Numerical erosion prediction of aluminum induced by cavitating jet using an Eulerian–Lagrangian method

Cavitation-induced erosion in pump machinery is a significant issue that leads to material loss and increased operating costs. This study develops a numerical method based on an energy balance approach to address the risk of cavitation erosion. Compared with other published methods, three improvements are made. Firstly, it assumes that the local instantaneous pressure is the driving force behind cavity collapse, rather than the far-away ambient pressure field. Secondly, it counts erosion only at the moment that the released shock wave is strong enough to cause surface damage. Thirdly, an Eulerian–Lagrangian method is introduced to simulate multiscale cavitation, in which the volume of fluid (VOF) method and a discrete bubble model (DBM) are combined to reproduce resolvable water-vapor interfaces and unresolvable discrete bubbles, respectively. The developed numerical modeling framework is validated by predicting the erosion of pure aluminum surface caused by a cavitating jet, and it is shown that the simulated erosion region fits well against the experimental results. Additionally, appropriate model coefficients of the erosion prediction method are introduced to achieve quantitative prediction of material mass loss, and the detailed erosion behavior is discussed.


Introduction
Cavitation erosion is a type of mechanical damage that occurs in liquid systems, particularly in those with high-speed flows.This phenomenon is caused by the formation and subsequent collapse of vapor bubbles in the fluid.When the bubbles collapse, they create high-pressure shock waves that can damage nearby surfaces, including metal [1,2], plastic [3,4], and even biological tissues [5,6].
Historically, the exploration of cavitation erosion stemmed from experimental approaches, wherein investigators executed controlled cavitation tests on diverse materials to examine erosion mechanisms [7,8] and assess the effectiveness of protective coatings and surface treatments [9][10][11].Nevertheless, these experimental techniques were costly and predominantly conducted during the latter phases of the design process.Consequently, computational approaches that enable the evaluation of cavitation erosion risks have emerged as a compelling substitute, given their capacity to be employed during the initial stages of design.
Due to the advancements in computational fluid dynamics (CFD) technology in recent decades, extensive research has been conducted to investigate cavitation phenomena, aiming to comprehend the mechanisms involved in bubble formation and collapse [12][13][14][15].Furthermore, researchers have sought to comprehend the effects of cavitation on the performance and durability of engineering systems.In this context, numerical simulation serves as a valuable tool for exploring and comprehending the underlying physics associated with cavitation erosion.Significant strides have been made in the development of numerical simulation methods for calculating cavitation erosion, resulting in more precise and efficient predictions of cavitation behavior.Nonetheless, current methods still possess certain limitations.
One of the primary challenges lies in the multiscale nature of cavitation flow, encompassing bubbles that vary in size from the microscopic to the macroscopic scale.The accurate prediction of cavitation behavior necessitates the capture of interactions occurring across these diverse scales.However, achieving this objective requires substantial computational resources and the utilization of advanced modeling techniques.Consequently, further research is imperative to incorporate multiscale features into the analysis of cavitating flow, with the ultimate goal of enhancing the accuracy of numerical simulation methods.
Drawing inspiration from prior research, this study endeavors to advance the field by developing a numerical technique grounded in the energy balance framework to effectively address the peril of cavitation erosion.In order to accurately capture the intricate multiscale attributes inherent in cavitating flows, an Eulerian-Lagrangian method is employed.Furthermore, a formula incorporating the physical properties of the specimen is introduced to furnish a quantitative forecast of cavitation mass loss.The efficacy of the numerical approach is duly ascertained through a meticulous comparison of the results with experimental data, thus substantiating the practicality of the erosion prediction method.Additionally, this investigation introduces appropriate model coefficients for the erosion prediction method and undertakes a comprehensive discourse on the intricate aspects of erosion behavior.

