**3. Numerical Analysis of the Erosion in Francis Turbines**

#### *3.1. Governing Equations*

#### 3.1.1. Liquid Phase Mathematical Model

Fluids were calculated using a Eulerian approach. The general form of the equations involved in the calculations is presented. The mass continuity equation has the following form:

$$\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u\_i)}{\partial x\_i} = 0 \tag{1}$$

where:


The momentum conservation equation is shown in Equation (2).

$$\frac{\rho \partial (u\_i)}{\partial t} + \frac{\rho \partial (u\_i u\_j)}{\partial x\_j} = -\frac{\partial p}{\partial x\_i} + \frac{\partial}{\partial x\_j} \left[ \mu \left( \frac{\partial u\_i}{\partial x\_j} + \frac{\partial u\_j}{\partial x\_i} \right) \right] + f\_i \tag{2}$$

where:

