**4. RANS models**

In 1972, Jones and Launder [17] proposed a model with two differential equations in addi‐ tion to the mass and momentum conservation equations. In that model, an equation approxi‐ mated the effect of the turbulent kinetic energy production rate *<sup>k</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> (*u*' ¯2 + *v*' ¯2 <sup>+</sup> *<sup>w</sup>*' ¯2 ) and the other one contained the effects of the turbulent kinetic energy dissipation rate "ε". Together, the use of both equations proved to be adequate to obtain good results in turbulent flows whose interest was focused on the study of the influence of average turbulent effects. Both equa‐ tions are as follows:

$$\frac{Dk}{Dt} \approx \frac{\partial}{\partial \mathbf{x}\_{\cdot}} \left| \frac{\nu\_{\cdot}}{\sigma\_{\mathbf{x}}} \frac{\partial k}{\partial \mathbf{x}\_{\cdot}} \right| + \nu\_{\cdot} \frac{\partial \overline{u}\_{i}}{\partial \mathbf{x}\_{\cdot}} \left( \frac{\partial \overline{u}\_{i}}{\partial \mathbf{x}\_{\cdot}} + \frac{\partial \overline{u}\_{i}}{\partial \mathbf{x}\_{\cdot}} \right) - \epsilon \tag{1.23}$$

Computational Fluid Dynamics in Turbulent Flow Applications http://dx.doi.org/10.5772/63831 333

$$\frac{D\epsilon}{Dt} \approx \frac{\partial}{\partial \mathbf{x}\_{\cdot}} \left[ \frac{\nu\_{\cdot}}{\sigma\_{\cdot}} \frac{\partial \epsilon}{\partial \mathbf{x}\_{\cdot}} \right] + C\_{\text{l}} \nu\_{\cdot} \frac{\epsilon}{k} \frac{\partial \overline{u}\_{\cdot}}{\partial \mathbf{x}\_{\cdot}} \left( \frac{\partial \overline{u}\_{\cdot}}{\partial \mathbf{x}\_{\cdot}} + \frac{\partial \overline{u}\_{\cdot}}{\partial \mathbf{x}\_{\cdot}} \right) - C\_{\text{l}} \frac{\epsilon^{2}}{k} \tag{1.24}$$

where *σK* and *σ* are the "effective Prandtl numbers", which relate the eddy diffusion of *K* and to the eddy viscosity: ν*<sup>t</sup>* /*ν<sup>K</sup>* **and** *σ* = *ν<sup>t</sup>* /*ν*. Eddy viscosity is modeled as follows:

$$\nu\_r \approx \frac{C\_\mu k^2}{\epsilon} \tag{1.25}$$

The five constants of this model are obtained in an experimental way for cases of boundary layer calculations in non-detached flows. They are as follows:

$$C\_{\mu} = 0.09 \quad C\_{1} = 1.44 \quad C\_{2} = 1.92 \quad C\_{\mu} = 0.09 \quad \sigma\_{K} = 1.0 \quad \sigma\_{\epsilon} = 1.3 \tag{1.26}$$

Unfortunately, the above presented values cannot be used for any kind of flows (they are not universal). They need to be modified to model the behavior of wakes, jets, and recirculating flows. To deal with this difficulty, different and more sophisticated models have been proposed, which try to reproduce these flow patterns. Some of them are as follows:

**1.** The *k* – model, d on the group renormalization, Yakhot y Smith [18].

