Investigation of the Computational Framework of Leading-Edge Erosion for Wind Turbine Blades

Abstract

Non-contact acoustic detection methods for blades have gained significant attention due to their advantages such as easy installation and immunity to mechanical noise interference. Numerical simulation investigations on the aerodynamic noise mechanism of blade erosion provide a theoretical basis for acoustic detection. However, constructing a three-dimensional erosion model remains a challenge due to the uncertainty in external natural environmental factors. This study investigates a leading-edge erosion calculation model for wind turbine blades subjected to rain erosion. A rain erosion distribution model based on the Weibull distribution of raindrop size is first constructed. Then, the airfoil modification scheme combined with the erosion distribution model is presented to calculate leading-edge erosion mass. Finally, for a sample National Renewable Energy Laboratory 5 MW wind turbine, a three-dimensional erosion model is investigated by analyzing erosion mass related to the parameter of the attack angle. The results indicate that the maximum erosion amount is presented at the pressure surface near the leading edge, and the decrease in erosion on the pressure surface is more rapid than the suction side from the leading edge to the trailing edge. With an increase in the attack angle, the erosion on the pressure side is more severe. Furthermore, a separation vortex appears at the leading edge of the airfoil under computational non-uniform erosion. For aerodynamic noise, a larger sound pressure level with significant fluctuation occurs at 400–1000 Hz.

1. Introduction

In recent years, with global climate warming and increasing energy demand, wind energy, as one of the most promising renewable energy sources, has attracted widespread attention [1]. Wind turbines are widely utilized as devices designed to harness wind energy and transform it into electrical power [2]. At the end of 2024, it was predicted that the total installed capacity in China could reach 510 GW. The blades, being key components of the wind turbines, are exposed to harsh external environments for extended periods, making them highly susceptible to surface erosion due to strong wind, rain, snow, hail, and other factors. The erosion can lead to a reduction in the aerodynamic performance of the wind turbine and a decrease in power generation [3]. Monitoring the operation condition of blades is of great importance to ensure wind turbine performance. To address this challenge, various automated monitoring methods, such as vibration analysis [4,5], acoustic emission [6], fiber grating [7], and acoustics signal [8], have been developed. Among these, the acoustic diagnosis method has the advantage of non-contact detection, flexible installation, and convenient maintenance. In particular, the clear noise mechanisms of blade erosion can provide theoretical support for acoustic detection and improve detection efficiency. Wang et al. [9] conducted an initial investigation on the aerodynamic noise characteristics of leading-edge erosion using a numerical simulation method. The leading-edge erosion was simulated with the equivalent sand-grain roughness by defining depth and length parameters. Their results indicated that the sound pressure level of the eroded blade is higher than that of a normal one.

For the investigation of aerodynamic noise, constructing an eroded blade geometric model becomes a critical issue. Some researchers have reported the empirical depth and chordwise length of the erosion form. Carraro et al. [10] considered severe erosion damage for the NACA0012 airfoil, where the erosion length and depth are 0.1 C and 0.014 C, respectively (C is the chord length), and investigated the aerodynamics characteristics. Papi et al. [11] simulated erosion through airfoil modifications, with the main parameters being maximum erosion depth and leading-edge erosion area (relative to the chord length). They also considered the angle of attack and operational state of the turbine blades, setting the erosion area on the pressure surface to be 1.3 times larger than that on the suction surface. Maniaci et al. [12] simulated leading-edge roughness with three heights and five surface area coverage levels, and they used three-dimensional printing to create rough leading-edge models to test the impact of erosion on the performance of the NACA63-418 airfoil (the upper surface extending 4.5% of the chord length to the lower surface at 12.5% of the chord length) and the National Renewable Energy Laboratory (NREL) S814 airfoil (the upper surface extending 4.5% of the chord length to the lower surface at 19.2% of the chord length).

