Comparative study of different cavitation erosion assessment methods in cavitating flow around a Delft Twist hydrofoil

Cavitation erosion is a common phenomenon in hydraulic machines, which severely threatens their stable operation. Although several methods for erosion risk assessment have been proposed, few researchers work on the comparative study of different cavitation erosion assessment methods. Therefore, this study utilized Large Eddy Simulation (LES) combined with Zwart-Gerber-Belamri (ZGB) cavitation model to conduct the numerical simulation for the cavitating flow around a Delft Twist hydrofoil. The Erosive Power Method (EPM), Improved Gray Level Method (IGLM) and Energy Conservative Method (ECM) are employed separately to evaluate the cavitation erosion. Among the three results, the EPM results cannot accurately predict erosion in the middle region and trailing edge region, where the U-shaped vortex collapse. The results of IGLM and ECM can predict the erosion in the middle region and erosion caused by the U-shaped vortex at the trailing edge. IGLM still needs artificial thresholds, while ECM emphasizes areas with higher cavitation risk by amplifying parameter n. The ECM method is more objective from the simulation results.


Introduction
Cavitation occurs when the local pressure drops below its vapor pressure [1,2].The attendant negative effects, especially the cavitation erosion, would greatly harm the stable operation of the hydraulic machines [3].Cavitation erosion means that when the cavitation bubbles enter the pressure recovery area, they are squeezed by the surrounding high-amplitude pressure, causing them to collapse violently [4][5][6].This process will generate high frequency and immense impact pressure, thus causing erosion damage on the surrounding solid material surface [7].
Currently, two cavitation mechanisms are widely accepted.Chapman et al. [8] investigated the nonspherical symmetric collapse of cavities due to minor disturbances in the bubble wall.They proposed that the wall leads to an asymmetric collapse of bubbles, which causes a microjet generated at the initial stage of the cavity collapse.Lauterbur et al. [9] recorded the collapse of a cavity close to a solid wall using high-speed photography and confirmed the presence of a microjet during the collapse.Fujikawa et al. [3] found that the pressure pulse of the shock wave could reach 10 3 ~10 4 MPa when measuring the radiation pressure during the collapse of a cavity.Shima et al. [10] carried out a detailed study of the bubble collapse process.They investigated the relationship between the ratio of the distance of the bubble from the wall to the maximum radius of the bubble, L/Rmax, and the mode of the high-pressure pulse for different bubbles.They pointed out that the high-pressure pulse is dominated by an impact wave when L/R max ≤0.3 and ≥1.5, dominated by a microjet when L/R max =0.6~0.8, while dominated by a combination of an impact wave and a microjet when L/Rmax ≥0.8 and R max ≤1.5.Huang et al. [11] simulated the collapse process of a cavity near a solid wall in a compressible liquid and the maximum pressure generated on the side wall is in qualitative agreement with Shima et al. [10].
Many cavitation erosion assessment methods have recently been proposed based on these two dominant mechanisms.However, few studies compare the performance of these methods.This paper compares the predicted cavitation erosion results of the Improved Gray Level Method (IGLM), Erosive Power Method (EPM) and Energy Conservative Method (ECM) on Delft Twist Hydrofoil.The results show that EPM cannot precisely predict the extent of cavitation erosion.Both IGLM and ECM can predict the main areas where cavitation erosion occurs.

Mathematical formulation and turbulence modelling
The numerical simulation in this study uses the large eddy simulation method.The multiphase flow model uses the Homogeneous equilibrium flow model (HEM), which treats the mixture of the vapour and liquid phases as a homogeneous fluid mixture and ignores the relative slip velocity between the vapour and liquid phases, whereby the control equations are as follows: () 0 where t is time, p is pressure, ui represents the velocity in coordinate direction i, and the mixed fluid density  and turbulent viscosity  are expressed as linear mixing relationships, respectively, as follows ) where the subscripts l, v denotes the liquid and vapour phases respectively.
Large Eddy Simulation (LES), the turbulence simulation method of choice, is based on the idea of solving the calculation directly for large-scale eddies after the filtering process and for small-scale eddies using a sub-grid-scale model to obtain the following control equations: () 0 where the quantity with the superscript is the filtered field variable and   is the SGS stresses, which characterizes the effect of the motion of the small-scale vortex on the solved equation of motion with the following expressions: Since   is an unknown quantity, the SGS model on   is introduced to close the system of equations: where   is the turbulent viscosity at the sub-grid scale and   is the strain rate tensor at the large scale.
The WALE model used in this study, which balances simplicity and accuracy, is formulated as follows: where Ls is where the SGS mixing length,  is the von Karman constant, d means the distance from the nearest wall, Cs denotes the WALE constant, and V means the local volume of the cell.

Cavitation model
The cavitation model used in this study is the Zwart-Gerber-Belamri(Z-G-B) cavitation model.This model is simpler than the full cavitation model and can better capture cavitation flow details.Many scholars have confirmed its accuracy, and it has been widely used.The transport equation describing the phase transition process is as follows where ̇+ and m − are the rate of evaporation or condensation of vapor.They are defined as follows where R is the bubble radius, αnuc is nucleation site volume fraction, Fvap is the evaporation coefficient, Fcond is the condensation coefficient.

