*2.3. Boundary Conditions*

The boundary conditions are given by the following equations:

$$\mu\_i|\_{\delta\Omega\_1} = \quad \mathcal{U}\_{\infty}x\_1. \tag{13}$$

$$p|\_{\delta\Omega\_2} = \ p\_{\infty\prime} \tag{14}$$

$$\frac{\partial \mathfrak{u}\_i}{\mathfrak{x}\_i} \mathfrak{n}\_i \vert\_{\delta \Omega\_{3/4,5,6}} = \begin{array}{c} 0, \\ \end{array} \tag{15}$$

$$\left.u\_i\right|\_{\text{wall}ls} = \begin{array}{rcl} 0, & & \end{array} \tag{16}$$

where *δ*Ω<sup>1</sup> is the inlet, *δ*Ω<sup>2</sup> is the outlet and *δ*Ω3,4,5,6 are the four other surfaces (bottom, up, front and back). *walls* represents solid structures, the blade surfaces here. *ni* is the normal vector of the surface on which it is applied. *p*∞ is the undisturbed pressure.

Equation (13) is a velocity inlet condition set to a constant value (*U*∞) in the flow direction. Equation (14) describes the pressure outlet value (usually chosen to avoid to over-constrain the system). A slip velocity condition is considered on the surrounding surfaces to limit the side effects Equations (15) and (16) is a no-slip condition applied on the blades. This condition is only valid in the solid surface datum.

Meanwhile, the initial conditions were set to:

$$
\mu\_i|\_{\Omega} = 0,\tag{17}
$$

$$p|\_{\Omega} = p\_{\infty} \tag{18}$$

where Ω is the computational domain.
