RNG K-Epsilon Model Reference

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.

RNG K-Epsilon Properties

Convection

Controls the convection scheme.

• 1st-order: Selects the first-order upwind convection scheme.
• 2nd-order: Selects the second-order upwind convection scheme.

Realizability Option

Controls whether to activate Durbin's realizability constraint on the turbulent time scale $T$ .

For guidelines when to activate Durbin's realizability constraint see Overcoming an Unexpectedly Large Growth of K.

Method	Corresponding Method Node
None Does not apply a constraint on the turbulent time scale.	None.
Durbin Scale Limiter Applies a minimum constraint on the turbulent time scale. See Eqn. (4062).	Realizability Coefficient Provides the Realizability Coefficient $C_T$ in Eqn. (4062).

Constitutive Relation

Controls the type of constitutive relation used.

Method	Corresponding Method Node
Linear Selects the linear constitutive relation as implied by the Boussinesq approximation (see Eqn. (1147). Use this selection for most simulations.	None.
Quadratic Selects the quadratic constitutive relation (see Eqn. (1203)). This relation requires the coefficients $C_{\mu }$ , $C_1$ , $C_2$ , and $C_3$ .	Non-Linear Cmu Coefficients Provides the parameters Ca0, Ca1, Ca2, and Ca3 to calculate $C_{\mu }$ . Non-Linear Quadratic Coefficients Provides the parameters Cnl1, Cnl2, Cnl3, Cnl6, and Cnl7 to calculate $C_1$ , $C_2$ , and $C_3$ .
Cubic Selects the cubic constitutive relation (see Eqn. (1204)). This relation requires the coefficients $C_{\mu }$ , $C_1$ , $C_2$ , $C_3$ , $C_4$ , and $C_5$ .	Non-Linear Cmu Coefficients As for Quadratic. Non-Linear Quadratic Coefficients As for Quadratic. Non-Linear Cubic Coefficients Provides the parameters Cnl4, Cnl5 to calculate $C_4$ and $C_5$ .

Buoyancy Production of Dissipation

Determines how the coefficient $C_{\epsilon 3}$ in the production term $P_{\epsilon }$ is calculated (see Eqn. (4067)).

• None: Sets $C_{\epsilon 3}$ to zero.
• Boundary Layer Orientation: Computes $C_{\epsilon 3}$ according to Eqn. (4068).
• Thermal Stratification: Computes $C_{\epsilon 3}$ according to Eqn. (4069).
• Constant Coefficient: Computes $C_{\epsilon 3}$ as a constant coefficient. This option requires the specification of $C_{\epsilon 3}$ in the corresponding child node C3e .

Cmu

The coefficient $C_{\mu }$ in the calculation of the turbulent viscosity ${\mu }_t$ and in the basic transport equations.

C1e

The coefficient $C_{\epsilon 1}$ in the basic transport equations.

C2e

The coefficient $C_{\epsilon 2}$ in the basic transport equations.

Ct

The coefficient $C_t$ used in the calculation of the turbulent time scale $T$ .

Sigma_k

The coefficient ${\sigma }_k$ in the basic transport equations.

Sigma_e

The coefficient ${\sigma }_e$ in the basic transport equations.

Sarkar

The coefficient $C_M$ in the compressibility modification ${\Upsilon }_y$ (see Eqn. (1185)).

Tke Minimum

The minimum value that the transported variable $k$ is permitted to have. An appropriate value is a small number that is greater than the floating point minimum of the computer.

Tdr Minimum

The minimum value that the transported variable $\epsilon$ is permitted to have. An appropriate value is a small number that is greater than the floating point minimum of the computer.

Secondary Gradients

Neglect or include the boundary secondary gradients for diffusion and/or the interior secondary gradients at mesh faces.

• On: Include both secondary gradients.
• Off: Exclude both secondary gradients.
• Interior Only: Include the interior secondary gradients only.
• Boundaries Only: Include the boundary secondary gradients only.

Normal Stress Term

This property is an explicit term that directly incorporates divergence and turbulent kinetic energy, $-\frac{2}{3}\mathrm{\rho k}\mathbf{\text{I}}$ , according to the full Boussinesq approximation.

When On, the stress tensor is modeled as:

${\mathbf{\text{T}}}_{\mathrm{RANS}}=2{\mu }_t\mathbf{\text{S}}-\frac{2}{3}({\mu }_t\nabla \cdot \mathbf{\text{v}}+\rho k)\mathbf{\text{I}}$

The turbulent production is modeled using:

$G_k={\mu }_t{S}^2-\frac{2}{3}\rho k\nabla \cdot \overline{v}-\frac{2}{3}{\mu }_t{(\nabla \cdot \overline{v})}^2$

When Off, the stress tensor is modeled as:

${\mathbf{\text{T}}}_{\mathrm{RANS}}=2{\mu }_t\mathbf{\text{S}}-\frac{2}{3}\left({\mu }_t\nabla \cdot \mathbf{\text{v}}\right)\mathbf{\text{I}}$

The turbulent production is given by:

$G_k={\mu }_t{S}^2-\frac{2}{3}{\mu }_t{(\nabla \cdot \overline{v})}^2$

This property is off by default, in which case the quantity $-\frac{2}{3}\rho k\mathbf{\text{I}}$ is absorbed into the pressure, and causes the pressure to be slightly different. In incompressible flow only the gradients of pressure matter, so this setting has no effect on the results. In compressible flow, however, the absolute value of pressure is used in the Ideal Gas Law (Eqn. (671)).

beta

The coefficient $\beta$ in Eqn. (4065).

eta0

The coefficient ${\eta }_0$ in Eqn. (4065).