Boundary condition and meshing
The three-dimensional symmetric delft twist hydrofoil is used in this study [12].The profile form of the twisted hydrofoil is symmetrical NACA0009, the chord length c = 150mm, and the spanwise length s = 300mm.Half of the hydrofoil s / 2 = 150mm is calculated in the numerical simulation.The angle of attack of the hydrofoil changes gradually along the spanwise direction, and there are:  (16) where  ̂ is the dimensionless spanwise length.It is the ratio between the span position and the hydrofoil chord length, with  ̂ ranging from 0 to 1. Set the overall angle of attack of the hydrofoil to -2°, i.e., the angle of attack of the hydrofoil varies from -2° to 9° along the spreading direction.
Due to the symmetry of the hydrofoil, half of the spanwise direction is calculated in the numerical simulation calculation in this paper.Set along the mainstream flow direction for the positive direction of the x-axis, along the spreading direction for the y-axis, vertical hydrofoil surface direction upwards for the z-axis direction.Considering the efficiency and accuracy, the specific calculation domain size and boundary conditions are set as shown in figure 1.The boundary of the inlet is the velocity inlet, with a velocity of 6.97m/s, and the outlet is the pressure outlet, with the outlet pressure determined by the cavitation number = 1.07.The hydrofoil surface is a non-slip wall surface; the top and bottom and side wall surfaces are free-slip walls.
The meshing of the model used in this study and the verification of irrelevance have been done in the previous work.See Ref. [13] for details.

Cavitation erosion assessment method
There are already a variety of cavitation erosion assessment methods, such as the most classical Intensity Function Method (IFM) [14] and Gray Level Method (GLM) [15], and Time-averaged Aggressiveness Indicators (TAIs) [16].Many studies have been conducted to improve these methods in recent years.Lei et al. [17] substituted the averaged pressure for the ambient pressure based on the GLM, and the results agreed well with the experiments.Usta et al. [18] combined IFM and GLM to propose a new cavitation prediction formula with a new threshold determination method.However, there are still few studies comparing different cavitation erosion assessment methods.In this study, the three methods, Gray Level Method (IGLM) [17], Erosive Power Method (EPM) [18] and Energy Conservative Method (ECM) [19], are adopted to assess the cavitation erosion risk.The authors will process the comparative study of the three typical methods in the following discussion.

Assessment of IGLM
The threshold of the IGLM is only material dependent and set as 3×10 7 which has been discussed by Lei et al. [17].The present study also follows the calculation setup, and the simulation results are shown in figure 2. IGLM can simulate the three central regions subject to erosion in the paint test: region 1 in the middle of the hydrofoil, regions 2 and 3 due to the collapse of the U-shaped vortex structure.

Assessment of EPM
Unlike IGLM, EPM does not require manually setting thresholds.It determines the threshold based on the cavitation erosion intensity indicator's time-space-averaged value.Erosion damage only occurs when the instantaneous indicator exceeds this threshold, as shown in figure 3(a).The EPM calculation shows that the cavitation mainly distributes between the hydrofoil's leading edge and middle.The erosion at regions 2 and 3 of the trailing edge due to the U-vortex is almost invisible, which is not ideal as this result differs significantly from the experimental results.Furthermore, to assess the effectiveness of the threshold calculated by the EPM method under the preceding assumptions, this study set a threshold of 0 as a comparison, and the results are shown in figure 3   The ECM method does not need an artificial selection of thresholds and generally amplifies extreme events by setting n.This study selected three values of n = 1, 1.5 and 2 for calculation and comparison.As shown in figure 4, The cavitation erosion indicators predicted by all three values of n mainly concentrate in the middle region of the hydrofoil and in the trailing edge, where the U-shaped vortex collapses.As n increases, the more minor erosion indicators at the leading edge of the hydrofoil are gradually invisible since the erosion indicators in these areas are small relative to those caused by extreme events.ECM applies a weighted average by n to amplify the extreme events, and the public events become less weighted.At n=2, the middle and rear regions of the hydrofoil have higher erosion risk, while the region between the leading edge and middle edge has lower erosion risk, which is also consistent with the results obtained from the paint test.

Results and discussions
The present work conducted a comparative study of different cavitation erosion assessment methods in cavitating flow around a Delft Twist hydrofoil.The main conclusions are as follows: (1) The results of the EPM calculation differ greatly from the experimental results, and the EPM has poor performance in determining the threshold.
(2) The results of the IGLM are consistent with the experimental results.Both the mid-hydrofoil erosion region and the cavitation erosion region related to the U-vortex at the trailing edge can be predicted.The results obtained by ECM also agree with the experimental results.The difference between the ECM and IGLM is mainly in the forward position of the middle of the hydrofoil, where the cavitation erosion intensity indicator predicted by the ECM is smaller than that of the trailing edge, i.e., fewer extreme impact events are forecasted in the middle forward position of the hydrofoil.
(3) Both the IGLM and the ECM have chosen the time-averaged pressure as an approximate proxy for pd.The results show that the driving pressure approximation as the average of local pressure can effectively improve the simulation results.

Figure 1 .
Figure 1.Calculation domain and boundary condition setting

Figure 4 .
Figure 4. EEPM results with different n (a) n=1 (b) n=1.5 (c) n=2 for the twisted hydrofoil and (d) the result from paint test.