Momentum equation and cavitation model
The Volume of Fluid (VOF) method is employed to characterize a two-phase flow domain as a unified medium exhibiting variable density, and the governing equations are solved in a consolidated manner.The primary equation of concern, denoted as the momentum equation, can be expressed as follows: where subscripts i, j, and k represent the coordinate directions.ρ, u and P are the density, velocity, and pressure, respectively.μm and μt are the dynamic and turbulent viscosities of the mixture.The density and viscosity of the mixed phase are expressed as follows:     where v, l, and m indicate the vapor, liquid, and mixture phases, respectively.As for mass transfer between water and vapor, the Schnerr-Sauer cavitation model is employed and the total mass transfer rate per unit volume in Schnerr-Sauer cavitation model can be expressed as

Discrete bubble model
In the present work, a discrete bubble model (DBM) has been utilized in the present work to account for sub-scale dynamics within a coarse-grid domain.The motion of a single discrete bubble is solved by Newton's second law: where M is the mass, the subscript B represents the discrete bubbles.virtual-mass and pressure gradient forces given by The spherical drag law [16] is applied to simulate the drag coefficient (CD).Additionally, the virtualmass force coefficient (CVM) is set at a constant value of 0.5.The relative Reynolds number (Re) is defined as: The simplified Rayleigh-Plesset equation is conducted to model the bubble growth and collapse:

Cavitation erosion model
The cavitation erosion model used in this paper is based on the EBA [17].The instantaneous potential power inside a vapor structure is given as where es is the instantaneous potential power.In order to address the cavitation erosion risk on the surface, the aggressiveness indicators proposed by Schenke and Terwisga [18] is used.

Mesh independence study
In the present study, three different mesh resolutions, namely coarse, medium, and fine, have been employed to investigate their impact on the results.The mesh configuration used in the simulations is depicted in Figure 1, consisting of a fully structured hexahedral mesh.To accurately capture the nearwall behavior, the cell height near the walls has been set to 0.05 mm, with a growth ratio of 1.1.
The simulations were conducted on workstations equipped with two Intel Xeon Platinum 8259L CPUs.The transient simulations utilized the Large Eddy Simulation (LES) technique, which was integrated with a bounded second-order implicit time formulation.For the transportation equation of the volume fraction, an implicit formulation was adopted, and a compressive scheme was employed for spatial discretization.To ensure numerical stability, the time step size was set to 1 × 10 -6 s, and a residual target of 1 × 10 -4 was used for all variables.Figure 2 depicts the time-averaged skin fraction coefficient observed on the surface of the specimen during 100 shedding cycles.The time-averaged skin fraction coefficient demonstrates an increased value in close proximity to the central area of the disc, where the impinging jet exerts its maximum force.As the cavitating jet moves radially outward from the central region, the friction coefficient gradually decreases, eventually reaching zero at the outer edge of the disc.The estimation of surface friction coefficients using medium and fine meshes reveals a significant similarity in their distribution, while the coarse grids result in a wider range of regions with high surface friction coefficients.

