kOmegaWilcox06 model
The kOmegaWilcox06 model is a simple two-equation RANS turbulence model by Wilcox (Wilcox 2006). It is an improved variant of the original Wilcox k-ω model (Wilcox 1994).
Model Equations
The transport equations for the kOmegaWilcox06 model are given by:
\[ \begin{aligned} \frac{\partial (\rho k)}{\partial t} + \frac{\partial (\rho k u_j)}{\partial x_j} &= P_k - \beta^* \rho k \omega + \frac{\partial}{\partial x_j} \left[ (\mu + \sigma_k \frac{\rho k}{\omega}) \frac{\partial k}{\partial x_j} \right], \\ \frac{\partial (\rho \omega)}{\partial t} + \frac{\partial (\rho \omega u_j)}{\partial x_j} &= P_\omega - \beta \rho \omega^2 + \frac{\partial}{\partial x_j} \left[ (\mu + \sigma_\omega \frac{\rho k}{\omega}) \frac{\partial \omega}{\partial x_j} \right] + C_D, \end{aligned} \] where: \[ \begin{aligned} P_k &= \tau_{ij} \frac{\partial u_i}{\partial x_j}, \\ \tau_{ij} &= \mu_t \left(2 S_{ij} - \frac{2}{3} \delta_{ij} \frac{\partial u_k}{\partial x_k}\right) - \frac{2}{3} \rho k \delta_{ij}, \\ S_{ij} &= \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right), \\ \overline{S}_{ij} &= S_{ij} - \frac{1}{3} \delta_{ij} S_{kk}, \\ P_\omega &= \frac{\gamma \omega}{k} P_k, \\ C_D &= \frac{\sigma_d \rho}{\omega} \frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}. \end{aligned} \]
Turbulent viscosity is computed as: \[ \mu_t = \rho \frac{k}{\hat{\omega}}, \] with \[ \hat{\omega} = \max\left[\omega, C_{lim} \sqrt{\frac{2 \overline{S}_{ij} \overline{S}_{ij}}{\beta^*}}\right]. \]
Model Coefficients
The model coefficients are set to the following values:
- \(\beta^* = 0.09\)
- \(\gamma = 13/25\)
- \(C_{lim} = 7/8\)
- \(\beta = \beta_0 f_\beta\,\) where \(\beta_0 = 0.0708\)
- \(f_\beta = \frac{1 + 85 \chi_\omega}{1 + 100 \chi_\omega}\)
- \(\chi_\omega = \left|\frac{\Omega_{ij} \Omega_{jk} \hat{S}_{ki}}{(\beta^* \omega)^3}\right|\)
- \(\Omega_{ij} = \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i}\right)\)
- \(\hat{S}_{ij} = S_{ij} - \frac{1}{2} \delta_{ij} S_{kk}\)
- \(\sigma_k = 0.6\)
- \(\sigma_\omega = 0.5\)
- \(\sigma_d = \frac{1}{8}\) for \(\frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j} > 0\), else 0.
Notes on OpenFOAM implementation
OpenFOAM implementation uses slightly different notation for some coefficients:
betaStaris used for \(\beta^*\)betais used for \(\beta_0\)gammais used for \(\alpha_\omega\)sigmaKis used for \(\sigma_k\)sigmaOmegais used for \(\sigma_\omega\)sigmaDis used for \(\sigma_d\)alphaOmegais used for \(\gamma\)Climis used for \(C_{lim}\)
Moreover, OpenFOAM works with kinematic turbulent viscosity nut multiplied by density instead of turbulent viscosity mu_t.