# Numerical Simulation of Cavitation Erosion Aggressiveness Induced by Unsteady Cloud Cavitation
## 2 Experiment Description and Numerical Model
### Numerical Model
For the current simulation, the turbulent viscosity, \(\mu_{t}\), in Equation (10) has been calculated with the _SST_\(k\)-\(\omega\) model according our previous investigation [31], and \(\mu_{t}\) has been defined as:  
\[\mu_{t}=\frac{f(\rho)a_{1}k^{2}}{\max(a_{1}\omega,SF_{2})} \tag{12}\]  
where \(a_{1}\) = 0.31, \(F_{2}\) is a blending function which restricts the limiter to the wall boundary layer, \(S\) is an invariant measure of the strain rate and \(k\) and \(\omega\) are the turbulence kinetic energy and frequency, respectively. In addition, \(\dot{f}(\rho)\) is the density correction to reduce the over-predicted turbulent viscosity as proposed by [PERSON] et al. [32]:  
\[f(\rho)=\rho_{v}+(1-\alpha_{v})^{n}(\rho_{l}-\rho_{v}) \tag{13}\]  
where \(n\) is the exponential coefficient which should be specified with a value of 10. Besides, the effects of the cavitation models developed by [PERSON] [33], [PERSON] [34] and [PERSON] [35] on the prediction of erosion were investigated. Table 2 lists the mathematical equations describing each of these cavitation models where \(R_{B}\) in the [PERSON] model is the vapor bubble radius with a constant value of \(10^{-6}\)\(m\). Meanwhile, in the [PERSON] model, it is a variable value that is function of the local vapor volume fraction. \(\alpha_{muc}\) is the nucleation site volume fraction with a default value of 0.0005. Then, \(t_{\infty}\) = \(C\)/\(L_{inf}\) is the mean time scale and \(C_{prod}\) and \(C_{dest}\) are the empirical coefficients for vaporization and condensation, respectively, which were taken as 50 and 0.01 for the [PERSON] model, and 100 and 100 for the [PERSON] model. The solution strategy adopted for the current simulations was based on our previous work [31] where the influence of the numerical setting was investigated in detail. Consequently, a medium-sized mesh of 29,749 elements with \(y+\) values in the boundary layers ranging from 0.1 to 3 with a mean value of 1.2 was used. A time step of 3\(\cdot 10^{-5}\) s corresponding to a Root Mean Square (RMS) Courant number of 1.6 was set as well as several successive iterations within each physical time step. A very small residual criterion of \(10^{-8}\) and a large iterative number were set to march the solution towards convergence in every time step. The pressure-velocity direct coupling method was used to solve the \begin{table}
\begin{tabular}{l c c} \hline \hline
**Model** & \(\dot{m}\) **(P \(<\) P\({}_{\mathbf{v}}\))** & \(\dot{m}\) **(P \(>\) P\({}_{\mathbf{v}}\))** \\ \hline Zwart & \(C_{prod}\frac{3\alpha_{muc}(1-\alpha_{v})\rho_{v}}{R_{B}}\sqrt{\frac{2}{3} \frac{(p_{v}-p)}{\rho_{v}}}\) & \(-C_{dest}\frac{3\alpha_{v}\rho_{v}}{R_{B}}\sqrt{\frac{3}{3}\frac{(p-p_{v})}{ \rho_{v}}}\) \\ [PERSON] & \(\frac{(\rho_{v}-p)}{\rho}\alpha_{v}(1-\alpha_{v})\frac{3}{R_{B}}\sqrt{\frac{3} {3}\frac{(p-p)}{\rho_{v}}}\) & \(-\frac{\rho_{v}-p}{\rho}\alpha_{v}(1-\alpha_{v})\frac{3}{R_{B}}\sqrt{\frac{3} {3}\frac{(p-p_{v})}{\rho_{v}}}\) \\ [PERSON] & \(\frac{C_{prod}\rho_{v}\alpha_{v}(1-\alpha_{v})^{2}}{t_{\infty}}+\frac{C_{prod} \rho_{v}(1-\alpha_{v})\min(0,p-p_{v})}{(0.5\rho_{v}(U_{dest}^{\prime}))t_{ \infty}}\) & \(-\frac{C_{dest}\rho_{v}\alpha_{v}(1-\alpha_{v})^{2}}{t_{\infty}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mathematical expressions of the cavitation models. governing equations. The high-resolution scheme was used for the convection terms. The second-order implicit time scheme was used for the transient term. To accelerate convergence, a series of transient simulations were run from previous steady state models. The simulations were carried out in parallel using twelve cores of an Intel(r)Core(tm) i7-8700 K CPU equipped with 32 GB of RAM. Every unsteady simulation was stopped after at least ten cavity shedding cycles were captured, which took around 12 h for each case. In addition, much effort was taken on building the relationship between the cavity structures and their corresponding erosion intensities during the postprocessing of the obtained results.