*2.3. CFDs Model*

Since the Navier–Stokes equations cannot be solved analytically, the assessment of the flow pattern was performed by means of ANSYS Fluent software. This software solves the Navier–Stokes Equations (1)–(7) [23] using the finite volume model.

Continuity equation:

$$
\frac{\partial \rho}{\partial t} + \nabla \cdot \left(\rho \ast \overrightarrow{v\_a}\right) = S\_m. \tag{1}
$$

Simplified to the problem:

$$
\frac{
\partial(\rho u)
}{
\partial \mathbf{x}
} + \frac{
\partial(\rho v)
}{
\partial y
} + \frac{
\partial(\rho w)
}{
\partial z
} = 0.
\tag{2}
$$

Momentum conservation equation:

$$\frac{\partial}{\partial t}(\rho \overrightarrow{\boldsymbol{v}\_{a}}) + \nabla \ast \left(\rho \overrightarrow{\boldsymbol{v}\_{a}} \overrightarrow{\boldsymbol{v}\_{a}}\right) = -\nabla P + \nabla \ast \left(\overrightarrow{\boldsymbol{\tau}}\right) + \rho \overrightarrow{\boldsymbol{g}} + \overrightarrow{F}.\tag{3}$$

Simplified to the problem and decomposed in three axes:

$$-\frac{\partial P}{\partial \mathbf{x}} + \frac{\partial (\tau \mathbf{x} \mathbf{x})}{\partial \mathbf{x}} + \frac{\partial (\tau y \mathbf{x})}{\partial y} + \frac{\partial (\tau z \mathbf{x})}{\partial z} = \nabla \ast \left(\rho \overrightarrow{\mathbf{u} \, \mathbf{u}} \, \mathbf{\hat{u}}\right), \tag{4}$$

$$-\frac{\partial P}{\partial y} + \frac{\partial (\tau xy)}{\partial \mathbf{x}} + \frac{\partial (\tau yy)}{\partial y} + \frac{\partial (\tau zy)}{\partial z} = \nabla \ast \left(\rho \stackrel{\rightharpoonup}{\upsilon} \stackrel{\rightharpoonup}{u}\right),\tag{5}$$

$$-\frac{\partial P}{\partial z} + \frac{\partial(\tau xz)}{\partial \mathbf{x}} + \frac{\partial(\tau yz)}{\partial y} + \frac{\partial(\tau zz)}{\partial z} = \nabla \ast \left(\rho \overrightarrow{w'u'}\right). \tag{6}$$

The stress vector was calculated as:

$$
\pi = \mu \Big[ \left( \overrightarrow{\nabla v\_a} + \overrightarrow{\nabla v\_a}^T \right) - \frac{2}{3} \overrightarrow{\nabla \ast \overrightarrow{v\_a} I} \Big]. \tag{7}
$$