In order to more accurately model the leading-edge erosion of wind turbine blades, the details of erosion patterns within pits, grooves, and delamination were proposed. Aird et al. [13] used convolutional neural networks to identify blade erosion images, automatically recognizing surface damage areas on the blades and quantifying the percentage of surface erosion area. Zhang et al. [14] detailed the sizes and distribution ranges of pits, grooves, and delamination erosion features, designed different degrees of erosion for an airfoil based on these characteristics, and conducted far-field noise measurement studies to assess the feasibility of non-contact acoustic monitoring of turbine blade erosion. Wang [15] proposed a pitting geometric model, using semicircles to represent pitting roughness elements, specifying the layout range, the distance between roughness elements, and the pitting depth. By combining chordwise depth and thickness parameters, a groove erosion model for the airfoil was defined. Ravishankara et al. [16] used a roughness model to simulate surface erosion, with pit and groove depths of 0.51 mm and 2.54 mm, respectively. The chordwise extensions on the pressure and suction surfaces accounted for 10% and 13% of the chord length, respectively.

Other researchers have begun to investigate the erosion mechanism, analyzing the evolution and distribution characteristics of blade erosion. Specially, among all the causes of leading-edge erosion for wind turbine blades, rain is the key factor due to its continuous effect from the initial stage of unit operation [17]. Compared with rain, hydrometeors, such as snow and hail, also participate in the erosion process, but the former has a soft texture, and the latter breaks down into raindrops, so the influence on leading-edge erosion is relatively weak. Javier [18] developed a computational framework to model leading-edge erosion evolution and its impact on wind turbine energy output over time. The approach quantifies rain-induced erosion through droplet impingement analysis, employing the Palmgren–Miner rule to simulate cumulative damage effects. By incorporating synthetic wind and rainfall data, the model enables long-term performance predictions, including 25-year projections of both structural degradation and power generation losses. Alessio et al. [19] innovatively integrated flow field calculation and particle cloud trajectory (PCT) tracking technology, analyzed the motion law of water particles through the turbulent diffusion closure model, and then established the erosion distribution prediction model. Campobasso [20] used the numerical simulation approach to predict rain erosion damage on wind turbine blades, considering the blade geometry, coating material, and the atmospheric conditions (wind and rain) expected at the installation site. Castorrini [21] constructed an assessment system of the blade erosion rate under specific working conditions by combining the analysis of two-dimensional cross-section impact characteristics with wind and rain flow field data.

The current erosion computational methods do not consider the effect of airfoil changes on the physical properties of particulate matter during erosion evolution. Tempelis [22,23] found that the predicted elevated stresses due to initial roughness or pre-existing damage could lead to faster damage growth and to erosion pits growing wider and deeper. Additionally, the local tangential velocity of the blade increased in proportion to the rotor radius, and the erosion was caused by high-speed particle impacts, so the severity of the erosion increased when moving toward the tip of the blade. Previous research studies mainly focused on leading-edge erosion of two-dimensional airfoils. Few studies investigated how to extend the wear calculation of a two-dimensional airfoil to the geometric model of three-dimensional blade abrasion. The primary objectives of this paper are to explore the computation method of leading-edge erosion for three-dimensional blades by combining the erosion model with the airfoil modification scheme. The contributions of this paper are summarized as follows: (1) A scale factor based on the Weibull distribution of raindrop size is constructed to express the rain erosion model. (2) By combining the rain erosion model, the airfoil modification scheme is designed to calculate the erosion amount in reality. (3) By analyzing the influence of the attack angle on the erosion amount, a three-dimensional blade erosion model is established. (4) The influence of computational non-uniform erosion on the aerodynamic and aeroacoustic performance of the three-dimensional blade is investigated.

The remainder of this paper is organized as follows. Section 2 outlines the methodology, which includes the rain erosion degradation model and the dispersed-phase model. Section 3 introduces the three-dimensional parametric model for leading-edge erosion and the numerical method. Section 4 provides the erosion distribution of the blade and the aeroacoustic characteristics results. The conclusion is presented in Section 5.

2. Methodology

Firstly, the rationale of computational framework to obtain the rain erosion distribution at the leading edge of the blade is defined. Figure 1 presents the overall flow diagram of the applied methodology.

