*2.1.1 k-ε models*

The first major effort to simulate turbulence in the context of CFD was the socalled k-ε model [5, 6]. This approach utilizes the fluctuating components of the turbulent velocity in the three coordinate directions to obtain a turbulent kinetic energy, from:

$$k = \frac{1}{2} \left( \overline{u'^2} + \overline{v'^2} + \overline{w'^2} \right) \tag{1}$$

That is, k is the additional turbulent energy that results from the timefluctuating turbulent motions. Accompanying the turbulent kinetic energy is a turbulent dissipation ε which can be calculated as

$$
\varepsilon = \frac{\kappa^{3/2}}{0.3D} \tag{2}
$$

for flows in pipes with diameter D [7, 8]. The connection of turbulence kinetic energy and turbulent dissipation will be provided, following the equations of motion. In essence, the governing equations of motion are conservation of mass, which under steady conditions is:

$$\frac{\partial u\_i}{\partial \mathbf{x}\_i} = \mathbf{0} \tag{3}$$

conservation of momentum, written as:

$$\rho \left( u\_i \frac{\partial u\_j}{\partial \mathbf{x}\_i} \right) = -\frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_i} \left( (\mu + \mu\_t) \frac{\partial u\_j}{\partial \mathbf{x}\_i} \right) j = 1, 2, 3 \tag{4}$$

and the closure equations for turbulence:

*Turbulence Models Commonly Used in CFD DOI: http://dx.doi.org/10.5772/intechopen.99784*

$$\rho \frac{\partial (u\_i k)}{\partial \mathbf{x}\_i} = P\_k + P\_b - \rho \varepsilon + \frac{\partial}{\partial \mathbf{x}\_i} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_i} \right] \tag{5}$$

$$\rho \frac{\partial (u\_i \varepsilon)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_i} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_i} \right] + \mathbf{C}\_1 \frac{\varepsilon}{k} (\mathbf{P}\_k + \mathbf{C}\_3 \mathbf{P}\_b) - \mathbf{C}\_2 \rho \frac{\varepsilon^2}{k} \tag{6}$$

The turbulent viscosity is calculated from

$$
\mu\_t = \rho \mathbf{C}\_{\mu} \frac{k^2}{\varepsilon} \tag{7}
$$

The Pk is the production of turbulent kinetic energy from the shear strain rate and Pb is the production of turbulent kinetic energy from buoyancy effects. The production of turbulent kinetic energy is obtained from the time-averaged velocity field from:

$$P\_{\kappa} = \mu\_t \left(\frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i}\right) \frac{\partial u\_i}{\partial \mathbf{x}\_j} - \frac{\partial u\_\kappa}{\partial \mathbf{x}\_\kappa} \left(3\mu\_t \frac{\partial u\_\kappa}{\partial \mathbf{x}\_\kappa} + \rho \kappa\right) \tag{8}$$

The σ terms are corresponding Prandtl numbers for the transported variables. The values of the constants and turbulent Prandtl numbers are specific to a particular k-ε model. The k-ε approach is likely the most widely used turbulent model, even today. It is generally sufficient for flows that are wall bounded, with limited adverse pressure gradients or separation.

Traditionally, the elements are not used to capture steep velocity and temperature gradients near the wall. Rather, wall functions are employed to interpolate to the wall. Of course, the accuracy of this approach depends on the suitability of a particular wall function to a problem. For example, wall functions often fail when the flow experiences adverse pressure gradients and/or separation. On the other hand, when small elements are deployed near the wall and/or when damping equations are used to limit fluid motion in the boundary layer, integration can be performed up to the wall. In our experience, if integration is to be performed up to the wall (and wall function interpolation is avoided), the near-wall element should have a size of y+�1 for models that resolve the boundary layer. This guidance is not used for models that use the law-of-the-wall to interpolate to the wall.

A popular modification of the traditional k-ε model is the RNG (Renormalization Group) model. It was developed by [9] in an effort to handle small flow phenomenon. The mechanism of multiple scale motions is achieved by modifying the turbulent dissipation equation production term. In our experience, it has somewhat better performance than the standard k-ε particularly for rotating flows. The differences between the RNG and standard models is in the relationship between the turbulent kinetic energy, turbulent dissipation, and turbulent viscosity. With the RNG approach the turbulent viscosity is found from:

$$
\mu\_t = \mathcal{C}\_{\mu \text{RNG}} \rho \frac{\kappa^2}{\varepsilon} \tag{9}
$$

and the new turbulent dissipation transport equation becomes:

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho u\_i \boldsymbol{\varepsilon}) = \frac{\partial}{\partial \mathbf{x}\_i} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_{\mathrm{rRNG}}} \right) \frac{\partial \boldsymbol{\varepsilon}}{\partial \mathbf{x}\_i} \right] + \frac{\boldsymbol{\varepsilon}}{\kappa} \left( \mathbf{C}\_{\mathrm{rMRG}} \mathbf{P}\_{\kappa} - \mathbf{C}\_{\mathrm{r2RNG}} \rho \boldsymbol{\varepsilon} \right) \tag{10}$$

With the following inputs

*Applications of Computational Fluid Dynamics Simulation and Modeling*

$$C\_{\epsilon \text{RNG}} = \mathbf{1.42} - \frac{\eta \left(\mathbf{1} - \frac{\eta}{4.38}\right)}{(\mathbf{1} + \beta\_{\text{RNG}} \eta^3)}\tag{11}$$

$$\eta = \sqrt{\frac{P\_{\kappa}}{\rho \mathbf{C}\_{\mu RNG} \mathbf{e}}} \tag{12}$$
