**3.4. Turbulent flows CFD**

combustion problems, or any problem involving not only fluid movements, but also a more

**Figure 5.** Overall stages in the SIMPLE algorithm, adapted with permission from Versteeg and Malalesekera [3].

The pressure implicit with splitting of operators (PISO) algorithm is an improved version of the SIMPLE algorithm. It was created for non-iterative calculations in the numerical solution of non-steady flows. It has been adopted for iterative solution process for steady and nonsteady flows, being especially useful for the last ones. This algorithm involves a stage where velocities and pressures are arbitrarily predicted, and two correcting stages. Therefore, it has an additional correcting stage with respect to SIMPLE and SIMPLEC. In the prediction and correction stages, the obtained pressures involve functions that contain within the continui‐

Even though the PISO algorithm requires additional memory to store the values of the added correction equations, and in general, it requires relaxation factors to stabilize the calculation

complex physical problem.

330 Numerical Simulation - From Brain Imaging to Turbulent Flows

ty equation [15].

process, it performs in a quick and efficient way.

Reynolds number (*Re* = *ρ Uℓ*/*μ*) is above a certain limit several events take place and cause that the flow behaves in a random manner, and its velocity components fluctuate along the three spatial directions. The flow will also present an unstable nature, promoting large-scale mixing and energy dissipation at small scales. Such regime is known as turbulent regime or turbulence. To include the effect of such fluctuations in the flow, it is necessary to substitute the flow variables that depend on the velocity vector *V* <sup>→</sup> <sup>=</sup> *<sup>f</sup>* (*u*, *<sup>v</sup>*, *<sup>w</sup>*), as well as the scalar *P* for the sum of an average component *i* ¯ and is fluctuating counterpart "*i*′" having then:

$$
\vec{V} = \overline{\vec{V}} + \vec{V}
\text{\u} = \overline{\mathbf{u}} + \mathbf{u}'; \quad \mathbf{v} = \overline{\mathbf{v}} + \mathbf{v}'; \quad \mathbf{w} = \overline{\mathbf{w}} + \mathbf{w}'; \quad P = \overline{P} + p' \tag{1.18}
$$

Such decomposition of the different flow variables has to be included in the governing equations, provoking the appearing of three normal stresses and three shear stresses, which, due to its units, are known as the Reynolds stresses, and are Represented as follows:

$$
\begin{aligned}
\tau'\_{\rm{xy}} &= \tau'\_{\rm{yx}} = -\overline{\rho u'^2}; & \tau'\_{\rm{yy}} &= -\overline{\rho u'^2}; & \tau'\_{zz} &= -\overline{\rho u'^2} \\\\ \tau'\_{\rm{xy}} &= \tau'\_{\rm{yx}} = -\overline{\rho u' v'}; & \tau'\_{zz} &= \tau'\_{zz} = -\overline{\rho u' w'}; & \tau'\_{\rm{yz}} &= \tau'\_{z\rm{y}} = -\overline{\rho v' w'}
\end{aligned}
\tag{1.19}
$$

The vector expression for the mass conservation equation and the Navier–Stokes equations are as follows:

$$\frac{\partial \mathcal{\rho}}{\partial t} + \nabla \cdot \left(\rho \overline{V}\right) = 0 \tag{1.20}$$

$$
\rho \frac{D\overline{V}}{Dt} = \rho \mathbf{g} - \nabla \overline{p} + \mathbf{\nabla} \cdot \mathbf{\tau}\_y \tag{1.21}
$$

where:

$$\mathbf{r}\_{\psi} = \mu \left( \frac{\partial \mathbf{u}\_{i}}{\partial \mathbf{x}\_{j}} - \frac{\partial \mathbf{u}\_{j}}{\partial \mathbf{x}\_{i}} \right) - \overline{\rho \mathbf{u}\_{i}' \mathbf{u}\_{j}'} \tag{1.22}$$

#### Laminar Turbulent

It can be noted that the mass conservation equation is practically the same for laminar and turbulent flows. However, it is necessary to point out that this equivalence is valid only for the average values with a significant number of samples, meaning that the average of the fluctuating quantities becomes zero.

Because these stresses vary in their frequencies and sizes as a function of the distance from the walls, turbulent models take into account such variations to solve the closure problem of these extended equations.
