**2.1 Multiphase model**

In this work, the following hypotheses are adopted:


The mass conservation equation of the mixture flow is [17]:

$$\frac{\partial}{\partial t}(\rho\_m) + \nabla. \left(\rho\_m \overrightarrow{U}\_m\right) = \mathbf{0} \tag{3}$$

where *U* ! *<sup>m</sup>* present the velocity of the mixture. The mixture momentum conservation equation is [17]:

$$\frac{\partial}{\partial t} \left( \rho\_m \overrightarrow{U}\_m \right) + \nabla \cdot \left( \rho\_m \overrightarrow{U}\_m \otimes \overrightarrow{U}\_m \right) = -\nabla p + \nabla \cdot \left[ (\mu\_t + \mu\_m) \left( \nabla \overrightarrow{U}\_m + \left( \nabla \overrightarrow{U}\_m \right)^T \right) \right] \tag{4}$$

where *<sup>μ</sup><sup>m</sup>* is the laminar viscosity of the mixture and *<sup>μ</sup><sup>t</sup>* <sup>¼</sup> *<sup>ρ</sup>mC<sup>μ</sup> <sup>k</sup>*<sup>2</sup> *<sup>ε</sup>* is the turbulent viscosity. C<sup>μ</sup> = 0.09, *k* is turbulent kinetic energy, *ε* is the dissipation rate.

It should be noted that the liquid and the vapor have the same velocity *U* ! *<sup>m</sup>*. Since the liquid is incompressible, we obtain *div U*! *m* � � <sup>¼</sup> 0. On the other hand, for the steady flow, the continuity equation becomes *div ρmU* ! *m* � � <sup>¼</sup> 0.

## **2.2 k-ε turbulence model**

Several experimental investigations have shown that turbulence has a significant effect on cavitating flows (e.g. [26]). Also, [27] studied the sensitivity of the cavitating flows to turbulent fluctuations. For the present computations, we use the standard k-ε turbulence [17, 23]:

$$\frac{\partial}{\partial t} \left( \rho\_m k \vec{U}\_m \right) + \nabla \cdot \left( \rho\_m k \vec{U}\_m \right) = \nabla \cdot \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \nabla k \right] + P - \rho \varepsilon \tag{5}$$

$$\frac{\partial}{\partial t} \left( \rho\_m \varepsilon \vec{U}\_m \right) + \nabla \cdot \left( \rho\_m \varepsilon \vec{U}\_m \right) = \nabla \cdot \left[ \left( \mu + \frac{\mu\_t}{\sigma\_\varepsilon} \right) \nabla \varepsilon \right] + \mathbf{C}\_{1\varepsilon} \frac{\varepsilon}{k} P - \mathbf{C}\_{2\varepsilon} \frac{\varepsilon^2}{k} \tag{6}$$

where *P* is the production term of turbulent kinetic energy given by:

$$P = \mu\_t \left[ \left( \nabla U\_m \right) + \left( \nabla U\_m \right)^T \right] \nabla U\_m \tag{7}$$

The standard values of the constants are: σ<sup>k</sup> = 1.0, σε = 1.3, C1<sup>ε</sup> = 1.44 and C2<sup>ε</sup> = 1.92 [17].