2.1. Dispersed-Phase Model

The Lagrangian reference frame and Euler reference frame methods are used to simulate the particle trajectories. Since the volume fraction of the particle phase in this case is less than 10%, the Lagrangian reference approach can be used. Furthermore, we apply a one-way approach [24] to simulate particle erosion. That is, we only consider the impact of the flow field on the particle without considering the inverse impact of the particle on the flow field. The Euler velocity field is modeled using steady-state Reynolds Average Navier–Stokes (RANS), while the instantaneous velocity is unknown, and the turbulent diffusion of the particles must be modeled. Consequently, the PCT model [25] is utilized to simulate a particle cloud with tracking of the center of mass. The particle cloud position $x_c$ is calculated by

$$x_c={\displaystyle \underset{0}{\overset{t}{\int }}{v}_c}d{t}^{\prime }+{\left(x_c\right)}_0$$

(1)

where $v_c$ represents the velocity of the cloud, ${\left(x_c\right)}_0$ corresponds to the initial position, and $t$ is the motion time of the cloud.

In this way, the Basset–Boussinesque–Oseen equation, which is the motion equation for computing the path of particle cloud, is defined as

$${\displaystyle \frac{d{v}_c}{dt}}={\tau }_R^{-1}\left(〈u〉-v_c\right)+\left(1-{\displaystyle \frac{\rho }{{\rho }_p}}\right)a_{GRAV}$$

(2)

with

$${\tau }_R^{-1}={\displaystyle \frac{3}{4d_p}}{C}_D{\displaystyle \frac{\rho }{{\rho }_p}}‖〈u〉-v_c‖$$

(3)

where $v_c$ and $u$ are the velocity of the particle cloud and free stream, respectively. ${\rho }_p$ and $\rho$ represent the particle density and fluid density. $a_{GRAV}$ is the gravitational acceleration. $C_D$ denotes the drag coefficient, which is a particle-Reynolds-number-based modification of the Stokes drag coefficient, and $d_p$ is the particle diameter.

The hypothesis of an independent statistical event is applied to present the ensemble average of the dispersed phase for a cloud; thus, an ensemble-averaged quantity $\theta$ can be written as

$$〈\theta 〉={\displaystyle \frac{{\displaystyle \underset{{\Omega }_c}{\int }\theta PDF\left(x,t\right)}d\Omega }{{\displaystyle \underset{{\Omega }_c}{\int }PDF\left(x,t\right)}d\Omega }}$$

(4)

where ${\Omega }_c$ denotes the cloud domain, which varies by time and flow condition. Assuming the particle position of a cloud is a Gaussian distribution, the probability density function $PDF\left(x,t\right)$ is shown as

$$PDF\left(x,t\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{1/2_{\sigma }}}}\exp \left(-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{‖x-x_c‖}{\sigma }}\right)}^2\right)$$

(5)

where $x$ is the particle position. $\sigma$ is the standard deviation of the individual particle position relative to the cloud center. The cloud radius is defined as three times the standard deviation. Under the Markovian approximation, this standard deviation of $m_{th}$ is determined based on the turbulence characteristics of the flow within the cloud’s volume and defined as:

$${\sigma }_m^2\left(t\right)=2〈v_m^2〉{\tau }_L^2\left[{\displaystyle \frac{t}{{\tau }_L}}-\left(1-e^{-t/{\tau }_L}\right)\right],\left(m=1,2,3\right)$$

(6)

where ${\tau }_L$ is the Lagrangian time scale, which is the maximum of the turbulence time scale and $1/{\tau }_L$.

2.2. Rain Erosion Degradation Model Based on the Weibull Distribution of the Particle Size

Due to the high-speed operation of the blades, during rainy weather, airflow carrying raindrops impacts the blade surface, causing rain erosion [26]. George Springer [27] proposed that erosion on the target surface begins after a threshold number of droplet impacts with a given size and speed, and the erosion rate can be approximated as linear. Most models use this method to calculate rain erosion, assuming that the raindrop diameter is uniform. However, the calculation of rain erosion on the blade surface coating needs to consider the physical properties of the raindrops and their characteristics as projectiles. The flow and size of raindrops change over time in actual operating environments. This paper introduces a random variation characteristic of particles to design a rain erosion degradation model, defining the erosion amount based on the mass of the particles impacting the blade surface per unit.

