# Numerical Simulation of Cavitation Erosion Aggressiveness Induced by Unsteady Cloud Cavitation
## 2 Experiment Description and Numerical Model
### Numerical Model  
The homogeneous mixture assumption has been used to model the two-phase flow. Thus, velocity and pressures are shared by water and vapor phases and the governing Navier-Stokes equations for the mixture are:  
\[\frac{\partial\rho_{m}}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho_{m}u _{i})=0 \tag{5}\]  
\[\frac{\partial}{\partial t}(\rho_{m}u_{i})+\frac{\partial}{\partial x_{j}}( \rho_{m}u_{i}u_{j})=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\!\left[\mu_{m}\!\left(\frac{\partial u_{i}}{\partial x_{j}}+ \frac{\partial u_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_ {k}}{\partial x_{k}}\right)\right] \tag{6}\]  
where \(u\) is the velocity, \(p\) is the pressure, \(t\) is the time, \(\mu_{m}\) is the mixture dynamic viscosity defined by \(\mu_{m}=\mu_{v}\alpha_{v}+\mu_{l}(I-\alpha_{v})\) and \(\rho_{m}\) is mixture density calculated by \(\rho_{m}=\rho_{v}\alpha_{v}+\rho_{l}(I-\alpha_{v})\). In the present simulation, the Reynolds Averaged Navier-Stokes (RANS) approach was used, in which the instantaneous quantities are decomposed into the mean and the fluctuating components, i.e., the instantaneous velocity \(u_{i}\) is equal to:  
\[u_{i}=\overline{u_{i}}+u_{i}^{\prime} \tag{7}\]  
Using Equation (7) to replace the instantaneous quantities of all the variables in Equations (5) and (6), the RANS equations are obtained:  
\[\frac{\partial\overline{\rho_{m}}}{\partial t}+\frac{\partial}{\partial x_{i }}(\overline{\rho_{m}u_{i}})=0 \tag{8}\]  
\[\frac{\partial}{\partial t}(\overline{\rho_{m}}\,\overline{u_{i}})+\frac{ \partial}{\partial x_{j}}(\overline{\rho_{m}}\,\overline{u_{i}}\,\overline{u _{j}})=-\frac{\partial\overline{p}}{\partial x_{i}}+\frac{\partial}{\partial x _{j}}\!\left[\mu_{m}\!\left(\frac{\partial\overline{u_{i}}}{\partial x_{j}}+ \frac{\partial\overline{u_{j}}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{ \partial\overline{u_{k}}}{\partial x_{k}}\right)\right]+\frac{\partial}{ \partial x_{j}}\!\left(-\overline{\rho_{m}}\overline{u_{i}^{\prime}u_{j}}^{ \prime}\right) \tag{9}\]  
Equations (8) and (9) have the same general form as the instantaneous Navier-Stokes equations, with the velocities and other solution variables now representing time-averaged values, and a series of new additional terms such as, \(\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\), which are known as the Reynolds stresses induced by the turbulence. In order to close this system of equations, there are two approaches: the first one is based on the eddy-viscosity hypothesis, which relates the Reynolds stresses to the mean flow; the second one  
Figure 2: Experimentally measured mean volume deformation rate (black left axis) and mean pitting rate (blue right axis). is based on solving the transport equation for all the components of the Reynolds stress tensor. In our case, the former method has been used, so a turbulent eddy viscosity, \(\mu_{t}\), has been introduced to model the effect of the Reynolds stresses, and the final momentum equation to be solved becomes:  
\[\frac{\partial}{\partial t}(\overline{\rho_{m}}\,\overline{u_{i}})+\frac{ \partial}{\partial x_{j}}(\overline{\rho_{m}}\,\overline{u_{i}}\,\overline{u_ {j}})=-\frac{\partial\overline{p}}{\partial x_{i}}+\frac{\partial}{\partial x _{j}}\bigg{[}(\mu_{m}+\mu_{t})\bigg{(}\frac{\partial\overline{u_{i}}}{ \partial x_{j}}+\frac{\partial\overline{u_{j}}}{\partial x_{i}}-\frac{2}{3} \delta_{ij}\frac{\partial\overline{u_{k}}}{\partial x_{k}}\bigg{)}\bigg{]} \tag{10}\]  
In addition, a vapor volume fraction transport equation (Equation (11)) is used to consider the mass transfer between vapor and water:  
\[\frac{\partial\alpha_{v}}{\partial t}+\
abla\cdot(\alpha_{v}u)=\frac{\dot{m} }{\rho_{v}} \tag{11}\]  
where the term \(\dot{m}\) represents the mass transfer rate between the two phases.