrm{d}}$ on the water runback direction at the boundary $nb$. Figure 2 displays the mass flow rate, length and other factors of a control volume in the calculation of active water.

Since only the movement of the water film is considered, the left side of Equation (9) represents the disparity in mass flow rate between the inflow and outflow of water runback. Thus, Equation (9) can be rewritten as

$${\dot{m}}_{\mathrm{in}}^n-{\dot{m}}_{\mathrm{out}}^n=-\mathrm{\Delta }\mathrm{t}\cdot {\dot{m}}_{act}^n{\displaystyle \sum _{nb}\frac{v_{nb}\cdot l_{nb}}{\tilde{S}}}$$

(10)

For an arbitrary direction $nb$ of the control volume, we can get

$${\dot{m}}_{\mathrm{in},nb}^n-{\dot{m}}_{\mathrm{out},nb}^n=-\mathrm{\Delta }\mathrm{t}\cdot {\displaystyle \frac{{\dot{m}}_{\mathrm{act}}^n}{\tilde{S}}}\cdot v_{nb}\cdot l_{nb}$$

(11)

It can be seen that when the velocity component $v_{nb}$ is positive, it indicates that the water flows out from this control volume at this direction. Otherwise, the water flows in from the adjacent control volume. Therefore, the mass flow rate of the outflow runback water at any direction of the control volume can be represent as

$${\dot{m}}_{\mathrm{out},nb}^n=\left\{\begin{array}{cc}\mathrm{\Delta }\mathrm{t}\cdot {\displaystyle \frac{{\dot{m}}_{\mathrm{act}}^n}{\tilde{S}}}\cdot v_{nb}\cdot l_{nb},& v_{nb}>0\hfill \\ 0,& v_{nb}\le 0\hfill \end{array}\right.$$

(12)

Actually, the water that flows out the control volume will be the water that flows in from the adjacent control volume that shares the same interface. Then we can get the mass flow rate of inflow water as

$${\dot{m}}_{\mathrm{in},nb}^n({\mathrm{{\rm X}}}_{nb})={\dot{m}}_{\mathrm{out},nb}^n(\mathrm{{\rm X}})$$

(13)

where $\mathrm{{\rm X}}$ is the current control volume and ${\mathrm{{\rm X}}}_{nb}$ is the adjacent control volume at direction $nb$ of $\mathrm{{\rm X}}$.

After the inflow and outflow of water mass flow rate is determined, the freezing and evaporated water can be obtained by solving Equations (1) and (2). Further, the active water mass flow rate is updated as follows

$${\dot{m}}_{act}^{n+1}={\dot{m}}_{act}^n+{\displaystyle \sum _{nb}{\dot{m}}_{\mathrm{in},nb}^n}-{\displaystyle \sum _{nb}{\dot{m}}_{\mathrm{out},nb}^n}-{\dot{m}}_{\mathrm{va}}^n-{\dot{m}}_{\mathrm{i}}^n$$

(14)

The algorithm for the runback characteristics of water film is given in Algorithm 1.

Algorithm 1 Numerical method for runback water characteristics

1. Initial ${\dot{m}}_{\mathrm{i}}=0$, ${\dot{m}}_{\mathrm{act}}=0$, ${\dot{m}}_{\mathrm{in}}=0$, and ${\dot{m}}_{\mathrm{out}}=0$. Choose certain $\mathrm{\Delta }\mathrm{t}$, $N$.   2. For $n=1,2,\dots ,N$   3.  Solve ${\dot{m}}_{\mathrm{i}}^n$ and ${\dot{m}}_{\mathrm{va}}^n$ according to Equations (1) and (2)   4.  Update outflow water mass flow rate ${\dot{m}}_{\mathrm{out}}^n$ according to Equation (12)   5.  Update inflow water mass flow rate ${\dot{m}}_{\mathrm{in}}^n$ according to Equation (13)   6.  Update active water mass flow rate ${\dot{m}}_{\mathrm{act}}^n$ according to Equation (14)   7. End

Remark 1.

