# Numerical Simulation of Cavitation Erosion Aggressiveness Induced by Unsteady Cloud Cavitation
## 2 Experiment Description and Numerical Model
### Cavitation Erosion Model  
The cavitation erosion model used in the calculations was developed by [PERSON] et al. [17,18]. In this model, the pressure waves emitted during the cavitation collapses that reach the solid wall are the main mechanism responsible for erosion damage. The potential energy of a vapor structure in the fluid domain is defined as:  
\[E_{pot}=\Delta pV_{nqp}=(p_{d}-p_{nqp})V_{nqp} \tag{14}\]  
where \(V_{nqp}\) is the volume of the vapor structure and \(p_{d}\) is the driving pressure which forces its collapse. Then an instantaneous potential power, \(P_{pot}\), can be defined with Lagrangian time derivatives as expressed with Equation (15):  
\[P_{pot}=\frac{DE_{pot}}{Dt}=(p_{d}-p_{nqp})\frac{DV_{vap}}{Dt}+\frac{Dp}{Dt}V_ {vap} \tag{15}\]  
Because the vapor volume is related to the vapor volume fraction, \(\alpha_{v}=V_{vap}/V_{cell}\), the potential power in each cell or the potential power density, \(P_{den}\), can be written as:  
\[P_{den}=\frac{P_{pot}}{V_{\text{cell}}}=(p_{d}-p_{vap})\frac{D\alpha_{v}}{Dt} +\frac{Dp}{Dt}\alpha_{v} \tag{16}\]  
[PERSON] et al., [36] found that the second term on the right-hand side of Equation (16) is negligible compared to the first term and they assumed that \(P_{pot}\) is released instantaneously only when condensation takes place. This implies that only the first term on the right-hand side of Equation (16) contributes to the radiated power and only if the time derivative of \(\alpha_{v}\) is positive. Therefore, \(P_{den}\) can be simplified to Equation (17):  
\[P_{den}=\frac{p_{pot}}{V_{cell}}=(p_{d}-p_{vap})\frac{D\alpha_{v}}{Dt} \tag{17}\]  
Note that \(P_{pot}\) is defined with Lagrangian time derivatives, which can be obtained substituting the sum of the first and second terms in the right-hand side of Equation (11) with the following relation:  
\[\frac{D\alpha_{v}}{Dt}=\frac{\partial\alpha_{v}}{\partial t}+\
abla\cdot( \alpha_{v}u)-\alpha_{v}\
abla\cdot u \tag{18}\]  
Besides, the divergence term, \(\
abla\cdot u\), is actually another form of continuity equation (Equation (5)) defined as \(\
abla\cdot u=\dot{m}\Big{(}\frac{1}{p_{v}}-\frac{1}{\rho_{l}}\Big{)}\)[26]. Hence, Equation (18) can be rewritten as:  
\[\frac{D\alpha_{v}}{Dt}=\frac{\dot{m}}{\rho_{v}}-\alpha_{v}\dot{m}\Bigg{(}\frac {1}{\rho_{v}}-\frac{1}{\rho_{l}}\Bigg{)}=\dot{m}\frac{\rho}{\rho_{v}\rho_{l}} \tag{19}\]Assuming that the positive time derivative contributes to erosion and combining Equations (17) and (19), \(P_{den}\) can be finally defined for a given cell as:  
\[P_{den}=\frac{p_{pot}}{V_{cell}}=(p_{d}-p_{up})\text{max}\bigg{(}\dot{m}\frac{ \rho}{p_{up}\rho_{l}},0\bigg{)} \tag{20}\]  
Equation (20) shows that \(P_{den}\) is determined by the driving pressure, \(p_{d}\), which can be defined as the pressure in a cell or the averaged pressure according to the references [18; 23] and [26; 27], respectively, and by the source term, \(\dot{m}\), which depends on the cavitation model being used. Therefore, the definition of \(p_{d}\) and the selection of the cavitation model should have an influence on the predicted values of \(P_{den}\). In order to calculate the potential power reaching a certain position \(j\) on the surface of the hydrofoil that has been induced by a collapse occurring at a given point source \(i\) on the fluid domain at instant \(t\), the method proposed by [PERSON] and [PERSON] [26] has been applied.