The open source CFD toolbox
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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