The threshold of raindrop impacts causing erosion on the blade surface is represented as follows:

$${\left(n_p\right)}_{thr}=a_1{\left({\displaystyle \frac{S_{eff}}{{\overline{\sigma }}_0}}\right)}^{a_2}/A_{pp}$$

(7)

where $A_{pp}$ is the projected area of the raindrop. $S_{eff}$ represents the parameter indicating the strength of the coating material. ${\overline{\sigma }}_0$ is the stress produced by the raindrop impact on the surface. $a_1=7.1\times {10}^{-6}$ and $a_2=5.7$ are the model constants defined in Ref. [27].

When the raindrop impact count reaches the threshold ${\left(n_p\right)}_{thr}$, the erosion amount produced by each impact is given by:

$$\alpha ={\mathrm{a}}_4{\displaystyle \frac{1}{{\left(A_{pp}{\left(n_p\right)}_{thr}\right)}^{a_5}}}{m}_p$$

(8)

where $m_p$ is the mass of the droplet and $a_4=0.023$ and $a_5=0.7$ are the empirical constants given in Ref. [27].

According to the Springer raindrop erosion rate model, further analysis was conducted to calculate the erosion damage under specific blade and wind–rain conditions. Specifically, the erosion model is applied to convert the impacts into erosion, and the erosion is scaled proportionally to obtain the erosion mass in a real external environment. The calculation formula is:

$$\eta =\alpha \left(F_{ACT}{n}_p-{(n_p)}_{thr}\right)$$

(9)

where $F_{ACT}$ represents the scaling factor, calculated as follows:

$$F_{ACT}={\displaystyle \frac{n_R}{n_{TOT}}}{\displaystyle \frac{T_R{f}_1\left(V_{vef}\right)}{T_{SIM}}}{\displaystyle \frac{1}{N_{step}}}$$

(10)

where $n_{TOT}$ represents the number of particles injected into the simulation domain, $T_R$ is the total operating time, $T_{SIM}$ is the total simulation time, and $n_R$ is the total particle flux. $f_1\left(V_{ref}\right)=\left({\displaystyle \frac{k}{c}}\right){\left({\displaystyle \frac{v}{c}}\right)}^{k-1}{e}^{-{\left({\displaystyle \frac{v}{c}}\right)}^k}$ denotes the Weibull probability density function of inflow velocity, in which the shape factor $c$ is 2 and the scale factor $k$ is 6.5 m/s. $N_{step}$ is the simulation time steps.

Typically, given the operating time and the local annual rainfall, the total number of incoming particles can be determined. Under average inflow conditions, the total particle flux $n_R$ is:

$$n_R={\displaystyle \frac{H_{BRF}}{f_2(V_d)}}$$

(11)

$$H_{BRF}={\displaystyle \frac{V_b}{V_r}}E(H_{RF})$$

(12)

where $V_d$ represents the volume of the droplets, $H_{RF}$ is the rainfall intensity, $V_b$ is the blade operational speed, and $V_r$ is the droplet impact velocity. For rainfall in the air, experimental data on raindrop size distribution have been analyzed, and the results indicate that, in many cases, the raindrop size distribution conforms to the following formula [28]:

$$f_2(V_d)={\displaystyle \frac{n}{a}}{\left({\displaystyle \frac{x}{a}}\right)}^{n-1}{e}^{-{(x/a)}^n}$$

(13)

where $a=1.79$ and $n=1.24$ are constants specific to the given rainfall amount.

When performing numerical simulations to analyze the evolution and distribution of erosion on the blade surface, the initial shape of the blade plays a crucial role in the simulation result. Additionally, the geometric change caused by erosion occurs over a very long time scale. Directly simulating the entire erosion cycle without accounting for the intermediate changes in the blade shape can lead to inaccurate distribution results. Therefore, a comprehensive method combining the rain erosion model with geometric updating is adopted to calculate the blade surface erosion. And the updating time step, which is considered a sufficiently long threshold operating cycle to alter the blade aerodynamics, is used to discrete the long-time erosion.