Erosion behavior
Figure 3 presents a comparative analysis between the eroded surface observed in an experimental photograph and the corresponding numerical outcomes.The experimental image highlights two distinct regions of erosion caused by the cavitating jet: an inner circular area and an outer ring-shaped secondary erosion region.An intermediate ring-like region, exhibiting less erosion, exists between the inner and outer erosion regions.This phenomenon can be attributed to the cushion effect, as discussed by Liu and Ma [19].It is imperative to acknowledge that the experimental photograph was obtained subsequent to a 300-second duration of jet propulsion.Performing numerical simulations for such an extended temporal span proves unfeasible.Nevertheless, owing to the consistent cumulative erosive effect exerted on the target surface during each cycle of jet activity, the resultant cavitation morphology retains comparable informative value to the numerically computed cycles.
The macroscale method effectively captures the inner circular erosion region, but fails to accurately represent the outer ring-like secondary erosion area due to the rapid collapse of vapor in that region.Consequently, the erosion near the outer edge of the disc is not adequately depicted.Conversely, the numerical results obtained through the implementation of a multiscale model align well with the experimental photographs, revealing the presence of two separate erosion areas.
Under normal jet impingement, cavitation vapors initially impact the inner ring of the disc, causing primary damage.Subsequently, the cavitation vapors propagate radially and subsequently collapse, resulting in a wider region of erosion along the outer boundary.The continued presence of discrete bubbles in the outermost region of the disc addresses the issue of inadequate anticipation of erosion risk in that specific area, which arises from the rapid collapse of macroscale cavities.
The prediction outcomes of the erosion prediction model based on an Eulerian-Lagrangian approach, as proposed in this study, exhibit a higher level of agreement with the experimental results.This substantiates the superiority of the proposed model in accurately predicting erosion, offering greater proximity to experimental observations.4 depicts the temporal evolution of bubbles and the spatial dispersion of an erosion indicator within a specific localized range over two consecutive time intervals.The visualization showcases a substantial presence of dispersed individual bubbles and large-scale voids, suggesting that a more comprehensive understanding of erosion dynamics can be achieved by monitoring their associated motion.In Figure 4(b), the distribution of the erosion indicator is presented for a particular time interval, revealing the absence of large-scale voids within the corresponding region.As a result, the observed erosion solely originates from the localized collapse of individual bubbles and the resulting changes in local pressure.The phenomenon in question involves the migration of individual bubbles towards a designated surface, followed by their subsequent collapse, which generates a pressure wave that poses a localized erosion risk.In cases where the bubbles fail to make contact with the surface of interest, despite experiencing a compressive force, our claim maintains that the emitted pressure wave is insufficient to induce cavitation erosion.In our study, the total mass loss serves as the baseline for experimental measurements, wherein a quartic polynomial is utilized to establish a fitting model for the experimental data.Moreover, we enforce the condition of zero mass loss at the initial moment (t=0 seconds).This approach enables us to estimate the amount of mass loss at a given time point by leveraging the physically computed time from numerical simulations.By considering the inherent physical properties of the specimen and adhering to the principle of dimensional consistency, we put forward the following formula as a means to predict the mass loss arising from cavitation erosion: where k denotes the model coefficient, while ρs, Ls and ζ represent the density, characteristic length and fatigue strength of the specimen, respectively.Moreover, S represents the surface area of the specimen.The present investigation utilizes a specimen comprising a homogeneous aluminum alloy (A91070) as the primary material under consideration.The density and fatigue strength of this specimen are ascertained to be 2710 kg/m3 and 22 MPa, respectively.The characteristic length utilized in the analysis refers to the diameter of the specimen's surface.By employing the fitting formula depicted in Figure 5, the coefficient k of the model is computed as 9.3777, based on the measured loss in mass.It is important to highlight that the applicability of this coefficient to other scenarios remains uncertain due to its strong dependence on the number of materials and discrete bubbles present.Consequently, further investigation is warranted to establish an erosion prediction formula that can be universally applied.

Conclusions
This study was centered on the anticipation of cavitation-induced erosion and aimed to develop a computational technique based on an energy balance approach to assess the risk of such erosion.The proposed approach incorporated an indicator of aggressiveness that takes into account variations in local instantaneous pressure and vapor volume fraction.To account for the intricate multiscale features observed in cavitating flows, an Eulerian-Lagrangian method was employed.Furthermore, a formula was presented that takes into consideration the physical properties of the specimen in order to provide a quantitative prediction of cavitation mass loss.The validity of the numerical simulations was confirmed through comparisons with experimental results, thus demonstrating the practical applicability of the erosion prediction method.The main conclusions can be drawn as follows: (1) The Eulerian-Lagrangian method exhibits enhanced longevity of discrete bubbles within the computational domain compared to the VOF method, particularly in areas where the macroscale cavity experiences complete collapse.This characteristic enables the Eulerian-Lagrangian method to capture more precise flow characteristics related to cavitation erosion.
where T is the total impact time, and es is given by 0

Figure 1 .
Figure 1.Mesh configuration of computational domain.Figure2depicts the time-averaged skin fraction coefficient observed on the surface of the specimen during 100 shedding cycles.The time-averaged skin fraction coefficient demonstrates an increased value in close proximity to the central area of the disc, where the impinging jet exerts its maximum force.As the cavitating jet moves radially outward from the central region, the friction coefficient gradually decreases, eventually reaching zero at the outer edge of the disc.The estimation of surface friction coefficients using medium and fine meshes reveals a significant similarity in their distribution, while the coarse grids result in a wider range of regions with high surface friction coefficients.