For rotating parts, we have taken the influence of flowing time and centrifugal forces of water film into account in order to solve or update ${\dot{m}}_{act}$ more accurately. The water film velocity satisfies the relationship ${\displaystyle \frac{\partial V_{\mathrm{d}}}{\partial t}}=r{\mathrm{\Omega }}^2$, thus we can get.

$$V_{\mathrm{d}}=V_{\mathrm{wr}}+t\cdot {\mathrm{\Omega }}^{2}r$$

(15)

where $r$ is radius vector, and $\mathrm{\Omega }$ is rotational speed, $V_{\mathrm{wr}}$ is the speed caused by air flow and can be computed by air shear force and water viscosity [19]. The second term represents centrifugal force and $t$ is the water motion time. During each iteration, the water in each control volume is treated as a whole and the corresponding mass is redistributed. Thus, we use the time step $\mathrm{\Delta }\mathrm{t}$ to replace $t$ when we solve the centrifugal force by Equation (15).

Remark 2.

The time step $\mathrm{\Delta }\mathrm{t}$ introduced in Equation (7) should not be excessively large, that may cause the runback water passing across the control volume. Meanwhile, $\mathrm{\Delta }\mathrm{t}$ should not be excessively small, as this would result in a reduction in calculation efficiency.

The final mass of frozen water can be acquired through the execution of Algorithm 1, which subsequently facilitates the determination of three-dimensional ice accretion. In our study, we have designated the direction of ice growth to be identical to the normal of the current grid. Subsequently, the ice height on each grid can be computed as follows

$$h={\displaystyle \frac{t_i\cdot {\dot{m}}_i}{\rho }}$$

where $t_i$ is icing time, $\rho$ is icing density.

3. Rapid Prediction Model of 3-D Ice Accretion

To efficiently predict the 3-D ice accretion on the helicopter rotor, we first construct a CFD dataset employing the methodology outlined in Section 2. Subsequently, the POD method is applied to effectively reduce the dimensionality of the high-dimensional 3-D ice dataset. Ultimately, the GA-BPNN method is introduced to establish a rapid prediction model, which serves as a substitute for the CFD method. The overall workflow is illustrated in Figure 3.

3.1. POD Dimensionality Reduction Method

The POD method is usually used to deal with problems with complex high-dimensional problems. Its main idea is to determine a set of orthogonal basis so that the sample set can be approximated by the linear combination of these bases. The POD dimension reduction method for 3-D ice accretion can be derived as follows.

Assuming that the set of $M$ ice accretion samples is $\{X_i|i=1,\dots ,M\}$, where $X_i$ is an $N$-dimensional column vector which is corresponding to ice height on the surface grid. We get a $N\times M$ samples matrix $X=(X_1,\dots ,X_M)$ and compute its mean as,

$$\overline{x}={\displaystyle \frac{1}{M}}{\displaystyle \sum _{i=1}^M{X}_i}$$

(16)

And then the standardized sample matrix $\overline{X}=(X_1-\overline{x},\dots ,X_M-\overline{x})$ is obtained to ensure that the mean of $\overline{X}$ is zero. Furthermore, the covariance matrix $C$ of $\overline{X}$ can be computed as

$$C={\displaystyle \frac{1}{N}}\overline{X}{\overline{X}}^T$$

(17)

According to Equation (17), the covariance matrix $C$ is a real symmetric matrix with size $N\times N$. The orthogonal basis can be obtained by eigenvalues and eigenvectors of matrix $C$.

Denote the non-empty eigenvalues of matrix $C$ as $\left\{{\mathrm{\lambda }}_1,\dots ,{\mathrm{\lambda }}_r\right\}$ where eigenvalues are arranged from largest to smallest. The corresponding eigenvectors are denoted as $\left\{{\xi }_1,\dots ,\left.{\xi }_r\right\}\right.$, where $r<N$ is the number of eigenvalues. Let the matrix formed by eigenvectors be $\zeta ={\left[{\xi }_1,\dots ,{\xi }_r\right]}^T$, which can be treated as a mapping, and the projection of the matrix $\overline{X}$ on it is computed as

$$A=\zeta \times \overline{X}$$

(18)

When the POD method is used to reduce dimensions, Ref. [28] points out that the values of the eigenvalues used correspond with the degrees of representation on samples. The energy ratio provided in Equation (19) is employed to determine which eigenvalues are actually used,

$$e={\displaystyle \sum _{i=1}^L{\mathrm{\lambda }}_i}/{\displaystyle \sum _{i=1}^r{\mathrm{\lambda }}_i}$$

(19)

Usually, energy ratio $e$ is chosen as 0.99 to obtain the number $L$.

