k-epsilon

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) [39],
    • 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, \(k\) [Eq. 2.2-2, [39]]:

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

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, \(\epsilon\) [Eq. 2.2-1, [39]]:

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

where

\(D_\epsilon \) = Effective diffusivity for \(\epsilon\) [-]
\(C_1 \) = Model coefficient [-]
\(C_2 \) = Model coefficient [-]

The turbulent viscosity equation, \(\nu_t\) [Eq. 2.2-3, [39]]:

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

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 [Table 2.1, [39];[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:

  • Fixed value
  • turbulentMixingLengthDissipationRateInlet

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 coefficieints
    Cmu             0.09;
    C1              1.44;
    C2              1.92;
    C3              0.0;
    sigmak          1.0;
    sigmaEps        1.3;
}

Further information

Source code:

References:

  • Standard model: Launder and Spalding (1974) [39]
  • Rapid Distortion Theory compression term: El Tahry (1983) [16]

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