Renormalization Group (RNG) k-ε Model

The RNG k-ε turbulence model (Yakhot and Orszag, 1986) deduces the behavior of large scale eddies from that of the smaller ones by utilizing the scale similarity properties that are inherent in the energy cascade (Bradshaw, 1997).

This model employs a modified coefficient in the dissipation rate equation to account for the interaction between the turbulent dissipation and mean shear. It results in better prediction of flows containing high streamline curvature, flows over a backward facing step (Yakhot et al, 1992), and flows in an expansion duct than the standard k-ε turbulence model. However, its performance worsens when predicting flows in a contraction duct (Hanjalic, 2004).

Transport Equations

Turbulent Kinetic Energy k

(1)

\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ ¯ ¯¯ ¯ u_{j} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}] + P_{k} + D_{k}

$\frac{\partial (ρ k)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} k)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}] + P_{k} + D_{k}$

Turbulent Dissipation Rate ε

(2)

$\frac{\partial (ρ ε)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} ε)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ε}}) \frac{\partial ε}{\partial x_{j}}] + P_{ε} + D_{ε}$

Production Modeling

Turbulent Kinetic Energy k

(3)

$P_{k} = μ_{t} S^{2}$

Turbulent Dissipation Rate ε

(4)

$P_{ε} = C_{ε 1} \frac{ε}{k} μ_{t} S^{2} = C_{ε 1} \frac{ε}{k} P_{k}$

Dissipation Modeling

Turbulent Kinetic Energy k

(5)

$D_{k} = - ρ ε$

Turbulent Dissipation Rate ε

(6)

$D_{ε} = - C'_{ε 2} ρ \frac{ε^{2}}{k}$

Modeling of Turbulent Viscosity $μ_{t}$

(7)

$μ_{t} = C_{μ} \frac{k^{2}}{ε}$

Model Coefficients

$C_{ε 1}$ =1.44, $C_{ε 2}$ =1.92, $C_{μ}$ =0.09, $σ_{k}$ =1.0, $σ_{ε}$ =1.3, $C'_{ε 2} = \tilde{C_{ε 2}} + \frac{C_{μ} λ^{3} (1 - λ / λ_{0})}{1 + β λ^{3}}$ , $\tilde{C_{ε 2}}$ = 1.68, $C_{μ}$ = 0.085, $λ$ = $\frac{k}{ε} S$ , $λ_{0}$ = 4.38, $β$ = 0.012, $σ_{k}$ = 0.72, $σ_{ε}$ = 0.72.

Renormalization Group (RNG) k-ε Model

Transport Equations

Production Modeling

Dissipation Modeling

Modeling of Turbulent Viscosity μt<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>μ</mi> <mi>t</mi> </msub> </mrow> </math>

Model Coefficients

Modeling of Turbulent Viscosity $μ_{t}$