In this way, the mapping ${\zeta }^{\prime }={\left[{\xi }_1,\dots ,{\xi }_L\right]}^T$ is obtained to reduce the dimension of samples. For a fixed ice accretion data $x$, its dimension will be reduced from $N$ to $L$ by $a={\zeta }^{\prime }\times x$ which is the projection of $x$ on ${\zeta }^{\prime }$. Conversely, we can map low dimensional data to high-dimensional data through the pseudo inverse of ${\zeta }^{\prime }$,

$$x={\zeta }^{\prime -1}\times a$$

(20)

where ${\zeta }^{\prime -1}$ is the inverse of ${\zeta }^{\prime }$.

3.2. GA-BPNN Model

The neural network aims to emulate the interconnections among neurons in the human brain, acquiring system information via memory and learning mechanisms. A widely adopted approach is the Back Propagation Neural Network (BPNN), which encompasses both forward and backward propagation phases. During backward propagation, the training error is conveyed, whereas in forward propagation, signals are transmitted. Figure 4 depicts the flowchart of the BPNN algorithm, illustrating dataflows and errors through its input, hidden, and output layers, and parameters modification.

It is well known that the three-layer neural network can simulate any nonlinear relationship. Assume that there are $l$ neurons $\left\{x_i|i=1,\dots ,l\right\}$ in the input layer, $m$ neurons in the hidden layer and $n$ neurons $\left\{y_i|i=1,\dots ,n\right\}$ in the output layer, then the output and input satisfy the following formula,

$$y_j=f_2\left[{\displaystyle \sum _{t=1}^m{w}_{j,t}}{f}_1\left({\displaystyle \sum _{i=1}^l{w}_{t,i}{x}_i}+b_1\right)+b_2\right]$$

(21)

where $w_{t,i}$ denotes the weight of the i-th neuron in the input layer to the t-th neuron in the hidden layer, $w_{j,t}$ denotes the weight of the t-th neuron in the hidden layer to the j-th neuron in the output layer, $b_1$ and $b_2$ denote the biases of the input layer to the hidden layer and the hidden layer to the output layer of the neural network, respectively, $f_1$ and $f_2$ are the activation functions of the hidden layer and the output layer.

Due to the randomness of initial weights and biases during the training process, the BPNN model may become trapped in a local minimum. To enhance the fitting performance and bolster the robustness of the BPNN model, the Genetic Algorithm (GA) is introduced to optimize the initial weights and biases within a constrained range. More details can be found in Ref. [29].

4. Ice Accretion Simulation Validation

To assess the accuracy of the proposed icing model, a 3-D ice accretion simulation is performed on the Bell UH-1H rotor. The rotor shape is the conventional NACA 0012 airfoil with a constant chord length of 0.5334 m. The radius is 7.3152 m and the solidity is about 0.0464. The rotor blade has a linear 10.9° twist along the span and with 10% root cutout. More details can be found in Figure 5. Due to the low gusts and ice accretion extending to 92% of the span, case ‘E’ from Ref. [8] is selected to be reproduced with numerical computation. The rotation speed is 33.9 rad/s, resulting in a tip Mach number of about 0.77. The test conditions for the hover flight simulation are shown in

Table 1. UH-1H hover test conditions8.

Collective Pitch Angle (°)	Rotational Speed (rad/s)	Static Temperature (°C)	MVD (μm)	LWC (g/m3)	Icing Time (min)
7.0	33.9	−19.0	30.0	0.7	3.0

.