Assuming $t_0$ is the initial time step, based on the initial blade shape and scale factor, the actual erosion amount ${\psi }_e$ of first step is obtained as follows:

$${\eta }^{\left(1\right)}={\alpha }^{(1)}\left({F_{ACT}}^{(1)}{n}_p^{\left(1\right)}-{\left(n_p\right)}_{thr}^{\left(1\right)}\right)$$

(14)

$$\left\{\begin{array}{l}{\psi }_{\mathrm{e}}=0\hspace{1em}\hspace{1em}{\lambda }^{\left(1\right)}=1,\hspace{1em}\hspace{1em}\mathrm{if} {\eta }^{\left(1\right)}\le 0\\ {\psi }_{\mathrm{e}}={\eta }^{\left(1\right)}\hspace{1em}{\lambda }^{\left(1\right)}=0,\hspace{1em}\mathrm{if} {\eta }^{\left(1\right)}>0\end{array}\right.$$

(15)

Then, the erosion amount for the $i\mathrm{th}$ time step is calculated using the following formula:

$$\left\{\begin{array}{l}{\psi }_{\mathrm{e}}^{\left(i\right)}=0, \hspace{1em}\hspace{1em}{\lambda }^{\left(i\right)}=1,\hspace{1em}\hspace{1em}\mathrm{if} {\eta }^{\left(i\right)}\le 0\\ {\psi }_{\mathrm{e}}^{\left(i\right)}={\eta }^{\left(i\right)},\hspace{1em}\hspace{1em}{\lambda }^{\left(i\right)}=0,\hspace{1em}\hspace{1em}\mathrm{if} {\eta }^{\left(i\right)}>0\end{array}\right.$$

(16)

$${\eta }^{\left(i\right)}={\alpha }^{(i)}\left(M^{\left(i-1\right)}+{F_{ACT}}^{\left(i\right)}{n}_p^{\left(i\right)}-{\lambda }^{\left(i-1\right)}{\left(n_p\right)}_{thr}^{\left(i\right)}\right)$$

(17)

where ${\lambda }^{\left(i\right)}$ is the marking variable of erosion occurrence. $M^{\left(i\right)}$ is a marking variable representing the current impact count of each cell on the blade surface, shown as:

$$M^{\left(i\right)}=M^{\left(i-1\right)}+{F_{ACT}}^{\left(i\right)}{n}_p^{\left(i\right)}$$

(18)

2.3. Three-Dimensional Erosion Computational Model

Erosion on wind turbine blades is influenced by several key factors, including the duration of the erosion event, the chord length of the airfoil, the angle of attack, and the wind speed. Erosion tends to increase with longer exposure times and is non-uniformly distributed along the blade length, as illustrated in Figure 2a. In practice, the outer section of the blade experiences greater erosion accumulation due to its larger swept area during rotation and the higher relative wind speeds at this location [29]. For this reason, this study focuses specifically on the erosion effects on the NACA 64 airfoil, which is positioned at the blade tip, where erosion is most pronounced.

On one blade, the duration of an erosion event is constant for each section toward the blade tip, while the different positions of blade with specific attack angle and chord length are influenced by different raindrop impact, as shown in Figure 2b. Consequently, the mass distribution is modeled by the influencing factors of the attack angle, and it defines an increasing mass distribution starting from zero to a value of ${\mu }_E$ at other parts. The formula for calculating the value of ${\mu }_E$ is as follows:

$${\mu }_E={\rho }_{E}k{f}_3(\varphi )$$

(19)

where ${\rho }_E$ is the material density, $k$ is constant parameter, and $f_3$ is the function of attack angle, respectively. The overall erosion amount of different attack angles is calculated by integrating the erosion distribution on the airfoil surface.