Initial Conditions

Turbulence Specification

Controls how you define the turbulence profile for initialization.

The RNG K-Epsilon model requires the turbulent kinetic energy $k$ and the turbulent dissipation rate $\epsilon$ . You can enter the corresponding values directly or have them derived from other turbulence quantities.

Method	Corresponding Value Nodes
K + Epsilon	Turbulent Kinetic Energy Scalar profile value to specify $k$ directly. Turbulent Dissipation Rate Scalar profile value to specify $\epsilon$ directly.
Intensity + Length Scale Calculation of $k$ and $\epsilon$ from a specified turbulence intensity $I$ , length scale $l$ , and velocity scale $v$ using Eqn. (1356) and Eqn. (1358). Do not use this method if the velocity field is initialized to zero.	Turbulence Intensity Scalar profile value to specify $I$ . Turbulent Length Scale Scalar profile value to specify $l$ . Turbulent Velocity Scale Scalar profile value to specify $v$ . For an initial value, use a representative velocity or a representative velocity scale. For example, for a pipe flow use the inlet velocity.
Intensity + Viscosity Ratio Calculation of $k$ and $\epsilon$ from a specified turbulence intensity $I$ , viscosity ratio ${\mu }_t/\mu$ , and velocity scale $v$ using Eqn. (1357) and Eqn. (1359). Do not use this method if the velocity field is initialized to zero.	Turbulence Intensity As for Intensity + Length Scale. Turbulent Viscosity Ratio Scalar profile value to specify the ratio of turbulent to laminar viscosity ${\mu }_t/\mu$ . Turbulent Velocity Scale As for Intensity + Length Scale.

Boundary Settings

Flow Boundaries

The following boundary conditions and values are equal to all boundaries of type:

• Free Stream
• Mass Flow Inlet
• Pressure Outlet
• Stagnation Inlet
• Velocity Inlet

Turbulence Specification

Controls how you define the turbulence profile at flow boundaries.

The RNG K-Epsilon model requires the turbulent kinetic energy $k$ and the turbulent dissipation rate $\epsilon$ . You can enter the corresponding values directly or have them derived from other turbulence quantities.

Method	Corresponding Value Nodes
K + Epsilon	Turbulent Kinetic Energy Scalar profile value to specify $k$ directly. Turbulent Dissipation Rate Scalar profile value to specify $\epsilon$ directly.
Intensity + Length Scale Calculation of $k$ and $\epsilon$ from a specified turbulence intensity $I$ and length scale $l$ using Eqn. (1356) and Eqn. (1358).	Turbulence Intensity Scalar profile value to specify $I$ . Turbulent Length Scale Scalar profile value to specify $l$ .
Intensity + Viscosity Ratio Calculation of $k$ and $\epsilon$ from a specified turbulence intensity $I$ and viscosity ratio ${\mu }_t/\mu$ using Eqn. (1357) and Eqn. (1359). Do not use this method if the velocity field is initialized to zero.	Turbulence Intensity As for Intensity + Length Scale. Turbulent Viscosity Ratio Scalar profile value to specify the ratio of turbulent to laminar viscosity ${\mu }_t/\mu$ .

Region Settings

Solvers

The following solvers and solver options are available:

Field Functions

Effective Viscosity

Scalar field that represents the sum of the laminar and turbulent viscosities $\mu +{\mu }_t$.

Kolmogorov Length Scale

Scalar field that represents the turbulent length scale $\eta$ , as defined in Eqn. (1484).

This field is only available when the Temporary Storage Retained property is ticked for the K-Epsilon Turbulence solver.

Kolmogorov Time Scale

Scalar field that represents the turbulent time scale ${\tau }_{\eta }$ , as defined in Eqn. (1485).

This field is only available when the Temporary Storage Retained property is ticked for the K-Epsilon Turbulence solver.

Reynolds Stresses

Scalar fields that represent the specific normal and shear stresses:

• Reynolds Stress uu
• Reynolds Stress uv
• Reynolds Stress uw
• Reynolds Stress vv
• Reynolds Stress vw
• Reynolds Stress ww

These fields are only available when a non-linear Constitutive Relation is used and when the Temporary Storage Retained property is ticked for the K-Epsilon Turbulence solver. If the simulation is two-dimensional, only the UU, VV, and UV stresses are exposed. All other stresses are equal to zero.

Strain Rate Tensor Modulus

Scalar field that represents the modulus of the mean strain rate tensor $S$ , as defined in Eqn. (1129).

This field is only available when the Temporary Storage Retained property is ticked for the K-Epsilon Turbulence solver.

Taylor Micro Scale

Scalar field that represents the turbulent length scale $\lambda$ , as defined in Eqn. (1486).

This field is only available when the Temporary Storage Retained property is ticked for the K-Epsilon Turbulence solver.

Turbulent Dissipation Rate

Scalar field that represents the transported variable $\epsilon$ .

Turbulent Kinetic Energy

Scalar field that represents the transported variable $k$ .

Turbulent Viscosity

Scalar field that represents the turbulent viscosity ${\mu }_t$.

Turbulent Viscosity Ratio

Scalar field that represents the ratio of turbulent to laminar viscosity ${\mu }_t/\mu$.