The simulations are conducted under hovering conditions, with the computational domain configured as a cylindrical region extending 10 diameters above, below, and radially outward from the rotor disk center, and far-field boundary conditions are adopted. A structured grid topology is employed for spatial discretization, with the grid topology diagram and local mesh refinement illustrated in Figure 6. The computational domain is composed of two parts. The ‘rotation’ part contains the blades and is refined. The y+ value for the first layer of boundary layers is about 1, the mesh growth ratio is 1.2, with 37 boundary layers used. The structured mesh has 53 blocks and contains approximately 12 million cells. Numerical computations are performed using the NNW-HeliX (China Aerodynamics Research and Development Center, Mianyang, China) [30], where the flow field is resolved through the Multi-Reference Frame (MRF) approach, with governing equations consisting of three-dimensional compressible Reynolds-averaged Navier-Stokes (RANS) equations, and spatial discretization is achieved via a second-order finite volume method, while turbulence closure is implemented using the $K-\omega$ SST (Shear Stress Transport) model. Furthermore, an unsteady simulation approach is adopted for water droplet trajectory analysis, where the physical time step is set to 0.125 ms with 25 inner iterations per time step to ensure numerical convergence and the total time steps is 10,000.

Under the above simulation conditions, we get the airflow and droplet collection efficiency of the UH-1H rotor (Bell Helicopter, Fort Worth, TX, USA). And then the water film on the blade to be frozen is solved by Algorithm 1 in Section 2, where $\mathrm{\Delta }\mathrm{t}$ is chosen as 0.001 s according to the grid size and water film speed. Figure 7a illustrates the impact and accumulation of water on the blade surface. It is evident that the water mass at the leading edge increases from the root to the tip. This is attributed to the fact that, as one moves from the root to the tip, the water droplets’ speed accelerates, resulting in a higher impact rate on the surface. When these droplets strike the surface, their kinetic energy and other factors contribute to achieving a state of equilibrium. In this test, the equilibrium state involves partial evaporation of the water and some of it flowing back out. Consequently, the icing area exceeds the area where water droplets initially impact, and only a fraction of the accumulated water freezes. This can be clearly observed from the distribution of frozen mass flow rate depicted in Figure 7b.

Furthermore, during the process of addressing the icing issue using Algorithm 1, we document the overall water mass on the blade at each step. This comprehensive data is then quantitatively presented in Figure 8. The results indicate that approximately 0.0205 kg of water mass is collected on the surface per second, with roughly 0.0017 kg (constituting 83% of the total accumulated water) solidifying. Notably, our algorithm demonstrates convergence at approximately 30 iterations, as evidenced by the negligible change in the solidified water mass beyond this point.

The solidified water, as depicted in Figure 7b, facilitates the formation of a 3-D ice shape. The ice accretion under the conditions outlined in Table 1 is illustrated in Figure 9. It is evident that the icing area expands from the root to the tip, maintaining a consistent distribution pattern with the solidification shown in Figure 7b. However, the maximum thickness at each radial section exhibits a trend of initially increasing and then decreasing from the root to the tip. This is attributed to the faster speed of droplets at the tip, which requires more kinetic energy to prevent immediate freezing.

Figure 10 presents a comparison of ice shapes between experimental [30] and simulated results at radial sections of 45% and 62%. Both experimental and simulated data exhibit similar trends in ice thickness and shape on the leading edge. Notably, at the radial section of 45%, the numerical ice shape aligns perfectly with the experimental data, demonstrating the high accuracy of the proposed model. Meanwhile, at the radial section of 62%, the leading-edge ice accretion geometry and ice thickness demonstrated quantitative agreement with experimental measurements. Figure 10c presents a quantitative comparison of the maximum ice thickness at cross-sectional regions spanning 30% to 60%. The absolute errors remain consistently below 3 mm across all analyzed sections, thereby providing robust validation of the methodology’s accuracy.

5. Rotor Icing Prediction Results

5.1. Rotor Model and Ice Accretions Dataset

In this section, a specific five-bladed rotor is chosen for analysis. The rotor radius is 14.96 m, and the solidity is about 0.0723. The collective pitch and taper angle are 7° and 3°, respectively. The computational domain is a cylinder that extends 20 rotor diameters axially above and below the rotor disk and 10 diameters radially from the disk periphery. Far-field boundary conditions are applied to all external boundaries. A structured mesh with 105 blocks is employed to generate the grid. The y+ value for the first layer of boundary layers is approximately 1, the mesh growth ratio is 1.2, and the number of boundary layer is set to 41. The final mesh comprises approximately 35 million cells. Figure 11 illustrates the grid near the blade surface. The computational methodology adopted in this study maintains consistency with the numerical framework delineated in Section 4.