Figure 2 .
Figure 2. The time-averaged skin friction coefficient simulated by (a) Coarse mesh, (b) Medium mesh and (c) Fine mesh.3.2.Erosion behaviorFigure3presents a comparative analysis between the eroded surface observed in an experimental photograph and the corresponding numerical outcomes.The experimental image highlights two distinct regions of erosion caused by the cavitating jet: an inner circular area and an outer ring-shaped secondary erosion region.An intermediate ring-like region, exhibiting less erosion, exists between the inner and outer erosion regions.This phenomenon can be attributed to the cushion effect, as discussed by Liu and Ma[19].It is imperative to acknowledge that the experimental photograph was obtained subsequent to a 300-second duration of jet propulsion.Performing numerical simulations for such an extended temporal span proves unfeasible.Nevertheless, owing to the consistent cumulative erosive effect exerted on the target surface during each cycle of jet activity, the resultant cavitation morphology retains comparable informative value to the numerically computed cycles.The macroscale method effectively captures the inner circular erosion region, but fails to accurately represent the outer ring-like secondary erosion area due to the rapid collapse of vapor in that region.Consequently, the erosion near the outer edge of the disc is not adequately depicted.Conversely, the

Figure 3 .
Figure 3. Erosion on specimen surface by (a) macroscale method and (b) multiscale model compared with (c) experimental result.Figure4depicts the temporal evolution of bubbles and the spatial dispersion of an erosion indicator within a specific localized range over two consecutive time intervals.The visualization showcases a substantial presence of dispersed individual bubbles and large-scale voids, suggesting that a more comprehensive understanding of erosion dynamics can be achieved by monitoring their associated motion.In Figure4(b), the distribution of the erosion indicator is presented for a particular time interval, revealing the absence of large-scale voids within the corresponding region.As a result, the observed erosion solely originates from the localized collapse of individual bubbles and the resulting changes in local pressure.The phenomenon in question involves the migration of individual bubbles towards a designated surface, followed by their subsequent collapse, which generates a pressure wave that poses a localized erosion risk.In cases where the bubbles fail to make contact with the surface of interest, despite experiencing a compressive force, our claim maintains that the emitted pressure wave is insufficient to induce cavitation erosion.

Figure
Figure 3. Erosion on specimen surface by (a) macroscale method and (b) multiscale model compared with (c) experimental result.Figure4depicts the temporal evolution of bubbles and the spatial dispersion of an erosion indicator within a specific localized range over two consecutive time intervals.The visualization showcases a substantial presence of dispersed individual bubbles and large-scale voids, suggesting that a more comprehensive understanding of erosion dynamics can be achieved by monitoring their associated motion.In Figure4(b), the distribution of the erosion indicator is presented for a particular time interval, revealing the absence of large-scale voids within the corresponding region.As a result, the observed erosion solely originates from the localized collapse of individual bubbles and the resulting changes in local pressure.The phenomenon in question involves the migration of individual bubbles towards a designated surface, followed by their subsequent collapse, which generates a pressure wave that poses a localized erosion risk.In cases where the bubbles fail to make contact with the surface of interest, despite experiencing a compressive force, our claim maintains that the emitted pressure wave is insufficient to induce cavitation erosion.

Figure 4 .
Figure 4. Bubble evolution and erosion indicator distribution in two consecutive time steps.In our study, the total mass loss serves as the baseline for experimental measurements, wherein a quartic polynomial is utilized to establish a fitting model for the experimental data.Moreover, we enforce the condition of zero mass loss at the initial moment (t=0 seconds).This approach enables us to estimate the amount of mass loss at a given time point by leveraging the physically computed time from numerical simulations.By considering the inherent physical properties of the specimen and adhering to the principle of dimensional consistency, we put forward the following formula as a means to predict the mass loss arising from cavitation erosion:

Figure 5 .
Figure 5.The fitted curve for experimental mass loss.

( 2 )
The erosion aggressiveness indicator, which is determined by variations in local instantaneous pressure and vapor volume fraction, is capable of accurately depicting the spatial distribution of cavitation susceptibility.Moreover, the indicator exhibits a higher level of sensitivity towards pressure fluctuations compared to alterations in the rate of vapor volume fraction change.(3)The formulation for the quantitative assessment of erosion-induced mass reduction, taking into account the inherent physical characteristics of materials and incorporating a model coefficient, demonstrates practical viability.The proposed model coefficient of 9.3777 is recommended for quantifying the erosion mass loss experienced by aluminum under the influence of a cavitating jet.