# Numerical Simulation of Cavitation Erosion Aggressiveness Induced by Unsteady Cloud Cavitation
## 3 Results
### Free Stream Velocity Effects  
Four different free stream water velocities, \(U_{inf}\), have been simulated with the same numerical model corresponding to 15, 20, 25 and 30 m/s. The time step has been set inversely proportional to \(U_{inf}\) in order to keep the Courant number constant between all the simulations. In addition, the outlet pressure has been adjusted correspondingly in order to get always the same cavity length around the 40% of the chord as in the experiment. The model setup main parameters and the shedding frequencies calculated numerically and measured experimentally have been listed in Table 3. It can be seen that the simulations predict with good accurately the shedding frequencies for any \(U_{inf}\) with a maximum percent deviation of 10.3%. Similarly to the results shown in Figure 4b, the time signals of the total vapor volume are repeatable from cycle to cycle for all the cases. Finally, the driving pressure has been based on the averaged instantaneous pressures over ten shedding cycles. To study the relationship between cavitation aggressiveness and flow velocity, the collapse efficiency proposed by [PERSON] et al., [39], \(\eta_{collapse}\), has been used which quantifies the effective energy transfer between the potential power of the vapor volume and the actually erosive power. The value of \(\eta_{collapse}\) is dependent on the initial gas pressure inside the bubble, \(P_{g0}\), and the environment pressure, \(P_{\infty}\), but it is weakly dependent on the initial vapor volume so it can be calculated as:  
\[\eta_{collapse}=0.029\big{(}P_{g0}/P_{\infty}\big{)}^{-0.54} \tag{25}\]  
Equation (25) shows that, for a given \(P_{g0}\), the efficiency is higher the higher is \(P_{\infty}\) which means that for a given cavitation number, the collapses will release more energy for higher \(U_{inf}\). In the present work, \(P_{g0}\) has been taken as 1500 Pa and \(P_{\infty}\) has been calculated with the numerical cavitation number as done in [39]. Therefore, \(\eta_{collapse}\) is a constant coefficient depending on \(U_{inf}\) as shown in Table 4. Then, the actual effective erosion power load, \(P_{off\_load}\), can be calculated as:  
\[P_{eff\_load}=P_{tot\_load}\eta_{collapse} \tag{26}\]  
Since the experimentally obtained Strouhal number at different \(U_{inf}\) is constant with a value around 0.28 (see Table 1), the shedding frequency will increase linearly with \(U_{inf}\) and the collapsing frequency on the hydrofoil will also increase. Therefore, the cavitation erosion aggressiveness per unit time, \(P_{agg}\), can defined as:  
\[P_{agg}=P_{eff\_load}/T=P_{eff\_load}\cdot f \tag{27}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(U_{inf}\) [m/s]} & \multirow{2}{*}{\(P_{tot\_load}\)} & \multirow{2}{*}{\(P_{\infty}\) [Pa]} & \multirow{2}{*}{\(P_{gg}\) [Pa]} & \multirow{2}{*}{\(P_{gg}\) [Pa]} & \multirow{2}{*}{\(P_{\infty}\) [\%]} & \multirow{2}{*}{\(P_{\infty}\) [\%]} & \multirow{2}{*}{\(P_{\infty}\) [\%]} & \multirow{2}{*}{\(P_{\infty}\) [\%]} \\  & & & & & & & & \\ \hline
15 & 5792 & 176,026 & 1500 & 38.0 & 2202 & 106 & 233,385 \\
20 & 13,963 & 311,380 & 1500 & 51.7 & 7222 & 139 & 1,003,915 \\
25 & 27,711 & 485,406 & 1500 & 65.7 & 18,216 & 177 & 3,224,249 \\
30 & 48,033 & 698,105 & 1500 & 80.0 & 38,421 & 216 & 8,298,901 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Numerically calculated indicators of cavitation erosion for different flow velocities. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(U_{inf}\) [m/s] & _Time Step_ [s] & \(\sigma_{exp}\) & \(\sigma_{CD}\) & \(f_{exp}\) [s\({}^{-1}\)] & \(f_{CFD}\) [s\({}^{-1}\)] & \(f_{dev}\) [\%] \\ \hline
15 & 0.00004 & 1.55 & 1.55 & 96.1 & 106 & 10.3 \\
20 & 0.00003 & 1.58 & 1.55 & 132.8 & 139 & 4.7 \\
25 All the calculated values of the various cavitation erosion indicators defined in this section have been listed in Table 4 and plotted in Figure 15, where several power law relations between them and \(U_{inf}\) have been found.