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, $$k$$ [Eq. 2.2-2, [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, $$\epsilon$$ [Eq. 2.2-1, [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, $$\nu_t$$ [Eq. 2.2-3, [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 [Table 2.1, [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:

• 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 coefficients
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) [36]
• Rapid Distortion Theory compression term: El Tahry (1983) [16]