Subsequently, the erosion shape model is parametrically constructed. Assume that the erosion occurs from points $A$ to $B$ on the airfoil section, and the midpoint of the straight-line $AB$ is marked as $O$. The radial distance $r_0(\phi )$ (shown in Figure 3a) between points on a clean airfoil and $O$ is defined to represent the clean airfoil shape, where $\phi$ is the angle between $AO$ and $BO$. Then, a new curve $r(\phi )$ (shown in Figure 3b) for an erosion shape, which is denoted by distance from the origin $O$ to a point on the eroded airfoil, can be expressed by:

$$r(\phi )=r_0(\phi )-f{\displaystyle \sum _j\left(\pm A_j\right)}\sin \left(j\phi \right)$$

(20)

$$f~N(\mu ,{\sigma }^2)$$

(21)

where $f$ is a scaling factor used to allocate the necessary erosion mass across the airfoil section. The sinusoid function is applied to define the irregular outline.

For the coefficients of the sinusoid function, an exponential curve defined by $A_j=e^{-s\left(j-1\right)/2}$ [30] is utilized to scale the $j\mathrm{th}$ sinusoid. Consequently, there are three parameters, including the scaling factor $f$, the order $j$ of the sinusoid function, and the value of $s$ in the $A_i$ function, to obtain the erosion shape $r(\phi )$ on an airfoil section. In order to make the computation process easier, the value of $s$ is chosen as 0.5. Furthermore, a cost function, shown in Equation (22), is defined to determine the factor $f$ by minimizing its cost value.

$${\displaystyle \sum _{\phi =0}^{\pi }\frac{1}{2}}\left\{{r_0}^2\left(\phi \right)-r^2\left(\phi \right)\right\}\Delta \phi ={\displaystyle \frac{m_{erosion}}{{\rho }_{erosion}}}$$

(22)

where $m_{erosion}$ is the erosion mass of per unit length of the blade, and ${\rho }_{erosion}$ is the density of the surface coating.

3. Numerical Simulation

To accurately calculate the flow field around the airfoil, the simulation domain for the flow field was constructed first, as shown in Figure 4a. The inlet boundary is a velocity inlet, and the outlet boundary is a pressure outlet, which are, respectively, 4D and 8D (D is the rotor diameter) away from the center of the airfoil to fully eliminate the influence of finite boundaries on the flow field calculation [18]. The outer domain of the flow field is a stationary domain, and the upper and lower boundaries are far-field symmetry, representing the infinity of the flow field space around the blade. The inner domain is a circular rotating domain to simulate the flow field characteristics at different angles of attack under the rotating motion of the blade.

Numerical simulation calculations require the discretization of the flow field calculation domain space. Regular and accurate spatial discretization is a key prerequisite for numerical calculations. Considering that the airfoil has an irregular shape, an unstructured mesh, which is well-suited for the discretization of large deformations and irregular spaces, is applied to divide the flow field calculation domain. In particular, the inner and outer domains have high-density and low-density grids, and the element sizes are set as 5 m and 1 m, respectively. For the airfoil surface, a boundary layer mesh consisting of 20 layers is designed, in which the height of the first layer is 5 mm, and the growth rate is 1.2. Figure 4b presents the grid of the inner and outer domains and the boundary layer grid around the airfoil.

The simulation was conducted using ANSYS Fluent 2021 R2 software, which can model the dispersed phase of particle erosion. Considering that the particles are affected by the mesh division when using face injection, and the number of mesh nodes determines the number of incident flows, which makes it difficult to meet the particle density in real scenarios, group injection was applied in this case. For the raindrop parameter, the characteristics of particle size affect the stress on the impact surface and the threshold of abrasion. Considering the impact time by the number of droplets impinging upon the unit area, the degree of abrasion was analyzed. The impingement rate was assumed to be sufficiently low so that all the effects produced by the impact of one droplet diminish before the impact of the next droplet. In this work, we considered an annual rainfall of 2000 mm and an average droplet size of 1 mm. Particles were released from the center of the particle cloud at a position 2 m from the leading edge of the airfoil along the chord line, with 5000 trajectories to simulate the motion of rain particles with the fluid in the wind field [19]. The flow field around the wind turbine blade is usually infinite, and the particles flow with the fluid without being restricted by the simulation calculation domain. Therefore, the far-field boundary and the outlet boundary are both escape boundaries for the particle phase. Additionally, the airfoil surface is a reflective boundary, indicating that particles hitting the blade surface will bounce off and continue to move under the combined action of the fluid. After the simulation operation, the pressure counter around the airfoil and overall particle trajectories are shown in Figure 5a,b. It can be seen that the raindrop particles along with the airflow come to the leading edge, where a part of the particles impacts the surface and then is reflected into the flow field, while the other part along the surface goes to the trailing edge of the airfoil.

4. Result and Discussion

4.1. Erosion Distribution and Validation

Figure 6 shows the impact count per unit surface over the curvilinear abscissa. The zero mark represents the position of the airfoil leading edge, with the negative direction indicating the pressure side and the positive direction indicating the suction side. It can be observed that the number of impacting particles reaches its maximum at the leading edge and decreases per unit surface area toward the trailing edge, with the area near the trailing edge being almost unaffected by erosion. Additionally, the impact count on the pressure surface is higher than that on the suction side, resulting in more severe erosion, which can be explained by the local angle of attack. Regarding the impact velocity acting on the airfoil surface, the impact velocity is minimal at the leading edge, and the pressure surface exhibits higher impact velocities compared to the suction side.

Figure 7a shows the surface erosion amount per unit over the curvilinear abscissa for a sample operation time of one year. It can be observed that the maximum erosion amount of 0.8 mm occurs near the leading edge of the airfoil. The erosion depth gradually decreases toward the trailing edge, with the erosion on the suction side decreasing more rapidly and disappearing at nearly 2.3% of the chord length from the leading edge, while erosion on the pressure side extends up to 3.1% of the chord length. This can be explained by the fact that the attack angle of the numerically simulated airfoil combined with the asymmetry in the impacting velocity. The inflow first reaches the stagnation point near the leading edge on the pressure side, with part of the flow continuing toward the pressure side and the other part flowing around the leading edge toward the suction side. This results in a reduction in the fluid velocity, thereby weakening the raindrop impacts on the suction side and causing lighter erosion compared to the pressure side. Field observations have shown that erosion damage is typically confined to the first 3% of the airfoil chord (from the leading edge), primarily on the pressure side, with a near-normal distribution centered around the maximum [31]. Studies [2,32] report damage depths ranging from 0.1 to 1.2 mm in turbines after five years of operation. The data in this study agree well with the field observation results. Furthermore, the erosion pattern and magnitude align with earlier numerical results from Castorrini et al. [21], which simulate typical operating conditions for this airfoil section. Based on the computational erosion model, the non-uniform eroded airfoil was constructed, as shown in Figure 7b.

In order to validate the simulation, the calculated lift–drag ratio of the airfoil was compared with the available experimental data, as shown in Figure 8. The results present good overall agreement at smaller angles of attack (0–6 degrees), with a very close match at the maximum lift-to-drag ratio. It is worth noting that as the angle of attack increases further, the numerical simulation values slightly exceed the experimental results. This may be because the turbulence model predicts flow separation later compared to the experiment. It also can be seen that the lift-to-drag ratio decreases when surface erosion occurs, leading to a reduction in aerodynamic performance.

In order to ensure the independence of the results on the time step of the airfoil modification, several cases with different time intervals were tested for a time step sensitivity study. The surface erosion distribution after 5 years of operation was calculated, and a comparison of the results is shown in Figure 9. It can be seen that when the time step is one year or three-quarters of a year, the simulation result of the annual production energy (APE) difference has a significant gap with the reference values (shown as the black columnar). However, when the time step is 1/2 year or 1/4 year, reducing the time step has little effect on the final error. And the results do not significantly change after employing less than 1/2 year. Therefore, this paper selects 1/2 year as the time step for airfoil modification.

