RNG K-Epsilon Turbulence

The RNG K-Epsilon model is a two-equation turbulence model that solves transport equations for the turbulent kinetic energy $k$ and the turbulent dissipation rate $ε$ to provide closure of the Reynolds-Averaged Navier-Stokes equations.

Yakhot and others [818] applied a statistical technique called Re-Normalisation Group (RNG) theory to the Navier-Stokes equations. The RNG theory accommodates the fact that eddies of different length scales contribute to turbulence. It accounts for these different scales in a global manner whilst calculating the dissipation rather than relying on a single turbulence scale.

The RNG model, as originally proposed, takes no explicit account of compressibility or buoyancy effects. In Simcenter STAR-CCM+, however, these effects are modeled as in the Standard K-Epsilon model.

Relation for Turbulent Viscosity

The turbulent eddy viscosity $μ_{t}$ is calculated as:

Figure 1. EQUATION_DISPLAY

μ_{t} = ρ C_{μ} k T

(4060)

where:

$ρ$ is the density.
$C_{μ}$ is a Model Coefficient.

The turbulent time scale $T$ is calculated as:


$T$	$T$ with Realizable Scale Option
Figure 2. EQUATION_DISPLAY $\max (T_{e}, C_{t} \sqrt{\frac{ν}{ε}})$ (4061)	Figure 3. EQUATION_DISPLAY $\max (\min (T_{e}, \frac{C_{T}}{C_{μ} f_{μ} S}), C_{t} \sqrt{\frac{ν}{ε}})$ (4062)

where:

$T_{e} = \frac{k}{ε}$ is the large-eddy time scale.
$C_{t}$ and $C_{T}$ are Model Coefficients.
$ν$ is the kinematic viscosity.
$S$ is given by Eqn. (1129).

Transport Equation

The transport equations for the kinetic energy $k$ and the turbulent dissipation rate $ε$ are:

Figure 4. EQUATION_DISPLAY

\begin{matrix} \frac{\partial}{\partial t} (ρ k) + \nabla\cdot (ρ k \bar{v}) = \end{matrix} \begin{matrix} \nabla\cdot [(μ + \frac{μ_{t}}{σ_{k}}) \nabla k] + P_{k} - ρ (ε - ε_{0}) + S_{k} \end{matrix}

(4063)

Figure 5. EQUATION_DISPLAY

\begin{matrix} \frac{\partial}{\partial t} (ρ ε) + \nabla\cdot (ρ ε \bar{v}) = \end{matrix} \begin{matrix} \nabla\cdot [(μ + \frac{μ_{t}}{σ_{ε}}) \nabla ε] + \frac{1}{T_{e}} C_{ε 1} P_{ε} - C_{ε 2} ρ (\frac{ε}{T_{e}} - \frac{ε_{0}}{T_{0}}) - C_{ε 4} ρ ε \nabla\cdot \bar{v} + S_{R N G} + S_{ε} \end{matrix}

(4064)

where:

$\bar{v}$ is the mean velocity.
$μ$ is the dynamic viscosity.
$σ_{k}$ , $σ_{ε}$ , $C_{ε 1}$ , $C_{ε 2}$ , and $C_{ε 4}$ are Model Coefficients.
$ε_{0}$ is the ambient turbulence value in the source terms that counteracts turbulence decay [316]. The possibility to impose an ambient source term also leads to the definition of a specific time-scale $T_{0}$ that is given by Eqn. (1170).
$P_{k}$ and $P_{ε}$ are Production Terms.
$S_{k}$ and $S_{ε}$ are the user-specified source terms.

$S_{R N G}$ in Eqn. (4064) represents the effect of mean flow distortion on turbulence and is defined as:

Figure 6. EQUATION_DISPLAY

S_{R N G} = \frac{- C_{μ} η^{3} (1 - η / η_{0})}{1 + β η^{3}} \frac{ρ ε^{2}}{k}

(4065)

where:

$η = S \frac{k}{ε}$
$β$ and $η_{0}$ are Model Coefficients.

Production Terms

The production terms $P_{k}$ and $P_{ε}$ are defined as:

Figure 7. EQUATION_DISPLAY

P_{k} = G_{k} + G_{n l} + G_{b} - ϒ_{M}

(4066)

Figure 8. EQUATION_DISPLAY

P_{ε} = G_{k} + G_{n l} + {C_{ε 3} G}_{b}

(4067)

where:

$G_{k}$ is the turbulent production given by Eqn. (1181).
$G_{b}$ is the buoyancy prodution given by Eqn. (1182).
$G_{n l}$ is the “non-linear” production given by Eqn. (1183).
$ϒ_{M}$ is the compressibility modification (Sarkar et al. [314]) given by Eqn. (1185).
$C_{ε 3}$ is a Model Coefficient.

Model Coefficients


$C_{M}$ (Sarkar)	$C_{t}$	$C_{T}$	$C_{ε 1}$	$C_{ε 2}$	$C_{ε 4}$	$C_{μ}$	$β$	$σ_{ε}$	$σ_{k}$
2	1	0.6	1.42	1.68	0.387	0.085	0.012	0.719	0.719

$C_{ε 3}$

The available literature is not clear as to the specification of this coefficient. By default, it is computed according to [305] as:

Figure 9. EQUATION_DISPLAY

C_{ε 3} = \tanh \frac{| v_{b} |}{| u_{b} |}

(4068)

where $v_{b}$ and $u_{b}$ are the velocity components parallel and perpendicular to the gravitational vector $g$ .

This formulation tends to set the coefficient to zero outside natural convection boundary layers.

Alternatively, $C_{ε 3}$ can be taken as constant everywhere, or specified depending on the buoyancy production term $G_{b}$ as follows:

Figure 10. EQUATION_DISPLAY

C_{ε 3} = {\begin{matrix} 1 for G_{b} \geq 0 \\ 0 for G_{b} < 0 \end{matrix}

(4069)