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 [ \(\text{m}^2 \text{s}^{-1}\)] |
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:
The model coefficients are [Table 2.1, [39];[16]]:
\[ C_\mu = 0.09; \quad C_1 = 1.44; \quad C_2 = 1.92; \quad C_{3, RDT} = 0.0; \quad \sigma_k = 1.0; \quad \sigma_\epsilon = 1.3 \]
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 [ \(\text{m}\)] |
Inlet:
Outlet:
Walls:
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; }
Source code:
References:
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