Properties🔗
- Two transport-equation linear-eddy-viscosity turbulence closure model:
- Turbulent kinetic energy, \(k\),
- Turbulent kinetic energy dissipation rate, \(\epsilon\).
- Based on:
- 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: