Particle erosioninduced damage to structural walls constitutes a critical concern in aerospace and petrochemical engineering. Over time, this phenomenon leads to surface roughening, thereby altering particle impingement dynamics, a facet often overlooked in current simulation methodologies. This study proposes a novel numerical approach for investigating erosion characteristics of flat specimens, integrating a corrective model for roughness angle to enhance accuracy in predicting erosion profiles. Experimental validation confirms the superior performance of the proposed correction method, achieving a remarkable enhancement of approximately 60.27% in accurately predicting maximum erosion depth compared to conventional techniques. Notably, the corrected erosion depth exhibits a nonlinear relationship with erosion duration, closely mirroring experimental observations. In regions characterized by intensified erosion rates, the present method discloses a notable decrease of approximately 1.8° in forecasted particle impact angles with advancing erosion stages, thereby achieving closer adherence to empirical trends. The refined simulation method effectively rectifies historical overestimations, offering a robust framework for future studies in erosion prediction and mitigation strategies.
I. INTRODUCTION
Erosion wear signifies a phenomenon in which sand particles, entrained within a fluid medium, impinge upon the surface of a component, leading to material ejection, surface mass, and thickness diminution.^{1} Erosion wear induced by solid particles represents a critical challenge across various fluidhandling sectors, notably aerospace and petrochemical industries.^{2,3} The degradation precipitated by erosion wear can lead to equipment dysfunction and eventual failure.^{4} Illustratively, it may cause blade fractures as well as compressor aerodynamic efficiency,^{5} malfunctions within transport infrastructure such as pipelines, valves, and pumps, which can pose safety hazards and substantial economic repercussions.^{6} Consequently, precise prognostication of the erosion rate and the locus of eroded surfaces constitutes a pivotal research foundation for enhancing equipment performance or lifespan.
Among the pertinent investigations into the particle erosion characteristics of materials, researchers predominantly employ experimental measurement and numerical simulation. Many have conducted experimental tests on erosion wear to gain direct and intuitive insights. Hamed et al.^{7} presented an experimental research to investigate the erosion characteristics of turbine blade materials under various impact conditions in an erosion wind tunnel. Their measured results indicate that both erosion rate and surface roughness increased with particle impact velocities and impingement angles, and larger particle size results in higher surface roughness. Oka^{8,9} employed experimental methods to explore the erosion behavior of SiC, SiO_{2}, and glass beads on aluminum, copper, carbon steel, and stainless steel and also examined erosion damage across different materials and impact angles. Their conclusion highlights the critical role of the target material's hardness, establishing predictive equations for estimated erosion damage across diverse industrial materials. Additionally, Oka^{10} identified a significant relationship between erosion rate and impact angle and investigated a dominant cutting mechanism at low impingement angles driven by particle sliding on the surface. Liu et al.^{11} experimentally studied the erosion wear of two types of steam turbine blade materials exposed to ferric oxide particles under varying operational conditions. They observed ductile behavior in both materials, with peak erosion rates occurring at approximately 25° and 20° impingement angles. One material exhibits greater sensitivity to impact angle variations compared to another, attributed to the absence of an oxide film layer that enhanced erosion resistance in the latter under elevated temperatures. Jing et al.^{12} proposed an erosion test on a 304SS steel square bend subjected to SiO_{2} sandblasting. Their study reveals that erosion induced by sand particles primarily affects the outer half of the bend, with the ratio of the mass loss between the outer and inner halves increasing with particle size. Furthermore, they found that particles with small mean diameter exert a greater effect on the dimensions of the eroded area but have less influence on wall thickness loss.
Computational fluid dynamics (CFD) has become integral in predicting erosion wear induced by solid particles, offering insights into flow field characteristics and detailed particle behavior beyond the reach of conventional experimental methods.^{13} Numerous numerical studies have endeavored to assess erosion rates across different components. For instance, Felten^{14} simulated solid particle erosion in a series of two 90° pipe elbows, observing that erosion in the first elbow appears independent of spacing and angular orientation relative to the second elbow. Conversely, the erosion dynamics in the second elbow depended on installation parameters, cautioning against a spacing ratio of L/D = 2 to mitigate potential puncture risks. In a similar vein, Peng and Cao^{15} conducted comprehensive numerical predictions on liquid–solid flow in pipe bends, highlighting factors such as pipe diameters, inlet velocities, bending angle, particle mass flow, particle diameter, R/D ratio, and bend orientation as critical in determining erosion rates. Their simulations indicate that for smaller Stokes numbers, maximum erosion typically occurred on the inner walls of the bend, contrasting with larger Stokes numbers where erosion was concentrated on the outermost wall. Aponte et al.^{16} employed CFD analysis to predict jet erosion from stainless steel hard particles impacting surfaces at angles ranging from 15° to 90°. They found that erosion rates are significantly affected by fluid flow, particularly for smaller particles, which more closely follow flow streamlines. Effective impact angles for small particles are lower compared to large particles, influencing the angle at which maximum erosion occurs. Further contributing to the field, Yang et al.^{17} carried out erosion wear tests using volcanic ash particles on TC4 alloy, exploring particle speeds, impact angles, and concentrations based on the E/CRC erosion model. Their findings underscored velocity as a critical factor in erosion rate, with a peak observed at approximately 30° angle of incidence. Notably, feed rate increases correlated with higher removal mass, indicating stable erosion rates under controlled conditions. Beyond these studies, Li et al.^{18} investigated the impact of SiO_{2} particle size on the erosion wear of compressor rotor and stator blades. Their simulations indicate heightened erosion near the blade tips compared to the base, with substantial concentration peaks observed. Specifically, 177 μm particles induced a 91% increase in maximum erosion rate on rotor blades and a 131% increase on stator blades compared to 423 μm particles, highlighting critical areas for protective measures in turboshaft engine design. Han et al.^{19} proposed a numerical research exploring the effect of particle parameters on erosion characteristics in Pelton turbines based on the Euler–Lagrange method; they found that large particles caused a broader area of erosion, while a greater degree of erosion was obtained through small particle size with higher impact velocity. Their investigation showcases a positive correlation that the erosion area of the nozzle tip, nozzle, and bucket increases with increment of the particle concentration. These investigations emphasize the pivotal role of CFDbased methodologies in elucidating erosion mechanisms and guiding protective strategies across diverse engineering applications, thereby advancing the frontier of wearresistant equipment design.
During the operational lifespan of equipment, erosion caused by solid particles within the transporting medium leads to gradual degradation of component surfaces. This degradation is prominently characterized by an increase in surface roughness over eroded areas, a phenomenon that intensifies over prolonged operating periods.^{20} As the surface roughness of the impacted wall accelerates with the severity of erosion, the particle–wall collision dynamics on the roughened surface deviate significantly from those observed on the initially smooth surface.^{21} The parameters governing the impact of particles represent critical factors influencing the distribution of erosion wear, further compounded by alterations in particle impact conditions due to the evolving surface roughness. Solnordal et al.^{22} performed erosion predictions employing the conventional Euler–Lagrange approach, coupled with an assumption of smooth wall for particle–wall collisions. However, it is observed that this method inaccurately predicted the maximum erosion depth, as it failed to account for the appropriate collision dynamics on the roughened surface during the simulation. Regrettably, there remains a dearth of literature discussing the feedback loop between changes in wall roughness, induced by particle erosion, and the resultant alterations in particle impact conditions. Such discussions are vital for understanding erosion wear characteristics comprehensively. With this in mind, it becomes essential to integrate the correction for roughness angles into the simulation procedure. This adjustment is poised to enhance the accuracy of wear distribution prediction on the wall by effectively accounting for the influence of surface roughness on particle impact conditions.
This study establishes a numerical simulationbased framework to elucidate the relationship between erosioninduced surface wear depth and the evolving roughness characteristics of impacted walls, thereby integrating these dynamics into the prediction of erosion wear distribution. The impact of roughened wall surfaces on particle impingement conditions is parameterized through roughness angles. Experimental validation employs measured erosion depths on flat specimens subjected to multiphase alumina sand–water flows to corroborate the accuracy of numerical predictions. Comparative analysis between correction methods and conventional approaches discerns enhancements in erosion depth prediction accuracy, elucidating the mechanisms underlying these improvements.
This investigation claims to propose an innovative numerical approach for predicting erosion wear on flat specimens more accurately, integrating the effect of roughness angle on particle collision dynamics into the modified model, validated through reliable experimental verification of erosion depth. The dynamically varying roughness angle as erosion accumulates leads to a nonlinear link between erosion depth and time. The substantial enhancement in erosion outcomes from the proposed model offers practical insights for future research, making a considerable contribution to aerospace and petrochemical engineering fields. The remainder of the manuscript is organized as follows: Sec. II presents descriptions of the numerical method, governing equations, definitions of variables, and the implementation steps of the correction method. In Sec. III, the geometry, mesh of the computational domain are demonstrated as well as the experimentally validated numerical simulation approach is established. In Sec. IV, the simulation results are analyzed with a focus on the predicted erosion characteristics of the flat specimen obtained from URANS. The difference of erosion wear results between the original method and the correction method is discussed. Conclusions follow in Sec. V.
II. NUMERICAL SIMULATION APPROACH
The erosion stage is delineated into three stages within the framework of CFDbased numerical investigation. This methodology encompasses the resolution of a continuous phase flow field, particle tracking, and erosion wear calculation based on particle impact information integrated with the erosion model. The Euler–Lagrange approach is used to solve the liquid–solid flow, maintaining consistency with Nguyen's experimental setup.^{23} Herein, the liquid is considered a continuous phase and is solved in the Eulerian reference frame, while sand particles are treated as discrete phase entities and tracked in the Lagrange coordinate. The rebound velocity of impacting particles is calculated using normal (e_{n}) and tangent (e_{t}) coefficients of restitution, derived from the rebound model developed by Forder.^{24} To predict erosion patterns on the test flat surface, the Oka erosion model^{8,9} is employed. Notably, the correction method incorporates the modified impact angle of particles due to surface roughness, integrated into both the rebound and erosion models for erosion prediction purposes.
A. Continuous phase model
In the present work, the incompressible nature of water is assumed, and no thermal exchange occurs between the water and sand particles. The governing equations for the continuous phase adhere to the ReynoldsAveraged Navier–Stokes equations, expressed as follows:
B. Discrete phase model
C. Roughness angle correction
The theoretical model proposed by Sommerfeld^{21} incorporates the shadow effect, which means that particles cannot collide with the lee side or shadow zone of the wall surface when particle impact angles are smaller than the absolute value of the negative roughness angle γ−, as depicted in Fig. 2. Consequently, a modification to α′ in the model is implemented accordingly, i.e., for the case corresponds to region 1 with γ−, the value of γ becomes 0 when $\alpha +\gamma \u22640$, which means α′ equals α in this case. The value of γ can be acquired by Eq. (9) corresponding to the situation of region 2 and 3 as $\alpha +\gamma >0$. Furthermore, Fig. 3 illustrates the schematic diagram of the roughness angle, specifically highlighting scenarios where γ assumes a negative value.
D. Particle–wall rebound model and erosion model
Coefficient .  s_{1} .  s_{2} .  q_{1} .  q_{2} .  K .  k_{1} .  k_{2} .  k_{3} . 

Value  0.71  2.4  0.14  −0.94  65  −0.12  2.3(Hv)^{0.038}  0.19 
Coefficient .  s_{1} .  s_{2} .  q_{1} .  q_{2} .  K .  k_{1} .  k_{2} .  k_{3} . 

Value  0.71  2.4  0.14  −0.94  65  −0.12  2.3(Hv)^{0.038}  0.19 
During the simulation of particle erosion damage on the flat specimen, the erosion rate is iteratively updated at each time step, yielding the erosion depth at the respective grid element. Subsequently, the wall roughness angle is calculated according to the erosion depth, thereby determining a revised impact angle that informs the erosion rate for the subsequent time step within the erosion model. These procedural steps are implemented in the Fluent solver utilizing userdefined functions, as depicted in Fig. 4.
III. VALIDATION OF NUMERICAL SIMULATION APPROACH
A. Geometry model and mesh generation
In this study, stainless steel SUS304 is selected for investigation using experimental data obtained from a water–sand erosion test rig. Details on the configuration of the wet erosion testing apparatus and experimental specifics can be found in Ref. 23. The erosion depth measurements on a representative eroded surface over various test durations are employed to validate the reliability of the established numerical simulation methodology. The experimental setup primarily comprises a stainless steel nozzle with a diameter of d_{n} = 0.0064 m, a cylindrical tank, and flat test specimens. Accordingly, a computational domain corresponding to the numerical simulation is delineated, preserving key geometric dimensions of the experimental setup. Figure 5 illustrates the computational domain encompassing the primary geometries, component dimensions, and pertinent boundary conditions.
To ensure the independence of flow field simulation results from mesh density, six sets of unstructured meshes are generated for the computational domain as detailed in Table II. Mesh refinement primarily focused on two key regions: the boundary layer adjacent to the planar surface (zone 1) and the region between the nozzle and the planar surface (zone 2), illustrated in Fig. 6. In zone 1, a boundary mesh consisting of 10 layers is generated with a growth ratio of 1.1, ensuring a wall Y^{+} value below 1 for all meshes except for mesh scheme 6 with first layer thickness of 300 μm. A finer grid resolution of 0.0003 m is applied to the upper surface of the flat specimen due to its susceptibility to erosive effects. Zone 2 underwent localized mesh refinement to capture the main flow region impacted by sedimentladen flow on the flat specimen. These strategies are employed to optimize mesh quality and resolution in critical areas of the computational domain, essential for robust and reliable numerical simulations of particle erosion dynamics.
Step .  Mesh scheme .  Zone 1—First layer thickness (μm) .  Zone 2—Refinement body size (mm) .  Total elements (× 10^{6}) . 

1  1  150  0.5  2.61 
2  150  0.4  3.43  
3  150  0.3  5.12  
2  4  90  0.4  3.58 
5  120  0.4  3.49  
2  150  0.4  3.43  
6  300  0.4  3.26 
Step .  Mesh scheme .  Zone 1—First layer thickness (μm) .  Zone 2—Refinement body size (mm) .  Total elements (× 10^{6}) . 

1  1  150  0.5  2.61 
2  150  0.4  3.43  
3  150  0.3  5.12  
2  4  90  0.4  3.58 
5  120  0.4  3.49  
2  150  0.4  3.43  
6  300  0.4  3.26 
The velocity profiles along two selected lines in zones 1 and 2 are analyzed to assess the mesh independence of the computational simulations. These lines, designated as line 1 (0.1 mm offset from the flat surface) and line 2 (5 mm offset), are chosen for evaluation. The Reynolds number for the flow is 1.9 × 10^{5}, and the nondimensional velocity u_{n} is defined as $un=u/uin$, where u represents the local velocity in the flow field and u_{in} is the inlet velocity of the nozzle, which is set as 30 m/s. Similarly, the nondimensional radius R_{n} relative to the center of the flat surface is computed using $Rn=R/dn$, with R denoting the radius at a specific location on the plate's wall surface. Figure 7(a) illustrates the nondimensional velocity distribution on line 2 using mesh schemes from step 1. It demonstrates that mesh 2 and mesh 3 exhibit highly consistent velocity profiles, suggesting that the grid size refinement in mesh 2 (0.4 mm) adequately captures the flow dynamics in zone 2. Conversely, Fig. 7(b) displays velocity profiles from step 2 meshes on line 1, revealing that three of the mesh sets exhibit consistent velocity distribution trends, except for scheme 6. Consequently, the boundary layer region with a first layer thickness of 150 μm is chosen for mesh refinement in zone 1. Thus, based on the mesh independence analysis of the computational domain, mesh scheme 2 is deemed optimal for subsequent numerical predictions. These findings underscore the critical role of mesh refinement strategies in ensuring accurate simulations of sedimentladen flow dynamics impacting flat specimens.
B. Numerical simulation of erosion wear
Utilizing the established methodology, a transient numerical investigation into erosion wear caused by sand particles on a flat specimen is conducted. The injection velocity of particle at the inlet of nozzle is consistent with the fluid velocity u_{in}, and the Stokes number of particles is 3.56. Pressure outlet conditions are applied to the upper and circumferential walls of the cylindrical tank, while a noslip velocity condition is enforced on the walls of the nozzle, the bottom of the tank, and the specimen's surface. At the nozzle inlet, particle velocities initially matched the fluid velocity, with escape and reflection conditions employed at the outlet and other walls of the computational domain, respectively. Detailed properties of the particles and the flat specimen, as well as the mathematical boundary conditions are presented in Tables III and IV, respectively.
.  Parameters .  Value . 

Particles (sand)  Density (kg/m^{3})  2400 
Mass flow (kg/s)  0.011 58  
Mean diameter (μm)  150  
Shape coefficient  0.58  
Flat specimen (SUS304)  Density (kg/m^{3})  7929 
Vickers hardness (Hv)  196 
.  Parameters .  Value . 

Particles (sand)  Density (kg/m^{3})  2400 
Mass flow (kg/s)  0.011 58  
Mean diameter (μm)  150  
Shape coefficient  0.58  
Flat specimen (SUS304)  Density (kg/m^{3})  7929 
Vickers hardness (Hv)  196 
Fluid medium .  Specific wall .  Boundary condition . 

Water  Inlet of the nozzle  Velocity inlet (30 m/s) 
Upper wall of the tank  Pressure outlet (101 325 pa)  
Circumferential wall of the tank  Pressure outlet (101 325 pa)  
Particle  Inlet of the nozzle  Inject with velocity (30 m/s) 
Outlet of the computational domain  Escape  
Other walls of the computational domain  Rebound (Forder's model) 
Fluid medium .  Specific wall .  Boundary condition . 

Water  Inlet of the nozzle  Velocity inlet (30 m/s) 
Upper wall of the tank  Pressure outlet (101 325 pa)  
Circumferential wall of the tank  Pressure outlet (101 325 pa)  
Particle  Inlet of the nozzle  Inject with velocity (30 m/s) 
Outlet of the computational domain  Escape  
Other walls of the computational domain  Rebound (Forder's model) 
The simulation employed the SIMPLEC algorithm to resolve the pressure and velocity fields, ensuring accurate modeling of the multiphase water–sand flow dynamics. Initial values for the transient simulation are derived from steadystate results of the continuous phase, adhering to a convergence criterion of 10^{−6}. Particle dynamics are synchronized with a nondimensional fluid flow time step $dt*=\Delta tfuin/dn=4.69\xd710\u22122$ ensuring a Courant number below 1, where Δt_{f} represents a specified fluid time step of 10^{−5} s. The discrete phase model parameters are updated every 20 iterations of the continuous phase. Detailed records of particle impingement events are meticulously gathered and archived using userdefined memory allocations.
It can be observed that the predicted erosion depth through the original Oka model is closest to the experimental results. In contrast, the other three models deviate significantly from the experimental outcomes, with deviations in maximum erosion depth of 139.5%, 225.2%, and 272.5%, respectively. This indicates that correcting the Oka model consistently provides superior accuracy compared to the other three models. Nguyen et al.^{37} also found that the predicted erosion results of flat sample using the Oka model more closely align with the experimental data as particle size varies. Pereira et al.^{38} and Liu et al.^{39} compared simulated erosion outcomes for the elbow under different erosion models, they concluded that the Oka model provides better agreement with the experimental results among several erosion models, as it is a complex erosion model that considers multiple factors compared to other models. Therefore, modifying the original Oka model is the most effective strategy to enhance precision in wear prediction over the original approaches.
It is evident from Fig. 9 that the predicted erosion depth closely approximated the experimental measurement, surpassing the results obtained from the other four configurations, notably when the acceleration coefficient AC is set to 2500. The errors ξ across the five AC schemes are summarized in Table V, with a minimal discrepancy of 5.8% observed at AC = 2500, the most favorable outcome among the tested configurations. Consequently, AC = 2500 is selected for subsequent phases of the numerical investigation.
$y\u2009exp\u2009*$(t1) .  $ycfd*$( $t1*$) .  ξ (%) . 

−18.09  −11.57  36.04 
−15.13  16.35  
−18.62  2.97  
−22.00  21.61  
−25.20  39.33 
$y\u2009exp\u2009*$(t1) .  $ycfd*$( $t1*$) .  ξ (%) . 

−18.09  −11.57  36.04 
−15.13  16.35  
−18.62  2.97  
−22.00  21.61  
−25.20  39.33 
The AC is considered in the calculation of erosion depth while carrying out the numerical erosion prediction on the flat with the original method and correction method. Figure 10 presents the erosion depth formed on the flat specimen at nondimensional numerical computational times t_{1}^{*}, t_{2}^{*}, and t_{3}^{*} of 4.69 × 10^{2}, 9.38 × 10^{2}, and 1.41 × 10^{3}, respectively, as well as its comparison with the experimental erosion depth corresponding to physical erosion time t_{1}, t_{2}, and t_{3}, which are 300, 600, and 900 s, respectively. It can be observed that the region of nondimensional maximum erosion depth, i.e., y^{*}_{max}, is unchanged, which is approximately within the R_{n} of 0.63–0.78 area and it remains the consistent with the position of maximum erosion depth measured from experiment. Compared with the results of the original method, the overall distribution of erosion depth predicted by the correction method is closer to the experimental results, which indicates that the prediction accuracy of the erosion wear on the flat specimen is improved while considering the correction of roughness angle formed with erosion accumulation.
The comparison of y^{*}_{max} predictions between the original Oka method and the corrected approach against experimental data at various erosion stages is depicted in Fig. 11. It illustrates that y^{*}_{max} calculated by the original method shows a linear correlation with erosion duration, whereas the corrected method exhibits a nonlinear trend, consistent with experimental y^{*}_{max} variations. As erosion progresses, y^{*}_{max} predicted by the original method increasingly diverges from experimental values, diminishing erosion depth prediction accuracy. Notably, the original method fails to establish a link between evolving roughness angles due to erosion and subsequent changes in particle impact dynamics over time, assuming static particle angles over erosion cycles, thus deviating from actual erosion dynamics. In contrast, the corrected method dynamically adjusts particle impact angles with erosion evolution, capturing nonlinear erosion depth changes. Evaluating y^{*}_{max} predictions against experimental benchmarks shows the corrected method enhances prediction accuracy by approximately 60.27% on average compared to the original method, significantly benefiting the prediction of wall erosion distribution in practical engineering contexts.
IV. NUMERICAL RESULTS AND DISCUSSION
A. Flow behavior
B. Erosion behavior
To investigate the erosion mechanisms affecting the flat specimen, two erosion prediction methods are employed to analyze numerical erosion outcomes over varying durations. Figure 14 illustrates the distribution of erosion rates and depths across the flat surface, as depicted in Fig. 15, using different prediction methods and erosion durations. Notably, a centrally located circular area exhibits minimal erosion rates, surrounded by an outer ring region characterized by significantly higher erosion rates across all erosion time intervals. Throughout the progression of erosion wear, the centrally symmetric erosion distribution pattern persists, while the depth of erosion in the ring region intensifies with prolonged erosion durations, thereby expanding the extent of erosion toward the periphery of the flat surface. Comparative analysis reveals that erosion rates and depths derived from the corrected method consistently register lower values compared to the original method at each erosion time point. This disparity amplifies with prolonged erosion durations, underscoring distinct differences between the two methodologies.
To elucidate the disparity in erosion results between the two erosion prediction methods, a series of 24 concentric rings, each 0.5 mm wide, are delineated on the flat specimen's surface within the radial span of 0–12 mm, as depicted in Fig. 16. For clarity, Fig. 16 displays twelve selected rings, highlighting their respective erosion depths. Observably, the fluid flow field proximate to the flat surface significantly influences the erosion distribution atop the specimen. Both the magnitude of fluid velocity and the erosion depth exhibit symmetric characteristics at the center of the flat surface. Erosion depth is minimal within the stagnation region at the flat center, contrasting sharply with the more severe erosion damage evident in the adjacent outer zones. Despite particle trajectories aligning predominantly with the fluid flow path at a 90° angle from the nozzle outlet, actual particle impact angles deviate due to altered water flow direction upon contact with the flat surface. Consequently, particle impingement parameters vary across different positions on the flat surface, influencing erosion dynamics accordingly.
Upon exiting the nozzle, particle velocities initially mirror those of the fluid carrier, gradually diminishing within the stagnation region as particle trajectories diverge outward. Upon encountering the flat surface, particle paths transition from vertical to radial orientations, resulting in higher impact angles conducive to the establishment of regions exhibiting minimal erosion rates. Meanwhile, fluid momentum intensifies radially along the stagnation zone peripheries, thereby augmenting particle velocities in the radial direction. Integration of observations from Figs. 10 and 16 suggests the formation of a Vshaped region characterized by heightened erosion rates within the flat specimen's highvelocity domain. Notably, particle impact angles diminish within this region due to fluid flow curvature and particle tendency to conform to streamline paths.
The impact angles and velocities of particles on the concentric rings are analyzed to investigate particle erosion dynamics on the flat specimen's surface. Figure 17 illustrates particle impingement data across the rings derived from two erosion prediction methods, with impact metrics aggregated using areaweighted averaging techniques.
Figure 17 highlights particle velocities ranging from 20 to 27 m/s within rings 8–10, accompanied by a notable decline in particle impingement angles, averaging approximately 5°–10° following correction. Experimental investigations into the surface topography of the flat specimen subjected to particle erosion reveal prominent cutting and plowing features in regions exhibiting high erosion rates, with cutting predominating as the primary erosion mechanism. Despite particle velocities reaching 25 m/s in the outer flat region, particle impingement angles remain exceptionally small, approximately 1°, underscoring erosion wear as a combined function of particle impact velocity and angle. Figure 17(b) demonstrates a decreasing trend in particle impact angles over erosion duration, particularly pronounced in rings 8–10, where angles reduced by up to 1.8°, reflecting the evolving characteristics of particle impingement with increasing erosion severity. The schematic diagram of fluid streamlines and particle trajectories within the semilateral region of the main computational domain are illustrated in Fig. 18. From the figure, it is evident that there are deviations between them, and particle impact angles varies across different regions of the plate.
Figure 19 presents a schematic depiction of the cutting mechanism when particles impact the flat specimen surface at low impingement angles. Erosion wear scars initially form due to particles with relatively shallow impact angles, primarily driven by cutting and plowing effects. Subsequent impacts in these eroded regions occur at reduced collision angles, following Sommerfeld's theory, which calculates the actual impact angle α′ considering the surface's roughness angle γ. Figure 20 illustrates the impact angle function curve, where the reduction in actual impact angle due to wear scar formation at lower impingement angles correlates with decreased erosion rates in these regions. As the particles' impact angles fall within the ascending range of the f(α) curve (i.e., α < 36.34°), the decrease in actual impact angle diminishes the function's value f(α). The correction method incorporates adjustments for roughness angle variations during erosion, thereby refining the particle impact angle function in the erosion model, as depicted in Fig. 20. Contrastingly, the original method neglects roughness angle adjustments, leading to overestimated predictions of erosion depth in highrate erosion areas.
In the experimental test platform, particles with mean size of 150 μm are suspended in water, whereas numerical simulation assumes uniformly sized particles of 150 μm to predict the representative erosion characteristics. The actual impact angle of particles varies from approximately 1° to 55°, thereby encompassing a broad range of particle impingement angles. This study validates proposed modification method through established flat erosion experiment, confirming the feasibility of incorporating roughness angle correction into conventional wear prediction methodologies. Nevertheless, despite the proposed modified model aims to enhance the alignment of simulated erosion outcomes with the results in practical erosion environment, discrepancies between them may present due to the correlation between the roughness angle and meshes, as well as boundary condition settings within the computational models. Furthermore, the proposed erosion model developed under ambient temperature condition and plastic materials is not recommended for brittle materials. Future efforts will focus on expanding the correction method to complex geometries, complemented by corresponding experimental verification.
V. CONCLUSION REMARKS
In this study, the roughness angle generated by the erosion of sand particleimpacted wall is utilized to refine the particle impact angle within the erosion approach. This correction is integrated into the Ansys Fluent solver using a userdefined function. Validation of the corrected erosion method is achieved through comparison with experimentally measured erosion depths of flat specimens. Numerical simulations encompassing erosion rates, erosion depths, and particle impact parameters are comprehensively analyzed. The main findings of this investigation are summarized as follows:

The prediction of the maximum erosion depth y^{*}_{max} region, achieved by both the original and correction methods, approximates within the radial distance Rn ranging from 0.63 to 0.78 relative to the flat specimen's center. From a crosssectional viewpoint, a distinctive Vshaped erosion profile is discerned in the semilateral domain of this region.

Deviations between y^{*}_{max} obtained via the original method and experimental findings escalated over the erosion duration. Conversely, predictions using the correction method closely align with experimental measurements.

The trajectory of y^{*}_{max} derived from the correction method exhibits nonlinearity, consistent with experimental trends. The correction method enhances prediction accuracy by an average margin of approximately 60.27%, pivotal for anticipatory models in diverse engineering scenarios.

In regions characterized by heightened erosion rates and annular geometries, erosion depths y^{*} derived from the correction method consistently undershoot those from the original method at successive erosion intervals, with disparities amplifying over time.

Predominantly manifesting in lower impingement angles within the ring region 8–10, the cutting mechanism demonstrates dominance. Concurrently, the correction method indicates a diminishing trend in actual particle impact angles with escalating erosion wear, thereby mitigating overestimations in predicted erosion depths within this area.
ACKNOWLEDGMENTS
This research is funded by the Basic Product Innovation Research Program (Grant No. 20221001).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Xing Li: Conceptualization (equal); Methodology (equal); Writing – original draft (equal). Ning Huang: Supervision (equal). Kan He: Writing – review & editing (equal). Ding Tong: Funding acquisition (equal). Yanli Zhang: Project administration (equal). Jie Zhang: Formal analysis (equal); Investigation (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Nomenclature
 AC

Acceleration coefficient
 C_{p}

Coefficient of pressure
 d_{n}

Diameter of the nozzle
 d_{t}^{*}

Nondimensional fluid flow time step
 L

Sampling length
 R

Local radius on the plate's wall
 R_{a}

Mean roughness of the wall
 R_{n}

Nondimensional radius
 R_{q}

Standard deviation of the roughness structure
 RS_{m}

Mean profile element width
 t_{1}

Physical erosion time
 t_{1}^{*}

Nondimensional computational duration time
 u_{in}

Local velocity of fluid
 u_{n}

Nondimensional velocity of fluid
 y^{*}

Nondimensional erosion depth
 y^{*}_{max}

Nondimensional maximum erosion depth
 α

Particle trajectory angle
 α′

Actual impact angle
 γ

Roughness angle
 Δt_{f}

Specified fluid time step
 ξ

Error of the maximum erosion depths
Nomenclature
 AC

Acceleration coefficient
 C_{p}

Coefficient of pressure
 d_{n}

Diameter of the nozzle
 d_{t}^{*}

Nondimensional fluid flow time step
 L

Sampling length
 R

Local radius on the plate's wall
 R_{a}

Mean roughness of the wall
 R_{n}

Nondimensional radius
 R_{q}

Standard deviation of the roughness structure
 RS_{m}

Mean profile element width
 t_{1}

Physical erosion time
 t_{1}^{*}

Nondimensional computational duration time
 u_{in}

Local velocity of fluid
 u_{n}

Nondimensional velocity of fluid
 y^{*}

Nondimensional erosion depth
 y^{*}_{max}

Nondimensional maximum erosion depth
 α

Particle trajectory angle
 α′

Actual impact angle
 γ

Roughness angle
 Δt_{f}

Specified fluid time step
 ξ

Error of the maximum erosion depths