The rotor is maintained in a hover state with a rotational speed of 130 rpm. The mean volume diameter (MVD) of the droplets is 50 μm. For the purpose of this analysis, we assume that temperature, liquid water content (LWC), and icing time are mutually independent variables. The temperature follows a uniform distribution ranging from −25 °C to −5 °C, the icing time spans a uniform distribution between 300 s and 960 s, and the LWC adheres to a normal distribution with a mean of 0.4 and a standard deviation of 0.16. To guarantee uniformity in both the input and projection spaces, 120 icing conditions are generated using the Latin Hypercube Sampling (LHS) technique. Subsequently, the corresponding ice accretions are produced according to the methodology outlined in Section 2. In this study, the primary characteristic of ice accretion is selected as the ice height within each grid.

5.2. Ice Accretion Prediction Validation

The surface mesh size of one blade is 85,330 (530 grids in span direction and 161 in chord direction), which means that the ice height $X_i$ in Equation (16) is a 85,330-dimensional column vector. Firstly, the POD method in Section 3.1 is employed to reduce the dimension of the data. To ensure high accuracy, we choose the energy ratio to 99.9% in Equation (19), resulting in the retention of 27 eigenvalues as shown in

Table 2. Eigenvalues determined by the POD method.

No.	$\mathbf{Eigenvalue}{\mathbf{\lambda }}_{\mathit{h}}$	No.	$\mathbf{Eigenvalue}{\mathbf{\lambda }}_{\mathit{h}}$
1	307.60309	15	0.09296
2	19.25679	16	0.08556
3	5.98227	17	0.07621
4	3.15939	18	0.06310
5	1.12216	19	0.05147
6	0.74467	20	0.04675
7	0.53879	21	0.04254
8	0.41088	22	0.03665
9	0.33444	23	0.03470
10	0.30189	24	0.03315
11	0.20751	25	0.02790
12	0.14947	26	0.02578
13	0.12135	27	0.02523
14	0.11986

. Ultimately, these 27 coefficients of the fundamental modal are selected as the output features for the training process of the GA-BPNN model.

The parameter settings of the GA-BPNN model are presented in

Table 3. Main parameters for GA-BPNN.

Hidden Layers	Number of Neurons in Each Layer	Learning Rate	Optimizer	Activation Function
1	3, 7, 27	0.01	Levenberg-Marquardt algorithm	tanh (for hidden layer) linear (for output layer)

. It is evident that the neural network employed is of the simplest type, containing only one hidden layer. Furthermore, a total of 120 samples are used in this study, with 70% allocated for training, 15% for validation, and the remaining 15% for testing.

When the training is finished, we choose a random icing condition to assess the accuracy and efficiency of the proposed prediction model. Specifically, this condition features a static temperature of −22.89 °C, LWC of 0.308 g/m³, and the icing time of 626.74 s. By acquiring the coefficient matrix ${\zeta }^{\prime }$ for this prediction case, we can reconstruct the icing height and accretion using the POD method. Figure 12 presents a comparison between the results obtained from CFD simulations and those predicted by the GA-BPNN model. It is evident that the predictions closely resemble the CFD results. Comparative analysis of cross-sectional ice profiles in Figure 12c–f reveals agreement between the CFD and predicted results, with localized discrepancies confined to regions exhibiting abrupt curvature transitions in ice morphology.

In order to analyze the accuracy and efficiency of the prediction model quantitatively, we choose two measurements, one is the maximum value of ice height difference and another is the relative error of the ice height. The measurements are defined as,

$$\mathrm{\Delta }{h}_{\max }={‖h_{\mathrm{pre}}-h_{\mathrm{CFD}}‖}_{\infty }$$

(22)

$$\mathrm{Error}={\displaystyle \frac{{‖\mathrm{abs}\left(h_{\mathrm{pre}}-h_{\mathrm{CFD}}\right)‖}_1}{{‖h_{\mathrm{CFD}}‖}_1}}$$

(23)

where $h_{\mathrm{pre}}$ and $h_{\mathrm{CFD}}$ are prediction and CFD simulation results, respectively, the operator ${‖\cdot ‖}_{\infty }$ is to calculate the maximum of the absolute value and ${‖\cdot ‖}_1$ is to summarize all elements in the matrix.