4.2. Blade Span Erosion Distribution and Aerodynamic Results

Figure 10 illustrates the distribution of impact counts per unit surface along the curvilinear abscissa for different angles of attack, considering a non-dimensional particle diameter of dp = 50 × 10⁻⁶ at a Reynolds number of Re = 12.5 × 10⁶. As anticipated, the highest impact density occurs near the leading edge at low angles of attack. The impact region moves to the trailing edge on the pressure surface and to the leading edge on the suction side by increasing the angle of attack. This phenomenon can be attributed to the fact that as the angle of attack increases, the area of the blade profile exposed to droplet impacts undergoes a change. Specifically, the exposed region on the suction side shrinks, while that on the pressure side expands. Consequently, the peak in the number of impacts shifts in response to these changes.

Figure 11a shows the erosion mass on the airfoil surface with different attack angles. It can be observed that the most severe erosion occurs close to the leading edge on the pressure side for each angle. Due to the influence of the angle of attack, the inflow first reaches the area near the leading edge on the pressure side, making this location experience the most erosion. And the erosion impact range and depth on the pressure side are greater than on the suction side. Moreover, as the angle of attack increases, the pressure side experiences more severe erosion, with the erosion depth increasing and extending toward the trailing edge, while the erosion on the suction side decreases as the angle of attack increases. This is consistent with the variation in the impact count on the airfoil surface and contrary to the surface force. Based on the erosion amount on different attack angles, the fitting curves for the impact angle function are illustrated in Figure 11b.

The non-uniform leading-edge erosion of the blade was modeled, and the surrounding flow field was analyzed. Figure 12 compares the velocity contours between a normal blade and an eroded blade under different erosion conditions at a wind speed of 11.4 m/s. The results show that the normal blade maintains attached flow, with only slight trailing edge separation. However, in the presence of erosion, substantial flow separation is observed, along with a noticeable forward movement of the separation point near the trailing edge. Furthermore, compared with the uniform eroded blade, the larger-scale separated flow appears at the trailing edge in the non-uniform erosion case. And there are obvious flow vortices present at the small groove of the leading edge, which leads to pressure fluctuations on the blade surface. The phenomena can be further explained by the fact that the irregular erosion shape at the leading edge causes the airflow to not attach to the surface at the middle part of the blade, and then an early separation occurs at the trailing edge.

Figure 13 shows the blade aerodynamic noise spectrum for the combination of thickness and loading noise between uniform and non-uniform erosion conditions at a receiver distance of 90 m. As the results show, the further away from the wind turbine, the lower the sound pressure level at the receiver position is observed. And there is also no obvious change in the distribution of sound energy and the location of tonal peaks. For both leading-edge erosion conditions of the blade, the tonal noise increases compared to that generated by a normal blade (43 dB) under an inflow velocity of U = 11.4 m/s. The variation trends are basically similar at 0–400 Hz for two erosion cases. When the frequency is at 400–1000 Hz, the sound pressure level shows a significant fluctuation in the non-uniform erosion condition, and the value of the sound pressure level is larger than the uniform one.

5. Conclusions

In this study, we investigated the calculation method of leading-edge erosion for three-dimensional wind turbine blades and analyzed the associated aerodynamic noise under non-uniform erosion conditions. Combined with the airfoil modification scheme, the erosion amount distribution was calculated based on the rain erosion model. Then, a three-dimensional leading-edge erosion geometric model was developed, taking into account the influencing factor of the attack angle on the blade section. The results indicate that the maximum erosion occurs near the leading edge, with the affected region extending toward the trailing edge by 34% of the chord length on the pressure side and 26% on the suction side. The impact velocity of raindrops is lowest at the leading edge and increases toward the trailing edge. For erosion mass, its maximum is 0.8 mm near the leading edge. On the pressure side, the erosion diminishes to zero at 3.1% of the chord length from the leading edge, while on the suction side, it decreases to zero at 2.3% of the chord length. The overall erosion profile follows a normal distribution centered at the leading edge. In addition, with an increase in the attack angle, the pressure-side erosion distribution moves to the trailing edge, the suction side moves to the leading edge, and the maximum amount of erosion is 3.0 mm at the blade tip. For non-uniform erosion on the blade, the shedding vortices appear at the leading edge and more separation flow occurs at the trailing edge. There is a larger sound pressure level with great fluctuations in the 400–100 Hz range on this condition.