k-ε

Properties

• Two transport-equation linear-eddy-viscosity turbulence closure model:
  • Turbulent kinetic energy, $k$,
  • Turbulent kinetic energy dissipation rate, $\epsilon$.
• Based on:
  • Standard model: Launder and Spalding (1974) 36,
  • Rapid Distortion Theory compression term: El Tahry (1983) 16.
• Extensively used with known performance,
• Over-prediction of turbulent kinetic energy at stagnation points,
• Requires near-wall treatment.

Model equations

The turbulent kinetic energy equation, $k$ 36]:

$$\Ddt{\rho k} = \div \left( \rho D_k \grad k \right) + P - \rho \epsilon$$

Where:

$k$ : Turbulent kinetic energy [$\text{m}^2 \text{s}^{-2}$]

$D_k$ : Effective diffusivity for $k$ [-]

$P$ : Turbulent kinetic energy production rate [$\text{m}^2 \text{s}^{-3}$]

$\epsilon$ : Turbulent kinetic energy dissipation rate [$\text{m}^2 \text{s}^{-3}$]

The turbulent kinetic energy dissipation rate equation, $\epsilon$ 36]:

$$\Ddt{\rho \epsilon} = \div \left( \rho D_{\epsilon} \grad \epsilon \right) + \frac{C_1 \epsilon}{k} \left( P + C_3 \frac{2}{3} k \div \u \right) - C_2 \rho \frac{\epsilon^2}{k}$$

Where:

$D_\epsilon$ : Effective diffusivity for $\epsilon$ [-]

$C_1$ : Model coefficient [-]

$C_2$ : Model coefficient [-]

The turbulent viscosity equation, $\nu_t$ 36]:

$$\nu_t = C_{\mu} \frac{k^2}{\epsilon}$$

Where:

$C_{\mu}$ : Model coefficient for the turbulent viscosity [-]

$\nu_t$ : Turbulent viscosity [$\text{m}^2 \text{s}^{-1}$]

OpenFOAM implementation

Equations

The turbulent kinetic energy dissipation rate, $\epsilon$:

$$\ddt{\alpha \rho \epsilon} + \div \left( \alpha \rho \u \epsilon \right) - \laplacian \left( \alpha \rho D_\epsilon \epsilon \right) = C_1 \alpha \rho G \frac{\epsilon}{k} - \left( \left( \frac{2}{3} C_1 - C_{3,RDT} \right) \alpha \rho \div \u \epsilon \right) - \left( C_2 \alpha \rho \frac{\epsilon}{k} \epsilon \right) + S_\epsilon + S_{\text{fvOptions}}$$

Where:

$\alpha$ : Phase fraction of the given phase [-]

$\rho$ : Density of the fluid [$\text{kg} \text{m}^{-3}$]

$G$ : Turbulent kinetic energy production rate due to the anisotropic part of the Reynolds-stress tensor [$\text{m}^2 \text{s}^{-3}$]

$D_\epsilon$ : Effective diffusivity for $\epsilon$ [-]

$C_1$ : Model coefficient [$s$]

$C_2$ : Model coefficient [-]

$C_{3,RDT}$ : Rapid-distortion theory compression term coefficient [-]

$S_\epsilon$ : Internal source term for $\epsilon$

$S_{\text{fvOptions}}$ : Source terms introduced by fvOptions dictionary for $\epsilon$

The turbulent kinetic energy equation, $k$:

$$\ddt{\alpha \rho k} + \div \left( \alpha \rho \u k \right) - \laplacian \left( \alpha \rho D_k k \right) = \alpha \rho G - \left( \frac{2}{3} \alpha \rho \div \u k \right) - \left( \alpha \rho \frac{\epsilon}{k} k \right) + S_k + S_{\text{fvOptions}}$$

Where:

$S_k$ : Internal source term for $k$

$S_{\text{fvOptions}}$ : Source terms introduced by fvOptions dictionary for $k$

Note that:

• buoyancy contributions are not included,
• the coefficient $C_3$ is not the same as $C_{3,RDT}$.

Default model coefficients

The model coefficients are 36;16]:

$$C_\mu = 0.09; \quad C_1 = 1.44; \quad C_2 = 1.92; \quad C_{3, RDT} = 0.0; \quad \sigma_k = 1.0; \quad \sigma_\epsilon = 1.3$$

Initial conditions

For isotropic turbulence, the turbulent kinetic energy can be estimated by:

$$k = \frac{3}{2} \left( I \mag{\u_{\mathit{ref}}} \right)^{2}$$

Where:

$I$ : Turbulence intensity [%]

$\u_{\mathit{ref}}$ : A reference flow speed [$\text{m} \text{s}^{-1}$]

For isotropic turbulence, the turbulence dissipation rate can be estimated by:

$$\epsilon = \frac{C_{\mu}^{0.75}k^{1.5}}{L}$$

Where:

$C_{\mu}$ : A model constant equal to 0.09 by default [-]

$L$ : A reference length scale [$\text{m}$]

Boundary conditions

Inlet:

• fixedValue
• turbulentMixingLengthDissipationRateInlet

Outlet:

• Zero gradient
• Constrained outlet

Walls:

• kLowReWallFunction
• kqRWallFunction
• epsilonWallFunction

Usage

The model can be enabled by using constant/turbulenceProperties dictionary:

RAS
{
    // Mandatory entries
    RASModel        kEpsilon;

    // Optional entries
    turbulence      on;
    printCoeffs     on;

    // Optional model coefficients
    Cmu             0.09;
    C1              1.44;
    C2              1.92;
    C3              0.0;
    sigmak          1.0;
    sigmaEps        1.3;
}

Further information

Source code:

• Foam::RASModels::kEpsilon

References:

• Standard model: Launder and Spalding (1974) 36
• Rapid Distortion Theory compression term: El Tahry (1983) 16