Table 4. Prediction error and time cost of the proposed model.

	$\mathbf{\Delta }{\mathit{h}}_{\mathbf{max}}$(mm)	$\mathbf{Error}$(%)	Time Cost (s)
GA-BPNN	7.2	3.6	2.66

presents the prediction error and time cost of the proposed GA-BPNN model. As indicated, the local height error of the GA-BPNN model is within 7.2 mm, with a relative error less than 5%, demonstrating the model’s high accuracy. Meanwhile, the time cost for this case is approximately 2.66 s, which facilitates rapid predictions and practical applications.

5.3. Multi-Parameters Prediction

Non-sampled data are not included in the POD method and neural network training, so they can used to assess the generalization ability of the prediction model. We conduct three distinct predictions, with the icing conditions detailed in

Table 5. Calculation conditions for multi-parameter icing.

	Static Temperature (°C)	LWC (g/m3)	Icing Time (s)
Case 1	−10.00	0.500	480.00
Case 2	−20.00	0.350	360.00
Case 3	−22.89	0.308	626.74

.

To evaluate the prediction effectiveness, we present comparisons of ice shapes at various radial sections between the CFD and prediction methods in Figure 13. Specifically, solid and dashed lines depict the CFD and prediction results, respectively, while red, blue, and green represent comparisons for Case 1, Case 2, and Case 3, respectively. The ice shape curves predicted by the GA-BPNN model align closely with the CFD results. However, a discrepancy is observed at the 95% radial section for Case 2, as illustrated in Figure 13d. This discrepancy may be attributed to the insufficiency of training data. To enhance the model’s accuracy in the future, we can consider expanding the scale of training samples.

To thoroughly investigate the rapid prediction methodology for rotor ice accretion profiles proposed in this study, particularly to establish the selection criteria for sample size requirements in model construction, numerical experiments are conducted under the icing condition Case 3 (Table 5) with varying sample sizes of 20, 40, 60, 80, 100, and 120. The comparative results are illustrated in Figure 14, where the black solid line represents the original airfoil contour, the black dashed line denotes the CFD reference solution, and six colored curves correspond to predictions from different sample sizes.

Critical observations reveal that the turquoise (20 samples) and green curves (40 samples) exhibit significant deviations from the CFD benchmark, attributable to insufficient sample sizes for proper feature extraction during POD-based dimensionality reduction. Specifically, limited samples constrain the completeness of basis functions representing ice accretion characteristics in the reduced-order space. Notably, predictions converge toward the CFD solution when sample counts exceed 60, with accuracy monotonically improving as sample size increases—a pattern consistent with data-driven modeling principles observed in neural networks and Kriging methods. This empirical evidence quantitatively validates the necessity of adequate sampling for robust reduced-order modeling of complex icing phenomena.

6. Conclusions

In this paper, we first introduce an improved 3-D ice accretion simulation method, whose high accuracy is confirmed through comparisons with experimental results. Based on this method, we present a rapid ice accretion prediction model designed to address the time-consuming nature of traditional CFD frameworks. Both error analysis and time cost assessments demonstrate the model’s efficient prediction capabilities, as well as its ability to handle multi-parameter predictions. The core innovation of this work lies in the enhanced physical modeling of runback water behavior—incorporating control volume size, water flow speed, and flow direction—which allows for a more precise calculation of both runoff and frozen ice mass.

The high-fidelity ice accretion dataset serves as the cornerstone of our rapid prediction model, underscoring the importance of the 3-D accretion simulation method within the prediction framework. In our proposed simulation method, we assume that the runback water exclusively moves along the surface without escaping into the air, representing an idealized scenario. However, in reality, some water may be ejected into the air, and icing shedding may occur due to the blade’s high-speed rotation and aeroelastic deformation [31]. To further enhance the simulation accuracy, future work will focus on incorporating water ejection and ice shedding phenomena into the model based on experimental observations and semi-empirical models.